# xcit_crosscovariance_image_transformers__c062b740.pdf

XCi T: Cross-Covariance Image Transformers

Alaaeldin El-Nouby1,2 Hugo Touvron1,3 Mathilde Caron1,2 Piotr Bojanowski1

Matthijs Douze1 Armand Joulin1 Ivan Laptev2 Natalia Neverova1

Gabriel Synnaeve1 Jakob Verbeek1 Hervé Jégou1

1Facebook AI 2Inria 3Sorbonne University

Following tremendous success in natural language processing, transformers have recently shown much promise for computer vision. The self-attention operation underlying transformers yields global interactions between all tokens, i.e. words or image patches, and enables flexible modelling of image data beyond the local interactions of convolutions. This flexibility, however, comes with a quadratic complexity in time and memory, hindering application to long sequences and highresolution images. We propose a transposed version of self-attention that operates across feature channels rather than tokens, where the interactions are based on the cross-covariance matrix between keys and queries. The resulting cross-covariance attention (XCA) has linear complexity in the number of tokens, and allows efficient processing of high-resolution images. Our cross-covariance image transformer (XCi T) built upon XCA combines the accuracy of conventional transformers with the scalability of convolutional architectures. We validate the effectiveness and generality of XCi T by reporting excellent results on multiple vision benchmarks, including (self-supervised) image classification on Image Net-1k, object detection and instance segmentation on COCO, and semantic segmentation on ADE20k.

1 Introduction

Transformers architectures [68] have provided quantitative and qualitative breakthroughs in speech and natural language processing (NLP). After a few attempts to incorporate wide-range self-attention in vision architectures [71, 82], Dosovitskiy et al. [21] established transformers as a viable architecture for learning visual representations, reporting competitive results for image classification while relying on large-scale pre-training. Touvron et al. [64] have shown on par or better accuracy/throughput compared to strong convolutional baselines such as Efficient Nets [58] when training transformers on Image Net-1k using extensive data augmentation and improved training schemes. Promising results have been obtained for other vision tasks, including image retrieval [22], object detection and semantic segmentation [44, 70, 81, 83], as well as video understanding [2, 7, 23].

One major drawback of transformers is the time and memory complexity of the core self-attention operation, that increases quadratically with the number of input tokens, or similarly number of patches in computer vision. For w h images, this translates to a complexity of O(w2h2), which is prohibitive for most tasks involving high-resolution images, such as object detection and segmentation. Various strategies have been proposed to alleviate this complexity, for instance using approximate forms of self-attention [44, 81], or pyramidal architectures which progressively downsample the feature maps [70]. However, none of the existing solutions are fully satisfactory, as they either trade complexity for accuracy, or their complexity remains excessive for processing very large images.

We replace the self-attention, as originally introduced by Vaswani et al. [68], with a transposed attention that we denote as cross-covariance attention (XCA). Cross-covariance attention substi-

Code: https://github.com/facebookresearch/xcit

35th Conference on Neural Information Processing Systems (Neur IPS 2021).

Local Patch Interaction (LPI)

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Figure 1: Our XCi T layer consists of three main blocks, each preceded by Layer Norm and followed by a residual connection: (i) the core cross-covariance attention (XCA) operation, (ii) the local patch interaction (LPI) module, and (iii) a feed-forward network (FFN). By transposing the query-key interaction, the computational complexity of XCA is linear in the number of data elements N, rather than quadratic as in conventional self-attention.

tutes the explicit full pairwise interaction between tokens by self-attention among features, where the attention map is derived from the cross-covariance matrix computed over the key and query projections of the token features. Importantly, XCA has a linear complexity in the number of patches. To construct our Cross-Covariance Image Transformers (XCi T), we combine XCA with local patch interaction modules that rely on efficient depth-wise convolutions and point-wise feedforward networks commonly used in transformers, see Figure 1. XCA can be regarded as a form of a dynamic 1 1 convolution, which multiplies all tokens with the same data-dependent weight matrix. We find that the performance of our XCA layer can be further improved by applying it on blocks of channels, rather than directly mixing all channels together. This block-diagonal shape of XCA further reduces the computational complexity with a factor linear in the number of blocks.

Given its linear complexity in the number of tokens, XCi T can efficiently process images with more than thousand pixels in each dimension. Notably, our experiments show that XCi T does not compromise the accuracy and achieves similar results to Dei T [64] and Cai T [67] in comparable settings. Moreover, for dense prediction tasks such as object detection and image segmentation, our models outperform popular Res Net [28] backbones as well as the recent transformer-based models [44, 70, 81]. Finally, we also successfully apply XCi T to the self-supervised feature learning using DINO [12], and demonstrate improved performance compared to a Dei T-based backbone [64].

