Thomas Gerald

Nina Miolane, Alice Le Brigant, Johan Mathe et al.

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 \url{geomstats.ai}.

Thomas Gerald, Aurélia Léon, Nicolas Baskiotis et al.

Classification with a large number of classes is a key problem in machine learning and corresponds to many real-world applications like tagging of images or textual documents in social networks. If one-vs-all methods usually reach top performance in this context, these approaches suffer from a high inference complexity, linear w.r.t the number of categories. Different models based on the notion of binary codes have been proposed to overcome this limitation, achieving in a sublinear inference complexity. But they a priori need to decide which binary code to associate to which category before learning using more or less complex heuristics. We propose a new end-to-end model which aims at simultaneously learning to associate binary codes with categories, but also learning to map inputs to binary codes. This approach called Deep Stochastic Neural Codes (DSNC) keeps the sublinear inference complexity but do not need any a priori tuning. Experimental results on different datasets show the effectiveness of the approach w.r.t baseline methods.