arXiv subjects

Max Welling

Publications and source records attributed to Max Welling.

At least 91 records · Page 5Linked to original sources

arXiv2022

Amortized Causal Discovery: Learning to Infer Causal Graphs from Time-Series Data

On time-series data, most causal discovery methods fit a new model whenever they encounter samples from a new underlying causal graph. However, these samples often share relevant information which is lost when following this approach. Specifically, different samples may share the dynamics which describe the effects of their causal relations. We propose Amortized Causal Discovery, a novel framework that leverages such shared dynamics to learn to infer causal relations from time-series data. This enables us to train a single, amortized model that infers causal relations across samples with different underlying causal graphs, and thus leverages the shared dynamics information. We demonstrate experimentally that this approach, implemented as a variational model, leads to significant improvements in causal discovery performance, and show how it can be extended to perform well under added noise and hidden confounding.

cs.LG↗

arXiv2022

E(n) Equivariant Graph Neural Networks

This paper introduces a new model to learn graph neural networks equivariant to rotations, translations, reflections and permutations called E(n)-Equivariant Graph Neural Networks (EGNNs). In contrast with existing methods, our work does not require computationally expensive higher-order representations in intermediate layers while it still achieves competitive or better performance. In addition, whereas existing methods are limited to equivariance on 3 dimensional spaces, our model is easily scaled to higher-dimensional spaces. We demonstrate the effectiveness of our method on dynamical systems modelling, representation learning in graph autoencoders and predicting molecular properties.

cs.LG↗

arXiv2022

E(n) Equivariant Normalizing Flows

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.

cs.LG↗

arXiv2022

Topographic VAEs learn Equivariant Capsules

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.

cs.LG↗

arXiv2021

Modeling Category-Selective Cortical Regions with Topographic Variational Autoencoders

Category-selectivity in the brain describes the observation that certain spatially localized areas of the cerebral cortex tend to respond robustly and selectively to stimuli from specific limited categories. One of the most well known examples of category-selectivity is the Fusiform Face Area (FFA), an area of the inferior temporal cortex in primates which responds preferentially to images of faces when compared with objects or other generic stimuli. In this work, we leverage the newly introduced Topographic Variational Autoencoder to model the emergence of such localized category-selectivity in an unsupervised manner. Experimentally, we demonstrate our model yields spatially dense neural clusters selective to faces, bodies, and places through visualized maps of Cohen's d metric. We compare our model with related supervised approaches, namely the Topographic Deep Artificial Neural Network (TDANN) of Lee et al., and discuss both theoretical and empirical similarities. Finally, we show preliminary results suggesting that our model yields a nested spatial hierarchy of increasingly abstract categories, analogous to observations from the human ventral temporal cortex.

q-bio.NC↗

arXiv2021

Deep Policy Dynamic Programming for Vehicle Routing Problems

Routing problems are a class of combinatorial problems with many practical applications. Recently, end-to-end deep learning methods have been proposed to learn approximate solution heuristics for such problems. In contrast, classical dynamic programming (DP) algorithms guarantee optimal solutions, but scale badly with the problem size. We propose Deep Policy Dynamic Programming (DPDP), which aims to combine the strengths of learned neural heuristics with those of DP algorithms. DPDP prioritizes and restricts the DP state space using a policy derived from a deep neural network, which is trained to predict edges from example solutions. We evaluate our framework on the travelling salesman problem (TSP), the vehicle routing problem (VRP) and TSP with time windows (TSPTW) and show that the neural policy improves the performance of (restricted) DP algorithms, making them competitive to strong alternatives such as LKH, while also outperforming most other 'neural approaches' for solving TSPs, VRPs and TSPTWs with 100 nodes.

cs.LG↗

arXiv2021

Particle Dynamics for Learning EBMs

Energy-based modeling is a promising approach to unsupervised learning, which yields many downstream applications from a single model. The main difficulty in learning energy-based models with the "contrastive approaches" is the generation of samples from the current energy function at each iteration. Many advances have been made to accomplish this subroutine cheaply. Nevertheless, all such sampling paradigms run MCMC targeting the current model, which requires infinitely long chains to generate samples from the true energy distribution and is problematic in practice. This paper proposes an alternative approach to getting these samples and avoiding crude MCMC sampling from the current model. We accomplish this by viewing the evolution of the modeling distribution as (i) the evolution of the energy function, and (ii) the evolution of the samples from this distribution along some vector field. We subsequently derive this time-dependent vector field such that the particles following this field are approximately distributed as the current density model. Thereby we match the evolution of the particles with the evolution of the energy function prescribed by the learning procedure. Importantly, unlike Monte Carlo sampling, our method targets to match the current distribution in a finite time. Finally, we demonstrate its effectiveness empirically compared to MCMC-based learning methods.

cs.LG↗

arXiv2021

Gauge Equivariant Mesh CNNs: Anisotropic convolutions on geometric graphs

A common approach to define convolutions on meshes is to interpret them as a graph and apply graph convolutional networks (GCNs). Such GCNs utilize isotropic kernels and are therefore insensitive to the relative orientation of vertices and thus to the geometry of the mesh as a whole. We propose Gauge Equivariant Mesh CNNs which generalize GCNs to apply anisotropic gauge equivariant kernels. Since the resulting features carry orientation information, we introduce a geometric message passing scheme defined by parallel transporting features over mesh edges. Our experiments validate the significantly improved expressivity of the proposed model over conventional GCNs and other methods.