Overall, we summarize our contributions as follows:

We introduce cross-covariance attention (XCA), which provides a transposed alternative to

conventional self-attention, attending over channels instead of tokens. Its complexity is linear in the number of tokens, allowing for efficient processing of high-resolution images, see Figure 2. XCA attends to a fixed number of channels, irrespective of the number of tokens. As a result, our

models are significantly more robust to changes in image resolution at test time, and are therefore more amenable to process variable-size images. For image classification, we demonstrate that our models are on par with state-of-the-art vision

transformers for multiple model sizes using a simple columnar architecture, i.e., in which we keep the resolution constant across layers. In particular, our XCi T-L24 model achieves 86.0% top-1 accuracy on Image Net, outperforming its Cai T-M24 [67] and NFNet-F2 [10] counterparts with comparable numbers of parameters. For dense prediction tasks with high-resolution images, our models outperform Res Net and multiple

transformer-based backbones. On the COCO benchmark, we achieve a strong performance of 48.5% and 43.7% m AP for object detection and instance segmentation respectively. Moreover, we report 48.4% m Io U for semantic segmentation on the ADE20k benchmark, outperforming the state-of-the-art Swin Transformer [44] backbones across all comparable model sizes. Finally, our XCi T model is highly effective in self-supervised learning setups, achieving 80.9%

top-1 accuracy on Image Net-1k using DINO [12].

2 Related work

Deep vision transformers. Training deep vision transformers can be challenging due to instabilities and optimization issues. Touvron et al. [67] successfully train models with up to 48 layers using Layer Scale, which weighs contributions of residual blocks across layers and improves optimization. Additionally, the authors introduce class attention layers which decouple the learning of patch features and the feature aggregation stage for classification.

Spatial structure in vision transformers. Yuan et al. [79] propose applying a soft split for patch projection with overlapping patches which is applied repeatedly across model layers, reducing the number of patches progressively. Han et al. [27] introduce a transformer module for intra-patch structure, exploiting pixel-level information and integrating with an inter-patch transformer to attain higher representation power. d Ascoli et al. [18] consider the initialization of self-attention blocks as a convolutional operator, and demonstrate that such initialization improves the performance of vision transformers in low-data regimes. Graham et al. [26] introduce Le Vi T, which adopts a multistage architecture with progressively reduced feature resolution similar to popular convolutional architectures, allowing for models with high inference speed while retaining a strong performance. Moreover, the authors adopt a convolution-based module for extracting patch descriptors. Yuan et al. [78] improve both the performance and the convergence speed of vision transformers by replacing the linear patch projection with convolutional layers and max-pooling, as well as modifying the feed-forward networks in each transformer layer to incorporate depth-wise convolutions.

Efficient attention. Numerous methods for efficient self-attention have been proposed in the literature to address the quadratic complexity of self-attention in the number of input tokens. These include restricting the span of the self-attention to local windows [48, 50], strided patterns [14], axial patterns [30], or an adaptive computation across layers [57]. Other methods provide an approximation of the self-attention matrix which can be achieved by a projection across the token dimension [69], or through a factorization of the softmax-attention kernel [15, 37, 56, 77], which avoids explicit computation of the attention matrix. While conceptually different, our XCA performs similar computations without being sensitive to the choice of the kernel. Similarly, Lee-Thorp et al. [41] achieve faster training by substituting self-attention with unparametrized Fourier Transform. Other efficient attention methods rely on local attention and adding a small number of global tokens, thus allowing interaction among all tokens only by hopping through the global tokens [1, 5, 34, 80]. Similarly, Goyal et al. [25] use a global workspace though which items interact, albeit one that is shared across layers.

Transformers for high-resolution images. Several works adopt visual transformers to highresolution image tasks beyond image classification, such as object detection and image segmentation. Wang et al. [70] design a model with a pyramidal architecture and address complexity by gradually reducing the spatial resolution of keys and values. Similarly, for video recognition Fan et al. [23] utilize pooling to reduce the resolution across the spatial and temporal dimensions to allow for an efficient computation of the attention matrix. Zhang et al. [81] adopt global tokens and local attention to reduce the model complexity, while Liu et al. [44] provide an efficient method for local attention with shifted windows. In addition, Zheng et al. [83] and Ranftl et al. [54] study problems like semantic segmentation and monocular depth estimation with the quadratic self-attention operation.

