# testtime_correlation_alignment__ad1c3baa.pdf Test-time Correlation Alignment Linjing You * 1 Jiabao Lu * 1 Xiayuan Huang 2 Deep neural networks often degrade under distribution shifts. Although domain adaptation offers a solution, privacy constraints often prevent access to source data, making Test-Time Adaptation (TTA) which adapts using only unlabeled test data increasingly attractive. However, current TTA methods still face practical challenges: (1) a primary focus on instance-wise alignment, overlooking CORrelation ALignment (CORAL) due to missing source correlations; (2) complex backpropagation operations for model updating, resulting in overhead computation and (3) domain forgetting. To address these challenges, we provide a theoretical analysis to investigate the feasibility of Test-time Correlation Alignment (TCA), demonstrating that correlation alignment between high-certainty instances and test instances can enhance test performances with a theoretical guarantee. Based on this, we propose two simple yet effective algorithms: Linear TCA and Linear TCA+. Linear TCA applies a simple linear transformation to achieve both instance and correlation alignment without additional model updates, while Linear TCA+ serves as a plug-and-play module that can easily boost existing TTA methods. Extensive experiments validate our theoretical insights and show that TCA methods significantly outperforms baselines across various tasks, benchmarks and backbones. Notably, Linear TCA achieves higher accuracy with only 4% GPU memory and 0.6% computation time compared to the best TTA baseline. It also outperforms existing methods on CLIP over 1.86%. Code: https://github.com/youlj109/TCA. *Equal contribution 1Institute of Automation, Chinese Academy of Sciences 2College of Science, Beijing Forestry University. Correspondence to: Xiayuan Huang , Linjing You . Proceedings of the 42 nd International Conference on Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s). Corruption Level Correlation Distance TTA Methods Memory (MB) Source Domain (Before) Target Domain Source Domain (After) Accuracy of Source Domain (%) Accuracy before Adaptation SOURCE BN T3A PLC SAR EATA TENT TSD TEA TIPI Ada NPC Linear TCA SOURCE BN T3A PLC SAR EATA TENT TSD TEA TIPI Ada NPC Linear TCA Accuracy of Target Domain (%) Figure 1. Illustration of key limitations in existing TTA methods. (a) Correlation distance increases with domain shifts. (b) Computation time and peak GPU memory usage on CIFAR-10-C, showing high overhead of existing methods. (c) Source domain performance after test-time adaptation, revealing challenges in retaining source knowledge. 1. Introduction Deep neural networks (DNNs) have significantly advanced numerous tasks in recent years (Le Cun et al., 2015; Jumper et al., 2021; Silver et al., 2016) when the training and test data are independent and identically distributed (i.i.d.). However, the i.i.d. condition rarely holds in practice as the data distributions are likely to change over time and space (Fang et al., 2020; Wang & Deng, 2018). This phenomenon, known as the out-of-distribution (OOD) problem or distribution shift, has been extensively investigated within the context of domain adaptation (DA) (You et al., 2019; Zhou et al., 2022; Liang et al., 2024). Among various DA methods, CORrelation ALignment (CORAL) (Sun et al., 2017; Sun & Saenko, 2016; Cheng et al., 2021a) has been proven to be an effective and frustratingly simple paradigm, which aligns the feature distributions of the source and target domains at a feature correlation level rather than merely aligning individual instances. However, DA methods are practically difficult when pre- Test-time Correlation Alignment trained models are publicly available but the training data and training process remain inaccessible due to privacy and resource restrictions (Liang et al., 2024). To address such a source-inaccessible domain shifts task at test time, test-time adaptation (TTA) (Gong et al., 2024; Su et al., 2024a;b; You et al., 2025) has emerged as a rapidly progressing research topic. Although some recent attempts have been made to handle this task, current TTA methods still face several limitations: Firstly, overlooking feature correlations: Most existing TTA methods focus on instance-wise alignment (Wang et al., 2023; Nguyen et al., 2023; Wang et al., 2020) that only capture central of the instances while neglecting the correlations between features. For example, relationships between edge and texture features can vary significantly across domains. Let s consider a simple test on the CIFAR-10-C dataset (Hendrycks & Dietterich, 2019) to show the relationship between feature correlation and domain shift. As shown in Figure 1a , the correlation distance (see Section 2.2) of Res Net-18 (He et al., 2016) embedding are computed with an increasing corruption level from 1 to 5. It illustrates that as domain shifts increase, the changes in feature correlation also increase. Secondly, overhead computation: Current TTA methods often rely on computationally expensive backpropagation for each test sample to update models (Sun et al., 2020; Wang et al., 2020; Goyal et al., 2022; Bartler et al., 2022). However, many applications are deployed on edge devices, such as smartphones and embedded systems (Niu et al., 2024), which typically lack the computational power and memory capacity required for such intensive calculations. As a result, backpropagation-based TTA methods are limited in their applicability on these edge devices. In Figure 1b, we illustrate the computation time and maximum GPU memory usage of different TTA methods on the CIFAR-10-C dataset. Compared to the non-adaptive source model (ERM(Vapnik, 1999)), most TTA methods show a dramatic increase in both items. Lastly, domain forgetting: Another drawback of backpropagation-based TTA methods is that they often lead to model updating, which gradually loses the prediction ability of the source or training domain (Niu et al., 2024; Zhang et al., 2023). As illustrated in Figure 1c, after adaptation on test domain, the performance of most methods declines when return to the source domain, indicating that existing TTA approaches struggle to retain knowledge of the source domain. To address the above challenges, applying the effective and frustratingly simple CORAL method to TTA appears intuitive but the lack of source data makes it highly challenging. We thus explore the feasibility of Test-time Correlation Alignment (TCA) by posing key questions: (1)Can we construct a pseudo-source correlation that approximates the true source correlation? (2) Can this enable effective TTA? We provide a theoretical analysis showing that aligning correlations between high-certainty and test instances improves test-time performance with guarantees. Based on this, we propose two simple yet effective methods: Linear TCA and Linear TCA+. Specifically, we first compute the pseudo-source correlation by using k high-certainty instances. Then, Linear TCA aligns correlation through simple linear transformations of embeddings without model updates, resulting in minimal computation and keeping source domain knowledge. While Linear TCA+ serves as a plugand-play module that can easily boost existing TTA methods. Main Findings and Contributions: (1) We introduce a novel and practical paradigm for TTA, termed Test-time Correlation Alignment (TCA). The construction of the pseudosource correlation and the adaptation effectiveness are theoretically guaranteed. (2) Based on our analysis, we develop two simple yet effective methods Linear TCA and Linear TCA+ to validate TCA s effectiveness and its plugand-play potential with other TTA approaches. (3)We conduct comprehensive experiments to validate our theoretical insights and compare performance across diverse benchmarks, backbones, and tasks, evaluating accuracy, efficiency, and resistance to forgetting. Results show that Linear TCA achieves outstanding performance, while Linear TCA+ robustly boosts other TTA methods under various conditions. (4) Further in-depth experimental analysis reveals the effective range of Linear TCA and provides valuable insights for future work. 2. Preliminary and Problem Statement We briefly revisit TTA and CORAL in this section for the convenience of further analyses, and put detailed related work discussions into Appendix A due to page limits. 2.1. Test Time Adaptation (TTA) In the test-time adaptation (TTA) (Tan et al., 2024; Yuan et al., 2023) scenario, it has access only to unlabeled data from the test domain and a pre-trained model from the source domain. Specifically, let Ds = {(xi s, yi s)}ns i=1 Ds represent the labeled source domain dataset, where (xi s, yi s) is sampled i.i.d from the distribution Ds and ns is the number of the total source instances. The model, trained on the source domain dataset and parameterized by θ, is denoted as hθ( ) = g(f( )) : Xs Ys, where f( ) is the backbone encoder and g( ) denotes the decoder head. During testing, hθ( ) will perform well on in-distribution (ID) test instances drawn from Ds. However, given a set of outof-distribution (OOD) test instances Dt = {xi t}nt i=1 Dt and Dt = Ds, the prediction performance of hθ( ) would Test-time Correlation Alignment decrease significantly. To this end, the goal of TTA is to adapt this model hθ( ) to Dt without access to Ds. For each instance xi t Xt, let the output of encoder f( ) and decoder g( ) be denoted as zi t = f(xi t) Rd and pi t = g(zi t) Rc, respectively, where d is the dimension of the embeddings and c is the number of classes in a classification task. When encountering an OOD test instance xi t, existing TTA methods (Wu et al., 2024; Sinha et al., 2023; Lee et al., 2024; Yuan et al., 2023) typically minimize an unsupervised or self-supervised loss function to align the embedding zi t or prediction pi t, thereby updating the model parameters θ: min θ L(zi t, pi t, θ), xi t Dt (1) where θ θ is a proper subset of θ involved in the update, such as the parameters of the batch normalization (BN) layers (Schneider et al., 2020; Su et al., 2024c) or all parameters. Generally, the TTA loss function L( ) can be formulated by nearest neighbor information (Zhang et al., 2023; Hardt & Sun, 2023; Jang et al., 2022), contrastive learning (Wang et al., 2023; Chen et al., 2022), entropy minimization (Wang et al., 2020; Niu et al., 2022), etc. 2.2. Correlation Alignment (CORAL) The aim of correlation alignment (CORAL) (Sun et al., 2017; Cheng et al., 2021a; Sun & Saenko, 2016; Sun et al., 2016; Das et al., 2021; Rahman et al., 2020b) is to minimize the distance of the second-order statistics (covariance) between the source and test features. Specifically, let Zs = {zi s}ns i=1 Rns d denotes the feature matrix from the source domain, and Zt = {zi t}nt i=1 Rnt d denotes the feature matrix from the test domain. CORAL computes the covariance matrices of the source features Zs and test features Zt, and aligns correlation by minimizing the Frobenius norm of their two covariance matrices. The covariance matrix is computed as below: Σ = 1 n 1(ZT Z 1 n1n ZT Z1n) (2) the correlation distance is then given by (Sun & Saenko, 2016): d(Σs, Σt) = 1 4d2 Σs Σt 2 F (3) where Σs and Σt are the covariance matrices of the source and test domains, respectively, and 1 is a column vector with all elements equal to 1 to perform mean-subtraction. F represents the Frobenius norm. 2.3. Problem Statement Existing TTA methods suffer from overlooking feature correlation, overhead computation and domain forgetting. Research and practice have demonstrated that CORAL is both effective and frustratingly easy to implement on DA. Since TTA is a subfield of DA, it is a natural extension to apply CORAL within TTA frameworks. However, due to privacy and resource constraints in TTA, it is impossible to compute the source correlation. This limitation hinders the application of CORAL in such real-world scenarios, i.e. test-time correlation alignment (TCA). 