ABSTRACT 

The problem of identifying geometric structure in heterogeneous, high-dimensional data is a cornerstone of Representation Learning. In this talk, we study this problem from the perspective of Discrete Geometry. We start by reviewing discrete notions of curvature with a focus on Ricci curvature. Then we discuss how curvature characterizations of graphs can be used to improve the efficiency of Graph Neural Networks. Specifically, we propose curvature-based rewiring and encoding approaches and study their impact on the Graph Neural Network’s downstream performance through theoretical and computational analysis. We further discuss applications of discrete Ricci curvature in Manifold Learning, where discrete-to-continuum consistency results allow for characterizing the geometry of a suitable embedding space both locally and in the sense of global curvature bounds.

ABSTRACT 

The performance of modern deep models relies on vast datasets to train, but in many domains data is scarce or difficult to collect. Incorporating symmetry constraints into neural networks has resulted in models called equivariant neural networks (ENNs) which have helped improve sample efficiency. As an application we consider equivariant policy learning which can be used to train robots using fewer iterations for reinforcement learning or fewer demonstrations for imitation learning.  We will also discuss the limitations of standard equivariant learning, which assumes a known group action and suggest methods to circumvent this assumption. 

ABSTRACT 

In this talk, we introduce a new class of problems related to integrating inertial measurements obtained from an IMU that play a significant role in navigation combined with visual data. While there have been tremendous technological advances in the precision of instrumentation, integrating acceleration and angular velocity still suffers from drift in the displacement estimates. Neural networks have come to the rescue in estimating displacement and the associated uncertainty covariance. However, such networks do not consider the physical roto reflective symmetries inherent in IMU data, leading to the need to memorize the same priors for every possible motion direction, which hinders generalization. In this work, we characterize these symmetries and show that the IMU data and the resulting displacement and covariance transform equivariantly when rotated around the gravity vector and reflected with respect to arbitrary planes parallel to gravity. We propose a network for predicting an equivariant gravity aligned frame from equivariant vectors and invariant scalars derived from IMU data, leveraging expressive linear and non-linear layers tailored to commute with the underlying symmetry transformation. Such a canonical frame can precede existing architectures that are end-to-end or filter-based. We will include an introduction to the inertial filtering problem and we will present results in real-world datasets.

ABSTRACT 

Invariant and equivariant networks have been the primary mechanism for imbuing machine learning with symmetry awareness. However, constraining architectures with symmetry may be infeasible or even undesirable. Infeasibility may be due to the inability of network design or lack of information on transformations, and undesirability may be due to the approximate nature of symmetries or suboptimal use of computing. In this talk, I'll briefly review several works from our group that use symmetries beyond equivariant networks. These examples explore symmetry in different learning paradigms ranging from reinforcement learning and self-supervised learning to physics-informed learning and generative modelling.  

ABSTRACT 

There is now undeniable evidence for traveling waves and other structured spatiotemporal recurrent neural dynamics in cortical structures. However, to date, these observations have been difficult to reconcile with the dominant conceptual notions of topographically organized selectivity and feedforward receptive fields. This conceptual conflict has lead to the frequent marginalization of these dynamics to the category of ‘noise’, often being removed prior to neural data analysis. Our research proposes a new ‘spacetime’ perspective on neural computation in which structured selectivity and dynamics are not contradictory but instead may be complimentary. Through the analysis of state of the art models in the machine learning literature, along with the construction of novel spatiotemporal artificial neural network architectures, we show that spatiotemporal dynamics may be a mechanism by which natural neural systems encode approximate visual, temporal, and abstract symmetries of the world as conserved quantities, thereby enabling improved generalization and long-term working memory. Ultimately, we advocate that spacetime may be the fundamental ‘fabric of neural computation’, and believe our research calls for further empirical and theoretical study in this direction.