Data-dependent layers. Our XCi T layer can be regarded as a dynamic 1 1 convolution, which multiplies all token features with the same data-dependent weight matrix, derived from the key and query cross-covariance matrix. In the context of convolutional networks, Dynamic Filter Networks [9] explore a related idea, using a filter generating subnetwork to produce convolutional filters based on features in previous layers. Squeeze-and-Excitation networks [32] use data dependent 1 1 convolutions in convolutional architectures. Spatially average-pooled features are fed to a 2-layer MLP which produces per channel scaling parameters. Closer in spirit to our work, Lambda layers propose a way to ensure global interaction in Res Net models [4]. Their content-based lambda function is computing a similar term as our cross-covariance attention, but differing in how the softmax and 2 normalizations are applied. Moreover, Lambda layers also include specific positionbased lambda functions, and Lambda Networks are based on Res Nets while XCi T follows the Vi T architecture. Recently data-independent analogues of self-attention have also been found to be an effective alternative to convolutional and self-attention layers for vision tasks [20, 46, 62, 66]. These

methods treat entries in the attention map as learnable parameters, rather than deriving the attention map dynamically from queries and keys, but their complexity remains quadratic in the number of tokens. Zhao et al. [82] consider alternative attention forms in computer vision.

In this section, we first recall the self-attention mechanism, and the connection between the Gram and covariance matrices, which motivated our work. We then propose our cross-covariance attention operation (XCA) which operates along the feature dimension instead of token dimension in conventional transformers and combine it with local patch interaction and feedforward layers to construct our Cross-Covariance Image Transformer (XCi T). See Figure 1 for an overview.

3.1 Background

Token self-attention. Self-attention, as introduced by Vaswani et al. [68], operates on an input matrix X 2 RN d, where N is the number of tokens, each of dimensionality d. The input X is linearly projected to queries, keys and values, using the weight matrices Wq 2 Rd dq, Wk 2 Rd dk and Wv 2 Rd dv, such that Q=XWq, K=XWk and V =XWv, where dq = dk. Keys and values are used to compute an attention map A(K, Q) = Softmax(QK>/pdk), and the output of the self-attention operation is defined as the weighted sum of N token features in V with the weights corresponding to the attention map: Attention(Q, K, V ) = A(K, Q)V . The computational complexity of self-attention scales quadratically in N, due to pairwise interactions between all N elements.

Relationship between Gram and covariance matrices. To motivate our cross-covariance attention operation, we recall the relation between Gram and covariance matrices. The unnormalised d d covariance matrix is obtained as C=X>X. The N N Gram matrix contains all pairwise innerproducts: G=XX>. The non-zero part of the eigenspectrum of the Gram and covariance matrix are equivalent, and the eigenvectors of C and G can be computed in terms of each other. If V are the eigenvectors of G, then the eigenvectors of C are given by U=XV . To minimise the computational cost, the eigendecomposition of either the Gram or covariance matrix can be obtained in terms of the decomposition of the other, depending on which of the two matrices is the smallest.1

We draw upon this strong connection between the Gram and covariance matrices to consider whether it is possible to avoid the quadratic cost to compute the N N attention matrix, which is computed from the analogue of the N N Gram matrix QK>=XWq W >

k X>. Below we consider how we can use the dk dq cross-covariance matrix, K>Q=W >

k X>XWq, which can be computed in linear time in the number of elements N, to define an attention mechanism.

3.2 Cross-covariance attention

We propose a cross-covariance based self-attention function that operates along the feature dimension, rather than along the token dimension as in token self-attention. Using the definitions of queries, keys and values from above, the cross-covariance attention function is defined as:

XC-Attention(Q, K, V ) = V AXC(K, Q), AXC(K, Q) = Softmax

where each output token embedding dimension is a convex combination of the dv features of its corresponding token embedding in V . The attention weights A are computed based on the crosscovariance matrix.

2-Normalization and temperature scaling. In addition to building our attention operation on the cross-covariance matrix, we make a second modification compared to token self-attention. We restrict the magnitude of the query and key matrices by 2-normalising them, such that each column of length N of the normalised matrices ˆQ and ˆK has unit norm, and every element in d d cross-covariance matrix ˆK> ˆQ is in the range [ 1, 1]. We observed that controlling the norm strongly enhances the

1For C to represent the covariance, X should be centered, i.e. X1=0. For the relation between C and G, however, centering is not required.

Figure 2: Inference memory usage of vision transformer variants. Our XCi T models scale linearly in the number of tokens, which makes it possible to scale to much larger image sizes, even in comparison to approaches employing approximate self-attention or a pyramidal design. All measurements are performed with a batch size of 64 on a single V100-32GB GPU.

Figure 3: Performance when changing the resolution at test-time for models with a similar number of parameters. All networks were trained at resolution 224, w/o distillation. XCi T is more tolerant to changes of resolution than the Gram-based Dei T and benefit more from the Fix Res effect [63] when inference is performed at a larger resolution than at train-time.

stability of training, especially when trained with a variable numbers of tokens. However, restricting the norm reduces the representational power of the operation by removing a degree of freedom. Therefore, we introduce a learnable temperature parameter which scales the inner products before the Softmax, allowing for sharper or more uniform distribution of attention weights.

