# inductive_moment_matching__f682056d.pdf

Inductive Moment Matching

Linqi Zhou 1 Stefano Ermon 2 Jiaming Song 1

Figure 1. Generated samples on Image Net-256 256 using 8 steps.

Diffusion models and Flow Matching generate high-quality samples but are slow at inference, and distilling them into few-step models often leads to instability and extensive tuning. To resolve these trade-offs, we propose Inductive Moment Matching (IMM), a new class of generative models for oneor few-step sampling with a single-stage training procedure. Unlike distillation, IMM does not require pre-training initialization and optimization of two networks; and unlike Consistency Models, IMM guarantees distribution-level convergence and remains stable under various hyperparameters and standard model architectures. IMM surpasses diffusion models on Image Net-256 256 with 1.99 FID using only 8 inference steps and achieves state-ofthe-art 2-step FID of 1.98 on CIFAR-10 for a model trained from scratch.

1Luma AI 2Stanford University. Correspondence to: Linqi Zhou <alexzhou@lumalabs.ai>.

Proceedings of the 42 nd International Conference on Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

1. Introduction

Generative models for continuous domains have enabled numerous applications in images (Rombach et al., 2022; Saharia et al., 2022; Esser et al., 2024), videos (Ho et al., 2022a; Blattmann et al., 2023; Open AI, 2024), and audio (Chen et al., 2020; Kong et al., 2020; Liu et al., 2023), yet achieving high-fidelity outputs, efficient inference, and stable training remains a core challenge a trilemma that continues to motivate research in this domain. Diffusion models (Sohl-Dickstein et al., 2015; Ho et al., 2020; Song et al., 2020b), one of the leading techniques, require many inference steps for high-quality results, while step-reduction methods, such as diffusion distillation (Yin et al., 2024; Sauer et al., 2025; Zhou et al., 2024; Luo et al., 2024a) and Consistency Models (Song et al., 2023; Geng et al., 2024; Lu & Song, 2024; Kim et al., 2023), often risk training collapse without careful tuning and regularization (such as pre-generating data-noise pair and early stopping).

To address the aforementioned trilemma, we introduce Inductive Moment Matching (IMM), a stable, single-stage training procedure that learns generative models from scratch for singleor multi-step inference. IMM operates on the time-dependent marginal distributions of stochastic interpolants (Albergo et al., 2023) continuous-time stochastic processes that connect two arbitrary probability

Inductive Moment Matching

density functions (data at t = 0 and prior at t = 1). By learning a (stochastic or deterministic) mapping from any marginal at time t to any marginal at time s < t, it can naturally support oneor multi-step generation (Figure 2).

IMM models can be trained efficiently from mathematical induction. For time s < r < t, we form two distributions at s by running a one-step IMM from samples at r and t. We then minimize their divergence, enforcing that the distributions at s are independent of the starting time-steps. This construction by induction guarantees convergence to the data distribution. To help with training stability, we model IMM based on certain stochastic interpolants and optimize the objective with stable sample-based divergence estimators such as moment matching (Gretton et al., 2012). Notably, we prove that Consistency Models (CMs) are a single-particle, first-moment matching special case of IMM, which partially explains the training instability of CMs.

On Image Net-256 256, IMM surpasses diffusion models and achieves 1.99 FID with only 8 inference steps using standard transformer architectures. On CIFAR-10, IMM similarly achieves state-of-the-art of 1.98 FID with 2-step generation for a model trained from scratch.

2. Preliminaries

2.1. Diffusion, Flow Matching, and Interpolants

For a data distribution q(x), Variance-Preserving (VP) diffusion models (Ho et al., 2020; Song et al., 2020b) and Flow Matching (FM) (Lipman et al., 2022; Liu et al., 2022) construct time-augmented variables xt as an interpolation between data x q(x) and prior ϵ N(0, I) such that xt = αtx + σtϵ where α0 = σ1 = 1, α1 = σ0 = 0. VP diffusion commonly chooses αt = cos π

2 t , σt = sin π

and FM chooses αt = 1 t, σt = t. Both v-prediction diffusion (Salimans & Ho, 2022) and FM are trained by matching the conditional velocity vt = α tx + σ tϵ such that a neural network Gθ(xt, t) approximates Ex,ϵ [vt|xt]. Samples can then be generated via probability-flow ODE (PF-ODE) dxt

dt = Gθ(xt, t) starting from ϵ N(0, I).

Stochastic interpolants. Unifying diffusion models and FM, stochastic interpolants (Albergo et al., 2023; Albergo & Vanden-Eijnden, 2022) construct a conditional interpolation qt(xt|x, ϵ) = N(It(x, ϵ), γ2 t I) between any data x q(x) and prior ϵ p(ϵ) and sets constraints I1(x, ϵ) = ϵ, I0(x, ϵ) = x, and γ1 = γ0 = 0. Similar to FM, a deterministic sampler can be learned by explicitly matching the conditional interpolant velocity vt = t It(x, ϵ) + γtz where z N(0, I) such that Gθ(xt, t) Ex,ϵ,z[vt|xt]. Sampling is performed following the PF-ODE dxt

dt = Gθ(xt, t) similarly starting from prior ϵ p(ϵ).

When γt 0 and It(x, ϵ) = αtx + σtϵ for αt, σt defined

Figure 2. Using an interpolation from data to prior, we define a one-step sampler that moves from any t to s < t, directly transforming qt(xt) to qs(xs). This can be repeated by jumping to an intermediate r < t before moving to s < r.

in FM, the intermediate variable xt = αtx + σtϵ becomes a deterministic interpolation and its interpolant velocity vt = α tx + σ tϵ reduces to FM velocity. Thus, its training and inference both reduce to that of FM. When ϵ N(0, I), stochastic interpolants reduce to v-prediction diffusion.

2.2. Maximum Mean Discrepancy

Maximum Mean Discrepancy (MMD, Gretton et al. (2012)) between distribution p(x), q(y) for x, y RD is an integral probability metric (M uller, 1997) commonly defined on Reproducing Kernel Hilbert Space (RKHS) H with a positive definite kernel k : RD RD R as

MMD2(p(x), q(y)) = Ex[k(x, )] Ey[k(y, )] 2 H (1)

where the norm is in H. Choices such as the RBF kernel imply an inner product of infinite-dimensional feature maps consisting of all moments of p(x) and q(y), i.e. E[xj] and E[yj] for integer j 1 (Steinwart & Christmann, 2008).

3. Inductive Moment Matching

We introduce Inductive Moment Matching (IMM), a method that trains a model of both high quality and sampling efficiency in a single stage. To do so, we assume a timeaugmented interpolation between data (distribution at t = 0) and prior (distribution at t = 1) and propose learning an implicit one-step model (i.e. a one-step sampler) that transforms the distribution at time t to the distribution at time s for any s < t (Section 3.1). The model enables direct onestep sampling from t = 1 to s = 0 and few-step sampling via recursive application from any t to any r < t and then to any s < r until s = 0; this allows us to learn the model from its own samples via bootstrapping (Section 3.2).

3.1. Model Construction via Interpolants

Given data x q(x) and prior ϵ p(ϵ), the timeaugmented interpolation xt defined in Albergo et al. (2023) follows xt qt(xt|x, ϵ). This implies a marginal interpolating distribution

qt(xt) = ZZ qt(xt|x, ϵ)q(x)p(ϵ)dxdϵ. (2)

Inductive Moment Matching

We learn a model distribution implicitly defined by a onestep sampler that transforms qt(xt) into qs(xs) for some s t. This can be done via a special class of interpolants, which preserves the marginal distribution qs(xs) while interpolating between x and xt. We term these marginalpreserving interpolants among a class of generalized interpolants. Formally, we define xs as a generalized interpolant between x and xt if, for all s [0, t], its distribution follows

qs|t(xs|x, xt) = N(Is|t(x, xt), γ2 s|t I) (3)

and satisfies constraints It|t(x, xt) = xt, I0|t(x, xt) = x, γt|t = γ0|t = 0, and qt|1(xt|x, ϵ) qt(xt|x, ϵ). When t = 1, it reduces to regular stochastic interpolants. Next, we define marginal-preserving interpolants. Definition 1 (Marginal-Preserving Interpolants). A generalized interpolant xs is marginal-preserving if for all t [0, 1] and for all s [0, t], the following equality holds:

qs(xs) = ZZ qs|t(xs|x, xt)qt(x|xt)qt(xt)dxtdx, (4)

qt(x|xt) = Z qt(xt|x, ϵ)q(x)p(ϵ)

qt(xt) dϵ. (5)

That is, this class of interpolants has the same marginal at s regardless of t. For all t [0, 1], we define our noisy model distribution at s [0, t] as

pθ s|t(xs) = ZZ qs|t(xs|x, xt)pθ s|t(x|xt)qt(xt)dxtdx (6)

where the interpolant is marginal preserving and pθ s|t(x|xt) is our clean model distribution implicitly parameterized as a one-step sampler. This definition also enables multistep sampling. To produce a clean sample x given xt qt(xt) in two steps via an intermediate s: (1) we sample ˆx pθ s|t(x|xt) followed by ˆxs qs|t(xs|ˆx, xt) and (2) if the marginal of ˆxs matches qs(xs), we can obtain x by x pθ 0|s(x|ˆxs). We are therefore motivated to minimize divergence between Eq. (4) and (6) using the objective below.

Na ıve objective. As one can easily draw samples from the model, it can be na ıvely learned by directly minimizing

L(θ) = Es,t h D(qs(xs), pθ s|t(xs)) i (7)

with time distribution p(s, t) and a sample-based divergence metric D( , ) such as MMD or GAN (Goodfellow et al., 2020). If an interpolant xs is marginal-preserving, then the minimum loss is 0 (see Lemma 3). One might also notice the similarity between right-hand sides of Eq. (4) and (6). However, qs(xs) = pθ s|t(xs) does not necessarily imply pθ s|t(x|xt) = qt(x|xt). In fact, the minimizer pθ s|t(x|xt) is not unique and, under mild assumptions, a deterministic minimizer exists (see Section 4).

3.2. Learning via Inductive Bootstrapping

While sound, the na ıve objective in Eq. (7) is difficult to optimize in practice because when t is far from s, the input distribution qt(xt) can be far from the target qs(xs). Fortunately, our interpolant construction implies that the model definition in Eq. (6) satisfies boundary condition qs(xs) = pθ s|s(xs) regardless of θ (see Lemma 4), which indicates that pθ s|t(xs) qs(xs) when t is close to s. Furthermore, the interpolant enforces pθ s|t(xs) pθ s|r(xs) for any r < t close to t as long as the model is continuous around t. Therefore, we can construct an inductive learning algorithm for pθ s|t(xs) by using samples from pθ s|r(xs).

For better analysis, we define a sequence number n for parameter θn and function r(s, t) where s r(s, t) < t such that pθn s|t(xs) learns to match pθn 1 s|r (xs).1 We omit r s arguments when context is clear and let r(s, t) be a finite decrement from t but truncated at s t (see Appendix B.3 for well-conditioned r(s, t)).

General objective. With marginal-preserving interpolants and mapping r(s, t), we learn θn in the following objective:

L(θn) = Es,t h w(s, t)MMD2(pθn 1 s|r (xs), pθn s|t(xs)) i

where w(s, t) is a weighting function. We choose MMD as our objective due to its superior optimization stability and show that this objective learns the correct data distribution.

Theorem 1. Assuming r(s, t) is well-conditioned, the interpolant is marginal-preserving, and θ n is a minimizer of Eq. (8) for each n with infinite data and network capacity, for all t [0, 1], s [0, t],

lim n MMD2(qs(xs), pθ n s|t(xs)) = 0. (9)

In other words, θn eventually learns the target distribution qs(xs) by parameterizing a one-step sampler pθn s|t(x|xt).

4. Simplified Formulation and Practice

We present algorithmic and practical decisions below.

4.1. Algorithmic Considerations

Despite theoretical soundness, it remains unclear how to empirically choose a marginal-preserving interpolant. First, we present a sufficient condition for marginal preservation.

Definition 2 (Self-Consistent Interpolants). Given s, t [0, 1], s t, an interpolant xs qs|t(xs|x, xt) is selfconsistent if for all r [s, t], the following holds:

qs|t(xs|x, xt) = Z qs|r(xs|x, xr)qr|t(xr|x, xt)dxr (10)

1Note that n is different from optimization steps. Advancing from n 1 to n can take arbitrary number of optimization steps.

Inductive Moment Matching

In other words, xs has the same distribution if one (1) directly samples it by interpolating x and xt and (2) first samples any xr (given x and xt) and then samples xs (given x and xr). Furthermore, self-consistency implies marginal preservation (Lemma 5).

DDIM interpolant. Denoising Diffusion Implicit Models (Song et al., 2020a) was introduced as a fast ODE sampler for diffusion models, defined as

DDIM(xt, x, s, t) = αs σs

and sample xs = DDIM(xt, Ex[x|xt], s, t) can be drawn when Ex[x|xt] is approximated by a network. We show in Appendix C.1 that DDIM as an interpolant, i.e. γs|t 0 and Is|t(x, xt) = DDIM(xt, x, s, t), is self-consistent. Moreover, with deterministic interpolants such as DDIM, there exists a deterministic minimizer pθ s|t(x|xt) of Eq. (7).

Proposition 1. (Informal) If γs|t 0 and Is|t(x, xt) satisfies mild assumptions, there exists a deterministic pθ s|t(x|xt) that attains 0 loss for Eq. (7).

See Appendix B.6 for formal statement and proof. This allows us to define pθ s|t(x|xt) = δ(x gθ(xt, s, t)) for a neural network gθ(xt, s, t) with parameter θ by default.

Eliminating stochasticity. We use DDIM interpolant, deterministic model, and prior p(ϵ) = N(0, σ2 d I) where σd is the data standard deviation (Lu & Song, 2024). As a result, one can draw xs from model via xs = f θ s,t(xt) := DDIM(xt, gθ(xt, s, t), s, t) where xt qt(xt).

Re-using xt for xr. Inspecting Eq. (8) and (6), one requires xr qr(xr) to generate samples from the target distribution. Instead of sampling xr given a new (x, ϵ) pair, we can reduce variance by reusing xt and x such that xr = DDIM(xt, x, r, t). This is justified because xr derived from xt preserves the marginal distribution qr(xr) (see Appendix C.2).

