Bispectral Neural Networks

We present a neural network architecture, Bispectral Neural Networks (BNNs) for learning representations that are invariant to the actions of compact commutative groups on the space over which a signal is defined. The model incorporates the ansatz of the bispectrum, an analytically defined group invariant that is complete -- that is, it preserves all signal structure while removing only the variation due to group actions. Here, we demonstrate that BNNs are able to simultaneously learn groups, their irreducible representations, and corresponding complete invariant maps purely from the symmetries implicit in data. Further, we demonstrate that the completeness property endows these networks with strong adversarial robustness. This work establishes Bispectral Neural Networks as a powerful computational primitive for robust invariant representation learning

Introduction to Geometric Learning in Python with Geomstats

International audienceThere is a growing interest in leveraging differential geometry in the machine learning community. Yet, the adoption of the associated geometric computations has been inhibited by the lack of a reference implementation. Such an implementation should typically allow its users: (i) to get intuition on concepts from differential geometry through a hands-on approach, often not provided by traditional textbooks; and (ii) to run geometric machine learning algorithms seamlessly, without delving into the mathematical details. To address this gap, we present the open-source Python package geomstats and introduce hands-on tutorials for differential geometry and geometric machine learning algorithms-Geometric Learning-that rely on it. Code and documentation: github.com/geomstats/geomstats and geomstats.ai

Geomstats: A Python Package for Riemannian Geometry in Machine Learning

We introduce Geomstats, an open-source Python toolbox for computations and statistics on nonlinear manifolds, such as hyperbolic spaces, spaces of symmetric positive definite matrices, Lie groups of transformations, and many more. We provide object-oriented and extensively unit-tested implementations. Among others, manifolds come equipped with families of Riemannian metrics, with associated exponential and logarithmic maps, geodesics and parallel transport. Statistics and learning algorithms provide methods for estimation, clustering and dimension reduction on manifolds. All associated operations are vectorized for batch computation and provide support for different execution backends, namely NumPy, PyTorch and TensorFlow, enabling GPU acceleration. This paper presents the package, compares it with related libraries and provides relevant code examples. We show that Geomstats provides reliable building blocks to foster research in differential geometry and statistics, and to democratize the use of Riemannian geometry in machine learning applications. The source code is freely available under the MIT license at http://geomstats.ai.Idex UCA JEDI3IA Côte d'AzurG-Statistics - Foundations of Geometric Statistics and Their Application in the Life Science

ICLR 2022 Challenge for Computational Geometry & Topology: Design and Results

International audienceThis paper presents the computational challenge on differential geometry and topology that was hosted within the ICLR 2022 workshop “Geometric and Topo- logical Representation Learning”. The competition asked participants to provide implementations of machine learning algorithms on manifolds that would respect the API of the open-source software Geomstats (manifold part) and Scikit-Learn (machine learning part) or PyTorch. The challenge attracted seven teams in its two month duration. This paper describes the design of the challenge and summarizes its main findings