3. Theoretical Studies In this section, we conduct an in-depth theoretical analysis of TCA based on domain adaptation and learning theory. We focus on two key questions: (1) Can we construct a pseudo-source correlation to approximate the original source correlation? (2) Can TCA based on this pseudosource correlation enable effective TTA? Before discussing the main results, we first state some necessary assumptions and concepts. Missing proofs and detailed explanations are provided in Appendix B. Definition 3.1. (Classification error and empirical error) Let H be a hypothesis class of VC-dimension dv. The error that an estimated hypothesis hθ H disagrees with the groundtruth labeling function l : Xt Yt according to distribution Dt is defined as: ϵ(hθ, l) = Ex Dt[|hθ(x) l(x)|] (4) which we also refer to as the error or risk ϵ(hθ). The empirical error of hθ H with respect to a labeled dataset Ds = {(xi s, yi s)}ns i=1 Ds is defined as: i=1 |hθ(xi s) yi s| (5) Assumption 3.2. (Strong density condition) Given the parameters µ , µ+, ct, c t , rt > 0, we assume that the distribution Ds and Dt are absolutely continuous with respect to the Lebesgue measure λ[ ] in Euclidean space. Let B(x, r) = {x0 : x0 x r} denote the closed ball centered at point x with radius r. We further assume that xt Dt and r (0, rt], the following conditions hold: λ[Ds B(xt, r)] ctλ[B(xt, r)] (6) λ[Dt B(xt, r)] c t λ[B(xt, r)] (7) λ < µ+; µ < Dt The strong density condition is commonly used when analyzing KNN classifiers (Audibert & Tsybakov, 2007; Cai & Wei, 2021). Recently, it has also been applied in the Test-time Correlation Alignment test-time adaptation (Zhang et al., 2023). Intuitively, Assumption 3.2 requires that the divergence between Ds and Dt is bounded. When ct = 1, for each xt Dt, the neighborhood ball B(xt, r) is completely contained within Ds. In contrast, when ct = 0, B(xt, r) and Ds are nearly disjoint. Assumption 3.3. (L-Lipschitz Continuity) Let hθ( ) = g(f( )) be a estimated hypothesis on H. We assume that there exists a constant L such that x1, x2 Ds Dt, the encoder f( ) satisfies the following condition: f(x1) f(x2) L x1 x2 (9) The assumption of L-Lipschitz continuity is frequently employed in the analysis of a model s adaptation capabilities (Mansour et al., 2009). It implies that the change rate of f( ) does not exhibit extreme fluctuations and is bounded by the constant L at any point. Assumption 3.4. (Taylor Approximation) Let hθ( ) = g(f( )) be a L-Lipschitz Continuous hypothesis on H. z = f(x) and p = g(z). We assume that there exists a constant r such that x1, x2 Ds Dt, if z1 z2 r , p2 = g(z2) can be approximated using the first-order Taylor expansion at z1 as follows: p2 = p1 + Jg(z1)(z2 z1) + o( z1 z2 ) (10) where p1 = g(z1), Jg(z1) is the Jacobian matrix of g evaluated at z1, and o( z1 z2 ) represents the higherorder terms in the expansion. It indicates that when the outputs z1 and z2 are close (i.e., their distance is within the radius r ), the decoder can be well-approximated by a linear function at z1. 3.1. Correlation of high-certainty test instances approximates the source correlation We characterize the divergence of correlation between the pseudo-source and the source correlation in the following Theorem 3.5. Theorem 3.5. Let hθ( ) = g(f( )) be an L-Lipschitz continuous hypothesis on H. Ω:= S x Dt B(x, r ) is the set of balls near the test data. We sample k source instances from Ds Ωand k test instances from Dt to obtain [Xs, Zs, Ps] and [Xt, Zt, Pt] by hθ( ), respectively. Per Assumption 3.2, Assumption 3.3 and Assumption 3.4, with a probability of at least 1 exp (ctµ πd I rd I ns 1)2 2ctµ πd I rd I ns + log k , we have Zt Zs Pt Ps + o(kr ) Jg(Zs) (11) where πd I = λ(B(0, 1)) is the volume of the d I dimension unit ball and d I is the dimension of input x. Furthermore, considering the true source correlation Σs = E[ Zs T Zs] and the pseudo-source correlation ˆΣs = Zt T Zt, where Zs and Zt are centered. With a probability of at least min(1 exp (ctµ πd I rd I ns 1)2 2ctµ πd I rd I ns + log k , 1 δ), the cor- relation distance Σs ˆΣs is bounded by: 2 Zs F ( ˆYt Pt F + A Jg(Zs) F ) + ( ˆYt Pt F + A Jg(Zs) F )2 + B where ˆYt is the one-hot encoding of Pt, A = o(kr ) + kϵ(hθ(Xt)) + kϵ(hθ(Xs)) represents the output error of the sampled instances, and B = q 2k is the sampling error. Theorem 3.5 implies the followings: (1) In Eq. (12), the terms Xs, Zs, and Jg(Zs) remain unchanged with the same source data. The primary factor influencing the correlation distance Σs ˆΣs is prediction uncertainty ˆYt Pt F and output error of the sampled instances ϵ(hθ(Xt)). (2) Intuitively, previous studies (Gui et al., 2024; Niu et al., 2022; Yuan et al., 2024) empirically suggest that instances with higher output certainty have less output error. In other words, with a smaller divergence between the prediction Pt and its one-hot encoding ˆYt, both uncertainty ˆYt Pt F and error ϵ(hθ(Xt)) will decrease, resulting in a smaller correlation distance. (3) Therefore, a reasonable pseudo-source construction method is to select the k test instances with the smallest ˆYt Pt F values (i.e. high-certainty test instances) and compute their correlation matrix as pseudosource correlation. 3.2. Test-time correlation alignment reduces test classification error In this section, we establish the TTA error bounds of hypothesis hθ when minimizing the empirical error in the source data (Theorem 3.6) and examine the influence of using the pseudo-source correlation (Corollary 3.7), which further indicates factors that affect the performance of hθ. Theorem 3.6. Let H be a hypothesis class of VCdimension dv. If ˆh H minimizes the empirical error ˆϵs(h) on Ds, and h t = arg minh H ϵt(h) is the optimal hypothesis on Dt, with the assumption that all hypotheses are L-Lipschitz continuous, then δ (0, 1), with probability with at least 1 δ the following inequality holds: Test-time Correlation Alignment ϵt(ˆh) ϵt(h t ) + O( q µs µt 2 F + Σs Σt 2 F ) + C where C = 2 q dvlog(2ns) log(δ) 2ns + 2γ and γ = minh H{ϵs(h(t)) + ϵt(h(t))}. µs, µt, Σs and Σt denote the means and correlations of the source and test embeddings, respectively. We use O( ) to hide the constant dependence. For fixed Ds and Dt, ϵt(h t ) and C are constants, indicating that the primary factors affecting the performance of hθ on the test data Dt (i.e., ϵt(ˆh)) are µs µt 2 F and Σs Σt 2 F . By aligning correlations during testing, which means reducing Σs Σt 2 F , we can effectively decrease the model s classification error on the test data. Combining Theorem 3.5 with Theorem 3.6, the following corollary provides a direct theoretical guarantee that TCA based on pseudo-source correlation can reduce the error bounds on test data. Corollary 3.7. Let Σs, ˆΣs and Σt denote the source, pseudo-source and test correlation, respectively. Theorem 3.5 establishes the error bound between ˆΣs and Σs, while Theorem 3.6 demonstrates that reducing the difference between Σt and Σs can decrease classification error on the test data. By applying the triangle inequality, we have: Σt Σs F = Σt ˆΣs + ˆΣs Σs F Σt ˆΣs F + ˆΣs Σs F (13) Therefore, Theorem 3.6 can be rewritten as: ϵt(h t ) + O( q µs µt 2 F + Σs Σt 2 F ) + C ϵt(h t ) + O(( µs µt 2 F + (2 Zs F ( ˆYt Pt F + A + ( ˆYt Pt F + A Jg(Zs) F )2 + B + Σt ˆΣs F )2)1/2) + C Corollary 3.7 indicates the followings: (1) Reducing the correlation distance between the test data and the pseudosource, i.e., Σt ˆΣs 2 F , can reduce the test classification error. The pseudo-source correlation ˆΣs is computed by selecting k instances from the test data with minimal uncertainty, measured by ˆYt Pt 2 F . (2) Updating model parameters to decrease ˆYt Pt 2 F can further reduce the test error. (3) Additionally, minimizing the instance-wise distance µs µt 2 2 can also contribute to reducing the test error, which is consistent with previous studies (Niu et al., 2022; Wang et al., 2023; 2020). Original Prediction Pseudo-Source Linear Transform Transformed Figure 2. The pipeline of our proposed Linear TCA method. During testing, we first obtain original embeddings and predictions using the source model. Based on the certainty of the original predictions, we select a subset embeddings to form a pseudo-source domain . A linear transformation is then applied to align the correlations of the original embeddings with those of the pseudo-source domain, ultimately producing the final predictions of Linear TCA. Notably, this process does not require updating any parameters of the original model. Remark. Section 3.1 answers the first question that the feature correlation of high-certainty test instances from the pre-trained model can approximate the feature correlation of the source domain. Section 3.2 provides a theoretical guarantee that conducting correlation alignment between pseudo-source correlation and test correlation during TTA can effectively reduce the test error bound. These theoretical findings are further validated in Section 5.2. 4. The Test-time Correlation Alignment Algorithms As illustrated in Figure 2, building on our theoretical findings, we propose two simple yet effective TCA methods: Linear TCA and Linear TCA+. We start with detailing the construction of the pseudo-source correlation, followed by the implementation of Linear TCA and Linear TCA+. 4.1. Pseudo-Source For each instance xi t arrives in test time, we first get embedding zi t = f(xi t) and prediction pi t = g(zi t). Per Theorem 3.5, we compute its prediction uncertainty ωi t = ˆyi t pi t 2 F , where ˆyi t = onehot(argmax(pi t)). We then temporarily store the pair (zi t, ωi t) in the Pseudo-Source bank M = M (zi t, ωi t). Subsequently, M is updated based on its element count and confidence. The update rule is as follows: ( M, if |M| k {(zi t, ωi t) | ωi t ωk min}, else (15) where ωk min represents k-th lowest uncertainty value in M. Finally, the Pseudo-Source correlation can be calculated as Test-time Correlation Alignment ˆΣs = 1 ˆns 1 ˆZT s ˆZs 1 ˆns 1ˆns ˆZT s ˆZs1ˆns where ˆZs = {zi t|zi t M} and ˆns = |M|. 4.2. Methods Linear TCA: During testing, given the embeddings Zt and ˆZs sampled from the test and pseudo-source domains, respectively, our objective is to minimize their correlation distance: LLinear TCA = Σt ˆΣs 2 To achieve this alignment, we aim to obtain a suitable linear transformation W as follows: W T Σt W ˆΣs 2 Setting W T Σt W = ˆΣs and applying eigenvalue decomposition, the closed-form solution for W can be derived as 1: W = UtΛ1/2 t ˆU T s ˆΛ 1/2 s (19) where ˆUs and Ut represent the eigenvector matrices, ˆΛs and Λt are the corresponding diagonal eigenvalue matrices, respectively. The transformed embeddings of the test domain can then be computed as: Z t = (Zt µt) W + ˆµs (20) where µt and ˆµs denote the mean embeddings of Zt and ˆZs, respectively. As shown in Eq. (20), we also align the instance-wise shift |µs µt| by using ˆµs. Finally, the predictions for the test domain after adaptation through Linear TCA are: P t = g(Z t) (21) Linear TCA+: Since Linear TCA does not require parameter updates to the model, it can serve as a plug-and-play boosting module for TTA methods. Specifically, during a TTA method optimizes the original model hθ to h θ via Eq. (1), we can obtain the resulting embeddings ZT T A and predictions PT T A. By applying the Linear TCA on ZT T A and PT T A with the same process from Eq. (15) to (21), the predictions of Linear TCA+ are obtained. More details on these methods are provided in Appendix C. 1To enhance the robustness of the results, we recommend using torch s automatic gradient descent method to mitigate potential instabilities associated with eigenvalue decomposition. For the following experiments, we implement this method with a fixed learning rate of 1e-3. 1 2 3 4 5 6 7 8 9 10 Group Average Uncertainty 0 1 2 3 4 5 Correlation Distance to the Pseudo-Source t s Accuracy (%) R2=0.96, =-1.00 R2=0.97, =1.00 Correlation Distance to the Source t s Correlation Distance to the Source t s Figure 3. Experimental validation of theories. (a) Average uncertainty and correlation distance to source domain of each group, groups with lower uncertainty exhibit smaller correlation distances. (b) Relationships between ACC, correlation distance to the source, and correlation distance to the pseudo-source, both ACC and Σt Σs are strongly linearly related to Σt ˆΣs . 