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The algebraic degree of the Wasserstein distance

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arxiv 2401.12735 v1 pith:DRN46RXG submitted 2024-01-23 math.AG math.GR

Chiara Meroni , Bernhard Reinke , Kexin Wang This is my paper · ORCID
classification math.AGmath.GR
keywords algebraicdistancewassersteindegreeassociatedbirkhoffcomplexcomplexity
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Given two rational univariate polynomials, the Wasserstein distance of their associated measures is an algebraic number. We determine the algebraic degree of the squared Wasserstein distance, serving as a measure of algebraic complexity of the corresponding optimization problem. The computation relies on the structure of a subpolytope of the Birkhoff polytope, invariant under a transformation induced by complex conjugation.

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  1. Algebra Unveils Deep Learning -- An Invitation to Neuroalgebraic Geometry

    cs.LG 2025-01 conditional novelty 4.0 of 10

    The paper argues that algebraic geometry offers a powerful dictionary for understanding deep learning models with polynomial or piecewise-polynomial activations.

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