Stop gradient. We set n to optimization step number, i.e. advancing from n 1 to n is a single optimizer step where θn is initialized from θn 1. Equivalently, we can omit n from θn and write θn 1 as the stop-gradient parameter θ .

Simplified objective. Let xt, x t be i.i.d. random variables from qt(xt) and xr, x r are variables obtained by reusing xt, x t respectively, the training objective can be derived from the MMD definition in Eq. (1) (see Appendix C.3) as

LIMM(θ) = Ext,x t,xr,x r,s,t h w(s, t) h k ys,t, y s,t (12)

+ k ys,r, y s,r k ys,t, y s,r k y s,t, ys,r ii

where ys,t = f θ s,t(xt), y s,t = f θ s,t(x t), ys,r = f θ s,r (xr), y s,r = f θ s,r (x r), k( , ) is a kernel function, and w(s, t) is a prior weighting function.

Figure 3. With self-consistent interpolants, IMM uses M particle samples (M = 2 is shown above) for moment matching. Samples from pθ s|t(xs) are obtained by drawing from pθ s|t(x|xt) followed by qs|t(xs|x, xt). Solid and dashed red lines indicate sampling with and without gradient propagation respectively. After M samples are drawn, sample xs is repulsed by x s and attracted towards samples of xs and x s through kernel function k( , ).

An empirical estimate of the above objective uses M particle samples to approximate each distribution indexed by t. In practice, we divide a batch of model output with size B into B/M groups within which share the same (s, t) sample, and the objective is approximated by instantiating B/M number of M M matrices. Note that the number of model passes does not change with respect to M (see Appendix C.4). A M = 2 version is visualized in Figure 3 and a simplified training algorithm is shown in Algorithm 1. A full training algorithm is shown in Appendix D.

4.2. Other Implementation Choices

We defer detailed analysis of each decision to Appendix C.

Flow trajectories. We investigate the two most used flow trajectories (Nichol & Dhariwal, 2021; Lipman et al., 2022),

Cosine. αt = cos 1

2πt , σt = sin 1

2πt . OT-FM. αt = 1 t, σt = t.

Network gθ(xt, s, t). We set gθ(xt, s, t) = cskip(t)xt + cout(t)Gθ(cin(t)xt, cnoise(s), cnoise(t)) with a neural network Gθ, following EDM (Karras et al., 2022). For all choices we let cin(t) = 1/ p

α2 t + σ2 t /σd (Lu & Song, 2024). Listed below are valid choices for other coefficients.

Identity. cskip(t) = 0, cout(t) = 1. Simple-EDM (Lu & Song, 2024). cskip(t) = αt/(α2 t + σ2 t ), cout(t) = σdσt/ p

α2 t + σ2 t . Euler-FM. cskip(t) = 1, cout(t) = tσd. This is specific to OT-FM schedule.

We show in Appendix C.5 that f θ s,t(xt) similarly follows the

Inductive Moment Matching

EDM parameterization of the form f θ s,t(xt) = cskip(s, t)xt+ cout(s, t)Gθ(cin(t)xt, cnoise(s), cnoise(t)).

Noise conditioning cnoise( ). We choose cnoise(t) = ct for some constant c 1. We find our model convergence relatively insensitive to c but recommend using larger c, e.g. 1000 (Song et al., 2020b; Peebles & Xie, 2023), because it enables sufficient distinction between nearby r and t.

Mapping function r(s, t). We find that r(s, t) via constant decrement in ηt = σt/αt works well where the decrement is chosen in the form of (ηmax ηmin)/2k for some appropriate k (details in Appendix C.7).

Kernel function. We use time-dependent Laplace kernels of the form ks,t(x, y) = exp( w(s, t) max( x y 2, ϵ)/D) for x, y RD, some ϵ > 0 to avoid undefined gradients, and w(s, t) = 1/|cout(s, t)|. We find Laplace kernels provide better gradient signals than RBF kernels. (see Appendix C.8).

Weighting w(s, t) and distribution p(s, t). We follow VDM (Kingma et al., 2021; Kingma & Gao, 2024) and define p(t) = U(ϵ, T) and p(s|t) = U(ϵ, t) for constants ϵ, T [0, 1]. Similarly, weighting is defined as

w(s, t) = 1

αa t α2 t + σ2 t (13)

where σ( ) is sigmoid function, λt denotes log-SNRt and a {1, 2}, σ( ), b R are constants (see Appendix C.9).

4.3. Sampling

Pushforward sampling. A sample xs can be obtained by directly pushing xt qt(xt) through f θ s,t(xt). This can be iterated for an arbitrary number of steps starting from ϵ N(0, σ2 d I) until s = 0. We note that, by definition, one application of f θ s,t(xt) is equivalent to one DDIM step using the learned network gθ(xt, s, t) as the x prediction. This sampler can then be viewed as a few-step sampler using DDIM where gθ(xt, s, t) outputs a realistic sample x instead of its expectation Ex[x|xt] as in diffusion models.

Restart sampling. Similar to Xu et al. (2023); Song et al. (2023), one can introduce stochasticity during sampling by re-noising a sample to a higher noise-level before sampling again. For example, a two-step restart sampler from xt requires s (0, t) for drawing sample ˆx = f θ 0,s(xs) where xs qs(xs|f θ 0,t(xt)).

Classifier-free guidance. Given a data-label pair (x, c), during inference time, classifier-free guidance (Ho & Salimans, 2022) with weight w replaces conditional model output Gθ(xt, s, t, c) by a reweighted model output via

w Gθ(xt, s, t, c) + (1 w)Gθ(xt, s, t, ) (14)

where denotes the null-token indicating unconditional out-

Algorithm 1 Training (see Appendix D for full version)

Input: parameter θ, DDIM(xt, x, s, t), B, M, p Output: learned θ while model not converged do

Sample data x, label c, and prior ϵ with batch size B and split into B/M groups. Each group shares a (s, r, t) sample.

For each group, xt DDIM(ϵ, x, t, 1). For each group, xr DDIM(xt, x, r, t). For each instance, set c = with prob. p. Minimize the empirical loss ˆLIMM(θ) in Eq. (67). end while

Algorithm 2 Pushforward Sampling (details in Appendix F)

Input: model f θ, {ti}N i=0, N(0, σ2 d I), (optional) w Output: xt0 Sample x N N(0, σ2 d I) for i = N, . . . , 1 do

xti 1 f θ ti 1,ti(xti) or f θ ti 1,ti,w(xti). end for

put. Similarly, we define our guided model as f θ s,t,w(xt) = cskip(s, t)xt+cout(s, t)Gw θ (xt, s, t, c) where Gw θ (xt, s, t, c) is as defined in Eq. (14) and we drop cin( ) and cnoise( ) for notational simplicity. We justify this decision in Appendix E. Similar to diffusion models, c is randomly dropped with probability p during training without special practices.

We present pushforward sampling in Algorithm 2 and detail both samplers in Appendix F.

5. Connection with Prior Works

Our work is closely connected with many prior works. Detailed analysis is found in Appendix G.

Consistency Models. Consistency models (CMs) (Song et al., 2023; Song & Dhariwal, 2023; Lu & Song, 2024) uses a network gθ(xt, t) that outputs clean data given noisy input xt. It requires point-wise consistency gθ(xt, t) = gθ(xr, r) for any r < t where xr is obatined via an ODE solver from xt using either pretrained model or groundtruth data. Discrete-time CM must satisfy gθ(x0, 0) = x0 and trains via loss Ext,x,t[d(gθ(xt, t), gθ (xr, r))] where d( , ) is commonly chosen as L2 or LPIPS (Zhang et al., 2018).

We show in the following Lemma that CM objective with L2 distance is a single-particle estimate of IMM objective with energy kernel.

Lemma 1. When xt = x t, xr = x r, k(x, y) = x y 2, and s > 0 is a small constant, Eq. (12) reduces to CM loss Ext,x,t h w(t) gθ(xt, t) gθ (xr, r) 2i for some

valid mapping r(t) < t.

This single-particle estimate ignores the repulsion force

Inductive Moment Matching

imposed by k( , ). Energy kernel also only matches the first moment, ignoring all higher moments. These decisions can be significant contributors to training instability and performance degradation of CMs.

Improved CMs (Song & Dhariwal, 2023) propose pseudohuber loss as d( , ) which we justify in the Lemma below.

Lemma 2. Negative pseudo-huber loss kc(x, y) = c q

x y 2 + c2 for c > 0 is a conditionally positive definite kernel that matches all moments of x and y where weights on higher moments depend on c.

From a moment-matching perspective, the improved performance is explained by the loss matching all moments of the distributions. In addition to pseudo-huber loss, many other kernels (Laplace, RBF, etc.) are all valid choices in the design space.

We also extend IMM loss to the differential limit by taking r(s, t) t, the result of which subsumes the continuoustime CM (Lu & Song, 2024) as a single-particle estimate (Appendix H). We leave experiments for this to future work.

Diffusion GAN and Adversarial Consistency Distillation. Diffusion GAN (Xiao et al., 2021) parameterizes the generative distribution as pθ s|t(xs|xt) = RR qs|t(xs|x, xt)δ(x Gθ(xt, z, t))p(z)dzdx for s as a fixed decrement from t and p(z) a noise distribution. It defines the interpolant qs|t(xs|x, xt) as the DDPM posterior distribution, which is self-consistent (see Appendix G.2) and introduces randomness to the sampling process to match qt(x|xt) instead of the marginal. Both Diffusion GAN and Adversarial Consistency Distillation (Sauer et al., 2025) use GAN objective, which shares similarity to MMD in that MMD is defined as an integral probability metric where the optimal discriminator is chosen in RKHS. This eliminates the need for explicit adversarial optimization of a neural-network discriminator.

Generative Moment Matching Network. GMMN (Li et al., 2015) directly applies MMD to train a generator Gθ(z) where z N(0, I) to match the data distribution. It is a special case of IMM in that when t = 1 and r(s, t) s = 0 our loss reduces to na ıve GMMN objective.

6. Related Works

Diffusion, Flow Matching, and stochastic interpolants. Diffusion models (Sohl-Dickstein et al., 2015; Song et al., 2020b; Ho et al., 2020; Kingma et al., 2021) and Flow Matching (Lipman et al., 2022; Liu et al., 2022) are widely used generative frameworks that learn a score or velocity field of a noising process from data into a simple prior. They have been scaled successfully for text-to-image (Rombach et al., 2022; Saharia et al., 2022; Podell et al., 2023; Chen et al., 2023; Esser et al., 2024) and text-to-video (Ho et al., 2022a; Blattmann et al., 2023; Open AI, 2024) tasks.

Stochastic interpolants (Albergo et al., 2023; Albergo & Vanden-Eijnden, 2022) extend these ideas by explicitly defining a stochastic path between data and prior, then matching its velocity to facilitate distribution transfer. While IMM builds on top of the interpolant construction, it directly learns one-step mappings between any intermediate marginal distributions.

Diffusion distillation. To resolve diffusion models sampling inefficiency, recent methods (Salimans & Ho, 2022; Meng et al., 2023; Yin et al., 2024; Zhou et al., 2024; Luo et al., 2024a; Heek et al., 2024) focus on distilling one-step or few-step models from pre-trained diffusion models. This two-stage approach currently has the best generation quality and efficiency. However, such methods usually require jointly optimizing two networks and training requires careful tuning in practice to avoid mode collapse (Yin et al., 2024). Another recent work (Salimans et al., 2024) explicitly matches the first moment of the data distribution available from pre-trained diffusion models. In contrast, our method implicitly matches all moments using MMD and can be trained from scratch with a single model.

Few-step generative models from scratch. Early onestep generative models primarily relied on GANs (Goodfellow et al., 2020; Karras et al., 2020; Brock, 2018) and MMD (Li et al., 2015; 2017) (or their combination) but scaling adversarial training remains challenging. Recent independent classes of few-step models, e.g. Consistency Models (CMs) (Song et al., 2023; Song & Dhariwal, 2023; Lu & Song, 2024), Consistency Trajectory Models (CTMs) (Kim et al., 2023; Heek et al., 2024) and Shortcut Models (SMs) (Frans et al., 2024) still face training instability and require specialized components (Lu & Song, 2024) (e.g., JVP for flash attention) or other special practices (e.g., high weight decay for SMs, combined LPIPS (Zhang et al., 2018) and GAN losses for CTMs, and special training schedules (Geng et al., 2024)) to remain stable. In contrast, our method trains stably with a single loss and achieves strong performance without special training practices.

7. Experiments

We evaluate IMM s empirical performance (Section 7.1), training stability (Section 7.2), sampling choices (Section 7.3), scaling behavior (Section 7.4) and ablate our practical decisions (Section 7.5).

7.1. Image Generation

We present FID (Heusel et al., 2017) results for unconditional CIFAR-10 and class-conditional Image Net-256 256 in Table 1 and 2. For CIFAR-10, we separate baselines into diffusion and flow models, distillation models, and few-step models from scratch. IMM belongs to the last category

Inductive Moment Matching

Family Method FID ( ) Steps ( )

Diffusion & Flow

DDPM (Ho et al., 2020) 3.17 1000 DDPM++ (Song et al., 2020b) 3.16 1000 NCSN++ (Song et al., 2020b) 2.38 1000 DPM-Solver (Lu et al., 2022) 4.70 10 i DDPM (Nichol & Dhariwal, 2021) 2.90 4000 EDM (Karras et al., 2022) 2.05 35 Flow Matching (Lipman et al., 2022) 6.35 142 Rectified Flow (Liu et al., 2022) 2.58 127

Few-Step via Distillation

PD (Salimans & Ho, 2022) 4.51 2 2-Rectified Flow (Salimans & Ho, 2022) 4.85 1 DFNO (Zheng et al., 2023) 3.78 1 KD (Luhman & Luhman, 2021) 9.36 1 TRACT (Berthelot et al., 2023) 3.32 2 Diff-Instruct (Luo et al., 2024a) 5.57 1 PID (LPIPS) (Tee et al., 2024) 3.92 1 DMD (Yin et al., 2024) 3.77 1 CD (LPIPS) (Song et al., 2023) 2.93 2 CTM (w/ GAN) (Kim et al., 2023) 1.87 2 Si D (Zhou et al., 2024) 1.92 1 Si M (Luo et al., 2024b) 2.06 1 s CD (Lu & Song, 2024) 2.52 2

Few-Step from Scratch

i CT (Song & Dhariwal, 2023) 2.83 1 2.46 2 ECT (Geng et al., 2024) 3.60 1 2.11 2 s CT (Lu & Song, 2024) 2.97 1 2.06 2 IMM (ours) 3.20 1 1.98 2

Table 1. CIFAR-10 results trained without label conditions.