Block-diagonal cross-covariance attention. Instead of allowing all features to interact among each other, we divide them into a h groups, or heads , in a similar fashion as multi-head token self-attention. We apply the cross-covariance attention separately per head where for each head, we learn separate weight matrices to project X to queries, keys and values, and collect the corresponding weight matrices in the tensors Wq 2 Rh d dq, Wk 2 Rh d dk and Wv 2 Rh d dv, where we set dk=dq=dv=d/h. Restricting the attention within heads has two advantages: (i) the complexity of aggregating the values with the attention weights is reduced by a factor h; (ii) more importantly, we empirically observe that the block-diagonal version is easier to optimize, and typically leads to improved results. This observation is in line with observations made for Group Normalization [73], which normalizes groups of channels separately based on their statistics, and achieves favorable results for computer vision tasks compared to Layer Normalization [3], which combines all channels in a single group. Figure 4 shows that each head learns to focus on semantically coherent parts of the image, while being flexible to change what type of features it attends to based on the image content.

Complexity analysis. The usual token self-attention with h heads has a time complexity of O(N 2d) and memory complexity of O(h N 2+Nd). Due to the quadratic complexity, it is problematic to scale token self-attention to images with a large number of tokens. Our cross-covariance attention overcomes this drawback as its computational cost of O(Nd2/h) scales linearly with the number of tokens, as does the memory complexity of O(d2/h+Nd). Therefore, our model scales much better to cases where the number of tokens N is large, and the feature dimension d is relatively small, as is typically the case, in particularly when splitting the features into h heads.

3.3 Cross-covariance image transformers

To construct our cross-covariance image transformers (XCi T), we adopt a columnar architecture which maintains the same spatial resolution across layers, similarly to [21, 64, 67]. We combine our cross-covariance attention (XCA) block with the following additional modules, each one being preceded by a Layer Norm [3]. See Figure 1 for an overview. Since in this section we specifically design the model for computer vision tasks, tokens correspond to image patches in this context.

Table 1: XCi T models. Design choices include model depth, patch embeddings dimensionality d, and the number of heads h used in XCA. By default our models are trained and tested at resolution 224 with patch sizes of 16 16. We also train with distillation using a convolutional teacher (denoted ) as proposed by Touvron et al. [64]. Finally, we report performance of our strongest models obtained with 8 8 patch size, fine-tuned (") and tested at resolution 384 384 (column @384/8), using distillation with a teacher that was also fine-tuned @384.

Model Depth d #heads #params GFLOPs Image Net-1k-val top-1 acc. (%) @224/16 @384/8 @224/16 @224/16 @384/8 "

XCi T-N12 12 128 4 3M 0.5 6.4 69.9 72.2 77.8 XCi T-T12 12 192 4 7M 1.2 14.3 77.1 78.6 82.4 XCi T-T24 24 192 4 12M 2.3 27.3 79.4 80.4 83.7 XCi T-S12 12 384 8 26M 4.8 55.6 82.0 83.3 85.1 XCi T-S24 24 384 8 48M 9.1 106.0 82.6 83.9 85.6 XCi T-M24 24 512 8 84M 16.2 188.0 82.7 84.3 85.8 XCi T-L24 24 768 16 189M 36.1 417.9 82.9 84.9 86.0

Local patch interaction. In the XCA block communication between patches is only implicit through the shared statistics. To enable explicit communication across patches we add a simple Local Patch Interaction (LPI) block after each XCA block. LPI consists of two depth-wise 3 3 convolutional layers with Batch Normalization and GELU non-linearity in between. Due to its depth-wise structure, the LPI block has a negligible overhead in terms of parameters, as well as a very limited overhead in terms of throughput and memory usage during inference.

Feed-forward network. As is common in transformer models, we add a point-wise feedforward network (FFN), which has a single hidden layer with 4d hidden units. While interaction between features is confined within groups in the XCA block, and no feature interaction takes place in the LPI block, the FFN allows for interaction across all features.

Global aggregation with class attention. When training our models for image classification, we utilize the class attention layers as proposed by Touvron et al. [67]. These layers aggregate the patch embeddings of the last XCi T layer through writing to a CLS token by one-way attention between the CLS tokens and the patch embeddings. The class attention is also applied per head, i.e. feature group.

Handling images of varying resolution. In contrast to the attention map involved in token selfattention, in our case the covariance blocks are of fixed size independent of the input image resolution. The softmax always operates over the same number of elements, which may explain why our models behave better when dealing with images of varying resolutions (see Figure 3). In XCi T we include additive sinusoidal positional encoding [68] with the input tokens. We generate them in 64 dimensions from the 2d patch coordinates and then linearly project to the transformer working dimension d. This choice is orthogonal to the use of learned positional encoding, as in Vi T [21]. However, it is more flexible since there is no need to interpolate or fine-tune the network when changing the image size.