5. Experiments 5.1. Experimental settings Following previous studies, we evaluate the adaptation performance on two main tasks: domain generalization (PACS (Li et al., 2017), Office Home (Venkateswara et al., 2017), and Domain Net (Peng et al., 2019) dataset) and image corruption (CIFAR-10-C,CIFAR-100-C, and Image Net-C (Hendrycks & Dietterich, 2019)). What s more, we also evaluate our method on multimodal tasks based on CLIP (Radford et al., 2021). The comparison methods include backpropagation-free (BN (Schneider et al., 2020), T3A (Iwasawa & Matsuo, 2021), Ada NPC (Zhang et al., 2023)) and backpropagation-based methods (TENT (Wang et al., 2020), PLC (Lee, 2013), EATA (Niu et al., 2022), SAR (Niu et al., 2023), TSD (Wang et al., 2023), TIPI (Nguyen et al., 2023), TEA (Yuan et al., 2024)). Backbone networks include Res Net-18/50 (He et al., 2016) and Vi T-B/16 (Dosovitskiy, 2020). Additionally, the evaluation encompasses multiple aspects, including accuracy, efficiency, and resistance to forgetting. For Linear TCA+, we report its results combined with the best baseline. Refer to Appendix D for more implement information. For further experimental results and analysis, please see Appendix E. 5.2. Experimental validation of theories For Theorem 3.5: Correlation of high-certainty test instances approximates the source correlation. We divide the test embeddings of CIFAR-10-C under Res Net-18 into 10 groups based on prediction uncertainty and calcu- Test-time Correlation Alignment Domain PACS Office Home Domain Net Method Res Net-18 Res Net-50 Vi T-B/16 AVG Res Net-18 Res Net-50 Vi T-B/16 AVG Res Net-18 Res Net-50 Vi T-B/16 AVG AVG SOURCE 81.84 84.78 87.02 84.54 62.01 67.01 76.11 68.37 39.13 43.58 50.29 44.33 65.75 BN 82.65 84.99 - - 62.05 66.30 - - 37.93 41.94 - - - T3A 83.20 85.71 88.06 85.66 63.26 67.85 78.87 69.99 40.62 44.92 53.94 46.49 67.38 Ada NPC 83.48 86.12 89.11 86.24 62.88 67.05 77.26 69.07 40.50 45.17 53.28 46.32 67.21 TENT 85.23 88.07 84.98 86.09 63.09 67.67 76.95 69.24 39.42 43.97 39.96 41.12 65.48 PLC 83.16 86.59 87.97 85.91 62.22 66.44 76.51 68.39 37.96 41.63 47.29 42.30 65.53 EATA 83.30 84.68 86.60 84.86 62.49 67.01 76.98 68.83 41.65 46.89 54.40 47.65 67.11 SAR 85.41 85.79 87.12 86.11 62.51 67.94 76.66 69.04 38.49 42.19 42.81 41.16 65.44 TIPI 87.39 88.01 87.98 87.79 63.25 68.36 77.09 69.57 36.05 44.08 39.70 39.94 65.77 TEA 87.19 88.75 87.37 87.77 63.43 68.56 76.15 69.38 39.43 43.48 48.41 43.78 66.98 TSD 87.83 89.99 83.43 87.08 62.47 68.63 75.49 68.87 38.59 42.12 48.72 43.14 66.36 Linear TCA 83.59 86.78 88.61 86.33 63.66 68.43 78.26 70.12 40.79 44.89 52.79 46.16 67.53 Ours Linear TCA+ 88.77 90.68 89.30 89.58 64.27 69.32 79.02 70.87 42.20 47.17 55.49 48.29 69.58 Img Cop CIFAR-10-C CIFAR-100-C Image Net-C Method Res Net-18 Res Net-50 Vi T-B/16 AVG Res Net-18 Res Net-50 Vi T-B/16 AVG Res Net-18 Res Net-50 Vi T-B/16 AVG AVG SOURCE 50.80 50.77 71.48 57.68 31.01 34.02 51.71 38.91 14.70 18.15 39.83 24.23 40.27 BN 73.70 72.24 - - 48.38 48.41 - - 27.59 32.06 - - - T3A 58.89 54.87 74.21 62.65 32.52 34.94 54.24 40.57 14.56 18.05 39.78 24.13 42.45 Ada NPC 57.72 54.75 74.60 62.36 29.70 32.27 53.21 38.39 11.93 15.62 36.78 21.44 40.73 TENT 75.21 72.33 71.48 73.01 50.82 50.12 52.72 51.22 35.39 41.32 48.01 41.57 55.27 PLC 73.72 72.34 71.46 72.51 48.35 48.38 51.71 49.48 27.59 32.06 38.74 32.80 51.59 EATA 73.86 72.38 73.67 73.30 49.71 49.89 62.40 54.00 39.19 48.17 64.36 50.58 59.29 SAR 73.97 73.37 71.48 72.94 51.60 50.25 54.29 52.05 38.55 46.30 57.94 47.60 57.53 TIPI 76.10 72.46 71.48 73.35 50.61 50.30 52.36 51.09 35.73 41.87 48.50 42.03 55.49 TEA 76.20 72.54 71.48 73.41 50.67 50.21 52.31 51.06 32.38 38.90 41.37 37.55 54.01 TSD 76.93 73.23 71.47 73.88 49.35 49.60 51.74 50.23 30.11 35.08 41.33 35.51 53.20 Linear TCA 60.96 60.27 77.26 66.16 35.03 37.28 55.42 42.58 16.07 19.34 41.37 25.60 44.78 Ours Linear TCA+ 77.13 73.53 79.55 76.74 52.08 51.17 63.71 55.65 39.21 48.22 64.71 50.71 61.04 Table 1. Accuracy comparison of different TTA methods based on Res Net-18/50 and Vi T-B/16 backbones. The upper part of the table corresponds to the domain generalization task, while the lower part corresponds to the image corruption task. The best results are highlighted in boldface, and the second ones are underlined. - indicates that Vi T-B/16 does not include any BN layers. late the correlation distance between each group and the original source. As shown in Figure 3a, groups with lower uncertainty exhibit smaller correlation distances, indicating a closer approximation to the source correlation. For Theorem 3.6 and Corollary 3.7: Test-time correlation alignment reduces test classification error. We iteratively optimize W and record the correlation distances between test domain and pseudo-source domain, Σt ˆΣs , as well as the true distances between test domain and source domain, Σt Σs , and ACC. As shown in Figure 3b, under a linear fit (R2 = 0.97), Σt ˆΣs is strongly positively related to Σt Σs (Spearman correlation coefficient = 1). Under R2 = 0.96, it is strongly negatively related to ACC (Spearman correlation coefficient = -1). This further validates that pseudo-source correlation alignment promotes alignment with the original source. Additionally, pseudo-source correlation alignment effectively reduces test classification error, thus improving the model s domain adaptation capability. 5.3. Comparison with TTA Methods Accuracy. Table 1 presents ACC comparisons between TCA methods and state-of-the-art TTA approaches across various benchmarks, backbones, and tasks. (1) As a plugand-play module, Linear TCA+ consistently enhances performance across all datasets and backbones, achieving a new state-of-the-art. Notably, on the CIFAR-10-C dataset with the Vi T-B/16 backbone, Linear TCA+ shows substantial improvements over the best-performing baseline, with an increase of 4.95%. (2) Across datasets, Linear TCA shows robust improvement compared to the source model, with average gains of 1.79%, 1.75%, 1.78%, 8.48%, 3.67% and 4.51%, respectively. Particularly, on the Office Home and Domain Net dataset, Linear TCA outperforms all baseline methods. However, on datasets such as CIFAR-10/100-C and Image Net-C, although Linear TCA yields ACC gains of 8.48%, 3.67% and 4.51% over the source model, it falls short of some advanced methods. (3) Across backbones, Linear TCA also demonstrates robust improvements compared to the source model, especially with the Vi T-B/16 backbone, surpassing the highest-performing baseline on most datasets. We provide a detailed analysis of these experimental results in Section 5.5 to further reveal the strengths and limitations of Linear TCA. Type Method Memory(MB) Time(s) Res Net-18 Res Net-50 Vi T-B/16 AVG Res Net-18 Res Net-50 Vi T-B/16 AVG SOURCE 920.61 878.87 917.02 905.50 3.92 9.16 3.98 5.69 BN +0.25 +48.57 - - +0.88 +4.80 - - T3A +1.00 +4.43 +2.02 +2.48 +1.98 +3.62 +12.22 +5.94 Ada NPC +2.04 +8.23 +2.96 +4.41 +1.73 +2.78 +12.08 +5.53 TENT +1883.63 +4788.93 +5246.53 +3973.03 +3.85 +11.52 +12.27 +9.22 PLC +1934.14 +4787.26 +8624.95 +5115.45 +5.94 +9.51 +25.86 +13.77 EATA +5332.44 +10838.53 +11172.56 +9114.51 +1.76 +4.20 +22.81 +9.59 SAR +2642.82 +5380.18 +5401.31 +4474.77 +11.23 +23.31 +54.08 +29.54 TSD +2025.07 +5162.55 +9280.69 +5489.44 +4.70 +13.47 +34.68 +17.62 TEA +7316.95 +15733.10 +16082.00 +13044.02 +123.14 +278.87 +596.28 +332.76 TIPI +2520.01 +10660.83 +12542.71 +8574.52 +26.54 +49.73 +45.25 +40.51 Ours TCA +0.00 +0.00 +0.00 +0.00 +0.06 +0.07 +0.08 +0.07 Table 2. Maximum GPU memory usage and running time of different TTA methods on CIFAR-10-C. Efficiency. We evaluate each method s efficiency in terms of peak GPU memory usage and total runtime. Table 2 reports results on the CIFAR-10-C dataset across different Test-time Correlation Alignment Type Method PACS Office Home CIFAR-10-C CIFAR-100-C AVG SOURCE 99.35 94.40 92.36 70.39 89.12 BN 98.90 (-0.44) 92.85 (-1.55) 62.98 (-29.38) 39.45 (-30.94) 73.55 (-15.58) T3A 99.33 (-0.01) 93.31 (-1.09) 91.95 (-0.41) 65.66 (-4.73) 87.56 (-1.56) Ada NPC 99.28 (-0.06) 93.31 (-1.09) 92.00 (-0.36) 63.88 (-6.51) 87.12 (-2.01) Linear TCA 99.42 (+0.08) 93.87 (-0.53) 91.16 (-1.20) 67.35 (-3.04) 87.95 (-1.17) Linear TCA w/o W 99.35 (0.00) 94.40 (0.00) 92.36 (0.00) 70.39 (0.00) 89.12 (0.00) TENT 96.74 (-2.61) 92.79 (-1.61) 90.26 (-2.10) 67.27 (-3.12) 86.76 (-2.36) PLC 97.12 (-2.23) 92.73 (-1.67) 63.05 (-29.31) 39.48 (-30.91) 73.09 (-16.03) EATA 98.33 (-1.02) 93.66 (-0.74) 90.24 (-2.12) 68.52 (-1.87) 87.69 (-1.44) SAR 97.12 (-2.23) 86.35 (-8.05) 90.31 (-2.05) 68.77 (-1.62) 85.63 (-3.49) TSD 95.10 (-4.24) 85.37 (-9.03) 67.78 (-24.58) 39.48 (-30.91) 71.93 (-17.19) TEA 90.22 (-9.13) 93.30 (-1.10) 90.60 (-1.76) 68.93 (-1.46) 85.76 (-3.36) TIPI 98.15 (-1.20) 92.79 (-1.61) 70.75 (-21.61) 46.03 (-24.36) 76.93 (-12.20) Linear TCA+ 99.03 (-0.31) 93.65 (-0.75) 90.68 (-1.68) 69.05 (-1.34) 88.10 (-1.02) Table 4. The accuracy of different TTA methods when returning to the source domain after adaptation. BP-Free indicates backpropagation-free TTA methods, while BP-Based denotes backpropagation-dependent ones. backbones. TCA consistently achieves the lowest memory and time cost. For memory, since we record peak memory consumption, Linear TCA exhibits minimal independent memory usage (as shown in Table 3) and thus does not impose additional memory constraints on the device. Method Res Net18 Res Net50 Vi T-B/16 AVG Linear TCA 118.56 448.64 452.11 339.77 Table 3. Independent maximum GPU memory usage of Linear TCA on CIFAR-10-C. In contrast, other methods are embedded within the model s forward and backward propagation processes, significantly increasing peak memory usage (e.g., TEA uses 15 the memory of the source model). For runtime, with a Vi T-B/16 backbone, Linear TCA requires only 0.6% of Ada NPC s time. These results highlight Linear TCA s high efficiency, making it well-suited for resource-constrained edge deployment. Forgetting resistance. Table 4 shows the change in accuracy when each method (using Res Net-18) returns to the source domain after adaptation. Linear TCA w/o W refers to the variant without the linear transformation, which is equivalent to the original source model and thus retains full source knowledge. Despite applying W, Linear TCA demonstrates much stronger resistance to forgetting than other methods especially on PACS, where it even improves source performance, showing positive backward transfer. Moreover, Linear TCA+ further enhances the forgetting robustness of existing TTA methods. 5.4. Performance on Closed-Source Foundation Models To validate TCA s effectiveness on closed-source foundation models, we conduct experiments with CLIP (Radford et al., 2021) on PACS, Office Home, and VLCS datasets, following the experimental setup in WATT (Osowiechi et al., 2024). As shown in Table 5, TCA achieving performance improvements of 1.28%, 2.08%, and 2.85% on the three datasets respectively. The superior results stem from our method s explicit alignment of embedding distributions with the source domain, which proves particularly effective for multi-modal models like CLIP that compute image-text similarity directly. While Linear TCA+ holds a slight advantage, Method PACS AVG A C P S CLIP 97.44 97.38 99.58 86.06 95.12 TENT 97.54 0.02 97.37 0.04 99.58 0.00 86.37 0.05 95.22 TPT 95.10 0.41 91.42 0.22 98.56 0.40 87.23 0.06 93.08 CLIPAr TT 97.64 0.02 97.37 0.02 99.58 0.00 86.79 0.04 95.35 WATT-P 97.49 0.08 97.47 0.04 99.58 0.00 89.73 0.16 96.07 WATT-S 97.66 0.08 97.51 0.02 99.58 0.00 89.56 0.14 96.08 Linear TCA 97.80 99.39 99.94 92.32 97.36 Linear TCA+ 97.87 0.06 99.20 0.02 99.94 0.00 92.36 0.06 97.34 Method Office Home AVG A C P R CLIP 79.30 65.15 87.34 89.31 80.28 TENT 79.26 0.14 65.64 0.05 87.49 0.02 89.50 0.04 80.47 TPT 81.97 0.17 67.01 0.21 89.00 0.06 89.66 0.06 81.91 CLIPAr TT 79.34 0.05 65.69 0.11 87.35 0.07 89.29 0.03 80.42 WATT-P 80.37 0.25 68.59 0.13 88.15 0.07 90.18 0.03 81.82 WATT-S 80.43 0.09 68.26 0.11 88.02 0.08 90.14 0.06 81.71 Linear TCA 85.55 68.70 90.26 90.58 83.77 Linear TCA+ 85.62 0.38 69.25 0.1 90.29 0.01 90.42 0.1 83.90 Method VLCS AVG C L S V CLIP 99.43 67.75 71.74 84.90 80.96 TENT 99.43 0.00 67.31 0.14 71.57 0.15 85.10 0.11 80.85 TPT 97.62 0.12 49.77 0.03 71.56 0.86 71.17 0.70 72.53 CLIPAr TT 99.43 0.00 67.74 0.10 71.67 0.01 84.73 0.08 80.89 WATT-P 99.36 0.00 67.55 0.39 74.75 0.07 82.53 0.10 81.05 WATT-S 99.36 0.00 68.59 0.25 75.16 0.12 83.24 0.05 81.59 Linear TCA 99.86 73.98 78.47 84.41 84.18 Linear TCA+ 99.88 0.03 74.39 0.1 79.44 0.22 84.06 0.14 84.44 Table 5. The accuracy comparison of different methods on PACS, Office Home, and VLCS datasets using CLIP-Vi T-B/16. : numbers are from WATT (Osowiechi et al., 2024). The best results are highlighted in boldface, and the second ones are underlined. both variants perform similarly, suggesting that even simple correlation alignment can notably enhance performance on popular models like CLIP. This underscores its effectiveness as a versatile plug-and-play module for improving diverse adaptation methods. 