Wasserstein Distance

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Wasserstein distance studies how much probability mass must move, and how far, to transform one distribution into another, making it sensitive to the geometry of the data. Methods include optimal transport couplings, entropically regularized Sinkhorn algorithms, sliced distances based on one-dimensional projections, and Gromov-Wasserstein distances for comparing spaces with different geometries. The distance is used to analyze and train generative models, compare graphs and shapes, estimate distribution shifts, and quantify convergence, with theory often focused on sample complexity and dimension dependence. A central tension is that accurate transport can be computationally expensive, while faster approximations or regularization may trade geometric fidelity for scalability and statistical efficiency.

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63 papers8 papers in the last four weeks, up 33% on the four weeks before. 0.0% of all new papers.

Improved Finite-Particle Convergence Rates for Stein Variational Gradient Descent
Sep 13, 2024Sayan Banerjee, Krishnakumar Balasubramanian, Promit GhosalKernel Methods Variational Inference
Department of Statistics and Operations Research, University of North Carolina, Chapel Hill. · Department of Statistics, University of California, Davis. · Department of Statistics, University of Chicago.
Structured Approximations of Measures
Oct 13, 2023Keaton Hamm, Varun KhuranaOptimal Transport Wasserstein Distance
DEPARTMENT OF MATHEMATICS, THE UNIVERSITY OF TEXAS AT ARLINGTON, ARLINGTON, TX · DIVISION OF DATA SCIENCE, THE UNIVERSITY OF TEXAS AT ARLINGTON, ARLINGTON, TX · DEPARTMENT OF APPLIED MATHEMATICS, BROWN UNIVERSITY, PROVIDENCE, RI 02906.
Linearized Wasserstein dimensionality reduction with approximation guarantees
Feb 14, 2023Alexander Cloninger, Keaton Hamm, Varun Khurana +1Dimensionality Reduction Manifold Learning
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CALIFORNIA, SAN DIEGO, CA · HALICIOĦGLU DATA SCIENCE INSTITUTE, UNIVERSITY OF CALIFORNIA, SAN DIEGO, CA · DEPARTMENT OF MATHEMATICS, UNIVERSITY OF TEXAS AT ARLINGTON, ARLINGTON, TX +1

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Improved Finite-Particle Convergence Rates for Stein Variational Gradient Descent

Sayan Banerjee, Krishnakumar Balasubramanian, Promit Ghosal

Department of Statistics and Operations Research, University of North Carolina, Chapel Hill. · Department of Statistics, University of California, Davis. · Department of Statistics, University of Chicago.

Kernel Methods Variational Inference Stein Variational Gradient Descent Wasserstein Distance

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We provide finite-particle convergence rates for the Stein Variational Gradient Descent (SVGD) algorithm in the Kernelized Stein Discrepancy (

\mathsf{KSD}

) and Wasserstein-2 metrics. Our key insight is that the time derivative of the relative entropy between the joint density of

N

particle locations and the

N

-fold product target measure, starting from a regular initial distribution, splits into a dominant negative part' proportional to $N$ times the expected $\mathsf{KSD}^2$ and a smaller positive part'. This observation leads to

\mathsf{KSD}

rates of order

1/\sqrt{N}

, in both continuous and discrete time, providing a near optimal (in the sense of matching the corresponding i.i.d. rates) double exponential improvement over the recent result by Shi and Mackey (2024). Under mild assumptions on the kernel and potential, these bounds also grow polynomially in the dimension

d

. By adding a bilinear component to the kernel, the above approach is used to further obtain Wasserstein-2 convergence in continuous time. For the case of `bilinear + Matérn' kernels, we derive Wasserstein-2 rates that exhibit a curse-of-dimensionality similar to the i.i.d. setting. We also obtain marginal convergence and long-time propagation of chaos results for the time-averaged particle laws.

Latest papers

Improved Finite-Particle Convergence Rates for Stein Variational Gradient Descent

Structured Approximations of Measures

Linearized Wasserstein dimensionality reduction with approximation guarantees