Model configurations. In Table 1 we list different variants of our model which we use in our experiments, with different choices for model width and depth. For the patch encoding layer, unless mentioned otherwise, we adopt the alternative used by Graham et al. [26] with convolutional patch projection layers. We also experimented with a linear patch projection as described in [21], see our ablation in Table 4. Our default patch size is 16 16, as in other vision transformer models including Vi T [21], Dei T [64] and Cai T [67]. We also experiment with smaller 8 8 patches, which has been observed to improve performance [12]. Note that this is efficient with XCi T as its complexity scales linearly which the number of patches, while Vi T, Dei T and Cai T scale quadratically.

4 Experimental evaluation

In this section we demonstrate the effectiveness and versatility of XCi T on multiple computer vision benchmarks, and present ablations providing insight on the importance of its different components. In the supplementary material we provide additional analysis, including the impact on performance of image resolution in Section A.1 and of multiple approximate attention baselines in Section A.2.

Table 2: Image Net classification. Number of parameters, FLOPs, image resolution, and top-1 accuracy on Image Net-1k and Image Net-V2. Training strategies vary across models, transformer-based models and the reported Reg Net mostly follow recipes from Dei T [64].

Model #params FLOPs Res. Im Net V2 Efficient Net-B5 RA [17] 30M 9.9B 456 83.7 _ Reg Net Y-4GF [53] 21M 4.0B 224 80.0 72.4 Dei T-S [64] 22M 4.6B 224 81.2 68.5 Swin-T [44] 29M 4.5B 224 81.3 _ Cai T-XS24 " [67] 26M 19.3B 384 84.1 74.1 XCi T-S12/16 26M 4.8B 224 83.3 72.5 XCi T-S12/16 " 26M 14.3B 384 84.7 74.1 XCi T-S12/8 " 26M 55.6B 384 85.1 74.8

Efficient Net-B7 RA [17] 66M 37.0B 600 84.7 _ NFNet-F0 [10] 72M 12.4B 256 83.6 72.6 Reg Net Y-8GF [53] 39M 8.0B 224 81.7 72.4 TNT-B [79] 66M 14.1B 224 82.8 _ Swin-S [44] 50M 8.7B 224 83.0 _ Cai T-S24 " [67] 47M 32.2B 384 85.1 75.4 XCi T-S24/16 48M 9.1B 224 83.9 73.3 XCi T-S24/16 " 48M 26.9B 384 85.1 74.6 XCi T-S24/8 " 48M 105.9B 384 85.6 75.7

Fix-Efficient Net-B8 [65] 87M 89.5B 800 85.7 75.9 Reg Net Y-16GF [53] 84M 16.0B 224 82.9 72.4 Swin-B" [44] 88M 47.0B 384 84.2 _ Dei T-B " [64] 87M 55.5B 384 85.2 75.2 Cai T-S48 " [67] 89M 63.8B 384 85.3 76.2 XCi T-M24/16 84M 16.2B 224 84.3 73.6 XCi T-M24/16 " 84M 47.7B 384 85.4 75.1 XCi T-M24/8 " 84M 187.9B 384 85.8 76.1

NFNet-F2 [10] 194M 62.6B 352 85.1 74.3 NFNet-F3 [10] 255M 114.8B 416 85.7 75.2 Cai T-M24 " [67] 186M 116.1B 384 85.8 76.1 XCi T-L24/16 189M 36.1B 224 84.9 74.6 XCi T-L24/16 " 189M 106.0B 384 85.8 75.8 XCi T-L24/8 " 189M 417.8B 384 86.0 76.6

Figure 4: Visualization of the attention map between the CLS token and individual patches in the class-attention stage. For each column, each row represents the attention map w.r.t. one head, corresponding to the image in the first row. Each head appears sensitive to semantically coherent regions. Heads are sensitive to similar features within the same or across images (e.g. people or bird faces). They are trigger by different concepts when such features are missing (e.g., cockpit for race cars).

4.1 Image classification

We use Image Net-1k [19] to train and evaluate our models for image classification. It consists of 1.28M training images and 50k validation images, labeled across 1,000 semantic categories. Our

training setup follows the Dei T recipe [64]. We train our model for 400 epochs with the Adam W optimizer [45] using a cosine learning rate decay. In order to enhance the training of larger models, we utilize Layer Scale [67] and adjust the stochastic depth [33] for each of our models accordingly (see the supplementary material for details). Following [67], images are cropped with crop ratio of 1.0 for evaluation. In addition to the Image Net-1k validation set, we report results for Image Net-V2 [55] which has a distinct test set. Our implementation is based on the Timm library [72].