5.5. Analysis Effective range of Linear TCA. As discussed in Section 5.3, although Linear TCA+ significantly improves all TTA methods, Linear TCA only achieves SOTA performance on part of datasets and backbones. The reasons may be: 1) Although the highest-certainty embeddings are selected as pseudo-source domains, if these embeddings still exhibit substantial differences from the true source domain (or if the backbone s feature extraction capacity is insufficient, e.g., Res Net-18 vs. Vi T-B/16), the performance ceiling of Linear TCA is limited. In contrast, other TTA methods update the model, thereby raising this ceiling and facilitating easier correlation alignment for Linear TCA+. 2) We only use a linear transformation W for alignment, which may work well for simple shifts; however, the true distribution shifts may not conform to linear transformations but exhibit complex nonlinear relationships. We design a demo experiment to validate this hypothesis. In Figure 4a and b, the test domain shifts are linear and nonlinear, respectively. As shown, the transformed embeddings in Figure 4a align well with the original distribution, while the perfor- Test-time Correlation Alignment Linear Shift Source Domain Xs Test Domain Xt Transformed Test Xt 15 10 5 0 5 10 Nonlinear Shift Source Domain Xs Test Domain Xt Transformed Test Xt Number of Pseudo-Source Instances k Accuracy (%) ACC of SOURCE Number of Sampling Test Instances k2 Figure 4. Analysis of TCA. (a) When the test domain (yellow) undergoes a nearly linear shift from the source domain (blue), after adaptation by Linear TCA, the transformed test domain (red) is well-aligned with the source. (b) In the case of a nonlinear shift, although partial alignment is achieved, it is still insufficient. (c) and (d) Ablation study examining the effect of pseudo-source domain size and test domain size. Backbone Method PACS AVG Office Home AVG A C P S A C P R Source 78.37 77.39 95.03 76.58 81.84 56.45 48.02 71.34 72.23 62.01 Linear TCA 80.91 81.02 95.69 76.74 83.59 59.46 50.40 72.02 72.78 63.66 Linear TCA+ 88.38 87.12 96.59 83.00 88.77 59.83 51.80 72.29 73.17 64.27 Linear TCA(MLP-2) 81.24 81.73 95.89 78.15 84.25 59.62 50.84 72.07 72.94 63.87 Linear TCA+(MLP-2) 88.68 87.15 96.68 83.19 88.93 59.83 51.80 72.79 73.46 64.47 Linear TCA(MLP-3) 81.62 81.81 96.03 79.35 84.70 59.62 50.65 72.07 73.02 63.84 Linear TCA+(MLP-3) 88.38 87.23 96.59 83.36 88.98 59.83 52.08 72.79 73.54 64.56 Table 6. Extending Linear TCA/Linear TCA+ by introducing MLPbased transformations with two (MLP-2) and three (MLP-3) layers The best results are highlighted in boldface, and the second ones are underlined. mance in Figure 4b shows partial alignment which is still insufficient. We further explore the utilization of nonlinear architecture (MLP) for calculating transformation W. As shown in Table 6, incorporating nonlinear activations with deeper architectures leads to further improvements. Ablation study. Our method involves only one hyperparameter the number of pseudo-source embeddings k. Since the total number of test samples is often unknown in practice, we also sample k2 embeddings from the test set to study its impact. As shown in Figure 4c,d, Linear TCA achieves the best accuracy on Office Home when k = 10 and k2 = 2400. Importantly, it consistently outperforms the source model across a wide range of k and k2, demonstrating strong practical applicability. Upper performance bound for TCA. To assess the upper bound of TCA, we conduct two additional experiments in Table 7: (a) fine-tuning directly on the target domain; (b) applying Linear TCA and Linear TCA+ with real source distributions. Compared to the original Linear TCA+, approach (b) further improves performance, by 0.38% on PACS and 1.03% on Office Home. Both (a) and (b) outperform the orig- Backbone Method PACS AVG Office Home AVG A C P S A C P R Source 78.37 77.39 95.03 76.58 81.84 56.45 48.02 71.34 72.23 62.01 Linear TCA 80.91 81.02 95.69 76.74 83.59 59.04 49.97 71.77 72.89 63.42 Linear TCA+ 88.38 87.12 96.59 83.00 88.77 59.33 51.18 72.20 71.72 63.61 TCA(a) 86.18 82.67 95.03 80.81 86.17 58.69 50.80 72.04 72.92 63.61 Linear TCA(b) 81.59 81.48 96.05 77.51 84.15 59.94 51.63 72.36 73.48 64.35 Linear TCA+(b) 88.98 87.57 96.74 83.30 89.15 60.03 52.29 72.55 73.87 64.64 Table 7. Upper performance bound for TCA. TCA(a): Finetuning directly on the target distribution. Linear TCA(b) and Linear TCA+(b): Applying Linear TCA and Linear TCA+ with real source distributions. Method Art Domian of Office Home AVG Batch Size 1 2 4 8 16 32 64 128 256 512 1024 Estimation error 2542 2414 2434 2430 2415 2417 2437 2415 2413 2424 2427 2433 Source 56.45 56.45 56.45 56.45 56.45 56.45 56.45 56.45 56.45 56.45 56.45 56.45 TEA 0.824 18.01 40.79 49.23 55.54 55.71 57.35 58.55 57.11 57.82 57.93 46.26 Linear TCA 58.61 58.61 58.57 58.77 58.94 58.86 59.06 59.46 59.27 59.35 59.56 59.05 Linear TCA+ 0.824 18.54 41.37 51.13 56.05 58.44 59.3 59.83 59.66 59.86 59.96 47.72 Table 8. Accuracy comparisons of different TTA methods on the Art domain of Office Home dataset with varying batch sizes based on Res Net-18. The best results are highlighted in boldface, and the second ones are underlined. inal Linear TCA in most domains. On Office Home, even the simpler Linear TCA with real source data (b) surpasses fine-tuning (a), highlighting the importance of source distribution and the effectiveness of approximating it in TCA. Performance under difference batch sizes. To study the impact of batch size, we evaluate TCA s performance and pseudo-source estimation error under varying batch sizes in Table 8. Even with batch size 1, Linear TCA outperforms the source model by 2.16%, and Linear TCA+ consistently improves over TEA across all settings. This robustness stems from TCA s incremental estimation of test-domain covariance, which converges over time. While small batch sizes mainly affect early predictions, their influence diminishes as more data is seen. Moreover, the pseudo-source estimation error remains unaffected by batch size, since it relies on a small set of high-confidence samples (Figure 4c) and benefits from the same incremental computation. 6. Conclusion and Future Work In this paper, we introduce the Test-time Correlation Alignment (TCA) to address the chanllenges in Test-Time Adaptation (TTA), such as overlooking feature correlation, overhead computation and domain forgetting. TCA is a novel paradigm that enhances test-time adaptation (TTA) by aligning the correlation of high-certainty instances and test instances and is demonstrated with a theoretical guarantee. Extensive experiments validate our theoretical insights and show that TCA methods significantly outperforms baselines on accuracy, efficiency, and forgetting resistance across various tasks, benchmarks and backbones. Future work may incorporate more nonlinear transformations for more effective TCA. Additionally, with the interesting positive backward transfer phenomenon in Table 4, we will further investigate the underlying mechanism. Test-time Correlation Alignment Impact Statement This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here. Arjovsky, M., Chintala, S., and Bottou, L. Wasserstein generative adversarial networks. 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In real-world scenarios, significant domain shifts often occur between training and test data, which can severely degrade the performance of conventional machine learning methods. To tackle this challenge, CORrelation ALignment (CORAL) (Cheng et al., 2021a) is introduced to align the feature-wise statistics of the source and target distributions through a linear transformation. Similar to CORAL, Maximum Mean Discrepancy (MMD) (Gretton et al., 2006) is another technique for mitigating domain gap by minimizing the mean discrepancy between different domains. Unlike CORAL, which focuses on feature-wise correlations, MMD match the instance-wise statistics of the domain distribution. Correlation Alignment has been extended and applied in several innovative ways. Deep CORAL (Sun & Saenko, 2016) extends CORAL to deep neural networks by employing a differentiable Correlation Alignment loss function. This enables end-to-end domain adaptation and facilitates more effective nonlinear transformations, thereby enhancing generalization performance on unsupervised target domains. Deer CORAL (Das et al., 2021) leverages CORAL loss in combination with synthetic data to address long-tailed distributions in real-world scenarios. High-order CORAL (Cheng et al., 2021b), which is inspired by MMD and CORAL, utilizes third-order correlation to capture more detailed statistical information and effectively characterize complex, non-Gaussian distributions. IJDA (Qian et al., 2023) introduces a novel metric that combines MMD and CORAL to improve distribution alignment and enhance domain confusion. In addition to these advancements, recent studies have explored the integration of CORAL into more complex models and settings. For example, CAADG (Rahman et al., 2020a) presents a domain generalization framework that combines CORAL with adversarial learning to jointly adapt features and minimize the domain disparity. Moreover, JCGNN (Wang et al., 2021) integrates CORAL into Graph Neural Network (GNN) to generate the domain-invariant features. Although CORAL has achieved significant success in domain adaptation (DA), its application in test-time adaptation (TTA) is constrained by privacy and resource limitations, which make it infeasible to compute the source correlation. This limitation significantly hampers the practicality of CORAL in more real-world scenarios, such as test-time correlation alignment (TCA). A.2. Test-Time Adaptation In real-world scenarios, test data often undergoes natural variations or corruptions, leading to distribution shifts between the training and testing domains. Recently, various Test-Time Adaptation (TTA) approaches have been proposed to adapt pre-trained models during testing. These methods can be broadly categorized into batch normalization calibration methods, pseudo-labeling methods, consistency training methods, and clustering-based training methods (Liang et al., 2024). For further discussion, we classify them into two groups based on their dependence on backpropagation, as outlined in (Niu et al., 2024). Test-time Correlation Alignment Backpropagation (BP)-Free TTA: This group includes batch normalization (BN) calibration methods (Wu et al., 2024; Schneider et al., 2020) and certain pseudo-labeling methods (Zhang et al., 2023) that do not update model parameters. BN-based methods posit that the statistics in BN layers capture domain-specific knowledge. To mitigate the domain gap, these methods replace training BN statistics with updated statistics computed from the target domain. Some pseudo-labeling methods such as T3A (Iwasawa & Matsuo, 2021) utilize prototype similarity and Ada NPC (Zhang et al., 2023) utilize k-nearest neighbor (k NN) to refine predictions. Although BP-Free TTA methods are computationally efficient, their image corruption adaptation capabilities are often limited. Backpropagation (BP)-Based TTA: This group encompasses certain pseudo-labeling methods (Zeng et al., 2024), consistency