Including covariant information, such as position, force, velocity or spin is important in many tasks in computational physics and chemistry. We introduce Steerable E(3) Equivariant Graph Neural Networks (SEGNNs) that generalise equivariant graph networks, such that node and edge attributes are not restricted to invariant scalars, but can contain covariant information, such as vectors or tensors. This model, composed of steerable MLPs, is able to incorporate geometric and physical information in both the message and update functions. Through the definition of steerable node attributes, the MLPs provide a new class of activation functions for general use with steerable feature fields. We discuss ours and related work through the lens of equivariant non-linear convolutions, which further allows us to pin-point the successful components of SEGNNs: non-linear message aggregation improves upon classic linear (steerable) point convolutions; steerable messages improve upon recent equivariant graph networks that send invariant messages. We demonstrate the effectiveness of our method on several tasks in computational physics and chemistry and provide extensive ablation studies.

We propose Hypernetwork Kalman Filter (HKF) for tracking applications with multiple different dynamics. The HKF combines generalization power of Kalman filters with expressive power of neural networks. Instead of keeping a bank of Kalman filters and choosing one based on approximating the actual dynamics, HKF adapts itself to each dynamics based on the observed sequence. Through extensive experiments on CDL-B channel model, we show that the HKF can be used for tracking the channel over a wide range of Doppler values, matching Kalman filter performance with genie Doppler information. At high Doppler values, it achieves around 2dB gain over genie Kalman filter. The HKF generalizes well to unseen Doppler, SNR values and pilot patterns unlike LSTM, which suffers from severe performance degradation.

In this work we seek to bridge the concepts of topographic organization and equivariance in neural networks. To accomplish this, we introduce the Topographic VAE: a novel method for efficiently training deep generative models with topographically organized latent variables. We show that such a model indeed learns to organize its activations according to salient characteristics such as digit class, width, and style on MNIST. Furthermore, through topographic organization over time (i.e. temporal coherence), we demonstrate how predefined latent space transformation operators can be encouraged for observed transformed input sequences -- a primitive form of unsupervised learned equivariance. We demonstrate that this model successfully learns sets of approximately equivariant features (i.e. "capsules") directly from sequences and achieves higher likelihood on correspondingly transforming test sequences. Equivariance is verified quantitatively by measuring the approximate commutativity of the inference network and the sequence transformations. Finally, we demonstrate approximate equivariance to complex transformations, expanding upon the capabilities of existing group equivariant neural networks.

We focus on the problem of efficient sampling and learning of probability densities by incorporating symmetries in probabilistic models. We first introduce Equivariant Stein Variational Gradient Descent algorithm -- an equivariant sampling method based on Stein's identity for sampling from densities with symmetries. Equivariant SVGD explicitly incorporates symmetry information in a density through equivariant kernels which makes the resultant sampler efficient both in terms of sample complexity and the quality of generated samples. Subsequently, we define equivariant energy based models to model invariant densities that are learned using contrastive divergence. By utilizing our equivariant SVGD for training equivariant EBMs, we propose new ways of improving and scaling up training of energy based models. We apply these equivariant energy models for modelling joint densities in regression and classification tasks for image datasets, many-body particle systems and molecular structure generation.

Motivated by the vast success of deep convolutional networks, there is a great interest in generalizing convolutions to non-Euclidean manifolds. A major complication in comparison to flat spaces is that it is unclear in which alignment a convolution kernel should be applied on a manifold. The underlying reason for this ambiguity is that general manifolds do not come with a canonical choice of reference frames (gauge). Kernels and features therefore have to be expressed relative to arbitrary coordinates. We argue that the particular choice of coordinatization should not affect a network's inference -- it should be coordinate independent. A simultaneous demand for coordinate independence and weight sharing is shown to result in a requirement on the network to be equivariant under local gauge transformations (changes of local reference frames). The ambiguity of reference frames depends thereby on the G-structure of the manifold, such that the necessary level of gauge equivariance is prescribed by the corresponding structure group G. Coordinate independent convolutions are proven to be equivariant w.r.t. those isometries that are symmetries of the G-structure. The resulting theory is formulated in a coordinate free fashion in terms of fiber bundles. To exemplify the design of coordinate independent convolutions, we implement a convolutional network on the M\"obius strip. The generality of our differential geometric formulation of convolutional networks is demonstrated by an extensive literature review which explains a large number of Euclidean CNNs, spherical CNNs and CNNs on general surfaces as specific instances of coordinate independent convolutions.

This paper introduces a generative model equivariant to Euclidean symmetries: E(n) Equivariant Normalizing Flows (E-NFs). To construct E-NFs, we take the discriminative E(n) graph neural networks and integrate them as a differential equation to obtain an invertible equivariant function: a continuous-time normalizing flow. We demonstrate that E-NFs considerably outperform baselines and existing methods from the literature on particle systems such as DW4 and LJ13, and on molecules from QM9 in terms of log-likelihood. To the best of our knowledge, this is the first flow that jointly generates molecule features and positions in 3D.

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 dynamical systems. We release our software library to enable researchers to construct equivariant layers for arbitrary matrix groups.

In this work, we explore adversarial attacks on the Variational Autoencoders (VAE). We show how to modify data point to obtain a prescribed latent code (supervised attack) or just get a drastically different code (unsupervised attack). We examine the influence of model modifications ($\beta$-VAE, NVAE) on the robustness of VAEs and suggest metrics to quantify it.

Unobserved confounding is one of the main challenges when estimating causal effects. We propose a novel causal reduction method that replaces an arbitrary number of possibly high-dimensional latent confounders with a single latent confounder that lives in the same space as the treatment variable without changing the observational and interventional distributions entailed by the causal model. After the reduction, we parameterize the reduced causal model using a flexible class of transformations, so-called normalizing flows. We propose a learning algorithm to estimate the parameterized reduced model jointly from observational and interventional data. This allows us to estimate the causal effect in a principled way from combined data. We perform a series of experiments on data simulated using nonlinear causal mechanisms and find that we can often substantially reduce the number of interventional samples when adding observational training samples without sacrificing accuracy. Thus, adding observational data may help to more accurately estimate causal effects even in the presence of unobserved confounders.

In this work we propose a batch Bayesian optimization method for combinatorial problems on permutations, which is well suited for expensive cost functions on permutations. We introduce LAW, a new efficient batch acquisition method based on the determinantal point process, using an acquisition weighted kernel. Relying on multiple parallel evaluations, LAW accelerates the search for the optimal permutation. We provide a regret analysis for our method to gain insight in its theoretical properties. We then apply the framework to permutation problems, which have so far received little attention in the Bayesian Optimization literature, despite their practical importance. We call this method LAW2ORDER. We evaluate the method on several standard combinatorial problems involving permutations such as quadratic assignment, flowshop scheduling and the traveling salesman, as well as on a structure learning task.