Visual Auto-Regressive modeling (VAR) has shown promise in bridging the speed and quality gap between autoregressive image models and diffusion models. VAR reformulates autoregressive modeling by decomposing an image into successive resolution scales. During inference, an image is generated by predicting all the tokens in the next (higher-resolution) scale, conditioned on all tokens in all previous (lower-resolution) scales. However, this formulation suffers from reduced image quality due to the parallel generation of all tokens in a resolution scale; has sequence lengths scaling superlinearly in image resolution; and requires retraining to change the sampling schedule.We introduce Hierarchical Masked AutoRegressive modeling (HMAR), a new image generation algorithm that alleviates these issues using next-scale prediction and masked prediction to generate high-quality images with fast sampling. HMAR reformulates next-scale prediction as a Markovian process, wherein the prediction of each resolution scale is conditioned only on tokens in its immediate predecessor instead of the tokens in all predecessor resolutions. When predicting a resolution scale, HMAR uses a controllable multi-step masked generation procedure to generate a subset of the tokens in each step. On ImageNet 256 × 256 and 512 × 512 benchmarks, HMAR models match or outperform parameter-matched VAR, diffusion, and autoregressive baselines. We develop efficient IO-aware block-sparse attention kernels that allow HMAR to achieve faster training and inference times over VAR by over 2.5× and 1.75× respectively, as well as over 3× lower inference memory foot-print. Finally, HMAR yields additional flexibility over VAR; its sampling schedule can be changed without further training, and it can be applied to image editing tasks in a zero-shot manner.

Recent advances in attention-free sequence models rely on convolutions as alternatives to the attention operator at the core of Transformers. In particular, long convolution sequence models have achieved state-of-the-art performance in many domains, but incur a significant cost during auto-regressive inference workloads -- naively requiring a full pass (or caching of activations) over the input sequence for each generated token -- similarly to attention-based models. In this paper, we seek to enable $\mathcal O(1)$ compute and memory cost per token in any pre-trained long convolution architecture to reduce memory footprint and increase throughput during generation. Concretely, our methods consist in extracting low-dimensional linear state-space models from each convolution layer, building upon rational interpolation and model-order reduction techniques. We further introduce architectural improvements to convolution-based layers such as Hyena: by weight-tying the filters across channels into heads, we achieve higher pre-training quality and reduce the number of filters to be distilled. The resulting model achieves 10x higher throughput than Transformers and 1.5x higher than Hyena at 1.3B parameters, without any loss in quality after distillation.

Group convolutional neural networks (G-CNNs) have been shown to increase parameter efficiency and model accuracy by incorporating geometric inductive biases. In this work, we investigate the properties of representations learned by regular G-CNNs, and show considerable parameter redundancy in group convolution kernels. This finding motivates further weight-tying by sharing convolution kernels over subgroups. To this end, we introduce convolution kernels that are separable over the subgroup and channel dimensions. In order to obtain equivariance to arbitrary affine Lie groups we provide a continuous parameterisation of separable convolution kernels. We evaluate our approach across several vision datasets, and show that our weight sharing leads to improved performance and computational efficiency. In many settings, separable G-CNNs outperform their non-separable counterpart, while only using a fraction of their training time. In addition, thanks to the increase in computational efficiency, we are able to implement G-CNNs equivariant to the $\mathrm{Sim(2)}$ group; the group of dilations, rotations and translations of the plane. $\mathrm{Sim(2)}$-equivariance further improves performance on all tasks considered, and achieves state-of-the-art performance on rotated MNIST.

Equivariances provide useful inductive biases in neural network modeling, with the translation equivariance of convolutional neural networks being a canonical example. Equivariances can be embedded in architectures through weight-sharing and place symmetry constraints on the functions a neural network can represent. The type of symmetry is typically fixed and has to be chosen in advance. Although some tasks are inherently equivariant, many tasks do not strictly follow such symmetries. In such cases, equivariance constraints can be overly restrictive. In this work, we propose a parameter-efficient relaxation of equivariance that can effectively interpolate between a (i) non-equivariant linear product, (ii) a strict-equivariant convolution, and (iii) a strictly-invariant mapping. The proposed parameterisation can be thought of as a building block to allow adjustable symmetry structure in neural networks. In addition, we demonstrate that the amount of equivariance can be learned from the training data using backpropagation. Gradient-based learning of equivariance achieves similar or improved performance compared to the best value found by cross-validation and outperforms baselines with partial or strict equivariance on CIFAR-10 and CIFAR-100 image classification tasks.

Group Convolutional Neural Networks (G-CNNs) constrain learned features to respect the symmetries in the selected group, and lead to better generalization when these symmetries appear in the data. If this is not the case, however, equivariance leads to overly constrained models and worse performance. Frequently, transformations occurring in data can be better represented by a subset of a group than by a group as a whole, e.g., rotations in $[-90^{\circ}, 90^{\circ}]$. In such cases, a model that respects equivariance partially is better suited to represent the data. In addition, relevant transformations may differ for low and high-level features. For instance, full rotation equivariance is useful to describe edge orientations in a face, but partial rotation equivariance is better suited to describe face poses relative to the camera. In other words, the optimal level of equivariance may differ per layer. In this work, we introduce Partial G-CNNs: G-CNNs able to learn layer-wise levels of partial and full equivariance to discrete, continuous groups and combinations thereof as part of training. Partial G-CNNs retain full equivariance when beneficial, e.g., for rotated MNIST, but adjust it whenever it becomes harmful, e.g., for classification of 6/9 digits or natural images. We empirically show that partial G-CNNs pair G-CNNs when full equivariance is advantageous, and outperform them otherwise. Our code is publicly available at www.github.com/merlresearch/partial_gcnn .

Although group convolutional networks are able to learn powerful representations based on symmetry patterns, they lack explicit means to learn meaningful relationships among them (e.g., relative positions and poses). In this paper, we present attentive group equivariant convolutions, a generalization of the group convolution, in which attention is applied during the course of convolution to accentuate meaningful symmetry combinations and suppress non-plausible, misleading ones. We indicate that prior work on visual attention can be described as special cases of our proposed framework and show empirically that our attentive group equivariant convolutional networks consistently outperform conventional group convolutional networks on benchmark image datasets. Simultaneously, we provide interpretability to the learned concepts through the visualization of equivariant attention maps.