training methods (Sinha et al., 2023), and clustering-based training methods (Lee et al., 2024). Some pseudo-labeling methods use filtering strategies, such as thresholding or entropy-based approaches, to generate reliable pseudo-labels, thereby reducing the discrepancy between predicted and pseudo-labels. For instance, PLC (Lee, 2013) updates classifier layer parameters with certain pseudo-labels during adaptation. TSD (Wang et al., 2023) filters unreliable features or predictions with high entropy, as lower entropy correlates with higher accuracy, and applies a consistency filter to refine instances further. Consistency training methods aim to enhance the stability of network predictions or features by addressing variations in input data, such as noise or perturbations, and changes in model parameters. TIPI (Nguyen et al., 2023), for example, simulates domain shifts via input transformations and employs regularizers to maintain model invariance. Clustering-based training methods leverage clustering techniques to group target features, and reduce uncertainty in predictions and improving model robustness. TENT (Wang et al., 2020) minimizes prediction entropy on target data, while EATA (Niu et al., 2022) selects reliable instances to minimize entropy loss and applies a Fisher regularizer. SAR (Niu et al., 2023) removes noisy instances with large gradients and encourages model weights to converge toward a flat minimum, enhancing robustness against residual noise. Generally, BP-Based TTA methods demonstrate superior domain adaptation capabilities compared to BP-Free methods, but they typically require multiple backward propagations for each test instance, leading to computational inefficiencies. Despite their strengths, both BP-Free and BP-Based TTA methods perform instance-wise alignment without considering feature correlation alignment. Our proposed method, TCA, is orthogonal to most existing TTA methods. It achieves both instance-wise and correlation alignment without backpropagation. TCA is a theoretically supported TTA paradigm that effectively addresses the challenges of efficiency and domain forgetting. By applying a simple linear transformation, TCA performs both instance and correlation alignment without requiring additional model updates. Moreover, it can function as a plug-and-play module to enhance the performance of existing TTA methods. B. Proof of Theoretical Statement B.1. Proof of Theorem 3.5 Here, we present Theorem 3.5 again for convenience. Theorem 3.5 Let hθ( ) = g(f( )) be an L-Lipschitz continuous hypothesis on H. Ω:= S x Dt B(x, r ) is the set of balls near the test data. We sample k source instances from Ds Ωand k test instances from Dt to obtain [Xs, Zs, Ps] and [Xt, Zt, Pt] by hθ( ), respectively. Per Assumption 3.2, Assumption 3.3 and Assumption 3.4, with a probability of at least1 exp (ctµ πd I rd I ns 1)2 2ctµ πd I rd I ns + log k , we have Zt Zs Pt Ps + o(kr ) Jg(Zs) (22) where πd I = λ(B(0, 1)) is the volume of the d I dimension unit ball and d I is the dimension of input x. Furthermore, considering the true source correlation Σs = E[ Zs T Zs] and the pseudo-source correlation ˆΣs = Zt T Zt, where Zs and Zt are centered. With a probability of at least min(1 exp (ctµ πd I rd I ns 1)2 2ctµ πd I rd I ns + log k , 1 δ), the correlation Test-time Correlation Alignment distance Σs ˆΣs is bounded by: 2 Zs F ( ˆYt Pt F + A Jg(Zs) F ) + ( ˆYt Pt F + A Jg(Zs) F )2 + B (23) where ˆYt is the one-hot encoding of Pt, A = o(kr ) + kϵ(hθ(Xt)) + kϵ(hθ(Xs)) represents the output error of the sampled instances, and B = q 2k is the sampling error. We begin by proving Equation (22). According to Assumption 3.3 and Assumption 3.4, and under the additional assumption that Zt = Zs + d Zs, where zs Zs, dzs r , the function g( ) can be expressed using a Taylor series: Pt = g(Zt) = g(Zs + d Zs) = Ps + Jg(Zs)d Zs + o(d Zs) (24) Pt Ps = Jg(Zs)d Zs + o(d Zs) (25) d Zs = Pt Ps o(d Zs) Jg(Zs) (26) d Zs F = Pt Ps o(d Zs) F + o(d Zs) Jg(Zs) F + o(kr ) Jg(Zs) Next, we examine the probability of the distance between zs and zt satisfying dzs r under Assumption 3.2. Following the result from (Zhang et al., 2023), for any xt Xt, and r < rt, the probability distribution of xs falling within a ball B(xt, r) of radius r centered at xt is given by: Ds(xs B(xt, r)) = Z dλ (xs) dxs µ λ(B(xt, r) Ds) ctµ πd Ird I (28) Let I(xs B(xt, r)) be an indicator function, where I(xs B(xt, r)) is independent and identically distributed Bernoulli random variables, representing the probability Ds(xs B(xt, r)). Let Sn(xt) = Pns i=1 I(xs B(xt, r)) denotes the number of source instances xs Ds that fall within B(xt, r). Then, Sn(xt) follows a Binomial distribution. Let W Binomial(ns, ctµ πd Ird I). By applying Chernoff s inequality, we obtain the probability that the number of source data points falling within B(xt, r) is less than m: P(Sn(xt) < m) = P(W < m) exp (E[W] m)2 = exp (ctµ πd Ird Ins m)2 2ctµ πd Ird Ins Let x(i) s denote the i-th nearest data point to xt within B(xt, r). The probability that the distance between x(i) s and xt is less than r is given by: P( x(m) s xt r) = P(Sn(xt) m) 1 exp (ctµ πd Ird Ins m)2 2ctµ πd Ird Ins For a fixed xt, it suffices to find a single nearest neighbor xs that lies within the ball B(xt, r), and thus we set m = 1. By applying the union bound, the desired probability can be expressed as follows: Test-time Correlation Alignment xt Xt P( x(1) s xt r) xt Xt P(Sn(xt) 1) xt Xt P(Sn(xt) < 1) 1 k exp (ctµ πd Ird Ins 1)2 2ctµ πd Ird Ins = 1 exp (ctµ πd Ird Ins 1)2 2ctµ πd Ird Ins + log k Thus, with at least the probability 1 exp (ctµ πd I rd I ns 1)2 2ctµ πd I rd I ns + log k , the distance satisfies dxs r rt. Finally, under Assumption 3.3, let r = r dzs F L dxs F r (32) Combining the above equations, with at least the probability: 1 exp (ctµ πd Ird Ins 1)2 2ctµ πd Ird Ins + log k d Zs F Pt Ps F + o(kr ) Jg(Zs) This completes the proof of Equation (22). Next, we prove Equation (23). Let Σ s denote the correlation matrix computed from k sampled source instances Zs, and let ˆΣs denote the pseudo-source correlation matrix computed from k sampled test instances Zt. These matrices are computed as follows: Σ s = ZT s Zs (34) ˆΣs = ZT t Zt = (Zs + d Zs)T (Zs + d Zs) = ZT s Zs + ZT s d Zs + (d Zs)T Zs + (d Zs)T d Zs (35) The change in the correlatione matrix is: ˆΣs Σ s = ZT s d Zs + (d Zs)T Zs + (d Zs)T d Zs (36) Using the Frobenius norm, we obtain: ˆΣs Σ s F ZT s d Zs + (d Zs)T Zs + (d Zs)T d Zs F 2 Zs F d Zs F + d Zs 2 F (37) Additionally, since Σ s is obtained from k source domain instances and contains statistical error relative to the true covariance matrix Σs = E[Σ s]. By Hoeffding s inequality, we have: P( Σ s E[Σ s] 2 F ϵ) 2 exp 2kϵ Test-time Correlation Alignment Here, d denotes the range of Σ s, which is set to 1. Let 2 exp 2kϵ d2 = σ, then: 2 ) 2k (39) With a probability of at least 1 σ, we have: Σ s Σs F < ϵ = By combining Equations (37) and (40), we obtain: Σs ˆΣs F Σs Σ s F + Σ s ˆΣs F 2k + 2 Zs F d Zs F + d Zs 2 F (41) We can further expand Equation (41) by applying Equation (33). However, since we cannot determine the true Ps in Equation (33), we scale Pt Ps F as follows: Pt Ps F = Pt ˆYt + ˆYt l + l Ps F Pt ˆYt F + ˆYt l F + l Ps F = Pt ˆYt F + ϵ(h(Xt)) + ϵ(h(Xs)) (42) where l is the true labels. Finally, combining Equations (33), (41) and (42), we derive the following proposition: with at least min(1 exp (ctµ πd I rd I ns 1)2 2ctµ πd I rd I ns + log k , 1 σ): Σs Σt F 2 Zs F ˆYt Pt F + A ˆYt Pt F + A where ˆYt is the one-hot encoding of Pt, A = o(kr ) F + ϵ(h(Xt)) + ϵ(h(Xs)) represents the output generalization error, 2k is the sampling error. B.2. Proof of Theorem 3.6 Here, we present Theorem 3.6 again for convenience. Theorem 3.6 Let H be a hypothesis class of VC-dimension dv. If ˆh H minimizes the empirical error ˆϵs(h) on Ds, and h t = arg minh H ϵt(h) is the optimal hypothesis on Dt, with the assumption that all hypotheses are L-Lipschitz continuous, then δ (0, 1), with probability with at least 1 δ the following inequality holds: ϵt(ˆh) ϵt(h t ) + O( q µs µt 2 F + Σs Σt 2 F ) + C where C = 2 q dvlog(2ns) log(δ) 2ns + 2γ and γ = minh H{ϵs(h(t)) + ϵt(h(t))}. µs, µt, Σs and Σt denote the means and correlations of the source and test embeddings, respectively. We use O( ) to hide the constant dependence. To complete the proof, we begin by introducing some necessary definitions and assumptions. Test-time Correlation Alignment Definition B.1. (Wasserstein Distance (Arjovsky et al., 2017)). The ρ-th order Wasserstein distance between two distributions Ds and Dt is defined as: Wρ(Ds, Dt) = inf γ Π[Ds,Dt] ZZ d(xs, xt)ρdγ(xs, xt) 1/ρ (44) where Π[Ds, Dt] is the set of all joint distributions on Xs Xt with marginal distributions Ds and Dt, and d(xs, xu) is the distance function between two instances xs and xu. The Wasserstein distance can be intuitively understood in terms of the optimal transport problem, where d(xs, xt)ρ represents the unit cost of transporting mass from xs Ds to xt Dt, and γ(xs, xt) is the transport plan that satisfies the marginal constraints. According to the Kantorovich-Rubinstein theorem, the dual representation of the second-order Wasserstein distance can be written as: = inf γ Π[Ds,Dt] ZZ d(xs, xt)2dγ(xs, xt) 1/2 = sup f L 1 ( µs µt 2 2 + tr(Σs + Σt 2(Σ1/2 s ΣtΣ1/2 s )1/2)1/2 (45) where µs and µt are the means of f(xs) and f(xt), respectively, and f L = sup |f(xs) f(xt)| d(xs,xt) is the Lipschitz semi-norm, which measures the rate of change of the function f relative to the distance between xs and xt. In this paper, we use W2 as the default and omit the subscript 2. For completeness, we present Theorem 1 from (Shen et al., 2018) as follows: Lemma B.2. (Theorem 1 in (Shen et al., 2018)) Let H be an L-Lipschitz continuous hypothesis class with VC-dimension dv. Given two domain distributions, Ds and Dt, let γ = minh H{ϵs(h(t)) + ϵt(h(t))}. The risk of hypothesis ˆh on the test domain is then bounded by: ϵt(ˆh) γ + ϵs(ˆh) + 2LW(Ds, Dt) (46) From Definition B.1 and Lemma B.2, the difference between the true error on the training domain ϵs(h(t)) and the true error on the test domain ϵt(h(t)) can be obtained: W(DS, DU) = q µs µt 2 2 + tr(Σs + Σt 2(Σ1/2 s ΣtΣ1/2 1 )1/2) q µs µt 2 F + Σs Σt 2 F (47) |ϵt(ˆh) ϵs(ˆh)| γ + 2L q µs µt 2 F + Σs Σt 2 F (48) we use O to hide the constant dependence. Thus, we have: |ϵt(ˆh) ϵs(ˆh)| γ + O( q µs µt 2 F + Σs Σt 2 F ) (49) Then, we provide an upper bound on the difference between the true error ϵs(h(t)) and the empirical error ˆϵs(h(t)) on the source domain. We apply Lemma 7 of (Gui et al., 2024): P[|ϵt(ˆh) ϵs(ˆh)| ϵ] (2ns)dv exp( 2nsϵ2) (50) For any δ (0, 1), set δ = (2ns)dv exp( 2nsϵ2), we have: dv log(2ns) log δ Test-time Correlation Alignment Therefore, with probability at least 1 δ, we have: |ˆϵs(ˆh) ϵs(ˆh)| dv log(2ns) log δ Combining Equations (49) and (52), let h j(t) = arg minh H ϵt(h), we obtain: ϵs(ˆh(t)) + γ + O q µs µt 2 2 + Σs Σt 2 F ˆϵs(ˆh(t)) + dv log(2ns) log δ 2ns + γ + O q µs µt 2 2 + Σs Σt 2 F ˆϵs(h t (t)) + dv log(2ns) log δ 2ns + γ + O q µs µt 2 2 + Σs Σt 2 F ϵs(h t (t)) + 2 dv log(2ns) log δ 2ns + γ + O q µs µt 2 2 + Σs Σt 2 F ϵt(h t (t)) + 2 dv log(2ns) log δ 2ns + 2γ + 2O q µs µt 2 2 + Σs Σt 2 F = ϵt(h t (t)) + O q µs µt 2 2 + Σs Σt 2 F + C (53) which completes the proof. C. Method Details In this section, we describe the steps involved in the TCA algorithms used for test-time adaptation. The algorithm aligns feature correlations between the test and pseudo-source domains, without requiring access to the source domain data. The steps of the algorithm are outlined in Algorithm 1. Algorithm 1 Linear TCA Algorithm 1: Input: Test instances Xt, source model hθ. 2: Output: Final predictions P T . 