Results on Image Net. We present a family of seven models in Table 1 with different operating points in terms of parameters and FLOPs. We observe that the performance of the XCi T models benefits from increased capacity both in depth and width. Additionally, consistent with [64, 67] we find that using hard distillation with a convolutional teacher improves the performance. Because of its linear complexity in the number of tokens, it is feasible to train XCi T at 384 384 resolution with small 8 8 patches, i.e. 2304 tokens, which provides a strong boost in performance across all configurations.

We compare to the state-of-the-art convolutional and transformer-based architectures [10, 44, 53, 58, 67] in Table 2. By varying the input image resolution and/or patch size, our models provide competitive or superior performance across model sizes and FLOP budgets. First, the models operating on 224 224 and 16 16 (e.g. XCi T-S12/16) enjoy high accuracy at relatively few FLOPs compared to their counterparts with comparable parameter count and FLOPs. Second, our models with 16 16 and 384 384 resolution images (e.g. XCi T-S12/16") yield an improved accuracy at the expense of higher FLOPs, and provide superior or on-par performance compared to state-of-the-art

Table 3: Self-supervised learning. Top-1 acc. on Image Net-1k. We report with a crop-ratio 0.875 for consistency with DINO. For the last row it is set to 1.0 (improves from 80.7% to 80.9%). All models are trained for 300 epochs.

SSL Method Model #params FLOPs Linear k-NN Mo BY [76] Swin-T [44] 29M 4.5B 75.0 DINO [12] Res Net-50 [28] 23M 4.1B 74.5 65.6 DINO [12] Vi T-S/16 [21] 22M 4.6B 76.1 72.8 DINO [12] Vi T-S/8 [21] 22M 22.4B 79.2 77.2 DINO [12] XCi T-S12/16 26M 4.9B 77.8 76.0 DINO [12] XCi T-S12/8 26M 18.9B 79.2 77.1

DINO [12] Vi T-B/16 [21] 87M 17.5B 78.2 76.1 DINO [12] Vi T-B/8 [21] 87M 78.2B 80.1 77.4 DINO [12] XCi T-M24/16 84M 16.2B 78.8 76.4 DINO [12] XCi T-M24/8 84M 64.0B 80.3 77.9 DINO [12] XCi T-M24/8"384 84M 188.0B 80.9 78.3

Table 4: Ablations of various architectural design choices on the task of Image Net-1k classification using the XCi T-S12 model. Our baseline model uses the convolutional projection adopted from Le Vit.

Model Ablation Im Net top-1 acc.

XCi T-S12/16 Baseline 82.0 XCi T-S12/8 83.4

XCi T-S12/16 Linear patch proj. 81.1 XCi T-S12/8 83.1

XCi T-S12/16 w/o LPI layer 80.8 w/o XCA layer 75.9

XCi T-S12/16 w/o 2-normal. failed w/o learned temp. 81.8

models with comparable computational requirements. Finally, the linear complexity of XCi T allows us to scale to process 384 384 images with 8 8 patch sizes (e.g. XCi T-S12/8"), achieving the highest accuracy across the board, albeit at a relatively high FLOPs count.

Class attention visualization. In Figure 4 we show the class attention map obtained in the feature aggregation stage. Each head focuses on different semantically coherent regions in the image (e.g. faces or umbrellas). Furthermore, heads tend to focus on similar patterns across images (e.g. bird head or human face), but adapts by focusing on other salient regions when such patterns are absent.

Robustness to resolution changes. In Figure 3 we report the accuracy of XCi T-S12, Dei T-S and Res Net-50 trained on 224 224 images and evaluated at different image resolutions. While Dei T outperforms Res Net-50 when train and test resolutions are similar, it suffers from a larger drop in performance as the image resolution deviates farther from the training resolution. XCi T displays a substantially increased accuracy when train and test resolutions are similar, while also being robust to resolution changes, in particular for the model with 8 8 patches.

Self-supervised learning. We train XCi T in a self-supervised manner using DINO [12] on Image Net-1k. In Table 3 we report performance using the linear and k-NN protocols as in [12]. Across model sizes XCi T obtains excellent accuracy with both protocols, substantially improving DINO with Res Net-50 or Vi T architectures, as well as over those reported for Swin-Transformer trained with Mo BY [76]. Comparing the larger models to Vi T, we also observed improved performance for XCi T achieving a strong 80.3% accuracy. For fair comparison, all reported models have been trained for 300 epochs. Further improved performance of small models is reported by Caron et al. [12] when training for 800 epochs, which we expect to carryover to XCi T based on the results presented here.