cs.LG↗

arXiv2021

An Expectation-Maximization Perspective on Federated Learning

Federated learning describes the distributed training of models across multiple clients while keeping the data private on-device. In this work, we view the server-orchestrated federated learning process as a hierarchical latent variable model where the server provides the parameters of a prior distribution over the client-specific model parameters. We show that with simple Gaussian priors and a hard version of the well known Expectation-Maximization (EM) algorithm, learning in such a model corresponds to FedAvg, the most popular algorithm for the federated learning setting. This perspective on FedAvg unifies several recent works in the field and opens up the possibility for extensions through different choices for the hierarchical model. Based on this view, we further propose a variant of the hierarchical model that employs prior distributions to promote sparsity. By similarly using the hard-EM algorithm for learning, we obtain FedSparse, a procedure that can learn sparse neural networks in the federated learning setting. FedSparse reduces communication costs from client to server and vice-versa, as well as the computational costs for inference with the sparsified network - both of which are of great practical importance in federated learning.

cs.LG↗

arXiv2021

Argmax Flows and Multinomial Diffusion: Learning Categorical Distributions

Generative flows and diffusion models have been predominantly trained on ordinal data, for example natural images. This paper introduces two extensions of flows and diffusion for categorical data such as language or image segmentation: Argmax Flows and Multinomial Diffusion. Argmax Flows are defined by a composition of a continuous distribution (such as a normalizing flow), and an argmax function. To optimize this model, we learn a probabilistic inverse for the argmax that lifts the categorical data to a continuous space. Multinomial Diffusion gradually adds categorical noise in a diffusion process, for which the generative denoising process is learned. We demonstrate that our method outperforms existing dequantization approaches on text modelling and modelling on image segmentation maps in log-likelihood.

stat.ML↗

arXiv2021

Neural Augmentation of Kalman Filter with Hypernetwork for Channel Tracking

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.

eess.SP↗

arXiv2021

Learning Equivariant Energy Based Models with Equivariant Stein Variational Gradient Descent

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.

cs.LG↗

arXiv2021

Federated Mixture of Experts

Federated learning (FL) has emerged as the predominant approach for collaborative training of neural network models across multiple users, without the need to gather the data at a central location. One of the important challenges in this setting is data heterogeneity, i.e. different users have different data characteristics. For this reason, training and using a single global model might be suboptimal when considering the performance of each of the individual user's data. In this work, we tackle this problem via Federated Mixture of Experts, FedMix, a framework that allows us to train an ensemble of specialized models. FedMix adaptively selects and trains a user-specific selection of the ensemble members. We show that users with similar data characteristics select the same members and therefore share statistical strength while mitigating the effect of non-i.i.d data. Empirically, we show through an extensive experimental evaluation that FedMix improves performance compared to using a single global model across a variety of different sources of non-i.i.d.-ness.

cs.LG↗

arXiv2021

Learning to Predict Error for MRI Reconstruction

In healthcare applications, predictive uncertainty has been used to assess predictive accuracy. In this paper, we demonstrate that predictive uncertainty estimated by the current methods does not highly correlate with prediction error by decomposing the latter into random and systematic errors, and showing that the former is equivalent to the variance of the random error. In addition, we observe that current methods unnecessarily compromise performance by modifying the model and training loss to estimate the target and uncertainty jointly. We show that estimating them separately without modifications improves performance. Following this, we propose a novel method that estimates the target labels and magnitude of the prediction error in two steps. We demonstrate this method on a large-scale MRI reconstruction task, and achieve significantly better results than the state-of-the-art uncertainty estimation methods.

cs.LG↗

arXiv2021

Deterministic Gibbs Sampling via Ordinary Differential Equations

Deterministic dynamics is an essential part of many MCMC algorithms, e.g. Hybrid Monte Carlo or samplers utilizing normalizing flows. This paper presents a general construction of deterministic measure-preserving dynamics using autonomous ODEs and tools from differential geometry. We show how Hybrid Monte Carlo and other deterministic samplers follow as special cases of our theory. We then demonstrate the utility of our approach by constructing a continuous non-sequential version of Gibbs sampling in terms of an ODE flow and extending it to discrete state spaces. We find that our deterministic samplers are more sample efficient than stochastic counterparts, even if the latter generate independent samples.

stat.CO↗

arXiv2021

Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds

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 Möbius 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.

cs.LG↗

arXiv2021

Self Normalizing Flows

Efficient gradient computation of the Jacobian determinant term is a core problem in many machine learning settings, and especially so in the normalizing flow framework. Most proposed flow models therefore either restrict to a function class with easy evaluation of the Jacobian determinant, or an efficient estimator thereof. However, these restrictions limit the performance of such density models, frequently requiring significant depth to reach desired performance levels. In this work, we propose Self Normalizing Flows, a flexible framework for training normalizing flows by replacing expensive terms in the gradient by learned approximate inverses at each layer. This reduces the computational complexity of each layer's exact update from $\mathcal{O}(D^3)$ to $\mathcal{O}(D^2)$, allowing for the training of flow architectures which were otherwise computationally infeasible, while also providing efficient sampling. We show experimentally that such models are remarkably stable and optimize to similar data likelihood values as their exact gradient counterparts, while training more quickly and surpassing the performance of functionally constrained counterparts.

cs.LG↗

arXiv2021

Orbital MCMC

Markov Chain Monte Carlo (MCMC) algorithms ubiquitously employ complex deterministic transformations to generate proposal points that are then filtered by the Metropolis-Hastings-Green (MHG) test. However, the condition of the target measure invariance puts restrictions on the design of these transformations. In this paper, we first derive the acceptance test for the stochastic Markov kernel considering arbitrary deterministic maps as proposal generators. When applied to the transformations with orbits of period two (involutions), the test reduces to the MHG test. Based on the derived test we propose two practical algorithms: one operates by constructing periodic orbits from any diffeomorphism, another on contractions of the state space (such as optimization trajectories). Finally, we perform an empirical study demonstrating the practical advantages of both kernels.

cs.LG↗