3: If use Linear TCA+: Update θ by Equation (1) 4: Obtain embeddings and predictions: ˆPt, Zt = hθ(Xt) 5: Select k high-certainty embeddings: ˆZs = {Zt[i] | ωi t ωk min} 6: Compute linear transformation matrix W: W = argmin W W T Σt W ˆΣs 2 7: Apply transformation to embeddings: Z t = (Zt µt) W + ˆµs 8: Generate final predictions: P t = g(Z t) Test-time Correlation Alignment D. Experimental Details D.1. Datasets The datasets used in this work consist of a variety of domain-shift challenges, enabling a comprehensive evaluation of test-time adaptation methods. The primary datasets employed include: PACS: The PACS dataset comprises 9,991 images across 7 distinct classes: {dog, elephant, giraffe, guitar, horse, house, person}. These images are drawn from four domains: {art, cartoons, photos, sketches }. Office Home: This dataset contains images from 4 different domains: {art, clipart, product, real-world}, with a total of 15,500 images. It includes 65 object categories, and the challenge lies in the significant domain shifts between the different visual styles. Office Home is widely used for evaluating domain generalization and adaptation methods due to its large number of categories and diverse image sources. Domain Net: The Domain Net dataset is a large-scale dataset used in transfer learning, consisting of 6 domains: {clipart, infograph, painting, quickdraw, real, and sketch}. It consists of a total of 586,575 images, with each domain containing 345 classes. CIFAR-10/100C: CIFAR-10 and CIFAR-100 are both foundational datasets in computer vision, containing 60,000 32x32 color images across 10 and 100 classes, respectively. The CIFAR-10/100C variants introduce additional corruptions (e.g., noise, blur, weather conditions) to simulate real-world distribution shifts, making them highly relevant for evaluating robustness under adversarial conditions. Image Net-C: Image Net-C is significantly larger compared to CIFAR10-C and CIFAR100-C. This dataset contains 1,281,167 training images and 50,000 test images, categorized into 1,000 classes. Like CIFAR10-C and CIFAR100-C, Image Net-C also includes 15 types of corruptions. D.2. Backbones The choice of backbone models is critical for the performance of domain adaptation algorithms, as they must efficiently extract features from images across various domains. For this work, we select the following backbone architectures: Res Net-18/50: Res Net-18 and Res Net-50 are used as backbone models in this study, where Res Net-18 offers a relatively lightweight model with fewer parameters, suitable for faster training and inference, while Res Net-50, with its deeper architecture, provides a more expressive feature representation that may improve performance on complex datasets. Vi T-B/16: The Vision Transformer (Vi T) is a more recent architecture that has demonstrated state-of-the-art performance in various vision tasks by treating images as sequences of patches. Vi T-B/16 refers to a Vi T model with a base configuration and a patch size of 16x16 pixels. Vi T models are especially useful in scenarios where large-scale data and diverse domains are involved. CLIP: Contrastive Language Image Pre-Training (CLIP), developed by Open AI, is a cutting-edge multimodal model that bridges visual and textual domains through contrastive learning. CLIP employs dual encoders (Res Net/Vi T for images and Transformer for text) to project both modalities into a shared semantic space, enabling zero-shot classification by matching image features with natural language prompts. Both Res Net and Vi T backbones are well-established in the literature and serve as strong candidates for evaluating domain adaptation techniques, with Res Net-18/50 being more computationally efficient and Vi T-B/16 being particularly effective in capturing complex relationships across domains. In this work, the zero-shot classification model CLIP is also included as a backbone to validate the effectiveness of our proposed methods on closed-source foundation models. D.3. Implementation Details Consistent with prior work (Wang et al., 2020; Niu et al., 2022; 2023; Nguyen et al., 2023; Yuan et al., 2024; Iwasawa & Matsuo, 2021; Wang et al., 2023; Zhang et al., 2023), hyperparameter tuning in our experimental setup is conducted Test-time Correlation Alignment across datasets. Specifically, in the Domain Generalization task, we first identify the optimal parameter set based on the highest accuracy achieved on the default domain (art paintings in PACS, art in Office Home and clipart in Domain Net). These parameters are then applied to other domains to assess their performance. Specifically, we conduct a search for the learning rate within the range {1e-7, 5e-7, 1e-6, 5e-6, 1e-5, 5e-5, 1e-4, 5e-4, 1e-3, 5e-3, 1e-2, 5e-2, 1e-1}. For methods that include an entropy filter component (e.g., TSD), we explore the entropy filter hyperparameter in the set {1, 5, 10, 15, 20, 50, 100, 200, 300}. For Ada NPC, we explore the hyperparameter k (the number of nearest samples used for voting) over {5, 10, 15, 20, 30, 40, 50}. For the Linear TCA method, we optimized the number of pseudo-source instances k within the range {5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 100, 200, 300}. For most datasets and backbones, smaller k values generally yield satisfactory results. For datasets with a substantial number of images per class, it is advisable to experiment with larger k values. For the Linear TCA+ method, we conducted an optimization of k values on the basis of other top-performing test-time adaptation method and its parameter settings. For the Image Corruption task, we experiment with each TTA method using learning rates from {1e-7, 5e-7, 1e-6, 5e-6, 1e-5, 5e-5, 1e-4, 5e-4, 1e-3, 5e-3, 1e-2, 5e-2, 1e-1} and the entropy filter hyperparameter in the set {1, 5, 10, 15, 20, 50, 100, 200, 300}. The parameter range for k in Ada NPC, Linear TCA/Linear TCA+ remains consistent with their respective selections in Domain Generalization task. The top-performing test-time adaptation approach on the Image Corruption is selected as the base method for Linear TCA+. The best performance results obtained for each method are selected as the final experimental outcomes. For the pre-trained model on Image Net-C dataset, we utilize the model provided by Torch Vision. During the Test-Time Adaptation phase, both the Domain Generalization and Image Corruption tasks utilize specific batch size for different backbones. Res Net-18 and Res Net-50 use a batch size of 128, whereas the Vi T-B/16 is configured with a batch size of 64. For the implementation of the TCA method, we first obtain the embeddings of all test data during the testing phase. Based on the inter-class proportion of the test data, we perform high-certainty filtering to select instances that match this proportion to construct the pseudo-source domain. Subsequently, we use the correlation distance between the pseudo-source domain and the test domain to compute the linear transformation matrix W. Finally, we apply this linear transformation to the previously retained embeddings of the test data and make final prediction. E. Additional Experimental Results E.1. Comparison Results Details Tables 9 to 17 provide the detailed results of our experimental results on Domain Generalization task, and Tables 18 to 26 offers a detailed overview of the outcomes from our Image Corruption task. These results demonstrate that our TCA method consistently outperforms other state-of-the-art TTA approaches across most domians and corruption types, effectively validating the TCA s capability to robustly enhance accuracy performance during the test phase. E.2. Analysis Details Figures 5 and 6 illustrate the adaptation process of Linear TCA to datasets with linear and nonlinear shifts, respectively. Figures (a) to (f) depict the gradual alignment process of linear and nonlinear shifts. Notably, Linear TCA demonstrates significantly better performance in adapting to linear shifts compared to nonlinear ones, which the Linear TCA s proficiency in handling simpler, linear distribution shifts while revealing its limitations when addressing more complex, nonlinear transformations. We also provide the code for generating source and target domain features with both linear and nonlinear distribution shifts. The features are generated using Py Torch and serve as synthetic examples. The source domain features (Xs, X(2) s ) consist of clusters sampled from normal distributions with fixed offsets. The target domain features (Xt, X(2) t ) are scaled and shifted versions of normal distributions to simulate linear and nonlinear domain shifts. The generated features can be visualized using 2D scatter plots for better understanding of the distributional changes. Test-time Correlation Alignment Linear Shift Code: # Linear Shift # Source domain features X_s = torch.cat((torch.randn(30, 2), torch.randn(30, 2) + 15, torch.randn(30, 2) + torch.tensor([0, 10])), dim=0) # Target domain features X_t = torch.cat((torch.randn(250, 2) * 2 + 7, torch.randn(250, 2) * 2.5 + torch.tensor([0, 20]), torch.randn(250, 2) * 3 + 21), dim=0) Nonlinear Shift Code: # Nonlinear Shift # Source domain features X_s_2 = torch.cat((torch.randn(30, 2), torch.randn(30, 2) + 10, torch.randn(30, 2) + torch.tensor([0, 10]), torch.randn(30, 2) + torch.tensor([-5, -10])), dim=0) # Target domain features X_t_2 = torch.cat((torch.randn(250, 2) * 3 + 5, torch.randn(250, 2) + 10, torch.randn(250, 2) * 2 + torch.tensor([0, 20]), torch.randn(250, 2) * 2.5 + torch.tensor([-9, 1])), dim=0) Backbone Method PACS Avg Hyper-parameters A C P S Source (He et al., 2016) 78.37 77.39 95.03 76.58 81.84 nan BN (Schneider et al., 2020) 80.91 80.80 95.09 73.81 82.65 nan T3A (Iwasawa & Matsuo, 2021) 80.27 79.56 95.57 77.40 83.20 fk=50 Ada NPC (Zhang et al., 2023) 80.81 79.14 96.17 77.81 83.48 fk=100 k=5 TENT (Wang et al., 2020) 82.86 82.12 96.11 79.82 85.23 lr=5e-3 PLC (Lee, 2013) 81.69 81.36 95.87 73.71 83.16 lr=1e-3 EATA (Niu et al., 2022) 82.71 81.36 94.79 74.34 83.30 lr=1e-2 SAR (Niu et al., 2023) 83.30 82.55 95.09 80.68 85.41 lr=1e-1 TIPI (Nguyen et al., 2023) 85.50 84.90 96.05 83.13 87.39 lr=5e-3 TEA (Yuan et al., 2024) 86.47 85.79 95.69 80.81 87.19 lr=5e-3 TSD (Wang et al., 2023) 86.96 86.73 96.41 81.22 87.83 lr=1e-4 fk=100 Linear TCA 80.91 81.02 95.69 76.74 83.59 fk TCA=30 Linear TCA + 88.38 87.12 96.59 83.00 88.77 TSD fk TCA=25 Table 9. Accuracy comparison of different TTA methods on PACS dataset based on Res Net-18 backbone. The best results are highlighted in boldface, and the second ones are underlined. Test-time Correlation Alignment Backbone Method PACS Avg Hyper-parameters A C P S Source (He et al., 2016) 83.89 81.02 96.17 78.04 84.78 nan BN (Schneider et al., 2020) 85.50 85.62 96.77 72.05 84.99 nan T3A (Iwasawa & Matsuo, 2021) 84.86 82.47 97.01 78.52 85.71 fk=100 Ada NPC (Zhang et al., 2023) 85.11 82.85 97.13 79.41 86.12 fk=200 k=10 TENT (Wang et al., 2020) 88.09 87.33 97.19 79.69 88.07 lr=1e-3 PLC (Lee, 2013) 86.52 84.94 97.01 77.88 86.59 lr=1e-3 EATA (Niu et al., 2022) 84.72 85.20 96.35 72.46 84.68 lr=5e-5 SAR (Niu et al., 2023) 85.55 85.62 96.77 75.24 85.79 lr=1e-2 TIPI (Nguyen et al., 2023) 88.18 87.93 97.13 78.80 88.01 lr=1e-3 TEA (Yuan et al., 2024) 88.67 87.80 97.54 80.99 88.75 lr=1e-3 TSD (Wang et al., 2023) 90.43 89.89 97.84 81.80 89.99 lr=1e-4 fk=100 Linear TCA 86.28 83.92 96.95 79.99 86.78 fk TCA=30 Linear TCA + 90.92 90.10 97.84 83.86 90.68 TSD fk TCA=30 Table 10. Accuracy comparison of different TTA methods on PACS dataset based on Res Net-50 backbone. The best results are highlighted in boldface, and the second ones are underlined. Backbone Method PACS Avg Hyper-parameters A C P S Source (He et al., 2016) 86.96 84.30 98.02 78.77 87.02 nan BN (Schneider et al., 2020) 0.00 0.00 0.00 0.00 0.00 nan T3A (Iwasawa & Matsuo, 2021) 88.23 85.96 98.86 79.18 88.06 fk=50 Ada NPC (Zhang et al., 2023) 89.01 87.37 98.98 81.06 89.11 fk=200 k=10 TENT (Wang et al., 2020) 89.60 73.08 97.90 79.33 84.98 lr=5e-3 PLC (Lee, 2013) 87.70 85.28 98.62 80.30 87.97 lr=5e-4 EATA (Niu et al., 2022) 87.45 84.17 97.84 76.92 86.60 lr=5e-3 SAR (Niu et al., 2023) 86.96 84.30 98.02 79.18 87.12 lr=5e-2 TIPI (Nguyen et al., 2023) 87.99 84.17 98.20 81.55 87.98 lr=5e-4 TEA (Yuan et al., 2024) 88.77 85.41 97.96 77.35 87.37 lr=1e-3 TSD (Wang et al., 2023) 90.72 85.41 97.96 59.63 83.43 lr=1e-5 fk=20 Linear TCA 88.57 86.52 98.26 81.09 88.61 fk TCA=15 Linear TCA + 88.96 86.90 98.26 83.05 89.30 TIPI fk TCA=30 Table 11. Accuracy comparison of different TTA methods on PACS