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What is Learnable in Valiant's Theory of the Learnable? Learning Perturbations to Extrapolate Your LLM Byzantine-Robust Distributed Sparse Learning Revisited The Sample Complexity of Multiple Change Point Identification under Bandit Feedback A proximal gradient algorithm for composite log-concave sampling Model-based Bootstrap of Controlled Markov Chains Approximation of Maximally Monotone Operators : A Graph Convergence Perspective Posterior Contraction Rates for Sparse Kolmogorov-Arnold Networks in Anisotropic Besov Spaces MIST: Reliable Streaming Decision Trees for Online Class-Incremental Learning via McDiarmid Bound A Spectral Framework for Closed-Form Relative Density Estimation Fast Rates for Offline Contextual Bandits with Forward-KL Regularization under Single-Policy Concentrability Higher-Order Equilibrium Tracking for EM-Compressible Online Estimation Scaling Limits of Long-Context Transformers A Note on Non-Negative $L_1$-Approximating Polynomials Susceptibilities and Patterning: A Primer on Linear Response in Bayesian Learning 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Multifractal Analysis of the Sinkhorn Algorithm: Unveiling the Intricate Structure of Optimal Transport Maps
Jose Rafael Espinosa Mena · 2024-05-25 · via math.ST updates on arXiv.org

The Sinkhorn algorithm has emerged as a powerful tool for solving optimal transport problems, finding applications in various domains such as machine learning, image processing, and computational biology. Despite its widespread use, the intricate structure and scaling properties of the coupling matrices generated by the Sinkhorn algorithm remain largely unexplored. In this paper, we delve into the multifractal properties of these coupling matrices, aiming to unravel their complex behavior and shed light on the underlying dynamics of the Sinkhorn algorithm. We prove the existence of the multifractal spectrum and the singularity spectrum for the Sinkhorn coupling matrices. Furthermore, we derive bounds on the generalized dimensions, providing a comprehensive characterization of their scaling properties. Our findings not only deepen our understanding of the Sinkhorn algorithm but also pave the way for novel applications and algorithmic improvements in the realm of optimal transport.

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