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Daniel Brooks

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2 papers

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JMLR Journal 2020 Journal Article

Geomstats: A Python Package for Riemannian Geometry in Machine Learning

Nina Miolane
Nicolas Guigui
Alice Le Brigant
Johan Mathe
Benjamin Hou
Yann Thanwerdas
Stefan Heyder
Olivier Peltre

We introduce Geomstats, an open-source Python package 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. 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. This paper presents the package, compares it with related libraries, and provides relevant code examples. We show that Geomstats provides reliable building blocks to both foster research in differential geometry and statistics and democratize the use of Riemannian geometry in machine learning applications. The source code is freely available under the MIT license at geomstats.ai. [abs] [ pdf ][ bib ] [ code ] &copy JMLR 2020. ( edit, beta )

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NeurIPS Conference 2019 Conference Paper

Riemannian batch normalization for SPD neural networks

Daniel Brooks
Olivier Schwander
Frederic Barbaresco
Jean-Yves Schneider
Matthieu Cord

Covariance matrices have attracted attention for machine learning applications due to their capacity to capture interesting structure in the data. The main challenge is that one needs to take into account the particular geometry of the Riemannian manifold of symmetric positive definite (SPD) matrices they belong to. In the con- text of deep networks, several architectures for these matrices have recently been proposed. In our article, we introduce a Riemannian batch normalization (batch- norm) algorithm, which generalizes the one used in Euclidean nets. This novel layer makes use of geometric operations on the manifold, notably the Riemannian barycenter, parallel transport and non-linear structured matrix transformations. We derive a new manifold-constrained gradient descent algorithm working in the space of SPD matrices, allowing to learn the batchnorm layer. We validate our proposed approach with experiments in three different contexts on diverse data types: a drone recognition dataset from radar observations, and on emotion and action recognition datasets from video and motion capture data. Experiments show that the Riemannian batchnorm systematically gives better classification performance compared with leading methods and a remarkable robustness to lack of data.

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