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0111 · Sep 202419922001200920172026
7 results for Hellinger-Kantorovich

Formula derived for curvature in measure spaces.

problem Deriving sectional curvature in measure spaces.
method Explicit formula derivation for sectional curvature in M(M){\cal M}(M) with metrics HKHK and W2W_2.
result Curvature analysis in M(M){\cal M}(M) reveals both negative and positive components.

Unified framework for analyzing gradient flows of measures with exponential decay of entropy.

problem Analyzing exponential decay of entropy functionals in gradient flows of measures.
method Characterization of global exponential decay behaviors using Hellinger-Kantorovich geometry, shape-mass decomposition, and Polyak-Łojasiewicz-type inequalities.
result Unified theoretical framework for gradient flows with complete analysis of exponential decay behaviors.

A new associative memory uses Sinkhorn divergence for efficient pattern retrieval.

problem Efficiently retrieving patterns from large datasets of weighted point clouds.
method Derived retrieval dynamics as a SHK gradient flow, discretized for a deterministic algorithm.
result Proved basin invariance, geometric convergence, and robust recovery from perturbations.

This work develops sampling methods for differential privacy using SHK geometry.

problem Approximating sampling for the exponential mechanism in differential privacy.
method Develops perturbation theory for SHK gradient flows and applies to differential privacy.
result Derives time-dependent Pure-DP guarantees and Approximate-DP certificates.

This paper uses UOT metrics for better dimensionality reduction and classification/clustering.

problem Improving dimensionality reduction and classification/clustering methods.
method Uses Hellinger--Kantorovich metric from unbalanced optimal transport (UOT).
result UOT outperforms Euclidean and OT-based methods in classification and clustering tasks.

The study analyzes the evolution of Gaussian measures under a specific gradient flow.

problem Analyzing the evolution of Gaussian measures under a specific gradient flow.
method Derives ordinary differential equations governing the evolution of mean, covariance, and mass under the HK-Boltzmann gradient flow.
result Exponential convergence to equilibrium demonstrated through Polyak-Lojasiewicz-type inequalities.