Family Method FID( ) Steps ( ) #Params

GAN Big GAN (Brock, 2018) 6.95 1 112M Giga GAN (Kang et al., 2023) 3.45 1 569M Style GAN-XL (Karras et al., 2020) 2.30 1 166M

Masked & AR

VQGAN (Esser et al., 2021) 26.52 1024 227M Mask GIT (Chang et al., 2022) 6.18 8 227M MAR (Li et al., 2024) 1.98 100 400M VAR-d20 (Tian et al., 2024a) 2.57 10 600M VAR-d30 (Tian et al., 2024a) 1.92 10 2B

Diffusion & Flow

ADM (Dhariwal & Nichol, 2021) 10.94 250 554M CDM (Ho et al., 2022b) 4.88 8100 - Sim Diff (Hoogeboom et al., 2023) 2.77 512 2B LDM-4-G (Rombach et al., 2022) 3.60 250 400M U-Di T-L (Tian et al., 2024b) 3.37 250 916M U-Vi T-H (Bao et al., 2023) 2.29 50 501M Di T-XL/2 (w = 1.0) (Peebles & Xie, 2023) 9.62 250 675M Di T-XL/2 (w = 1.25) (Peebles & Xie, 2023) 3.22 250 675M Di T-XL/2 (w = 1.5) (Peebles & Xie, 2023) 2.27 250 675M Si T-XL/2 (w = 1.0) (Ma et al., 2024) 9.35 250 675M Si T-XL/2 (w = 1.5) (Ma et al., 2024) 2.15 250 675M

Few-Step from Scratch

i CT (Song et al., 2023) 34.24 1 675M 20.3 2 675M Shortcut (Frans et al., 2024) 10.60 1 675M 7.80 4 675M 3.80 128 675M IMM (ours) (XL/2, w = 1.25) 7.77 1 675M 5.33 2 675M 3.66 4 675M 2.77 8 675M IMM (ours) (XL/2, w = 1.5) 8.05 1 675M 3.99 2 675M 2.51 4 675M 1.99 8 675M

Table 2. Class-conditional Image Net-256 256 results.

50k 100k 150k 200k 250k Training Steps

Positional emb. Fourier emb. w/ scale=16

Figure 4. FID convergence for different embeddings (left). CIFAR10 samples from Fourier embedding (scale = 16) (right).

in which it achieves state-of-the-art performance of 1.98 using pushforward sampler. For Image Net-256 256, we use the popular Di T (Peebles & Xie, 2023) architecture because of its scalability, and compare it with GANs, masked and autoregressive models, diffusion and flow models, and few-step models trained from scratch.

We observe decreasing FID with more steps and IMM achieves 1.99 FID with 8 steps (with w = 1.5), surpassing Di T and Si T (Ma et al., 2024) using the same architecture except for trivially injecting time s (see Appendix I). Notably, we also achieve better 8-step FID than the 10step VAR (Tian et al., 2024a) of comparable size. At 16 steps, IMM also achieves 1.90 FID outperforming VAR s 2B variant (see Appendix I.4). However, different from VAR, IMM grants flexibility of variable number of inference steps. Lastly, we similarly surpass Shortcut models (Frans et al., 2024) best performance with only 8 steps2. We defer inference details to Section 7.3 and Appendix I.2.

2NFE is twice the number of steps.

50k 100k 150k 200k 250k 300k 350k 400k Training Steps

M=1 M=2 M=4 M=8 M=16

Figure 5. More particles indicate more stable training on Image Net-256 256.

1 2 4 8 Inference Steps

Pushforward/Uniform

Pushforward/EDM

Restart/Uniform

Restart/EDM

Figure 6. Image Net256 256 FID with different sampler types.

7.2. IMM Training is Stable

We show that IMM is stable and achieves reasonable performance across a range of parameterization choices.

Positional vs. Fourier embedding. A known issue for CMs (Song et al., 2023) is its training instability when using Fourier embedding with scale 16, which forces reliance on positional embeddings for stability. We find that IMM does not face this problem (see Figure 4). For Fourier embedding we use the standard NCSN++ (Song et al., 2020b) architecture and set embedding scale to 16; for positional embeddings, we adopt DDPM++ (Song et al., 2020b). Both embedding types converge reliably, and we include samples from the Fourier embedding model in Figure 4.

Particle number. Particle number M for estimating MMD is an important parameter for empirical success (Gretton et al., 2012; Li et al., 2015), where the estimate is more accurate with larger M. In our case, na ıvely increasing M can slow down convergence because we have a fixed batch size B in which the samples are grouped into B/M groups

Inductive Moment Matching

109 1010 1011 1012 Training Compute (GFLOPs)

101 102 103 Sampling Compute (GFLOPs)

101 102 Transformer GFLOPs

Correlation: -0.92

XL/2,1-step

XL/2,2-step

XL/2,4-step

XL/2,8-step

Figure 7. IMM scales with both training and inference compute, and exhibits strong correlation between model size and performance.

Figure 8. Sample visual quality increases with increase in both model size and sampling compute.

0.0 0.2 0.4 0.6 0.8 1.0 t

log abs. mean diff.

cnoise(t) = 1t

cnoise(t) = 32t

cnoise(t) = 256t

cnoise(t) = 512t

cnoise(t) = 1000t

50k 100k 150k 200k 250k 300k 350k 400k Training Steps

c = 1 c = 32 c = 512 c = 1000

Figure 9. Log distance in embedding space for cnoise(t) = ct (left). Similar Image Net-256 256 convergence across different c (right).

of M where each group shares the same t. The larger M means that fewer t s are sampled. On the other hand, using extremely small numbers of particles, e.g. M = 2, leads to training instability and performance degradation, especially on a large scale with Di T architectures. We find that there exists a sweet spot where a few particles effectively help with training stability while further increasing M slows down convergence (see Figure 5). We see that in Image Net256 256, training collapses when M = 1 (which is CM) and M = 2, and achieves lowest FID under the same computation budget with M = 4. We hypothesize M < 4 does not allow sufficient mixing between particles and larger M

means fewer t s are sampled for each step, thus slowing convergence. A general rule of thumb is to use a large enough M for stability, but not too large for slowed convergence.

Noise embedding cnoise( ). We plot in Figure 9 the log absolute mean difference of t and r(s, t) in the positional embedding space. Increasing c increases distinguishability of nearby distributions. We also observe similar convergence on Image Net-256 256 across different c, demonstrating the insensitivity of our framework w.r.t. noise function.

7.3. Sampling

We investigate different sampling settings for best performance. One-step sampling is performed by simple pushforward from T to ϵ (concrete values in Appendix I.2). On CIFAR-10 we use 2 steps and set intermediate time t1 such that ηt1 = 1.4, a choice we find to work well empirically. On Image Net-256 256 we go beyond 2 steps and, for simplicity, investigate (1) uniform decrement in t and (2) EDM (Karras et al., 2024) schedule (detailed in Appendix I.2). We plot FID of all sampler settings in Figure 6 with guidance weight w = 1.5. We find pushforward sam-

Inductive Moment Matching

id/cos id/FM s EDM/cos s EDM/FM e FM

CIFAR-10 3.77 3.45 2.39 2.10 2.53

Image Net-256 256 46.44 47.32 27.33 28.67 27.01

Table 3. FID results with different flow schedules and network parameterization.

plers with uniform schedule to work the best on Image Net256 256 and use this as our default setting for multi-step generation. Additionally, we concede that pushforward combined with restart samplers can achieve superior results. We leave such experiments to future works.

7.4. Scaling Behavior

Similar to diffusion models, IMM scale with training and inference compute as well as model size on Image Net256 256. We plot in Figure 7 FID vs. training and inference compute in GFLOPs and we find strong correlation between compute used and performance. We further visualize samples in Figure 8 with increasing model size, i.e. Di T-S, Di T-B, Di T-L, Di T-XL, and increasing inference steps, i.e. 1, 2, 4, 8 steps. The sample quality increases along both axes as larger transformers with more inference steps capture more complex distributions. This also explains that more compute can sometimes yield different visual content from the same initial noise as shown in the visual results.

7.5. Ablation Studies

All ablation studies are done with DDPM++ architecture for CIFAR-10 and Di T-B for Image Net-256 256. FID comparisons use 2-step samplers by default.

Flow schedules and parameterization. We investigate all combinations of network parameterization and flow schedules: Simple-EDM + cosine (s EDM/cos), Simple-EDM + OT-FM (s EDM/FM), Euler-FM + OT-FM (e FM), Identity + cosine (id/cos), Identity + OT-FM (id/FM). Identity parameterization consistently fall behind other types of parameterization, which all show similar performance across datasets (see Table 3). We see that on smaller scale (CIFAR-10), s EDM/FM works the best but on larger scale (Image Net-256 256), e FM works the best, indicating that OT-FM schedule and Euler paramaterization may be more scalable than other choices.

Mapping function r(s, t). Our choices for ablation are (1) constant decrement in ηt, (2) constant decrement in t, (3) constant decrement in λt = log α2 t/σ2 t , (4) constant increment in 1/ηt (see Appendix C.6). For fair comparison, we choose the decrement gap so that the minimum t r(s, t) is 10 3 and use the same network parameterization. FID progression in Figure 11 show that (1) consistently outperforms other choices. We additionally ablate the mapping gap using M = 4 in (1). The constant decrement is in

50k 100k 150k 200k 250k 300k 350k 400k Training Steps

k = 11 k = 12 k = 13 k = 14

Figure 10. Image Net-256 256 FID progression with different t, r gap with M = 4.

w(s, t) = 1 40.19

+ ELBO weight 96.43

+ 1/(α2 t + σ2 t ) 27.43

Table 4. Ablation of weighting function w(s, t) on Image Net-256 256.

50k 100k 150k 200k Training Steps

constant dec. in t constant dec. in t constant dec. in t constant inc. in 1/ t

50k 100k 150k 200k 250k 300k 350k 400k Training Steps

constant dec. in t constant dec. in t constant dec. in t constant inc. in 1/ t

Figure 11. FID progression on different types of mapping function r(s, t) for CIFAR-10 (left) and Image Net-256 256 (right).

the form of (ηmax ηmin)/2k for an appropriately chosen k. We show in Figure 10 that the performance is relatively stable across k {11, 12, 13} but experiences instability for k = 14. This suggests that, for a given particle number, there exists a largest k for stable optimization.

Weighting function. In Table 4 we first ablate the weighting factors in three groups: (1) the VDM ELBO factors 1 2σ(b λt)( d

dtλt), (2) weighting αt (i.e. when a = 1), and (3) weighting 1/(α2 t + σ2 t ). We find it necessary to use αt jointly with ELBO weighting because it converts v-pred network to a ϵ-pred parameterization (see Appendix C.9), consistent with diffusion ELBO-objective. Factor 1/(α2 t + σ2 t ) upweighting middle time-steps further boosts performance, a helpful practice also known for FM training (Esser et al., 2024). We leave additional study of the exponent a to Appendix I.5 and find that a = 2 emphasizes optimizing the loss when t is small while a = 1 more equally distributes weights to larger t. As a result, a = 2 achieves higher quality multi-step generation than a = 1.

8. Conclusion

We present Inductive Moment Matching, a framework that learns a few-step generative model from scratch. It trains by first leveraging self-consistent interpolants to interpolate between data and prior and then matching all moments of its own distribution interpolated to be closer to that of data. Our method guarantees convergence in distribution and generalizes many prior works. Our method also achieves stateof-the-art performance across benchmarks while achieving orders of magnitude faster inference. We hope it provides a new perspective on training few-step models from scratch and inspire a new generation of generative models.

Inductive Moment Matching

Impact Statement

This paper advances research in diffusion models and generative AI, which enable new creative possibilities and democratize content creation but also raise important considerations. Potential benefits include expanding artistic expression, assisting content creators, and generating synthetic data for research. However, we acknowledge challenges around potential misuse for deepfakes, copyright concerns, and impacts on creative industries. While our work aims to advance technical capabilities, we encourage ongoing discussion about responsible development and deployment of these technologies.

Acknowledgement

We thank Wanqiao Xu, Bokui Shen, Connor Lin, and Samrath Sinha for additional technical discussions and helpful suggestions.

Albergo, M. S. and Vanden-Eijnden, E. Building normalizing flows with stochastic interpolants. ar Xiv preprint ar Xiv:2209.15571, 2022.

Albergo, M. S., Boffi, N. M., and Vanden-Eijnden, E. Stochastic interpolants: A unifying framework for flows and diffusions. ar Xiv preprint ar Xiv:2303.08797, 2023.

Auffray, Y. and Barbillon, P. Conditionally positive definite kernels: theoretical contribution, application to interpolation and approximation. Ph D thesis, INRIA, 2009.

Bao, F., Nie, S., Xue, K., Cao, Y., Li, C., Su, H., and Zhu, J. All are worth words: A vit backbone for diffusion models. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 22669 22679, 2023.

Berthelot, D., Autef, A., Lin, J., Yap, D. A., Zhai, S., Hu, S., Zheng, D., Talbott, W., and Gu, E. Tract: Denoising diffusion models with transitive closure time-distillation. ar Xiv preprint ar Xiv:2303.04248, 2023.

Blattmann, A., Dockhorn, T., Kulal, S., Mendelevitch, D., Kilian, M., Lorenz, D., Levi, Y., English, Z., Voleti, V., Letts, A., et al. Stable video diffusion: Scaling latent video diffusion models to large datasets. ar Xiv preprint ar Xiv:2311.15127, 2023.

Brock, A. Large scale gan training for high fidelity natural image synthesis. ar Xiv preprint ar Xiv:1809.11096, 2018.

Chang, H., Zhang, H., Jiang, L., Liu, C., and Freeman, W. T. Maskgit: Masked generative image transformer. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 11315 11325, 2022.

Chen, J., Yu, J., Ge, C., Yao, L., Xie, E., Wu, Y., Wang, Z., Kwok, J., Luo, P., Lu, H., et al. Pixart-α: Fast training of diffusion transformer for photorealistic text-to-image synthesis. ar Xiv preprint ar Xiv:2310.00426, 2023.

Chen, N., Zhang, Y., Zen, H., Weiss, R. J., Norouzi, M., and Chan, W. Wavegrad: Estimating gradients for waveform generation. ar Xiv preprint ar Xiv:2009.00713, 2020.