Analysis and ablations. In Table 4 we provide ablation experiments to analyse the impact of different design choices for our XCi T-S12 model. First, we observe the positive effect of using the convolutional patch projection as compared to using linear patch projection, for both 8 8 and 16 16 patches. Second, while removing the LPI layer reduces the accuracy by only 1.2% (from 82.0 to 80.8), removing the XCA layer results in a large drop of 6.1%, underlining the effectiveness of XCA. We noticed that the inclusion of two convolutional components convolutional patch projection and LPI not only brings improvements in accuracy, but also accelerates training. Third, although we were able to ensure proper convergence without 2-normalization of queries and keys by tweaking the hyper-parameters, we found that it provides stability across model size (depth and width) and other hyper-parameters. Finally, while the learnable softmax temperature parameter is not critical, removing it drops accuracy by 0.2%. Additional ablations are provided in the supplementary material.

Table 5: COCO object detection and instance segmentation performance on the mini-val set. All backbones are pre-trained on Image Net-1k, use Mask R-CNN model [29] and are trained with the same 3x schedule.

Backbone #params APb APb

75 Res Net18 [28] 31.2M 36.9 57.1 40.0 33.6 53.9 35.7 PVT-Tiny [70] 32.9M 39.8 62.2 43.0 37.4 59.3 39.9 Vi L-Tiny [81] 26.9M 41.2 64.0 44.7 37.9 59.8 40.6 XCi T-T12/16 26.1M 42.7 64.3 46.4 38.5 61.2 41.1 XCi T-T12/8 25.8M 44.5 66.4 48.8 40.3 63.5 43.2

Res Net50 [28] 44.2M 41.0 61.7 44.9 37.1 58.4 40.1 PVT-Small [70] 44.1M 43.0 65.3 46.9 39.9 62.5 42.8 Vi L-Small [81] 45.0M 43.4 64.9 47.0 39.6 62.1 42.4 Swin-T [44] 47.8M 46.0 68.1 50.3 41.6 65.1 44.9 XCi T-S12/16 44.3M 45.3 67.0 49.5 40.8 64.0 43.8 XCi T-S12/8 43.1M 47.0 68.9 51.7 42.3 66.0 45.4

Res Net101 [28] 63.2M 42.8 63.2 47.1 38.5 60.1 41.3 Res Ne Xt101-32 62.8M 44.0 64.4 48.0 39.2 61.4 41.9 PVT-Medium [70] 63.9M 44.2 66.0 48.2 40.5 63.1 43.5 Vi L-Medium [81] 60.1M 44.6 66.3 48.5 40.7 63.8 43.7 Swin-S [44] 69.1M 48.5 70.2 53.5 43.3 67.3 46.6 XCi T-S24/16 65.8M 46.5 68.0 50.9 41.8 65.2 45.0 XCi T-S24/8 64.5M 48.1 69.5 53.0 43.0 66.5 46.1

Res Ne Xt101-64 [75] 101.9M 44.4 64.9 48.8 39.7 61.9 42.6 PVT-Large [70] 81.0M 44.5 66.0 48.3 40.7 63.4 43.7 Vi L-Large [81] 76.1M 45.7 67.2 49.9 41.3 64.4 44.5 XCi T-M24/16 101.1M 46.7 68.2 51.1 42.0 65.6 44.9 XCi T-M24/8 98.9M 48.5 70.3 53.4 43.7 67.5 46.9

Table 6: ADE20k semantic segmentation performance using Semantic FPN [38] and Uper Net [74] (in comparable settings). We do not include comparisons with other state-of-the-art models that are pre-trained on larger datasets [44, 54, 83].

Backbone Semantic FPN Uper Net

#params m Io U #params m Io U

Res Net18 [28] 15.5M 32.9 - - PVT-Tiny [70] 17.0M 35.7M - - XCi T-T12/16 8.4M 38.1 33.7M 41.5 XCi T-T12/8 8.4M 39.9 33.7 43.5

Res Net50 [28] 28.5M 36.7 66.5M 42.0 PVT-Small [70] 28.2M 39.8 - - Swin-T [44] - - 59.9M 44.5 XCi T-S12/16 30.4M 43.9 52.4M 45.9 XCi T-S12/8 30.4M 44.2 52.3M 46.6

Res Net101 [28] 47.5M 38.8 85.5M 43.8 Res Ne Xt101-32 [75] 47.1M 39.7 - - PVT-Medium [70] 48.0M 41.6 - - Swin-S [44] - - 81.0M 47.6 XCi T-S24/16 51.8M 44.6 73.8M 46.9 XCi T-S24/8 51.8M 47.1 73.8M 48.1

Res Ne Xt101-64 [75] 86.4M 40.2 - - PVT-Large [70] 65.1M 42.1 - - Swin-B [44] - - 121.0M 48.1 XCi T-M24/16 90.8M 45.9 109.0M 47.6 XCi T-M24/8 90.8M 46.9 108.9M 48.4

4.2 Object detection and instance segmentation

Our XCi T models can efficiently process high-resolution images (see Figure 2). Additionally, XCi T has a better adaptability to varying image resolutions compared to Vi T models (see Figure 3). These two properties make XCi T a good fit for dense prediction tasks including detection and segmentation.

