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Analyse de données géométriques, au delà des convolutions

Abstract : Geometric data analysis, beyond convolutionsTo model interactions between points, a simple option is to rely on weighted sums known as convolutions. Over the last decade, this operation has become a building block for deep learning architectures with an impact on many applied fields. We should not forget, however, that the convolution product is far from being the be-all and end-all of computational mathematics.To let researchers explore new directions, we present robust, efficient and principled implementations of three underrated operations: 1. Generic manipulations of distance-like matrices, including kernel matrix-vector products and nearest-neighbor searches.2. Optimal transport, which generalizes sorting to spaces of dimension D > 1.3. Hamiltonian geodesic shooting, which replaces linear interpolation when no relevant algebraic structure can be defined on a metric space of features.Our PyTorch/NumPy routines fully support automatic differentiation and scale up to millions of samples in seconds. They generally outperform baseline GPU implementations with x10 to x1,000 speed-ups and keep linear instead of quadratic memory footprints. These new tools are packaged in the KeOps (kernel methods) and GeomLoss (optimal transport) libraries, with applications that range from machine learning to medical imaging. Documentation is available at: www.kernel-operations.io/keops and /geomloss.
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Submitted on : Tuesday, September 22, 2020 - 5:22:08 PM
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Jean Feydy. Analyse de données géométriques, au delà des convolutions. Mathématiques générales [math.GM]. Université Paris-Saclay, 2020. Français. ⟨NNT : 2020UPASN017⟩. ⟨tel-02945979⟩

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