Dhariwal, P. and Nichol, A. Diffusion models beat gans on image synthesis. Advances in neural information processing systems, 34:8780 8794, 2021.

Esser, P., Rombach, R., and Ommer, B. Taming transformers for high-resolution image synthesis. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 12873 12883, 2021.

Esser, P., Kulal, S., Blattmann, A., Entezari, R., M uller, J., Saini, H., Levi, Y., Lorenz, D., Sauer, A., Boesel, F., et al. Scaling rectified flow transformers for high-resolution image synthesis. In Forty-first International Conference on Machine Learning, 2024.

Frans, K., Hafner, D., Levine, S., and Abbeel, P. One step diffusion via shortcut models. ar Xiv preprint ar Xiv:2410.12557, 2024.

Geng, Z., Pokle, A., Luo, W., Lin, J., and Kolter, J. Z. Consistency models made easy. ar Xiv preprint ar Xiv:2406.14548, 2024.

Goodfellow, I., Pouget-Abadie, J., Mirza, M., Xu, B., Warde Farley, D., Ozair, S., Courville, A., and Bengio, Y. Generative adversarial networks. Communications of the ACM, 63(11): 139 144, 2020.

Gretton, A., Borgwardt, K. M., Rasch, M. J., Sch olkopf, B., and Smola, A. A kernel two-sample test. The Journal of Machine Learning Research, 13(1):723 773, 2012.

Heek, J., Hoogeboom, E., and Salimans, T. Multistep consistency models. ar Xiv preprint ar Xiv:2403.06807, 2024.

Heusel, M., Ramsauer, H., Unterthiner, T., Nessler, B., and Hochreiter, S. Gans trained by a two time-scale update rule converge to a local nash equilibrium. Advances in neural information processing systems, 30, 2017.

Ho, J. and Salimans, T. Classifier-free diffusion guidance. ar Xiv preprint ar Xiv:2207.12598, 2022.

Ho, J., Jain, A., and Abbeel, P. Denoising diffusion probabilistic models. Advances in neural information processing systems, 33:6840 6851, 2020.

Ho, J., Chan, W., Saharia, C., Whang, J., Gao, R., Gritsenko, A., Kingma, D. P., Poole, B., Norouzi, M., Fleet, D. J., et al. Imagen video: High definition video generation with diffusion models. ar Xiv preprint ar Xiv:2210.02303, 2022a.

Ho, J., Saharia, C., Chan, W., Fleet, D. J., Norouzi, M., and Salimans, T. Cascaded diffusion models for high fidelity image generation. Journal of Machine Learning Research, 23(47): 1 33, 2022b.

Hoogeboom, E., Heek, J., and Salimans, T. simple diffusion: Endto-end diffusion for high resolution images. In International Conference on Machine Learning, pp. 13213 13232. PMLR, 2023.

Kang, M., Zhu, J.-Y., Zhang, R., Park, J., Shechtman, E., Paris, S., and Park, T. Scaling up gans for text-to-image synthesis. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 10124 10134, 2023.

Inductive Moment Matching

Karras, T., Laine, S., Aittala, M., Hellsten, J., Lehtinen, J., and Aila, T. Analyzing and improving the image quality of stylegan. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 8110 8119, 2020.

Karras, T., Aittala, M., Aila, T., and Laine, S. Elucidating the design space of diffusion-based generative models. Advances in neural information processing systems, 35:26565 26577, 2022.

Karras, T., Aittala, M., Lehtinen, J., Hellsten, J., Aila, T., and Laine, S. Analyzing and improving the training dynamics of diffusion models. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 24174 24184, 2024.

Kim, D., Lai, C.-H., Liao, W.-H., Murata, N., Takida, Y., Uesaka, T., He, Y., Mitsufuji, Y., and Ermon, S. Consistency trajectory models: Learning probability flow ode trajectory of diffusion. ar Xiv preprint ar Xiv:2310.02279, 2023.

Kingma, D. and Gao, R. Understanding diffusion objectives as the elbo with simple data augmentation. Advances in Neural Information Processing Systems, 36, 2024.

Kingma, D., Salimans, T., Poole, B., and Ho, J. Variational diffusion models. Advances in neural information processing systems, 34:21696 21707, 2021.

Kong, Z., Ping, W., Huang, J., Zhao, K., and Catanzaro, B. Diffwave: A versatile diffusion model for audio synthesis. ar Xiv preprint ar Xiv:2009.09761, 2020.

Li, C.-L., Chang, W.-C., Cheng, Y., Yang, Y., and P oczos, B. Mmd gan: Towards deeper understanding of moment matching network. Advances in neural information processing systems, 30, 2017.

Li, T., Tian, Y., Li, H., Deng, M., and He, K. Autoregressive image generation without vector quantization. ar Xiv preprint ar Xiv:2406.11838, 2024.

Li, Y., Swersky, K., and Zemel, R. Generative moment matching networks. In International conference on machine learning, pp. 1718 1727. PMLR, 2015.

Lipman, Y., Chen, R. T., Ben-Hamu, H., Nickel, M., and Le, M. Flow matching for generative modeling. ar Xiv preprint ar Xiv:2210.02747, 2022.

Liu, H., Chen, Z., Yuan, Y., Mei, X., Liu, X., Mandic, D., Wang, W., and Plumbley, M. D. Audioldm: Text-to-audio generation with latent diffusion models. ar Xiv preprint ar Xiv:2301.12503, 2023.

Liu, X., Gong, C., and Liu, Q. Flow straight and fast: Learning to generate and transfer data with rectified flow. ar Xiv preprint ar Xiv:2209.03003, 2022.

Lu, C. and Song, Y. Simplifying, stabilizing and scaling continuous-time consistency models. ar Xiv preprint ar Xiv:2410.11081, 2024.

Lu, C., Zhou, Y., Bao, F., Chen, J., Li, C., and Zhu, J. Dpm-solver: A fast ode solver for diffusion probabilistic model sampling in around 10 steps. Advances in Neural Information Processing Systems, 35:5775 5787, 2022.

Luhman, E. and Luhman, T. Knowledge distillation in iterative generative models for improved sampling speed. ar Xiv preprint ar Xiv:2101.02388, 2021.

Luo, W., Hu, T., Zhang, S., Sun, J., Li, Z., and Zhang, Z. Diffinstruct: A universal approach for transferring knowledge from pre-trained diffusion models. Advances in Neural Information Processing Systems, 36, 2024a.

Luo, W., Huang, Z., Geng, Z., Kolter, J. Z., and Qi, G.-j. One-step diffusion distillation through score implicit matching. ar Xiv preprint ar Xiv:2410.16794, 2024b.

Ma, N., Goldstein, M., Albergo, M. S., Boffi, N. M., Vanden Eijnden, E., and Xie, S. Sit: Exploring flow and diffusion-based generative models with scalable interpolant transformers. ar Xiv preprint ar Xiv:2401.08740, 2024.

Meng, C., Rombach, R., Gao, R., Kingma, D., Ermon, S., Ho, J., and Salimans, T. On distillation of guided diffusion models. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 14297 14306, 2023.

M uller, A. Integral probability metrics and their generating classes of functions. Advances in applied probability, 29(2):429 443, 1997.

Nichol, A. Q. and Dhariwal, P. Improved denoising diffusion probabilistic models. In International conference on machine learning, pp. 8162 8171. PMLR, 2021.

Open AI. Video generation models as world simulators. https: //openai.com/sora/, 2024.

Peebles, W. and Xie, S. Scalable diffusion models with transformers. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 4195 4205, 2023.

Podell, D., English, Z., Lacey, K., Blattmann, A., Dockhorn, T., M uller, J., Penna, J., and Rombach, R. Sdxl: Improving latent diffusion models for high-resolution image synthesis. ar Xiv preprint ar Xiv:2307.01952, 2023.

Rombach, R., Blattmann, A., Lorenz, D., Esser, P., and Ommer, B. High-resolution image synthesis with latent diffusion models. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 10684 10695, 2022.

Saharia, C., Chan, W., Saxena, S., Li, L., Whang, J., Denton, E. L., Ghasemipour, K., Gontijo Lopes, R., Karagol Ayan, B., Salimans, T., et al. Photorealistic text-to-image diffusion models with deep language understanding. Advances in neural information processing systems, 35:36479 36494, 2022.

Salimans, T. and Ho, J. Progressive distillation for fast sampling of diffusion models. ar Xiv preprint ar Xiv:2202.00512, 2022.

Salimans, T., Mensink, T., Heek, J., and Hoogeboom, E. Multistep distillation of diffusion models via moment matching. ar Xiv preprint ar Xiv:2406.04103, 2024.

Sauer, A., Lorenz, D., Blattmann, A., and Rombach, R. Adversarial diffusion distillation. In European Conference on Computer Vision, pp. 87 103. Springer, 2025.

Sohl-Dickstein, J., Weiss, E., Maheswaranathan, N., and Ganguli, S. Deep unsupervised learning using nonequilibrium thermodynamics. In International conference on machine learning, pp. 2256 2265. PMLR, 2015.

Inductive Moment Matching

Song, J., Meng, C., and Ermon, S. Denoising diffusion implicit models. ar Xiv preprint ar Xiv:2010.02502, 2020a.

Song, Y. and Dhariwal, P. Improved techniques for training consistency models. ar Xiv preprint ar Xiv:2310.14189, 2023.

Song, Y., Sohl-Dickstein, J., Kingma, D. P., Kumar, A., Ermon, S., and Poole, B. Score-based generative modeling through stochastic differential equations. ar Xiv preprint ar Xiv:2011.13456, 2020b.

Song, Y., Dhariwal, P., Chen, M., and Sutskever, I. Consistency models. ar Xiv preprint ar Xiv:2303.01469, 2023.

Steinwart, I. and Christmann, A. Support vector machines. Springer Science & Business Media, 2008.

Tee, J. T. J., Zhang, K., Yoon, H. S., Gowda, D. N., Kim, C., and Yoo, C. D. Physics informed distillation for diffusion models. ar Xiv preprint ar Xiv:2411.08378, 2024.

Tian, K., Jiang, Y., Yuan, Z., Peng, B., and Wang, L. Visual autoregressive modeling: Scalable image generation via nextscale prediction. ar Xiv preprint ar Xiv:2404.02905, 2024a.

Tian, Y., Tu, Z., Chen, H., Hu, J., Xu, C., and Wang, Y. U-dits: Downsample tokens in u-shaped diffusion transformers. ar Xiv preprint ar Xiv:2405.02730, 2024b.

Xiao, Z., Kreis, K., and Vahdat, A. Tackling the generative learning trilemma with denoising diffusion gans. ar Xiv preprint ar Xiv:2112.07804, 2021.

Xu, Y., Deng, M., Cheng, X., Tian, Y., Liu, Z., and Jaakkola, T. Restart sampling for improving generative processes. Advances in Neural Information Processing Systems, 36:76806 76838, 2023.

Yin, T., Gharbi, M., Zhang, R., Shechtman, E., Durand, F., Freeman, W. T., and Park, T. One-step diffusion with distribution matching distillation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 6613 6623, 2024.

Zhang, R., Isola, P., Efros, A. A., Shechtman, E., and Wang, O. The unreasonable effectiveness of deep features as a perceptual metric. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 586 595, 2018.

Zheng, H., Nie, W., Vahdat, A., Azizzadenesheli, K., and Anandkumar, A. Fast sampling of diffusion models via operator learning. In International conference on machine learning, pp. 42390 42402. PMLR, 2023.

Zhou, M., Zheng, H., Wang, Z., Yin, M., and Huang, H. Score identity distillation: Exponentially fast distillation of pretrained diffusion models for one-step generation. In Forty-first International Conference on Machine Learning, 2024.

Inductive Moment Matching

A. Background: Properties of Stochastic Interpolants

We note some relevant properties of stochastic interpolants for our exposition.

Boundary satisfaction. For an interpolant distribution qt(xt|x, ϵ) defined in Albergo et al. (2023), and the marginal qt(xt) as defined in Eq. (2), we can check that q1(x1) = p(x1) and q0(x0) = q(x0) so that x1 = ϵ and x0 = x.

q1(x1) = ZZ q1(x1|x, ϵ)q(x)p(ϵ)dxdϵ (15)

= ZZ δ(x1 ϵ)q(x)p(ϵ)dxdϵ (16)

= ZZ q(x)p(x1)dx (17)

= p(x1) (18)

q0(x0) = ZZ q0(x0|x, ϵ)q(x)p(ϵ)dxdϵ (19)

= ZZ δ(x0 x)q(x)p(ϵ)dxdϵ (20)

= ZZ q(x0)p(ϵ)dx (21)

= q(x0) (22)

Joint distribution. The joint distribution of x and xt is written as

qt(x, xt) = Z qt(xt|x, ϵ)q(x)p(ϵ)dϵ (23)

Indepedence of joint at t = 1.

q1(x, x1) = Z q1(x1|x, ϵ)q(x)p(ϵ)dϵ (24)

= Z δ(x1 ϵ)q(x)p(ϵ)dϵ (25)

= q(x)p(x1) (26)

= q(x)p(ϵ) (27)

in which case x1 = ϵ.

B. Theorems and Derivations

B.1. Divergence Minimizer

Lemma 3. Assuming marginal-preserving interpolant and metric D( , ), a minimizer θ of Eq. (7) exists, i.e. pθ s|t(x|xt) = qt(x|xt), and the minimum is 0.

Proof. We directly substitute qt(x|xt) into the objective to check. First,

pθ s|t(xs) = ZZ qs|t(xs|x, xt)pθ s|t(x|xt)qt(xt)dxtdx (28)

= ZZ qs|t(xs|x, xt)qt(x|xt)qt(xt)dxtdx (29)

(a) = qs(xs) (30)

Inductive Moment Matching

where (a) is due to definition of marginal preservation. So the objective becomes

Es,t h w(s, t)D(qs(xs), pθ s|t(xs)) i = Es,t [w(s, t)D(qs(xs), qs(xs))] = 0 (31)

In general, the minimizer qt(x|xt) exists. However, this does not show that the minimizer is unique. In fact, the minimizer is not unique in general because a deterministic minimizer can also exist under certain assumptions on the interpolant (see Appendix B.6).

