Symmetries and equivariance are fundamental to the generalization of neural networks on domains such as images, graphs, and point clouds. Existing work has primarily focused on a small number of groups, such as the translation, rotation, and permutation groups. In this work we provide a completely general algorithm for solving for the equivariant layers of matrix groups. In addition to recovering solutions from other works as special cases, we construct multilayer perceptrons equivariant to multiple groups that have never been tackled before, including $\mathrm{O}(1,3)$, $\mathrm{O}(5)$, $\mathrm{Sp}(n)$, and the Rubik's cube group. Our approach outperforms non-equivariant baselines, with applications to particle physics and modeling dynamical systems. We release our software library to enable researchers to construct equivariant layers for arbitrary

Most neural network pruning methods, such as filter-level and layer-level prunings, prune the network model along one dimension (depth, width, or resolution) solely to meet a computational budget. However, such a pruning policy often leads to excessive reduction of that dimension, thus inducing a huge accuracy loss. To alleviate this issue, we argue that pruning should be conducted along three dimensions comprehensively. For this purpose, our pruning framework formulates pruning as an optimization problem. Specifically, it first casts the relationships between a certain model's accuracy and depth/width/resolution into a polynomial regression and then maximizes the polynomial to acquire the optimal values for the three dimensions. Finally, the model is pruned along the three optimal dimensions accordingly. In this framework, since collecting too much data for training the regression is very time-costly, we propose two approaches to lower the cost: 1) specializing the polynomial to ensure an accurate regression even with less training data; 2) employing iterative pruning and fine-tuning to collect the data faster. Extensive experiments show that our proposed algorithm surpasses state-of-the-art pruning algorithms and even neural architecture search-based algorithms.

Although ubiquitous in the sciences, histogram data have not received much attention by the Deep Learning community. Whilst regression and classification tasks for scalar and vector data are routinely solved by neural networks, a principled approach for estimating histogram labels as a function of an input vector or image is lacking in the literature. We present a dedicated method for Deep Learning-based histogram regression, which incorporates cross-bin information and yields distributions over possible histograms, expressed by $\tau$-quantiles of the cumulative histogram in each bin. The crux of our approach is a new loss function obtained by applying the pinball loss to the cumulative histogram, which for 1D histograms reduces to the Earth Mover's distance (EMD) in the special case of the median ($\tau = 0.5$), and generalizes it to arbitrary quantiles. We validate our method with an illustrative toy example, a football-related task, and an astrophysical computer vision problem. We show that with our loss function, the accuracy of the predicted median histograms is very similar to the standard EMD case (and higher than for per-bin loss functions such as cross-entropy), while the predictions become much more informative at almost no additional computational cost.

Existing deep learning methods for solving mean-field games (MFGs) with common noise fix the sampling common noise paths and then solve the corresponding MFGs. This leads to a nested loop structure with millions of simulations of common noise paths in order to produce accurate solutions, which results in prohibitive computational cost and limits the applications to a large extent. In this paper, based on the rough path theory, we propose a novel single-loop algorithm, named signatured deep fictitious play (Sig-DFP), by which we can work with the unfixed common noise setup to avoid the nested loop structure and reduce the computational complexity significantly. The proposed algorithm can accurately capture the effect of common uncertainty changes on mean-field equilibria without further training of neural networks, as previously needed in the existing machine learning algorithms. The efficiency is supported by three applications, including linear-quadratic MFGs, mean-field portfolio game, and mean-field game of optimal consumption and investment. Overall, we provide a new point of view from the rough path theory to solve MFGs with common noise with significantly improved efficiency and an extensive range of applications. In addition, we report the first deep learning work to deal with extended MFGs (a mean-field interaction via both the states and controls) with common noise.

Message passing neural networks have become a method of choice for learning on graphs, in particular the prediction of chemical properties and the acceleration of molecular dynamics studies. While they readily scale to large training data sets, previous approaches have proven to be less data efficient than kernel methods. We identify limitations of invariant representations as a major reason and extend the message passing formulation to rotationally equivariant representations. On this basis, we propose the polarizable atom interaction neural network (PaiNN) and improve on common molecule benchmarks over previous networks, while reducing model size and inference time. We leverage the equivariant atomwise representations obtained by PaiNN for the prediction of tensorial properties. Finally, we apply this to the simulation of molecular spectra, achieving speedups of 4-5 orders of magnitude compared to the electronic structure reference.

In this work, we focus on the ability of graph neural networks (GNNs) to learn long-range patterns in graphs with edge features. Learning patterns that involve longer paths in the graph, requires using deeper GNNs. However, GNNs suffer from a drop in performance with increasing network depth. To improve the performance of deeper GNNs, previous works have investigated normalization techniques and various types of skip connections. While they are designed to improve depth-wise backpropagation between the representations of the same node in successive layers, they do not improve breadth-wise backpropagation between representations of neighbouring nodes. To analyse the consequences, we design synthetic datasets serving as a testbed for the ability of GNNs to learn long-range patterns. Our analysis shows that several commonly used GNN variants with only depth-wise skip connections indeed have problems learning long-range patterns. They are clearly outperformed by an attention-based GNN architecture that we propose for improving both depth- and breadth-wise backpropagation. We also verify that the presented architecture is competitive on real-world data.

Face recognition is an important yet challenging problem in computer vision. A major challenge in practical face recognition applications lies in significant variations between profile and frontal faces. Traditional techniques address this challenge either by synthesizing frontal faces or by pose invariant learning. In this paper, we propose a novel method with Lie algebra theory to explore how face rotation in the 3D space affects the deep feature generation process of convolutional neural networks (CNNs). We prove that face rotation in the image space is equivalent to an additive residual component in the feature space of CNNs, which is determined solely by the rotation. Based on this theoretical finding, we further design a Lie Algebraic Residual Network (LARNet) for tackling pose robust face recognition. Our LARNet consists of a residual subnet for decoding rotation information from input face images, and a gating subnet to learn rotation magnitude for controlling the strength of the residual component contributing to the feature learning process. Comprehensive experimental evaluations on both frontal-profile face datasets and general face recognition datasets convincingly demonstrate that our method consistently outperforms the state-of-the-art ones.