dataset based on Vi T-B/16 backbone. The best results are highlighted in boldface, and the second ones are underlined. Backbone Method Office Home Avg Hyper-parameters A C P R Source (He et al., 2016) 56.45 48.02 71.34 72.23 62.01 nan BN (Schneider et al., 2020) 55.62 49.32 70.60 72.66 62.05 nan T3A (Iwasawa & Matsuo, 2021) 56.61 50.06 73.39 72.99 63.26 fk=20 Ada NPC (Zhang et al., 2023) 55.95 49.42 73.10 73.05 62.88 fk=20 k=5 TENT (Wang et al., 2020) 56.94 50.65 71.86 72.92 63.09 lr=1e-3 PLC (Lee, 2013) 55.95 49.37 70.83 72.73 62.22 lr=5e-5 EATA (Niu et al., 2022) 56.41 49.62 71.66 72.27 62.49 lr=1e-3 SAR (Niu et al., 2023) 57.15 50.31 70.24 72.34 62.51 lr=5e-2 TIPI (Nguyen et al., 2023) 57.03 50.61 72.07 73.28 63.25 lr=1e-3 TEA (Yuan et al., 2024) 58.55 50.47 71.75 72.94 63.43 lr=5e-4 TSD (Wang et al., 2023) 58.06 49.81 71.37 70.67 62.47 lr=1e-4 fk=10 Linear TCA 59.46 50.40 72.02 72.78 63.66 fk TCA=10 Linear TCA + 59.83 51.80 72.29 73.17 64.27 TEA fk TCA=10 Table 12. Accuracy comparison of different TTA methods on Office Home dataset based on Res Net-18 backbone. The best results are highlighted in boldface, and the second ones are underlined. Test-time Correlation Alignment Backbone Method Office Home Avg Hyper-parameters A C P R Source (He et al., 2016) 64.85 52.26 75.04 75.88 67.01 nan BN (Schneider et al., 2020) 63.54 52.71 73.89 75.05 66.30 nan T3A (Iwasawa & Matsuo, 2021) 65.02 53.31 76.10 76.96 67.85 fk=100 Ada NPC (Zhang et al., 2023) 63.74 52.33 75.72 76.43 67.05 fk=200 k=5 TENT (Wang et al., 2020) 64.65 54.85 75.04 76.15 67.67 lr=5e-4 PLC (Lee, 2013) 63.82 52.83 74.09 75.03 66.44 lr=5e-5 EATA (Niu et al., 2022) 63.95 53.95 74.57 75.56 67.01 lr=1e-3 SAR (Niu et al., 2023) 64.77 55.92 75.24 75.81 67.94 lr=1e-2 TIPI (Nguyen et al., 2023) 64.73 56.24 75.47 77.00 68.36 lr=1e-3 TEA (Yuan et al., 2024) 65.97 57.57 74.72 75.97 68.56 lr=1e-3 TSD (Wang et al., 2023) 65.51 56.54 76.17 76.31 68.63 lr=1e-4 fk=1 Linear TCA 66.50 54.39 75.76 77.07 68.43 fk TCA=5 Linear TCA + 67.16 56.22 76.86 77.05 69.32 TSD fk TCA=10 Table 13. Accuracy comparison of different TTA methods on Office Home dataset based on Res Net-50 backbone. The best results are highlighted in boldface, and the second ones are underlined. Backbone Method Office Home Avg Hyper-parameters A C P R Source (He et al., 2016) 73.51 63.18 82.68 85.06 76.11 nan BN (Schneider et al., 2020) 0.00 0.00 0.00 0.00 0.00 nan T3A (Iwasawa & Matsuo, 2021) 77.79 65.57 85.92 86.18 78.87 fk=5 Ada NPC (Zhang et al., 2023) 75.57 63.76 84.30 85.43 77.26 fk=200 k=5 TENT (Wang et al., 2020) 74.58 64.15 83.74 85.36 76.95 lr=1e-3 PLC (Lee, 2013) 74.41 63.51 82.81 85.31 76.51 lr=1e-4 EATA (Niu et al., 2022) 74.17 64.81 83.58 85.38 76.98 lr=1e-3 SAR (Niu et al., 2023) 74.95 63.07 83.58 85.06 76.66 lr=1e-1 TIPI (Nguyen et al., 2023) 74.50 64.47 83.92 85.49 77.09 lr=1e-3 TEA (Yuan et al., 2024) 73.71 63.23 82.74 84.92 76.15 lr=1e-4 TSD (Wang et al., 2023) 75.94 55.95 84.75 85.33 75.49 lr=1e-5 fk=20 Linear TCA 76.02 67.35 84.12 85.56 78.26 fk TCA=5 Linear TCA + 77.21 68.36 84.64 85.88 79.02 TIPI fk TCA=5 Table 14. Accuracy comparison of different TTA methods on Office Home dataset based on Vi T-B/16 backbone. The best results are highlighted in boldface, and the second ones are underlined. Backbone Method Domain Net Avg Hyper-parameters C I P Q R S Source (He et al., 2016) 57.30 16.86 45.03 12.69 56.89 46.00 39.13 nan BN (Schneider et al., 2020) 57.26 11.55 43.32 11.77 56.58 47.09 37.93 nan T3A (Iwasawa & Matsuo, 2021) 58.44 18.57 46.80 14.54 57.66 47.72 40.62 fk=100 Ada NPC (Zhang et al., 2023) 57.61 15.83 44.89 18.44 59.72 46.53 40.50 fk=100 k=10 TENT (Wang et al., 2020) 58.41 13.09 45.17 13.02 57.89 48.94 39.42 lr=1e-4 PLC (Lee, 2013) 57.45 12.60 44.77 10.01 55.74 47.20 37.96 lr=1e-5 EATA (Niu et al., 2022) 59.18 16.22 46.65 18.04 59.59 50.21 41.65 lr=1e-3 SAR (Niu et al., 2023) 59.13 13.10 45.75 4.88 58.25 49.83 38.49 lr=5e-3 TIPI (Nguyen et al., 2023) 58.42 11.68 42.53 5.37 50.76 47.58 36.05 lr=5e-4 TEA (Yuan et al., 2024) 58.01 12.83 45.10 14.33 57.55 48.80 39.43 lr=5e-5 TSD (Wang et al., 2023) 57.73 12.19 44.58 12.78 55.94 48.31 38.59 lr=1e-5 fk=100 Linear TCA 58.67 18.60 46.85 14.88 57.93 47.80 40.79 fk TCA=10 Linear TCA + 59.95 16.89 47.68 18.35 59.67 50.66 42.20 EATA fk TCA=10 Table 15. Accuracy comparison of different TTA methods on Domain Net dataset based on Res Net-18 backbone. The best results are highlighted in boldface, and the second ones are underlined. Test-time Correlation Alignment Backbone Method Domain Net Avg Hyper-parameters C I P Q R S Source (He et al., 2016) 63.68 20.93 50.35 12.95 62.16 51.42 43.58 nan BN (Schneider et al., 2020) 63.30 14.84 48.54 10.83 62.02 52.12 41.94 nan T3A (Iwasawa & Matsuo, 2021) 63.76 21.06 49.82 18.46 64.05 52.39 44.92 fk=100 Ada NPC (Zhang et al., 2023) 64.38 20.12 51.07 17.34 65.59 52.51 45.17 fk=200 k=10 TENT (Wang et al., 2020) 64.95 17.46 51.58 11.28 64.04 54.51 43.97 lr=1e-4 PLC (Lee, 2013) 63.56 14.89 49.09 8.83 60.83 52.61 41.63 lr=1e-5 EATA (Niu et al., 2022) 65.89 19.88 52.67 20.36 66.58 55.99 46.89 lr=5e-4 SAR (Niu et al., 2023) 65.44 14.63 50.68 3.94 63.94 54.49 42.19 lr=5e-3 TIPI (Nguyen et al., 2023) 64.97 17.47 51.63 11.67 64.03 54.69 44.08 lr=1e-4 TEA (Yuan et al., 2024) 64.87 16.95 51.40 11.48 61.90 54.31 43.48 lr=1e-4 TSD (Wang et al., 2023) 64.31 16.53 50.75 8.52 58.97 53.63 42.12 lr=5e-5 fk=5 Linear TCA 64.58 23.79 50.06 14.10 63.60 53.21 44.89 fk TCA=5 Linear TCA + 66.46 21.04 51.61 20.47 66.86 56.57 47.17 EATA fk TCA=5 Table 16. Accuracy comparison of different TTA methods on Domain Net dataset based on Res Net-50 backbone. The best results are highlighted in boldface, and the second ones are underlined. Backbone Method Domain Net Avg Hyper-parameters C I P Q R S Source (He et al., 2016) 71.62 25.59 57.34 18.07 71.90 57.24 50.29 nan BN (Schneider et al., 2020) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 nan T3A (Iwasawa & Matsuo, 2021) 73.56 26.95 59.77 27.34 75.77 60.24 53.94 fk=100 Ada NPC (Zhang et al., 2023) 73.76 25.52 59.86 24.42 75.99 60.14 53.28 fk=200 k=5 TENT (Wang et al., 2020) 72.65 18.00 35.08 4.20 74.03 35.82 39.96 lr=1e-4 PLC (Lee, 2013) 72.29 19.27 56.69 5.00 72.45 58.04 47.29 lr=5e-5 EATA (Niu et al., 2022) 73.91 28.76 61.71 24.79 75.39 61.84 54.40 lr=1e-3 SAR (Niu et al., 2023) 73.06 17.42 40.94 11.37 73.67 40.39 42.81 lr=5e-2 TIPI (Nguyen et al., 2023) 72.71 17.62 33.37 4.54 73.98 35.96 39.70 lr=1e-4 TEA (Yuan et al., 2024) 71.96 24.17 55.31 8.83 72.20 58.00 48.41 lr=5e-5 TSD (Wang et al., 2023) 72.40 23.47 59.20 4.12 73.54 59.60 48.72 lr=1e-6 fk=50 Linear TCA 73.37 28.42 60.88 20.66 73.39 60.03 52.79 fk TCA=5 Linear TCA + 75.02 30.11 63.33 25.14 76.05 63.31 55.49 EATA fk TCA=5 Table 17. Accuracy comparison of different TTA methods on Domain Net dataset based on Vi T-B/16 backbone. The best results are highlighted in boldface, and the second ones are underlined. Method CIFAR-10-C Avg Gau. Sho. Imp. Def. Gla. Mot. Zoo. Sno. Fro. Fog Bri. Con. Ela. Pix. Jpe. Source (He et al., 2016) 27.43 33.56 21.57 43.64 40.48 51.26 51.29 68.18 54.52 66.65 87.50 27.59 67.06 48.86 72.37 50.80 BN (Schneider et al., 2020) 66.05 68.22 56.83 82.34 57.86 79.78 82.32 74.99 74.30 78.85 87.22 81.80 70.31 73.61 71.00 73.70 T3A (Iwasawa & Matsuo, 2021) 44.16 50.32 29.64 56.98 49.02 60.85 62.29 70.20 60.83 70.75 87.23 37.68 71.60 58.59 73.22 58.89 Ada NPC (Zhang et al., 2023) 40.96 47.54 27.01 54.84 46.47 59.93 61.91 70.12 60.59 71.18 87.16 35.13 71.06 58.15 73.73 57.72 TENT (Wang et al., 2020) 65.09 72.78 58.93 82.78 59.02 81.01 83.92 77.82 75.83 79.34 88.10 82.77 72.10 76.47 72.26 75.21 PLC (Lee, 2013) 66.06 68.25 56.92 82.66 57.69 79.78 82.29 74.84 74.33 78.91 87.07 81.82 70.49 73.63 71.00 73.72 EATA (Niu et al., 2022) 66.89 68.21 56.76 82.49 57.59 80.10 82.09 74.90 74.35 78.82 87.13 82.04 70.66 74.16 71.73 73.86 SAR (Niu et al., 2023) 66.28 68.23 58.30 82.34 59.20 79.78 82.32 74.99 74.53 78.85 87.22 82.51 70.32 73.61 71.00 73.97 TIPI (Nguyen et al., 2023) 67.69 73.21 59.54 83.80 62.36 81.29 84.15 78.15 76.90 79.91 88.63 82.99 72.46 77.34 73.11 76.10 TEA (Yuan et al., 2024) 70.76 72.46 61.44 83.40 60.45 81.56 84.05 77.57 76.12 81.07 87.97 82.82 72.51 76.51 74.26 76.20 TSD (Wang et al., 2023) 72.33 75.73 64.84 83.24 61.45 82.49 83.92 78.29 75.79 81.96 87.55 79.43 73.07 78.48 75.36 76.93 Linear TCA 52.17 55.61 36.34 57.08 48.18 62.25 62.26 71.94 67.17 73.09 87.23 41.70 70.28 56.43 72.68 60.96 Linear TCA + 73.11 75.93 65.30 83.23 62.13 82.21 83.87 78.41 76.25 82.12 87.42 79.32 73.48 78.60 75.62 77.13 Table 18. Accuracy comparisons of different TTA methods on CIFAR-10-C dataset at damage level of 5, based on Res Net-18 backbone. The best results are highlighted in boldface, and the second ones are underlined. Test-time Correlation Alignment Method CIFAR-10-C Avg Gau. Sho. Imp. Def. Gla. Mot. Zoo. Sno. Fro. Fog Bri. Con. Ela. Pix. Jpe. Source (He et al., 2016) 30.81 37.09 24.71 38.07 41.66 51.97 51.17 68.49 60.52 66.79 86.19 28.25 65.19 38.95 71.66 50.77 BN (Schneider et al., 2020) 61.98 63.05 56.25 82.58 54.49 80.11 82.61 74.16 72.36 79.28 87.04 81.06 67.16 71.27 70.22 72.24 T3A (Iwasawa & Matsuo, 2021) 45.34 49.51 36.76 39.10 46.88 56.85 53.52 65.88 57.84 68.82 84.61 33.13 68.20 46.08 70.49 54.87 Ada NPC (Zhang et al., 2023) 41.93 47.29 32.50 41.60 45.70 56.09 56.38 67.25 59.62 69.90 85.51 32.24 68.27 46.08 70.88 54.75 TENT (Wang et al., 2020) 62.04 63.30 56.26 82.66 54.52 80.09 82.68 74.40 72.43 79.20 87.21 81.11 67.34 71.39 70.32 72.33 PLC (Lee, 2013) 62.35 62.71 56.09 82.57 54.07 80.24 82.91 74.54 72.26 79.37 87.20 81.09 67.62 71.39 70.71 72.34 EATA (Niu et al., 2022) 62.61 63.63 56.13 82.34 54.71 79.97 82.16 74.89 72.16 79.27 87.66 81.32 67.76 70.81 70.28 72.38 SAR (Niu et al., 2023) 65.12 66.49 58.49 82.58 55.65 80.12 82.61 75.10 73.60 79.63 87.04 81.56 68.49 72.63 71.47 73.37 TIPI (Nguyen et al., 2023) 62.02 63.61 55.37 82.80 54.43 80.29 83.11 74.81 72.77 78.96 87.52 81.35 67.49 71.72 70.70 72.46 TEA (Yuan et al., 2024) 63.92 65.15 55.73 82.32 52.34 80.54 83.14 74.99 73.17 80.08 87.58 80.90 67.57 70.47 70.26 72.54 TSD (Wang et al., 2023) 64.42 65.56 56.16 83.06 53.95 80.88 83.32 75.18 73.58 80.17 87.84 81.49 68.38 72.91 71.61 73.23 Linear TCA 52.05 55.76 43.06 51.79 49.06 61.68 62.03 71.53 67.67 72.83 86.04 37.62 69.92 50.28 72.69 60.27 Linear TCA + 65.27 66.63 59.15 82.87 56.37 80.78 82.80 75.05 72.69 79.61 86.85 80.97 69.10 72.74 72.05 73.53 Table 19. Accuracy comparisons of different TTA methods on CIFAR-10-C dataset at damage level of 5, based on Res Net-50 backbone. The best results are highlighted in boldface, and the second ones are underlined. Method CIFAR-10-C Avg Gau. Sho. Imp. Def. Gla. Mot. Zoo. Sno. Fro. Fog Bri. Con. Ela. Pix. Jpe. Source (He et al., 2016) 37.25 44.31 39.94 83.16 70.31 83.54 85.80 87.15 85.06 79.19 92.75 29.73 84.73 84.68 84.58 71.48 BN (Schneider et al., 2020) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 T3A (Iwasawa & Matsuo, 2021) 47.84 52.78 51.52 83.16 73.06 83.35 85.66 87.04 84.96 79.53 92.72 36.44 84.49 85.72 84.81 74.21 Ada NPC (Zhang et al., 2023) 48.70 54.14 50.95 83.31 74.10 83.46 85.86 87.20 85.31 80.09 92.70 37.92 84.70 85.84 84.70 74.60 TENT (Wang et al., 2020) 37.25 44.31 39.94 83.16 70.31 83.54 85.80 87.15 85.05 79.19 92.75 29.73 84.73 84.68 84.58 71.48 PLC (Lee, 2013) 37.18 44.27 39.84 83.15 70.31 83.54 85.77 87.15 85.07 79.19 92.75 29.72 84.74 84.69 84.59 71.46 EATA (Niu et al., 2022) 46.55 48.34 31.91 86.30 69.31 84.78 86.56 88.62 87.25 80.32 93.05 45.84 84.87 86.99 84.29 73.67 SAR (Niu et al., 2023) 37.25 44.31 39.94 83.16 70.31 83.54 85.80 87.15 85.06 79.19 92.75 29.73 84.73 84.68 84.58 71.48 TIPI (Nguyen et al., 2023) 37.24 44.32 39.93 83.17 70.30 83.55 85.77 87.16 85.07 79.18 92.74 29.73 84.75 84.69 84.58 71.48 TEA (Yuan et al., 2024) 37.23 44.31 39.92 83.17 70.30 83.56 85.79 87.15 85.06 79.20 92.75 29.73 84.74 84.69 84.58 71.48 TSD (Wang et al., 2023) 37.17 44.22 39.80 83.18 70.35 83.58 85.80 87.16 85.08 79.20 92.75 29.70 84.74 84.70 84.59 71.47 Linear TCA 56.10 60.11 55.13 85.21 76.10 84.90 87.50 87.89 87.00 82.26 92.86 45.61 85.64 87.20 85.37 77.26 Linear TCA + 64.74 64.97 54.15 87.24 75.39 