Failure Case without Marginal Preservation. We additionally show that marginal-preservation property of the interpolant qs|t(xs|x, xt) is important for the na ıve objective in Eq. (7) to attain 0 loss (Lemma 3). Consider the failure case below where the constructed interpolant is a generalized interpolant but not necessarily marginal-preserving. Then we show that there exists a t such that pθ s|t(xs) can never reach qs(xs) regardless of θ.

Proposition 2 (Example Failure Case). Let q(x) = δ(x), p(ϵ) = δ(ϵ 1), and suppose an interpolant Is|t(x, xt) = (1 s

t xt and γs|t = s

t 1(t < 1), then D(qs(xs), pθ s|t(xs)) > 0 for all 0 < s < t < 1 regardless of the learned distribution pθ s|t(x|xt) given any metric D( , ).

Proof. This example first implies the learning target

qs(xs) = ZZ qs(xs|x, ϵ)q(x)p(ϵ)dxdϵ (32)

= ZZ δ(xs ((1 s)x + sϵ))δ(x)δ(ϵ 1)dxdϵ (33)

= δ(xs s) (34)

is a delta distribution. However, we show that if we select any t < 1 and 0 < s < t, pθ s|t(xs) can never be a delta distribution.

pθ s|t(xs) = ZZ qs|t(xs|x, xt)pθ s|t(x|xt)qt(xt)dxtdx (35)

= ZZ N((1 s

t2 )I)pθ s|t(x|xt)δ(xt t)dxtdx (36)

t )x + s, s2

t2 )I)pθ s|t(x|xt = t)dx (37)

Now, we show the model distribution has non-zero variance under these choices of t and s. Expectations are over pθ s|t(xs) or conditional interpolant qs|t(xs|x, xt) for all equations below.

Var(xs) = E x2 s E [xs]2 (38)

= ZZ E x2 s|x, xt pθ s|t(x|xt)qt(xt)dxdxt E [xs]2 (39)

= ZZ E x2 s|x, xt pθ s|t(x|xt)qt(xt)dxdxt E [xs]2 (40)

= ZZ h Var(xs|x, xt) + E [xs|x, xt]2i pθ s|t(x|xt)qt(xt)dxdxt 2E [xs]2 + E [xs]2 (41)

= ZZ h Var(xs|x, xt) + E [xs|x, xt]2i pθ s|t(x|xt)qt(xt)dxdxt (42)

ZZ E [xs] E [xs|xxt] pθ s|t(x|xt)qt(xt)dxdxt + E [xs]2

= ZZ h Var(xs|x, xt) + E [xs|x, xt]2 E [xs] E [xs|x, xt] + E [xs]2i

pθ s|t(x|xt)qt(xt)dxdxt (43)

Inductive Moment Matching

where (a) can be simplified as

Var(xs|x, xt) + E [xs|x, xt]2 E [xs] 2 > 0 (44)

because Var(xs|x, xt) > 0 for all 0 < s < t < 1 due to its non-zero Gaussian noise. Therefore, Var(xs) > 0, implying pθ s|t(xs) can never be a delta function regardless of model pθ s|t(x|xt). A valid metric D( , ) over probability pθ s|t(xs) and qs(xs) implies D(qs(xs), pθ s|t(xs)) = 0 pθ s|t(xs) = qs(xs)

which means pθ s|t(xs) = qs(xs) = D(qs(xs), pθ s|t(xs)) > 0

B.2. Boundary Satisfaction of Model Distribution

The operator output pθ s|t(xs) satisfies boundary condition.

Lemma 4 (Boundary Condition). For all s [0, 1] and all θ, the following boundary condition holds.

qs(xs) = pθ s|s(xs) (45)

pθ s|s(xs) = ZZ qs,s(xs|x, xs) | {z } δ(xs xs)

pθ s|s(x| xs)qs,1( xs)d xsdx

= Z pθ s|s(x|xs)qs(xs)dx

= qs(xs) Z pθ s|s(x|xs)dx

B.3. Definition of Well-Conditioned r(s, t)

For simplicity, the mapping function r(s, t) is well-conditioned if

r(s, t) = max(s, t (t)) (46)

where (t) ϵ > 0 is a positive function such that r(s, t) is increasing for t c0(s) where c0(s) is the largest t that is mapped to s. Formally, c0(s) = sup{t : r(s, t) = s}. For t c0(s), the inverse w.r.t. t exists, i.e. r 1(s, ) and r 1(s, r(s, t)) = t. All practical implementations follow this general form, and are detailed in Appendix C.6.

B.4. Main Theorem

Theorem 1. Assuming r(s, t) is well-conditioned, the interpolant is marginal-preserving, and θ n is a minimizer of Eq. (8) for each n with infinite data and network capacity, for all t [0, 1], s [0, t],

lim n MMD2(qs(xs), pθ n s|t(xs)) = 0. (9)

Proof. We prove by induction on sequence number n. First, r(s, t) is well-conditioned by following the definition in Eq. (46). Furthermore, for notational convenience, we let r 1 n (s, ) := r 1(s, r 1(s, r 1(s, . . . ))) be n nested application of r 1(s, ) on the second argument. Additionally, r 1 0 (s, t) = t.

Inductive Moment Matching

Base case: n = 1. Given any s 0, r(s, u) = s for all s < u c0(s), implying

MMD2(pθ0 s|s(xs), pθ 1 s|u(xs)) (a) = MMD2(qs(xs), pθ 1 s|u(xs)) (b) = 0 (47)

for u c0(s) where (a) is implied by Lemma 4 and (b) is implied by Lemma 3.

Inductive assumption: n 1. We assume p θ n 1 s|u (xs) = qs(xs) for all s u r 1 n 2(s, c0(s)).

We inspect the target distribution p θ n 1 s|r(s,u)(xs) in Eq. (8) if optimized on s u r 1 n 1(s, c0(s)). On this interval, we can apply r(s, ) to the inequality and get s = r(s, s) r(s, u) r(s, r 1 n 1(s, c0(s))) = r 1 n 2(s, c0(s)) since r(s, ) is

increasing. And by inductive assumption p θ n 1 s|r(s,u)(xs) = qs(xs) for s r(s, u) r 1 n 2(s, c0(s)), this implies minimizing

Es,u h w(s, u)MMD2(p θ n 1 s|r(s,u)(xs), pθn s|u(xs)) i

on s u r 1 n 1(s, c0(s)) is equivalent to minimizing

Es,u h w(s, u)MMD2(qs(xs), pθn s|u(xs)) i (48)

for s u r 1 n 1(s, c0(s)). Lemma 3 implies that its minimum achieves pθ n s|u(xs) = qs(xs).

Lastly, taking n implies limn r 1 n (s, c0(s)) = 1 and thus the induction covers the entire [s, 1] interval given each s. Therefore, limn MMD(qs(xs), pθ n s|t(xs)) = 0 for all 0 s t 1.

B.5. Self-Consistency Implies Marginal Preservation

Without assuming marginal preservation, it is important to define the marginal distribution of xs under generalized interpolants qs|t(xs|x, xt) as

qs|t(xs) = ZZ qs|t(xs|x, xt)qt(x|xt)qt(xt)dxtdx (49)

and we show that with self-consistent interpolants, this distribution is invariant of t, i.e. qs|t(xs) = qs(xs). Lemma 5. If the interpolant qs|t(xs|x, xt) is self-consistent, the marginal distribution qs|t(xs) as defined in Eq. (49) satisfies qs(xs) = qs|t(xs) for all t [s, 1].

Proof. For t [s, 1],

qs|t(xs) = ZZ qs|t(xs|x, xt))qt(x|xt)qt(xt)dxtdx (50)

= ZZ qs|t(xs|x, xt))qt(xt) Z qt(xt|x, ϵ)q(x)p(ϵ)

qt(xt) dϵdxtdx (51)

= ZZ qs|t(xs|x, xt)q(x) Z qt(xt|x, ϵ)p(ϵ)dϵdxtdx (52)

= ZZ q(x)p(ϵ) Z qs|t(xs|x, xt)qt(xt|x, ϵ)dxt

= ZZ q(x)p(ϵ) Z qs|t(xs|x, xt)qt,1(xt|x, ϵ)xt

(a) = ZZ qs,1(xs|x, ϵ)q(x)p(ϵ)dϵdx (55)

(b) = ZZ qs(xs|x, ϵ)q(x)p(ϵ)dϵdx (56)

= qs(xs) (57)

where (a) uses definition of self-consistent interpolants and (b) uses definition of our generalized interpolant.

Inductive Moment Matching

We show in Appendix C.1 that DDIM is an example self-consistent interpolant. Furthermore, DDPM posteior (Ho et al., 2020; Kingma et al., 2021) is also self-consistent (see Lemma 6).

B.6. Existence of Deterministic Minimizer

We present the formal statement for the deterministic minimizer. Proposition 3. If for all t [0, 1], s [0, t], γs|t 0, Is|t(x, xt) is invertible w.r.t. x, and there exists C1 < such that It|1(x, ϵ) < C1 x ϵ , then there exists a function hs|t : RD RD such that

qs(xs) = ZZ qs|t(xs|x, xt)δ(x hs|t(xt))qt(xt)dxdxt. (58)

Proof. Let I 1 s|t ( , xt) be the inverse of Is|t w.r.t. x such that I 1 s|t (Is|t(x, xt), xt) = x and Is|t(I 1 s|t (y, xt), xt) = y for all x, xt, y RD. Since there exists C1 < such that It|1(x, ϵ) < C1 x ϵ for all t [0, 1], the PF-ODE of the original interpolant It|1(x, ϵ) = It(x, ϵ) exists for all t [0, 1] (Albergo et al., 2023). Then, for all t [0, 1], s [0, t], we let

ˆhs|t(xt) = xt + Z s

u Iu(x, ϵ) xu

which pushes forward the measure qt(xt) to qs(xs). We define:

hs|t(xt) = I 1 s|t (ˆhs|t(xt), xt). (60)

Then, since γs|t 0, qs|t(xs|x, xt) = δ(xs Is|t(x, xt)) where x δ(x hs|t(xt)). Therefore,

xs = Is|t(hs|t(xt), xt) = Is|t(I 1 s|t (ˆhs|t(xt), xt), xt) = ˆhs|t(xt) (61)

whose marginal follows qs(xs) due to it being the result of PF-ODE trajectories starting from qt(xt).

Concretely, DDIM interpolant satisfies all of the deterministic assumption, the regularity condition, and the invertibility assumption because it is a linear function of x and xt. Therefore, any diffusion or FM schedule with DDIM interpolant will enjoy a deterministic minimizer pθ s|t(x|xt).

C. Analysis of Simplified Parameterization

C.1. DDIM Interpolant

We check that DDIM interpolant is self-consistent. By definition, qs|t(xs|x, xt) = δ(xs DDIM(xt, x, t, s)). We check that for all s r t, Z qs|r(xs|x, xr)qr|t(xr|x, xt)dxr = Z δ(xs DDIM(xr, x, r, s))δ(xr DDIM(xt, x, t, r))dxr

= δ(xs DDIM(DDIM(xt, x, t, r), x, r, s))

DDIM(DDIM(xt, x, t, r), x, r, s) = αsx + (σs/σr)([αrx + (σr/σt)(xt αtx)] αrx)

= αsx + (σs/σr)(σr/σt)(xt αtx)

= αsx + (σs/σt)(xt αtx)

= DDIM(xt, x, t, s)

Therefore, δ(xs DDIM(DDIM(xt, x, t, r), x, r, s)) = δ(xs DDIM(xt, x, t, s)). So DDIM is self-consistent.

It also implies a Gaussian forward process qt(xt|x) = N(αtx, σ2 t σ2 d I) as in diffusion models. By definition,

qt,1(xt|x) = qt(xt|x) = Z qt(xt|x, ϵ)p(ϵ)dϵ

so that xt is a deterministic transform given x and ϵ, i.e., xt = DDIM(ϵ, x, t, 1) = αtx + σtϵ, which implies qt(xt|x) = N(αtx, σ2 t σ2 d I).

Inductive Moment Matching

C.2. Reusing xt for xr

We propose that instead of sampling xr via forward flow αrx + σrϵ we reuse xt such that xr = DDIM(xt, x, r, t) to reduce variance. In fact, for any self-consistent interpolant, one can reuse xt via xr qr|t(xr|x, xt) and xr will follow qr(xr) marginally. We check

qr(xr) (a) = qr|t(xr) = ZZ qr|t(xr|x, xt)qt(x|xt)qt(xt)dxdxt = ZZ qr|t(xr|x, xt) Z qt(xt|x, ϵ)q(x)p(ϵ)dϵdxdxt

where (a) is due to Lemma 5. We can see that sampling x, xt first then xr qr|t(xr|x, xt) respects the marginal distribution qr(xr).

C.3. Simplified Objective

We derive our simplified objective. Given MMD defined in Eq. (1), we write our objective as

LIMM(θ) = Es,t h w(s, t) Ext[k(f θ s,t(xt), )] Exr[k(f θ s,r (xr), )] 2

(a) = Es,t h w(s, t) Ext,xr[k(f θ s,t(xt), ) k(f θ s,r (xr), )] 2

= Es,t h w(s, t) D Ext,xr[k(f θ s,t(xt), ) k(f θ s,r (xr), )], Ex t,x r[k(f θ s,t(x t), ) k(f θ s,r (x r), )] Ei (64)

= Es,t h w(s, t) Ext,xr,x t,x r h k(f θ s,t(xt), ), k(f θ s,t(x t), ) + k(f θ s,r (xr), ), k(f θ s,r (x r), ) (65)

k(f θ s,t(xt), ), k(f θ s,r (x r), ) k(f θ s,t(x t), ), k(f θ s,r (xr), ) ii

= Ext,xr,x t,x r,s,t h w(s, t) h k(f θ s,t(xt), f θ s,t(x t)) + k(f θ s,r (xr), f θ s,r (x r)) (66)

k(f θ s,t(xt), f θ s,r (x r)) k(f θ s,t(x t), f θ s,r (xr)) ii

where , is in RKHS, (a) is due to the correlation between xr and xt by re-using xt.