We evalutate XCi T for object detection and instance segmentation using the COCO benchmark [42] which consists of 118k training and 5k validation images including bounding boxes and mask labels for 80 categories. We integrate XCi T as backbone in the Mask R-CNN [29] detector with FPN [43]. Since the XCi T architecture is inherently columnar, we make it FPN-compatible by extracting features from different layers, e.g., layers 4, 6, 8, and 12 for XCi T-S12. All features have a constant stride of 8 or 16 based on the patch size, and the feature resolutions are adjusted to have strides of 4, 8, 16, and 32, similar to Res Net-FPN backbones, where the downsampling is achieved by max pooling and the upsampling is obtained using a single transposed convolution layer (see the supplementary material for details). The model is trained for 36 epochs (3x schedule) using the Adam W optimizer with learning rate of 10 4, 0.05 weight decay and 16 batch size. We adopt the multiscale training and augmentation strategy of DETR [11]. Our implementation is based on the mmdetection library [13].

Results on COCO. In Table 5 we report object detection and instance segmentation results of four variants of XCi T using 16 16 and 8 8 patches. We compare to Res Nets [28] and concurrent efficient vision transformers [44, 70, 81]. All models are trained using the 3x schedule after Image Net-1k pretraining. Note that other results with higher absolute numbers have been achieved when pre-training on larger datasets [44] or with longer schedules [4], and are therefore not directly comparable to the reported results. First, across all model sizes XCi T outperforms the convolutional Res Net [28] and Res Ne Xt [75] by a large margin with either patch size. Second, we observe a similar increase in accuracy compared to PVT [70] and Vi L [81] backbones. Finally, XCi T provides a competitive performance with Swin [44].2 For relatively small models, XCi T-S12/8 outperforms its Swin-T counterpart with a decent margin. On the other hand, Swin-S provides slightly stronger results compared to XCi T-S24/8. Utilizing smaller 8 8 patches leads to a consistent gain across all models.

2We use report the results provided by the authors in their open-sourced code https://github.com/ Swin Transformer/Swin-Transformer-Object-Detection.

4.3 Semantic segmentation

We further show transferability of our models with semantic segmentation experiments on the ADE20k dataset [84], which consists of 20k training and 5k validation images with labels over 150 semantic categories. We integrate our backbones in two segmentation methods: Semantic FPN [38] and Uper Net [74]. We train for 80k and 160k iterations for Semantic FPN and Uper Net respectively. Following [44], the models are trained using batch size 16 and an Adam W optimizer with learning rate of 6 10 5 and 0.01 weight decay. We apply the same method of extracting FPN features as explained in Section 4.2. We report the performance using the standard single scale protocol (without multi-scale and flipping). Our implementation is based on the mmsegmentation library [16].

Results on ADE20k. We present the semantic segmentation performance using XCi T backbones in Table 6. First, for Semantic FPN [38], XCi T provides a superior performance compared to Res Net, Res Ne Xt and PVT backbones using either option of patch size. Second, compared to Swin Transformers using the same Uper Net decoder [74], XCi T with 8 8 patches consistently achieves a higher m Io U for different models. XCi T with 16 16 patches provides a strong performance especially for smaller models where XCi T-S12/16 outperforms Swin-T.

5 Conclusion

Contributions. We present an alternative to token self-attention which operates on the feature dimension, eliminating the need for expensive computation of quadratic attention maps. We build our XCi T models with the cross-covariance attention as its core component and demonstrate the effectiveness and generality of our models on various computer vision tasks. In particular, it exhibits a strong image classification performance on par with state-of-the-art transformer models while similarly robust to changing image resolutions as convnets. XCi T is effective as a backbone for dense prediction tasks, providing excellent performance on object detection, instance and semantic segmentation. Finally, we showed that XCi T can be a strong backbone for self-supervised learning, matching the state-of-the-art results with less compute. XCi T is a generic architecture that can readily be deployed in other research domains where self-attention has shown success.

Limitations. Our models enable training with smaller patches and on higher-resolution images, which leads to clear performance gains. However, for tasks like image classification this gain comes at a cost of relatively high number of FLOPs. In order to address this issue, other components, like FFN, could also be re-examined. Another point is that XCi T models seem to overfit more than their Cai T counterparts, see Table 2. They are more similar to some convnets in that respect.

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