85.88 88.35 88.94 88.24 83.10 93.09 60.32 85.72 88.16 84.96 79.55 Table 20. Accuracy comparisons of different TTA methods on CIFAR-10-C dataset at damage level of 5, based on Vi T-B/16 backbone. The best results are highlighted in boldface, and the second ones are underlined. Method CIFAR-100-C Avg Gau. Sho. Imp. Def. Gla. Mot. Zoo. Sno. Fro. Fog Bri. Con. Ela. Pix. Jpe. Source (He et al., 2016) 10.46 12.49 3.36 34.44 23.63 38.10 42.67 39.25 33.01 32.84 55.78 11.55 46.48 34.88 46.15 31.01 BN (Schneider et al., 2020) 39.78 39.81 29.95 56.18 40.92 54.71 58.68 48.52 49.59 46.79 61.89 48.63 50.26 54.61 45.37 48.38 T3A (Iwasawa & Matsuo, 2021) 10.51 11.59 3.93 36.77 26.94 40.54 45.08 39.49 34.68 35.63 56.05 13.23 47.61 40.63 45.15 32.52 Ada NPC (Zhang et al., 2023) 10.01 10.69 3.64 33.57 24.38 37.00 41.06 35.88 31.45 32.28 52.19 12.29 43.32 37.21 40.50 29.70 TENT (Wang et al., 2020) 43.19 44.38 31.70 58.86 43.29 56.57 61.00 51.19 50.66 50.75 64.02 47.77 52.08 57.74 49.11 50.82 PLC (Lee, 2013) 39.65 39.47 30.25 56.31 40.70 54.50 58.88 48.56 49.37 46.73 62.04 48.68 50.44 54.20 45.45 48.35 EATA (Niu et al., 2022) 41.95 41.87 31.96 57.55 42.62 55.94 59.00 49.47 50.43 48.48 62.54 49.57 51.12 55.64 47.50 49.71 SAR (Niu et al., 2023) 44.07 45.12 33.37 59.80 43.69 57.21 61.15 51.70 51.97 51.49 63.90 50.46 52.64 57.97 49.52 51.60 TIPI (Nguyen et al., 2023) 44.04 45.11 32.86 57.89 43.85 55.87 60.08 52.16 51.69 49.38 63.40 44.24 51.43 57.42 49.76 50.61 TEA (Yuan et al., 2024) 43.78 43.43 32.68 58.20 42.62 56.30 60.67 50.84 51.32 50.16 63.87 49.95 51.78 56.60 47.83 50.67 TSD (Wang et al., 2023) 41.77 42.52 32.16 57.88 41.38 56.08 59.84 49.30 50.43 49.65 62.83 43.52 50.49 55.23 47.20 49.35 Linear TCA 13.98 16.45 5.42 38.96 29.15 42.56 46.30 42.40 39.41 39.56 56.78 15.33 49.51 42.56 47.07 35.03 Linear TCA + 44.70 45.77 33.76 59.77 44.45 57.41 61.49 52.25 52.52 51.92 64.25 51.18 53.28 58.68 49.81 52.08 Table 21. Accuracy comparisons of different TTA methods on CIFAR-100-C dataset at damage level of 5, based on Res Net-18 backbone. The best results are highlighted in boldface, and the second ones are underlined. Test-time Correlation Alignment Method CIFAR-100-C Avg Gau. Sho. Imp. Def. Gla. Mot. Zoo. Sno. Fro. Fog Bri. Con. Ela. Pix. Jpe. Source (He et al., 2016) 17.23 19.42 9.77 35.34 31.87 39.15 41.98 41.99 38.68 32.00 54.56 11.18 47.57 42.51 47.02 34.02 BN (Schneider et al., 2020) 42.09 42.22 31.37 56.23 42.36 54.61 57.22 48.43 49.61 45.29 60.06 45.07 50.52 55.09 45.96 48.41 T3A (Iwasawa & Matsuo, 2021) 18.46 20.58 10.98 37.34 34.81 40.71 44.04 40.74 39.09 33.48 53.89 10.94 47.23 45.59 46.20 34.94 Ada NPC (Zhang et al., 2023) 17.41 18.85 10.53 35.26 31.43 37.49 40.99 37.25 35.78 30.65 49.75 10.68 43.74 41.78 42.53 32.27 TENT (Wang et al., 2020) 43.96 44.24 31.76 58.87 43.16 56.70 59.49 50.64 50.86 49.07 60.81 43.55 52.37 57.94 48.39 50.12 PLC (Lee, 2013) 41.80 42.50 31.57 55.98 42.62 54.60 57.27 48.35 48.83 45.28 60.03 44.77 50.47 55.23 46.34 48.38 EATA (Niu et al., 2022) 44.69 44.76 34.96 57.10 43.49 56.26 58.80 49.86 50.29 47.29 61.00 45.32 51.65 56.05 46.81 49.89 SAR (Niu et al., 2023) 44.59 44.64 34.57 58.26 43.55 56.41 58.62 50.08 50.74 47.77 61.39 46.76 51.49 56.85 48.07 50.25 TIPI (Nguyen et al., 2023) 46.12 46.31 34.13 57.48 43.46 55.63 58.51 51.32 52.45 48.56 61.05 40.80 51.28 57.93 49.48 50.30 TEA (Yuan et al., 2024) 44.64 45.79 34.71 57.63 43.66 56.11 58.37 50.18 50.21 48.86 61.11 45.59 51.21 56.46 48.61 50.21 TSD (Wang et al., 2023) 45.37 46.18 34.51 57.85 42.44 55.98 58.50 50.33 50.54 49.66 60.61 36.94 50.92 56.05 48.19 49.60 Linear TCA 21.90 24.46 12.80 39.80 36.53 42.66 45.80 43.03 42.66 36.47 55.13 12.97 49.49 47.41 48.09 37.28 Linear TCA + 47.29 48.95 36.13 57.60 44.46 55.68 58.80 53.31 52.11 48.68 61.78 41.87 51.49 58.48 50.99 51.17 Table 22. Accuracy comparisons of different TTA methods on CIFAR-100-C dataset at damage level of 5, based on Res Net-50 backbone. The best results are highlighted in boldface, and the second ones are underlined. Method CIFAR-100-C Avg Gau. Sho. Imp. Def. Gla. Mot. Zoo. Sno. Fro. Fog Bri. Con. Ela. Pix. Jpe. Source (He et al., 2016) 21.71 24.74 19.53 62.41 43.14 61.13 67.65 66.34 67.48 54.03 77.43 33.26 60.09 60.48 56.17 51.71 BN (Schneider et al., 2020) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 T3A (Iwasawa & Matsuo, 2021) 24.32 27.25 23.66 65.04 47.67 63.22 69.56 67.46 69.16 57.09 78.16 36.55 62.36 63.61 58.53 54.24 Ada NPC (Zhang et al., 2023) 22.75 25.77 22.65 63.54 46.46 62.02 68.99 66.78 68.50 56.45 77.60 35.34 61.63 62.87 56.82 53.21 TENT (Wang et al., 2020) 10.95 13.94 4.40 66.79 45.92 67.13 71.28 67.83 69.92 59.26 78.42 49.29 62.18 66.26 57.29 52.72 PLC (Lee, 2013) 21.66 24.71 19.49 62.42 43.16 61.15 67.61 66.34 67.53 54.04 77.44 33.29 60.10 60.49 56.17 51.71 EATA (Niu et al., 2022) 50.06 52.96 44.88 70.07 54.45 69.01 70.21 66.45 70.10 62.13 78.08 60.10 62.59 66.26 58.61 62.40 SAR (Niu et al., 2023) 16.59 18.07 9.89 67.86 47.37 67.31 71.48 67.99 70.19 60.58 78.17 52.90 61.29 66.11 58.56 54.29 TIPI (Nguyen et al., 2023) 7.95 9.85 3.77 67.08 45.89 66.96 71.98 68.01 70.63 59.47 78.24 47.70 62.37 67.37 58.17 52.36 TEA (Yuan et al., 2024) 10.99 17.39 8.09 66.54 45.55 65.24 70.78 67.06 69.09 58.30 76.44 45.15 61.60 64.82 57.56 52.31 TSD (Wang et al., 2023) 21.53 24.49 19.03 62.61 43.25 61.34 67.72 66.34 67.67 54.15 77.46 33.36 60.10 60.73 56.26 51.74 Linear TCA 27.46 30.02 25.33 65.29 47.98 64.26 69.91 68.32 70.01 58.49 78.16 39.42 62.74 65.09 58.82 55.42 Linear TCA + 51.98 54.92 46.74 71.00 56.07 69.73 71.06 67.56 71.01 63.93 78.61 62.35 63.42 67.73 59.49 63.71 Table 23. Accuracy comparisons of different TTA methods on CIFAR-100-C dataset at damage level of 5, based on Vi T-B/16 backbone. The best results are highlighted in boldface, and the second ones are underlined. Method Image Net-C Avg Gau. Sho. Imp. Def. Gla. Mot. Zoo. Sno. Fro. Fog Bri. Con. Ela. Pix. Jpe. Source (He et al., 2016) 1.54 2.27 1.48 11.44 8.68 11.12 17.62 10.64 16.21 14.02 51.52 3.44 16.49 23.35 30.67 14.70 BN (Schneider et al., 2020) 13.65 14.84 14.17 11.95 13.04 23.34 33.89 29.18 28.42 40.80 58.11 12.09 38.92 44.35 37.08 27.59 T3A (Iwasawa & Matsuo, 2021) 1.61 2.35 1.65 10.57 8.20 10.12 17.38 11.03 16.14 15.19 49.23 3.24 18.00 23.37 30.31 14.56 Ada NPC (Zhang et al., 2023) 1.42 2.01 1.42 8.23 6.49 7.64 13.82 8.50 12.08 11.97 42.81 2.77 15.41 19.92 24.49 11.93 TENT (Wang et al., 2020) 23.45 25.71 24.08 18.79 20.90 33.54 42.85 39.64 32.95 50.36 60.13 10.68 48.81 51.96 46.98 35.39 PLC (Lee, 2013) 13.64 14.85 14.16 11.96 13.02 23.36 33.91 29.18 28.43 40.78 58.12 12.08 38.91 44.35 37.08 27.59 EATA (Niu et al., 2022) 28.24 30.16 28.88 25.30 25.74 36.61 43.71 41.80 36.42 50.87 59.12 31.75 49.10 52.33 47.82 39.19 SAR (Niu et al., 2023) 28.04 29.59 27.88 23.66 23.90 36.16 43.40 40.94 36.71 51.01 60.18 27.38 48.95 52.47 47.98 38.55 TIPI (Nguyen et al., 2023) 24.45 26.52 24.75 20.37 22.25 33.65 42.46 39.31 33.47 49.93 59.44 12.53 48.41 51.51 46.92 35.73 TEA (Yuan et al., 2024) 18.82 20.50 19.00 16.27 17.68 28.51 39.17 35.19 32.26 46.92 59.16 15.42 44.39 48.81 43.64 32.38 TSD (Wang et al., 2023) 15.60 16.99 16.13 15.59 15.41 28.69 38.07 32.92 30.01 45.90 58.69 7.62 41.06 47.47 41.52 30.11 Linear TCA 2.22 3.05 2.15 11.44 9.11 11.56 19.46 13.19 18.71 17.07 52.18 3.70 19.56 25.43 32.30 16.07 Linear TCA + 28.25 30.20 28.80 25.34 25.74 36.50 43.73 41.82 36.52 50.91 59.14 31.79 49.17 52.37 47.88 39.21 Table 24. Accuracy comparisons of different TTA methods on Image Net-C dataset at damage level of 5, based on Res Net-18 backbone. The best results are highlighted in boldface, and the second ones are underlined. Test-time Correlation Alignment Method Image Net-C Avg Gau. Sho. Imp. Def. Gla. Mot. Zoo. Sno. Fro. Fog Bri. Con. Ela. Pix. Jpe. Source (He et al., 2016) 3.00 3.70 2.64 17.91 9.74 14.71 22.45 16.60 23.06 24.01 59.12 5.38 16.51 20.87 32.63 18.15 BN (Schneider et al., 2020) 16.32 17.09 16.97 15.23 15.54 26.64 39.38 34.46 33.45 48.43 65.67 17.08 44.62 49.49 40.47 32.06 T3A (Iwasawa & Matsuo, 2021) 2.97 3.38 2.65 17.05 9.37 13.69 22.63 16.98 22.83 25.34 57.68 5.05 18.41 20.31 32.36 18.05 Ada NPC (Zhang et al., 2023) 2.61 3.00 2.36 14.19 7.92 11.14 18.88 14.22 18.65 22.08 52.65 4.21 16.57 17.73 28.04 15.62 TENT (Wang et al., 2020) 26.70 28.90 28.01 25.01 24.72 38.66 48.84 46.04 40.49 57.04 67.96 24.42 53.73 57.72 51.54 41.32 PLC (Lee, 2013) 16.32 17.09 16.96 15.24 15.54 26.64 39.39 34.44 33.45 48.44 65.66 17.07 44.61 49.50 40.47 32.06 EATA (Niu et al., 2022) 35.38 37.75 36.34 33.29 32.74 47.51 53.07 52.53 46.30 60.54 68.06 43.82 58.42 61.15 55.72 48.17 SAR (Niu et al., 2023) 34.39 35.42 35.77 32.27 31.15 45.34 51.58 50.03 44.20 59.27 68.14 35.58 56.82 60.06 54.52 46.30 TIPI (Nguyen et al., 2023) 27.96 31.36 31.37 24.74 24.39 42.68 49.93 48.01 37.40 57.94 66.51 16.66 56.19 58.94 53.90 41.87 TEA (Yuan et al., 2024) 22.51 23.94 22.70 20.70 21.00 36.06 46.93 44.91 39.37 56.06 67.01 22.79 52.65 56.86 49.99 38.90 TSD (Wang et al., 2023) 18.90 18.80 19.63 18.31 17.49 31.12 44.27 39.74 36.02 53.38 67.00 13.06 47.97 54.54 45.95 35.08 Linear TCA 3.35 4.39 3.08 17.67 10.01 15.12 22.89 19.21 25.61 27.30 59.50 5.81 20.39 21.72 34.11 19.34 Linear TCA + 35.44 37.78 36.29 33.37 32.80 47.63 53.32 52.45 46.32 60.55 68.04 43.90 58.61 61.08 55.71 48.22 Table 25. Accuracy comparisons of different TTA methods on Image Net-C dataset at damage level of 5, based on Res Net-50 backbone. The best results are highlighted in boldface, and the second ones are underlined. Method Image Net-C Avg Gau. Sho. Imp. Def. Gla. Mot. Zoo. Sno. Fro. Fog Bri. Con. Ela. Pix. Jpe. Source (He et al., 2016) 35.09 32.16 35.88 31.42 25.31 39.45 31.55 24.47 30.13 54.74 64.48 48.98 34.20 53.17 56.45 39.83 BN (Schneider et al., 2020) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 T3A (Iwasawa & Matsuo, 2021) 27.87 28.15 30.27 32.42 27.00 40.66 33.52 25.95 30.76 56.32 64.85 50.20 37.99 53.75 57.01 39.78 Ada NPC (Zhang et al., 2023) 30.01 26.86 30.98 28.19 23.40 36.38 29.65 21.18 26.59 52.78 61.24 44.53 34.44 50.50 54.98 36.78 TENT (Wang et al., 2020) 51.19 50.00 52.48 47.35 42.95 54.40 45.19 7.43 16.27 64.87 70.90 64.35 25.97 63.35 63.48 48.01 PLC (Lee, 2013) 29.10 29.61 31.58 31.19 25.10 39.33 31.45 24.58 30.18 54.33 64.47 48.44 34.17 52.52 55.11 38.74 EATA (Niu et al., 2022) 56.63 56.20 57.49 56.13 57.13 62.26 62.89 64.02 62.77 73.71 77.07 70.34 67.39 71.42 69.96 64.36 SAR (Niu et al., 2023) 54.90 55.82 56.68 55.94 55.61 62.47 58.11 17.20 34.16 71.85 77.14 63.48 65.75 71.45 68.52 57.94 TIPI (Nguyen et al., 2023) 56.76 56.70 58.45 55.90 56.10 61.75 19.67 1.88 4.51 64.18 75.91 69.24 6.37 70.05 69.97 48.50 TEA (Yuan et al., 2024) 39.46 38.72 41.90 24.47 28.03 42.01 33.46 13.43 33.46 53.89 66.09 58.66 34.95 55.65 56.41 41.37 TSD (Wang et al., 2023) 36.78 33.68 37.61 32.28 26.36 40.93 32.65 25.20 31.49 56.17 65.65 54.27 35.10 54.61 57.17 41.33 Linear TCA 30.45 30.77 32.83 33.50 27.47 42.14 34.91 26.98 32.85 57.65 64.88 56.65 38.18 54.08 57.22 41.37 Linear TCA + 56.92 56.47 57.63 56.52 57.56 62.65 63.62 64.53 63.30 74.06 77.11 70.64 67.82 71.65 70.19 64.71 Table 26. Accuracy comparisons of different TTA methods on Image Net-C dataset at damage level of 5, based on Vi T-B/16 backbone. The best results are highlighted in boldface, and the second ones are underlined. Test-time Correlation Alignment Source Domain Xs Test Domain Xt Transformed Test Xt Linear Shift Source Domain Xs Test Domain Xt Transformed Test Xt Source Domain Xs Test Domain Xt Transformed Test Xt Source Domain Xs Test Domain Xt Transformed Test Xt Source Domain Xs Test Domain Xt Transformed Test Xt Source Domain Xs Test Domain Xt Transformed Test Xt Figure 5. Adaptation process of Linear TCA to datasets with linear shifts. Test-time Correlation Alignment Source Domain Xs Test Domain Xt Transformed Test Xt Nonlinear Shift Source Domain Xs Test Domain Xt Transformed Test Xt Source Domain Xs Test Domain Xt Transformed Test Xt Source Domain Xs Test Domain Xt Transformed Test Xt Source Domain Xs Test Domain Xt Transformed Test Xt Source Domain Xs Test Domain Xt Transformed Test Xt Figure 6. Adaptation process of Linear TCA to datasets with nonlinear shifts