C.4. Empirical Estimation

As proposed in Gretton et al. (2012), MMD is typically estimated with V-statistics by instantiating a matrix of size M M such that a batch of B x samples, {x(i)}B i=1, is separated into groups of M (assume B is divisible by M) particles {x(i,j)}B/M,M i=1,j=1 where each group share a (si, ri, ti) sample. The Monte Carlo estimate becomes

ˆLIMM(θ) = 1 B/M

i=1 w(si, ti) 1

h k(f θ si,ti(x(i,j) ti ), f θ si,ti(x(i,k) ti )) + k(f θ si,ri(x(i,j) ri ), f θ si,ri(x(i,k) ri )) (67)

2k(f θ si,ti(x(i,j) ti ), f θ si,ri(x(i,k) ri )) i

Computational efficiency. First we note that regardless of M, we require only 2 model forward passes - one with and one without stop gradient, since the model takes in all B instances together within the batch and produce outputs for the entire batch. For the calculation of our loss, although the need for M particles may imply inefficient computation, the cost of this matrix computation is negligible in practice compared to the complexity of model forward pass. Suppose a forward pass for a single instance is O(K), then the total computation for computation loss for a batch of B instances is O(BK) + O(BM). Deep neural networks often has K M, so O(BK) dominates the computation.

C.5. Simplified Parameterization

We derive f θ s,t(xt) for each parameterization, which now generally follows the form

f θ s,t(xt) = cskip(s, t)xt + cout(s, t)Gθ(cin(s, t)xt, cnoise(s), cnoise(t))

Inductive Moment Matching

Identity. This is simply DDIM with x-prediction network.

f θ s,t(xt) = (αs σs

α2 t + σ2 t , cnoise(s), cnoise(t)

Simple-EDM.

f θ s,t(xt) = (αs σs

" αt α2 t + σ2 t xt σt p

α2 t + σ2 t σd Gθ

α2 t + σ2 t , cnoise(s), cnoise(t)

= αsαt + σsσt

α2 t + σ2 t xt αsσt σsαt p

α2 t + σ2 t σd Gθ

α2 t + σ2 t , cnoise(s), cnoise(t)

When noise schedule is cosine, f θ s,t(xt) = cos 1

2π(t s) xt sin 1

2π(t s) σd Gθ

α2 t +σ2 t , cnoise(s), cnoise(t) .

And similar to Lu & Song (2024), we can show that predicting xs = DDIM(xt, x, s, t) with L2 loss is equivalent to v-prediction with cosine schedule.

w(s, t) f θ s,t(xt) (αs σs

σt αt)x + σs

αsαt + σsσt

α2 t + σ2 t xt αsσt σsαt p

α2 t + σ2 t σd Gθ

α2 t + σ2 t , cnoise(s), cnoise(t)

σt αt)x + σs

α2 t + σ2 t , cnoise(s), cnoise(t)

α2 t + σ2 t αsσt σsαt

σt αt)x + σs

σt xt αsαt + σsσt

α2 t + σ2 t xt

| {z } Gtarget

α2 t + σ2 t αsσt σsαt

σt αt)x + σs

σt xt αsαt + σsσt

α2 t + σ2 t xt

α2 t + σ2 t αsσt σsαt

σt αt)x + σsα2 t + σsσ2 t αsαtσt σsσ2 t σt(α2 t + σ2 t ) (αtx + σtϵ)

α2 t + σ2 t αsσt σsαt

! (αsσt σsαt)(α2 t + σ2 t ) + σsα3 t αsα2 tσt σt(α2 t + σ2 t ) x + σsα2 t αsαtσt α2 t + σ2 t ϵ

α2 t + σ2 t αsσt σsαt

! (αsσt σsαt)(α2 t + σ2 t ) (αsσt σsαt)α2 t α2 t + σ2 t x (αsσt σsαt)αt

σt(α2 t + σ2 t ) ϵ

= αtϵ σtx p

α2 t + σ2 t

This reduces to v-target if cosine schedule is used, and it deviates from v-target if FM schedule is used instead.

Inductive Moment Matching

Euler-FM. We assume OT-FM schedule.

f θ s,t(xt) = ((1 s) s

α2 t + σ2 t , cnoise(s), cnoise(t)

α2 t + σ2 t , cnoise(s), cnoise(t)

= xt (t s)σd Gθ

α2 t + σ2 t , cnoise(s), cnoise(t)

This results in Euler ODE from xt to xs. We also show that the network output reduces to v-prediction if matched with xs = DDIM(xt, x, s, t). To see this,

w(s, t) f θ s,t(xt) ((1 s) s

t (1 t))x + s

= w(s, t) f θ s,t(xt) (1 s

xt (t s)σd Gθ

α2 t + σ2 t , cnoise(s), cnoise(t)

α2 t + σ2 t , cnoise(s), cnoise(t)

α2 t + σ2 t , cnoise(s), cnoise(t)

| {z } Gtarget

Gtarget = 1 (t s)

t 1)((1 t)x + tϵ)

t s 1 + t)x + (s t)ϵ

((t s)x + (s t)ϵ)

which is v-target under OT-FM schedule. This parameterization naturally allows zero-SNR sampling and satisfies boundary condition at s = 0, similar to Simple-EDM above. This is not true for Identity parametrization using Gθ as it satisfies boundary condition only at s > 0.

C.6. Mapping Function r(s, t)

We discuss below the concrete choices for r(s, t). We use a constant decrement ϵ > 0 in different spaces.

Constant decrement in η(t) := ηt = σt/αt. This is the choice that we find to work better than other choices in practice. First, let its inverse be η 1( ),

r(s, t) = max s, η 1(η(t) ϵ)

We choose ϵ = (ηmax ηmin)/2k for some k. We generally choose ηmax 160 and ηmin 0. We find k = {10, . . . , 15} works well enough depending on datasets.

Inductive Moment Matching

Constant decrement in t.

r(s, t) = max (s, t ϵ)

We choose ϵ = (T ϵ)/2k.

Constant decrement in λ(t) := log-SNRt = 2 log(αt/σt).

Let its inverse be λ 1( ), then

r(s, t) = max s, λ 1(λ(t) ϵ))

We choose ϵ = (λmax λmin)/2k. This choice comes close to the first choice, but we refrain from this because r(s, t) becomes close to t both when t 0 and t 1 instead of just t 1. This gives more chances for training instability than the first choice.

Constant increment in 1/η(t).

r(s, t) = max s, η 1 1 1/η(t) + ϵ

We choose ϵ = (1/η(t)min 1/η(t)max)/2k.

C.7. Time Distribution p(s, t)

In all cases we choose p(t) = U(ϵ, T) and p(s|t) = U(ϵ, t) for some ϵ 0 and T 1. The decision for time distribution is coupled with r(s, t). We list the constraints on p(s, t) for each r(s, t) choice below.

Constant decrement in η(t). We need to choose T < 1 because, for example, assuming OT-FM schedule, ηt = t/(1 t), one can observe that constant decrement in ηt when t 1 results in r(s, t) that is too close to t due to ηt s exploding gradient around 1. We need to define T < 1 such that r(s, T) is not too close to T for s reasonably far away. With ηmax 160, we can choose T = 0.994 for OT-FM and T = 0.996 for VP-diffusion.

Constant decrement in t. No constraints needed. T = 1, ϵ = 0.

Constant decrement in λt. One can similarly observe exploding gradient causing r(s, t) to be too close to t at both t 0 and t 1, so we can choose ϵ > 0, e.g. 0.001, in addition to choosing T = 0.994 for OT-FM and T = 0.996 for VP-diffusion.

Constant increment in 1/ηt. This experience exploding gradient for t 0, so we require ϵ > 0, e.g. 0.005. And T = 1.

C.8. Kernel Function

For our Laplace kernel k(x, y) = exp( w(s, t) max( x y 2, ϵ)/D), we let ϵ > 0 be a reasonably small constant, e.g. 10 8. Looking at its gradient w.r.t. x,

xe w(s,t) max( x y 2,ϵ)/D =

D e w(s,t) x y 2/D x y x y 2 , if x y 2 > ϵ

0, otherwise

one can notice that the gradient is self-normalized to be a unit vector, which is helpful in practice. In comparison, the gradient of RBF kernel of the form k(x, y) = exp 1

2 w(s, t) x y 2/D ,

xe w(s,t) x y 2/D = w(s, t)

D e w(s,t) 1

2 x y 2/D(x y) (68)

whose magnitude can vary a lot depending on how far x is from y.

For w(s, t), we find it helpful to write out the L2 loss between the arguments. For simplicity, we denote ˆGθ(xt, s, t) =

Inductive Moment Matching

α2 t +σ2 t , cnoise(s), cnoise(t)

w(s, t) f θ s,t(xt) f θ s,r (x r) 2

= w(s, t) cskip(s, t)xt + cout(s, t) ˆGθ(xt, s, t) cskip(s, r)x r + cout(s, r) ˆG(x r, s, r) 2

= w(s, t)cout(s, t) ˆGθ(xt, s, t) 1 cout(s, t)

cskip(s, r)x r + cout(s, r) ˆG(x r, s, r) cskip(s, t)xt 2

We simply set w(s, t) = 1/cout(s, t) for the overall weighting to be 1. This allows invariance of magnitude of kernels w.r.t. t.

C.9. Weighting Function w(s, t)

To review VDM (Kingma et al., 2021), the negative ELBO loss for diffusion model is

LELBO(θ) = 1

ϵθ(xt, t) ϵ 2 (69)

where ϵθ is the noise-prediction network and λt = log-SNRt. The weighted-ELBO loss proposed in Kingma & Gao (2024) introduces an additional weighting function w(t) monotonically increasing in t (monotonically decreasing in log-SNRt) understood as a form of data augmentation. Specifically, they use sigmoid as the function such that the weighted ELBO is written as

Lw-ELBO(θ) = 1

ϵθ(xt, t) ϵ 2 (70)

where σ( ) is sigmoid function.

The αt is tailored towards the Simple-EDM and Euler-FM parameterization as we have shown in Appendix C.5 that the networks σd Gθ amounts to v-prediction in cosine and OT-FM schedules. Notice that ELBO diffusion loss matches ϵ instead of v. Inspecting the gradient of Laplace kernel, we have (again, for simplicity we let ˆGθ(xt, s, t) = Gθ( xt σd

α2 t +σ2 t , cnoise(s), cnoise(t)))

θe w(s,t) f θ s,t(xt) f θ s,r (xr) 2/D

D e w(s,t) f θ s,t(xt) f θ s,r (xr) 2/D f θ s,t(xt) f θ s,r (xr) f θ s,t(xt) f θ s,r (xr) 2

θf θ s,t(xt)

D e w(s,t) f θ s,t(xt) f θ s,r (xr) 2/D ˆGθ(xt, s, t) ˆGtarget ˆGθ(xt, s, t) ˆGtarget 2

θf θ s,t(xt)

for some constant ˆGtarget. We can see that gradient θf θ s,t(xt) is guided by vector ˆGθ(xt, s, t) ˆGtarget. Assuming ˆGθ(xt, s, t) is v-prediction, as is the case for Simple-EDM parameterization with cosine schedule and Euler-FM parameterization with OT-FM schedule, we can reparameterize vto ϵ-prediction with ϵθ as the new parameterization. We omit arguments to network for simplicity.

We show below that for both cases ϵθ ϵtarget = αt( ˆGθ ˆGtarget) for some constants ϵtarget and ˆGtarget. For Simple-EDM, we know x-prediction from v-prediction parameterization (Salimans & Ho, 2022), xθ = αtxt σt ˆGθ, and we also know

Inductive Moment Matching

x-prediction from ϵ-prediction, xθ = (xt σtϵθ)/αt. We have

αt = αtxt σt ˆGθ (71)

xt σtϵθ = α2 txt αtσt ˆGθ (72)

(1 α2 t)xt + αtσt ˆGθ = σtϵθ (73)

σ2 t xt + αtσt ˆGθ = σtϵθ (74)

ϵθ = σtxt + αt ˆGθ (75)

ϵθ ϵtarget = σtxt + αt ˆGθ σtxt + αt ˆGtarget (76)

ϵθ ϵtarget = αt( ˆGθ ˆGtarget) (77)

For Euler-FM, we know x-prediction from v-prediction parameterization, xθ = xt t ˆGθ and we also know x-prediction from ϵ-prediction, xθ = (xt tϵθ)/(1 t). We have

1 t = xt t ˆGθ (78)

xt tϵθ = (1 t)xt t(1 t) ˆGθ (79)

txt + t(1 t) ˆGθ = tϵθ (80)

ϵθ = xt + (1 t) ˆGθ (81)

ϵθ = xt + αt ˆGθ (82)

ϵθ ϵtarget = xt + αt ˆGθ xt + αt ˆGtarget (83)

ϵθ ϵtarget = αt( ˆGθ ˆGtarget) (84)

In both cases, ( ˆGθ(xt, s, t) ˆGtarget) can be rewritten to (ϵθ(xt, s, t) ϵtarget) by multiplying a factor αt, and the guidance vector now matches that of the ELBO-diffusion loss. Therefore, we are motivated to incorporate αt into w(s, t) as proposed.

The exponent a for αa t takes a value of either 1 or 2. We explain the reason for each decision here. When a = 1, we guide the gradient θf θ s,t(xt), with score difference (ϵθ(xt, s, t) ϵtarget). To motivate a = 2, we first note that the weighted gradient

θf θ s,t(xt) = 1 |cout(s, t)| θf θ s,t(xt) =

θ ˆGθ(xt, s, t)

and as shown above that ϵ-prediction is parameterized as

ϵθ(xt, s, t) = xt + αt ˆGθ(xt, s, t).

Multiplying an additional αt to θ ˆGθ(xt, s, t) therefore implicitly reparameterizes our model into an ϵ-prediction model. The case of a = 2 therefore implicitly reparameterizes our model into an ϵ-prediction model guided by the score difference (ϵθ(xt, s, t) ϵtarget). Empirically, α2 t additionally downweights loss for larger t compared to αt, allowing the model to train on smaller time-steps more effectively.

Lastly, the division of α2 t + σ2 t is inspired by the increased weighting for middle time-steps (Esser et al., 2024) for Flow Matching training. This is purely an empirical decision.

D. Training Algorithm

We present the training algorithm in Algorithm 3.

E. Classifier-Free Guidance

We refer readers to Appendix C.5 for analysis of each parameterization. Most notably, the network Gθ in both (1) Simple EDM with cosine diffusion schedule and (2) Euler-FM with OT-FM schedule are equivalent to v-prediction parameterization

Inductive Moment Matching

Algorithm 3 IMM Training

Input: model f θ, data distribution q(x) and label distribution q(c|x) (if label is used), prior distribution N(0, σ2 d I), time distribution p(t) and p(s|t), DDIM interpolator DDIM(xt, x, s, t) and its flow coefficients αt, σt, mapping function r(s, t), kernel function k( , ), weighting function w(s, t), batch size B, particle number M, label dropout probability p Output: learned model f θ

Initialize n 0, θ0 θ while model not converged do

Sample a batch of data, label, and prior, and split into B/M groups, {(x(i,j), c(i,j), ϵ(i,j))}B/M,M i=1,j=1 For each group, sample {(si, ti)}B/M i=1 and ri = r(si, ti) for each i. This results in a tuple {(si, ri, ti)}B/M i=1 x(i,j) ti DDIM(ϵ(i,j), x(i,j), ti, 1) = αtix(i,j) + σtiϵ(i,j), (i, j) x(i,j) ri DDIM(x(i,j) ti , x(i,j), ri, ti), (i, j) (Optional) Randomly drop each label c(i,j) to be null token with probability p θn+1 optimizer step by minimizing ˆLIMM(θn) using model f θn (see Eq. (67)) (optionally inputting c(i,j) into network) end while

in diffusion (Salimans & Ho, 2022) and FM (Lipman et al., 2022). When conditioned on label c during sampling, it is customary to use classifier-free guidance to reweight this v-prediction network via

Gw θ (cin(t)xt, cnoise(s), cnoise(t), c) (85)

= w Gθ(cin(t)xt, cnoise(s), cnoise(t), c) + (1 w)Gθ(cin(t)xt, cnoise(s), cnoise(t), )

with guidance weight w so that the classifier-free guided f θ s,t,w(xt) is

f θ s,t,w(xt) = cskip(s, t)xt + cout(s, t)Gw θ (cin(t)xt, cnoise(s), cnoise(t), c) (86)

F. Sampling Algorithms

Pushforward sampling. See Algorithm 4. We assume a series of N time steps {ti}N i=0 with T = t N > t N 1 > > t2 > t1 > t0 = ϵ for the maximum time T and minimum time ϵ. Denote σd as data standard deviation.

Restart sampling. See Algorithm 5. Different from pushforward sampling, N time steps {ti}N i=0 do not need to be strictly decreasing for all time steps, e.g. T = t N t N 1 t2 t1 t0 = ϵ (assuming T > ϵ). Different from pushforward sampling, restart sampling first denoise a clean sample before resampling a noise to be added to this clean sample. Then a clean sample is predicted again. The process is iterated for N steps.

G. Connection with Prior Works

G.1. Consistency Models

Consistency models explicitly match PF-ODE trajectories using a network gθ(xt, t) that directly outputs a sample given any xt qt(xt). The network explicitly uses EDM parameterization to satisfy boundary condition gθ(x0, 0) = x0 and trains

via loss Ext,x,t h gθ(xt, t) gθ (xr, r) 2i where xr is a deterministic function of xt from an ODE solver.

We show that CM loss is a special case of our simplified IMM objective.

Lemma 1. When xt = x t, xr = x r, k(x, y) = x y 2, and s > 0 is a small constant, Eq. (12) reduces to CM loss

Ext,x,t h w(t) gθ(xt, t) gθ (xr, r) 2i for some valid mapping r(t) < t.

Proof. Since xt = x t, xr = x r, we have f θ s,t(xt) = f θ s,t(x t) and f θ s,r(xr) = f θ s,r(x r). So k(f θ s,t(xt), f θ s,t(x t)) = k(f θ s,r(xr), f θ s,r(x r)) = 0 by definition. Since k(x, y) = x y 2, it is easy to see Eq. (12) reduces to

Ext,t h w(s, t) f θ s,t(xt) f θ s,r(xr) 2i (87)

Inductive Moment Matching

Algorithm 4 Pushforward Sampling

Input: model f θ, time steps {ti}N i=0, prior distribution N(0, σ2 d I), (optional) guidance weight w Output: xt0 Sample x N N(0, σ2 d I) for i = N, . . . , 1 do

(Optional) w 1 if N = 1 // can optionally discard unconditional branch for N = 1 xti 1 f θ ti 1,ti(xti) or f θ ti 1,ti,w(xti) end for

Algorithm 5 Restart Sampling

Input: model f θ, time steps {ti}N i=0, prior distribution N(0, σ2 d I), DDIM interpolant coefficients αt and σt, (optional) guidance weight w Output: xt0 Sample x N N(0, σ2 d I) for i = N, . . . , 1 do

(Optional) w 1 if N = 1 // can optionally discard unconditional branch for N = 1 x f θ t0,ti(xti) or f θ t0,ti,w(xti) if i = 1 then

ϵ N(0, σ2 d I) xti 1 αti 1 x + σti 1 ϵ // or more generally xti 1 qt(xti 1| x, ϵ) else

xt0 = x end if end for

where w(s, t) is a weighting function. If s is a small positive constant, we further have f θ s,t(xt) gθ(xt, t) where we drop s as input. If gθ(xt, t) itself satisfies boundary condition at s = 0, we can directly take s = 0 in which case f θ 0,t(xt) = gθ(xt, t). And under these assumptions, our loss becomes

Ext,x,t h w(t) gθ(xt, t) gθ (xr, r) 2i , (88)

which is simply a CM loss using ℓ2 distance.

However, one can notice that from a moment-matching perspective, this loss significantly deviates from a proper divergence between distributions, and is problematic in two aspects. First, it assumes single-particle estimate, which now ignores the entropy repulsion term in MMD that arises only during multi-particle estimation. This can contribute to mode collapse and training instability of CM. Second, the choice of energy kernel is not a proper positive definite kernel required by MMD. At best, it only matches the first moment (its Taylor expansion cannot cover all moments as in RBF kernels), which is insufficient for matching two complex distributions! We should use kernels that match higher moments in practice. In fact, we show in the following Lemma that the pseudo-huber loss proposed in Song & Dhariwal (2023) matches higher moments as a kernel.

Lemma 2. Negative pseudo-huber loss kc(x, y) = c q

x y 2 + c2 for c > 0 is a conditionally positive definite kernel that matches all moments of x and y where weights on higher moments depend on c.

Proof. We first check that negative pseudo-huber loss c q

x y 2 + c2 is a conditionally positive definite kernel (Auffray & Barbillon, 2009). By definition, k(x, y) is conditionally positive definite if for x1, , xn RD and c1, , cn RD

with Pn i=1 ci = 0

j=1 cicjk(xi, xj) 0 (89)

Inductive Moment Matching

We know that negative L2 distance x y is conditionally positive definite. We prove this below for completion. Due to triangle inequality, x y x y . Then

j=1 cicj ( xi xj )

j=1 cicj ( xi xj ) (90)

(a) = 0 (92)

where (a) is due to Pn i=1 ci = 0. Now since c q

z 2 + c2 z for all c > 0, we have

xi xj 2 + c2

j=1 cicj ( xi xj ) 0 (93)

So negative pseudo-huber loss is a valid conditionally positive definite kernel.

Next, we analyze pseudo-huber loss s effect on higher-order moments by directly Taylor expanding q

z 2 + c2 c at z = 0 q

z 2 + c2 c = 1

8c3 z 4 + 1 16c5 z 6 5 128c7 z 8 + O( z 9) (94)

8c3 x y 4 + 1 16c5 x y 6 5 128c7 x y 8 + O( x y 9) (95)

where we substitute z = x y. Each higher order x y k for k > 2 expands to a polynomial containing up to k-th moments, i.e., {x, x2, . . . xk}, {y, y2, . . . yk}, thus the implicit feature map contains all higher moments where c contributes to the weightings in front of each term.

Furthermore, we extend our finite difference (between r(s, t) and t) IMM objective to the differential limit by taking r(s, t) t in Appendix H. This results in a new objective that similarly subsumes continuous-time CM (Song et al., 2023; Lu & Song, 2024) as a single-particle special case.

G.2. Diffusion GAN and Adversarial Consistency Distillation

Diffusion GAN (Xiao et al., 2021) parameterizes its generative distribution as

pθ s|t(xs|xt) = Z qs|t(xs|Gθ(xt, z), xt)p(z)dz

where Gθ is a neural network, p(z) is standard Gaussian distribution, and qs|t(xs|x, xt) is the DDPM posterior

qs|t(xs|x, xt) = N(µs,t, σ2 s,t I) (97)

µQ = αtσ2 s αsσ2 t xt + αs(1 α2 t α2s

σ2 s σ2 t )x

σ2 Q = σ2 s(1 α2 t α2s

σ2 s σ2 t )

Note that DDPM posterior is a stochastic interpolant, and more importantly, it is self-consistent, which we show in the Lemma below.

Lemma 6. For all 0 s < t 1, DDPM posterior distribution from t to s as defined in Eq. (97) is a self-consistent Gaussian interpolant between x and xt.

Inductive Moment Matching

Proof. Let xr qr|t(xr|x, xt) and xs qs|r(xs|x, xr), we show that xs follows qs|t(xs|x, xt).

xr = αtσ2 r αrσ2 t xt + αr(1 α2 t α2r

σ2 r σ2 t )x + σr

σ2r σ2 t ϵ1 (98)

xs = αrσ2 s αsσ2r xr + αs(1 α2 r α2s

σ2 r σ2 t )x + σs

σ2s σ2 t ϵ2 (99)

where ϵ1, ϵ2 N(0, I) are i.i.d. Gaussian noise. Directly expanding

xs = αrσ2 s αsσ2r

" αtσ2 r αrσ2 t xt + αr(1 α2 t α2r

σ2 r σ2 t )x + σr

σ2r σ2 t ϵ1

+ αs(1 α2 r α2s

σ2 r σ2 t )x + σs

σ2s σ2 t ϵ2 (100)

= αtσ2 s αsσ2 t xt + α2 rσ2 s αsσ2r (1 α2 t α2r

σ2 r σ2 t ) + αs(1 α2 r α2s

σ2 r σ2 t ) x + αrσ2 s αsσ2r

σ2r σ2 t ϵ1

σ2s σ2 t ϵ2 (101)

= αtσ2 s αsσ2 t xt + αs(1 α2 t α2s

σ2 s σ2 t )x + αrσ2 s αsσr

σ2r σ2 t ϵ1 + σs

σ2s σ2 t ϵ2 (102)

(a) = αtσ2 s αsσ2 t xt + αs(1 α2 t α2s

σ2 s σ2 t )x + σs

σ2s σ2 t ϵ3 (103)

where (a) is due to the fact that sum of two independent Gaussian variables with variance a2 and b2 is also Gaussian with variance a2 + b2, and ϵ3 N(0, I) is another independent Gaussian noise. We show the calculation of the variance:

σ4 s σ2r (1 α2 t α2r

σ2 r σ2 t ) + σ2 s(1 α2 r α2s

σ2 s σ2 t ) = α2 r α2s

σ4 s σ2r α2 tσ4 s σ2 t α2s + σ2 s α2 r α2s

= σ2 s(1 α2 t α2s

σ2 s σ2 t )

This shows xs follows qs|t(xs|x, xt) and completes the proof.

This shows another possible design of the interpolant that can be used, and diffusion GAN s formulation generally complies with our design of the generative distribution, except that it learns this conditional distribution of x given xt directly while we learn a marginal distribution. When they directly learn the conditional distribution by matching pθ s|t(x|xt) with qt(x|xt), the model is forced to learn qt(x|xt) and there only exists one minimizer. However, in our case, the model can learn multiple different solutions because we match the marginals instead.

GAN loss and MMD loss. We also want to draw attention to similarity between GAN loss used in Xiao et al. (2021); Sauer et al. (2025) and MMD loss. MMD is an integral probability metric over a set of functions F in the following form

DF(P, Q) = sup f F

Ex P (x)f(x) Ey Q(y)f(y)

where a supremum is taken on this set of functions. This naturally gives rise to an adversarial optimization algorithm if F is defined as the set of neural networks. However, MMD bypasses this by selecting F as the RKHS where the optimal f can be analytically found. This eliminates the adversarial objective and gives a stable minimization objective in practice. However, this is not to say that RKHS is the best function set. With the right optimizers and training scheme, the adversarial objective may achieve better empirical performance, but this also makes the algorithm difficult to scale to large datasets.

G.3. Generative Moment Matching Network

It is trivial to check that GMMN is a special parameterization. We fix t = 1, and due to boundary condition, r(s, t) s = 0 implies training target pθ s|r(s,t)(xs) = qs(xs) is the data distribution. Additionally, pθ s|t(xs) is a simple pushforward of prior p(ϵ) through network gθ(ϵ) where drop dependency on t and s since they are constant.

Inductive Moment Matching

H. Differential Inductive Moment Matching

Similar to the continuous-time CMs presented in (Lu & Song, 2024), our MMD objective can be taken to the differential limit. Consider the simplifed loss and parameterization in Eq. (12), we use the RBF kernel as our kernel of choice for simplicity.

Theorem 2 (Differential Inductive Moment Matching). Let f θ s,t(xt) be a twice continuously differentiable function with bounded first and second derivatives, let k( , ) be RBF kernel with unit bandwidth, x, x q(x), xt qt(xt|x), x t qs(x t|x ), xr = DDIM(xt, x, t, r) and x r = DDIM(x t, x , t, r), the following objective

LIMM- (θ, t) = lim r t 1 (t r)2 Ext,x t,xr,x r h k f θ s,t(xt), f θ s,t(x t) + k f θ s,r(xr), f θ s,r(x r) (104)

k f θ s,t(xt), f θ s,r(x r) k f θ s,t(x t), f θ s,r(xr) i

can be analytically derived as

2 f θ s,t(x t) f θ s,t(xt) 2 df θ s,t(xt)

df θ s,t(x t) dt (105)

df θ s,t(xt)

f θ s,t(xt) f θ s,t(x t) f θ s,t(xt) f θ s,t(x t) df θ s,t(x t) dt

Proof. Firstly, the limit can be exchanged with the expectation due to dominated convergence theorem where the integrand consists of kernel functions which can be assumed to be upper bounded by 1, e.g. RBF kernels are upper bounded by 1, and thus integrable. It then suffices to check the limit of the integrand. Before that, let us review the first and second-order Taylor expansion of e x y 2. We let a, b RD be constants to be expanded around. We note down the Taylor expansion to second-order below for notational convenience.

e x b 2/2 at x = a:

e a b 2/2 e a b 2/2(a b) (x a) + 1

2(x a) e a b 2/2 (a b)(a b) I (x a)

e y a 2/2 at y = b:

e b a 2/2 e b a 2/2(b a) (y b) + 1

2(y b) e b a 2/2 (b a)(b a) I (y b)

e x y 2/2 at x = a, y = b:

e a b 2/2 e a b 2/2(a b) (x a) e b a 2/2(b a) (y b)

2(y b) e b a 2/2 (b a)(b a) I (y b)

2(y b) e b a 2/2 (b a)(b a) I (y b)

+ (x a) e b a 2/2 I (a b)(a b) (y b)

Putting it together, the above results imply

e x y 2/2 + e a b 2/2 e x b 2/2 e y a 2/2

(x a) e b a 2/2 I (a b)(a b) (y b)

since it is easy to check that the remaining terms cancel.

Inductive Moment Matching

Substituting x = f θ s,t(xt), a = f θ s,r(xr), y = f θ s,t(x t), b = f θ s,r(x r), we furthermore have

lim r t 1 t r(x a) = lim r t 1 t r f θ s,t(xt) f θ s,r(xr) (106)

= lim r t 1 t r

(t r)df θ s,t(xt)

dt + O(|t r|2)

= df θ s,t(xt)

lim r t 1 t r(y b) = lim r t 1 t r f θ s,t(x t) f θ s,r(x r) = df θ s,t(x t) dt (109)

Therefore, LIMM- (θ, t) can be derived as

2 f θ s,t(x t) f θ s,t(xt) 2 df θ s,t(xt)

I f θ s,t(xt) f θ s,t(x t) f θ s,t(xt) f θ s,t(x t) df θ s,t(x t) dt

2 f θ s,t(x t) f θ s,t(xt) 2 df θ s,t(xt)

df θ s,t(x t) dt (112)

df θ s,t(xt)

f θ s,t(xt) f θ s,t(x t) f θ s,t(xt) f θ s,t(x t) df θ s,t(x t) dt

H.1. Pseudo-Objective

Due to the stop-gradient operation, we can similarly find a pseudo-objective whose gradient matches the gradient of LIMM- (θ, t) in the limit of r t.

Theorem 3. Let f θ s,t(xt) be a twice continuously differentiable function with bounded first and second derivatives, k( , ) be RBF kernel with unit bandwidth, x, x q(x), xt qt(xt|x), x t qt(x t|x ), xr = DDIM(xt, x, t, r) and x r = DDIM(x t, x , t, r), the gradient of the following pseudo-objective

θL IMM- (θ, t) = lim r t 1 (t r) θExt,x t,xr,x r h k f θ s,t(xt), f θ s,t(x t) + k f θ s,r (xr), f θ s,r (x r) (113)

k f θ s,t(xt), f θ s,r (x r) k f θ s,t(x t), f θ s,r (xr) i

can be used to optimize θ and can be analytically derived as

2 f θ s,t (x t) f θ s,t (xt) 2

df θ s,t (x t) dt

h f θ s,t (xt) f θ s,t (x t) i df θ s,t (x t) dt

h f θ s,t (xt) f θ s,t (x t) i !

θf θ s,t(xt)+

df θ s,t (xt)

h f θ s,t (xt) f θ s,t (x t) i df θ s,t (xt)

h f θ s,t (xt) f θ s,t (x t) i !

θf θ s,t(x t)

Inductive Moment Matching

Proof. Similar to the derivation of LIMM- (θ, t), let x = f θ s,t(xt), a = f θ s,r (xr), y = f θ s,t(x t), b = f θ s,r (x r), we have

lim r t 1 (t r) θ(x a) e b a 2/2 I (a b)(a b) (y b)

= lim r t 1 (t r)e b a 2/2 "

(y b) (a b) (y b) (a b) !

(x a) (a b) (x a) (a b) !

lim r t 1 (t r)(x a) = df θ s,t (xt)

dt , lim r t 1 (t r)(y b) = df θ s,t (x t) dt

Note that df θ s,t (xt)

dt is now parameterized by θ instead of θ because the gradient is already taken w.r.t. f θ s,t(xt) outside of the brackets, so (x a) and (y b) merely require evaluation at current θ with no gradient information, which θ satisfies. The objective can be derived as

θL IMM- (θ, t)

2 f θ s,t (x t) f θ s,t (xt) 2

df θ s,t (x t) dt

h f θ s,t (xt) f θ s,t (x t) i df θ s,t (x t) dt

h f θ s,t (xt) f θ s,t (x t) i !

θf θ s,t(xt)+

df θ s,t (xt)

h f θ s,t (xt) f θ s,t (x t) i df θ s,t (xt)

h f θ s,t (xt) f θ s,t (x t) i !

θf θ s,t(x t)

H.2. Connection with Continuous-Time CMs

Observing Eq. (105) and Eq. (110), we can see that when xt = x t and xr = x r, s being a small positive constant, then f θ s,t(x t) = f θ s,t(xt), and exp 1

2 f θ s,t(x t) f θ s,t(xt) 2 = 1, and f θ s,t(x t) gθ(xt, t) where since s is fixed we discard the dependency on s as input. Then, Eq. (105) reduces to

" df θ s,t(xt)

which is the same as differential consistency loss (Song et al., 2023; Geng et al., 2024). And Eq. (110) reduces to

" df θ s,t (xt)

θf θ s,t(xt)

which is the pseudo-objective for continuous-time CMs (Song et al., 2023; Lu & Song, 2024) (minus a weighting function of choice).

I. Experiment Settings

I.1. Training & Parameterization Settings

We summarize our best runs in Table 5. Specifically, for Image Net-256 256, we adopt a latent space paradigm for computational efficiency. For its autoencoder, we follow EDM2 (Karras et al., 2024) and pre-encode all images from

Inductive Moment Matching

CIFAR-10 Image Net-256 256

Parameterization Setting

Architecture DDPM++ Di T-S Di T-B Di T-L Di T-XL Di T-XL GFlops 21.28 6.06 23.01 80.71 118.64 118.64 Params (M) 55 33 130 458 675 675 cnoise(t) 1000t 1000t 1000t 1000t 1000t 1000t 2nd time conditioning s s s s s t s Flow Trajectory OT-FM OT-FM OT-FM OT-FM OT-FM OT-FM gθ(xt, s, t) Simple-EDM Euler-FM Euler-FM Euler-FM Euler-FM Euler-FM σd 0.5 0.5 0.5 0.5 0.5 0.5 Training iter 400K 1.2M 1.2M 1.2M 1.2M 1.2M

Training Setting

Dropout 0.2 0 0 0 0 0 Optimizer RAdam Adam W Adam W Adam W Adam W Adam W Optimizer ϵ 10 8 10 8 10 8 10 8 10 8 10 8

β1 0.9 0.9 0.9 0.9 0.9 0.9 β2 0.999 0.999 0.999 0.999 0.999 0.999 Learning Rate 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 Weight Decay 0 0 0 0 0 0 Batch Size 4096 4096 4096 4096 4096 4096 M 4 4 4 4 4 4 Kernel Laplace Laplace Laplace Laplace Laplace Laplace r(s, t) max(s, η 1(ηt (ηmax ηmin)

215 )) max(s, η 1(ηt (ηmax ηmin)

212 )) Minimum t, r gap - - - - - 10 4

p(t) U(0.006, 0.994) U(0, 0.994) U(0, 0.994) U(0, 0.994) U(0, 0.994) U(0, 0.994) a 1 1 1 1 1 2 b 5 4 4 4 4 4 EMA Rate 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 Label Dropout - 0.1 0.1 0.1 0.1 0.1 x-flip True False False False False False EDM augment (Karras et al., 2022) True False False False False False

Inference Setting

Sampler Type Pushforward Pushforward Pushforward Pushforward Pushforward Pushforward Number of Steps 2 8 8 8 8 8 Schedule Type t1 = 1.4 Uniform Uniform Uniform Uniform Uniform FID-50K (w = 0) 1.98 - - - - - FID-50K (w = 1.0, i.e. no guidance) - 42.28 26.02 9.33 7.25 6.69 FID-50K (w = 1.5) - 20.36 9.69 2.80 2.13 1.99

Table 5. Experimental settings for different architectures and datasets.

Image Net into latents without flipping, and calculate the channel-wise mean and std for normalization. We use Stable Diffusion VAE3 and rescale the latents by channel mean [0.86488, 0.27787343, 0.21616915, 0.3738409] and channel std [4.85503674, 5.31922414, 3.93725398, 3.9870003]. After this normalization transformation, we further multiply the latents by 0.5 so that the latents roughly have std 0.5. For Di T architecture of different sizes, we use the same hyperparameters for all experiments.

Choices for T and ϵ. By default assuming we are using mapping function r(s, t) by constant decrement in ηt, we keep ηmax 160. This implies that for time distribution of the form U(ϵ, T), we set T = 0.996 for cosine diffusion and T = 0.994 for OT-FM. For ϵ, we set it differently for pixel-space and latent-space model. For pixel-space on CIFAR-10, we follow Nichol & Dhariwal (2021) and set ϵ to a small positive constant because pixel quantization makes smaller noise imperceptible. We find 0.006 to work well. For latent-space on Image Net-256 256, we have no such intuition as in pixel-space. We simply set ϵ = 0 in this case.

Exceptions occur when we ablate other choices of r(s, t), e.g. constant decrement in λt in which case we set ϵ = 0.001 to prevent r(s, t) for being too close to t when t is small.

Injecting time s. The design for additionally injecting s can be categorized into 2 types injecting s directly and injecting stride size (t s). In both cases, architectural designs exactly follow the time injection of t. We simply extract positional time embedding of s (or t s) fed through 2-layer MLP (same as for t) before adding this new embedding to the embedding of t after MLP. The summed embedding is then fed through all the Transformer blocks as in standard Di T architecture.

3https://huggingface.co/stabilityai/sd-vae-ft-mse

Inductive Moment Matching

10-step 16-step 32-step

FID w/ w = 1.5 1.98 1.90 1.89

Table 6. FID of Image Net-256 256 beyond 8 steps presented in the main text. At 16 steps, IMM already outperforms VAR with 2B parameters (1.92 FID) and its performance saturates beyond that, with 32 steps performing marginally better.

1-step 2-step 4-step 8-step

a = 1 7.97 4.01 2.61 2.13

a = 2 8.28 4.08 2.60 2.01

Table 7. Ablation of exponent a in the weighting function on Image Net256 256. We see that a = 2 excels at multi-step generation while lagging slightly behind in 1-step generation.

1-step 2-step 4-step 8-step

TF32 w/ a = 1 7.97 4.01 2.61 2.13

FP16 w/ a = 1 8.73 4.54 3.03 2.38

FP16 w/ a = 2 8.05 3.99 2.51 1.99

Table 8. Ablation of lower precision training on Image Net-256 256. For lower precision training, we employ both minimum gap = 10 4 and (t r) conditioning.

Improved CT baseline. For Image Net-256 256, we implement i CT baseline by using our improved parameterization with Simple-EDM and OT-FM schedule. We use the proposed pseudo-huber loss for training but find training often collapses using the same r(s, t) schedule as ours. We carefully tune the gap to achieve reasonable performance without collapse and present our results in Table 2.

I.2. Inference Settings

Inference schedules. For all one-step inference, we directly start from ϵ N(0, σ2 d I) at time T to time ϵ through pushforward sampling. For all 2-step methods, we set the intermediate timestep t1 such that ηt1 = 1.4; this choice is purely empirical which we find to work well. For N 4 steps we explore two types of time schedules: (1) uniform decrement in t with η0 < η1 < ηN where

ti = T + N i

N (ϵ T) (117)

and (2) EDM (Karras et al., 2022) time schedule. EDM schedule specifies η0 < η1 < ηN where

1 ρ max + N i

1 ρ max) ρ and ρ = 7 (118)

We slightly modify the schedule so that η0 = ηmin is the endpoint instead of η1 = ηmin and η0 = 0 as originally proposed, since our η0 can be set to 0 without numerical issue.

We also specify the time schedule type used for our best runs in Table 5 and their results.

I.3. Scaling Settings

Model GFLOPs. We reuse numbers from Di T (Peebles & Xie, 2023) for each model architecture.

Training compute. Following Peebles & Xie (2023), we use the formula model GFLOPs batch size training steps 4 for training compute where, different from Di T, we have constant 4 because for each iteration we have 2 forward pass and 1 backward pass, which is estimated as twice the forward compute.

Inference compute. We calculate inference compute via model GFLOPs number of steps.

I.4. Scaling Beyond 8 Steps

We investigate when the performance saturates for Image Net-256 256 in Table 6. We see continued improvement beyond 8 steps and at 16 steps our method already outperforms VAR with 2B parameters (1.92 FID).

I.5. Ablation on exponent a

We compare the performance between a = 1 and a = 2 on full Di T-XL architecture in Table 7, which shows how a affects results of different sampling steps. We observe that a = 2 causes slightly higher 1-step sampling FID but outperforms a = 1 in the multi-step regime.

Inductive Moment Matching

I.6. Caveats for Lower-Precision Training

For all experiments, we follow the original works (Song et al., 2020b; Peebles & Xie, 2023) and use the default TF32 precision for training and evaluation. When switching to lower precision such as FP16, we find that our mapping function, i.e. constant decrement in ηt, can cause indistinguishable time embedding after some MLP layers when t is large. To mitigate this issue, we simply impose a minimum gap between any t and r, for example, = 10 4. Our resulting mapping function becomes

r(s, t) = max

s, min t , η 1(η(t) ϵ) !

Optionally, we can also increase distinguishability between nearby time-steps inside the network by injecting (r s) instead of s as our second time condition. We use this as default for FP16 training. With these simple changes, we observe minimal impact on generation performance.

Lastly, if training from scratch with lower precision, we recommend FP16 instead of BF16 because of higher precision that is needed to distinguish between nearby t and r.

We show results in Table 8. For FP16, a = 1 causes slight performance degradation because of the small gap issue at large t. This is effectively resolved by a = 2 which downweights losses at large t by focusing on smaller t instead. At lower precision, while not necessary, a = 2 is an effective solution to achieve good performance that matches or even surpasses that of TF32.

J. Additional Visualization

We present additional visualization results in the following page.

Inductive Moment Matching

Figure 12. Uncurated samples on CIFAR-10, unconditional, 2 steps.

Inductive Moment Matching

Figure 13. Uncurated samples on Image Net-256 256 using Di T-XL/2 architecture. Guidance w = 1.5, 8 steps.

Inductive Moment Matching

Figure 14. Uncurated samples on Image Net-256 256 using Di T-XL/2 architecture. Guidance w = 1.5, 8 steps.