# Accelerating optimization over the space of probability measures

Source: [Accelerating optimization over the space of probability measures](<https://www.sota2.com/research/paper/68fa5c881e209131ce9d1e16>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T17:42:27.264Z

Authors: Shi Chen, Qin Li, Oliver Tse, Stephen J. Wright

Published: 2023-10-06T04:32:15.000Z

## Abstract

The acceleration of gradient-based optimization methods is a subject of significant practical and theoretical importance, particularly within machine learning applications. While much attention has been directed towards optimizing within Euclidean space, the need to optimize over spaces of probability measures in machine learning motivates exploration of accelerated gradient methods in this context too. To this end, we introduce a Hamiltonian-flow approach analogous to momentum-based approaches in Euclidean space. We demonstrate that, in the continuous-time setting, algorithms based on this approach can achieve convergence rates of arbitrarily high order. We complement our findings with numerical examples.

## Recorded benchmark results

A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage.

- [Non-rigid Surface Registration on Liver (ASSD, HD90)](<https://www.sota2.com/research/sota/non-rigid-surface-registration-on-liver-assd-hd90.md>): Non-rigid Surface Registration; Liver; ASSD (mm): 0.749.

- [Non-rigid Surface Registration on Pancreas (ASSD, HD90)](<https://www.sota2.com/research/sota/non-rigid-surface-registration-on-pancreas-assd-hd90.md>): Non-rigid Surface Registration; Pancreas; ASSD (mm): 0.416.

- [Non-rigid Surface Registration on Left ventricle (ASSD, HD90)](<https://www.sota2.com/research/sota/non-rigid-surface-registration-on-left-ventricle-assd-hd90.md>): Non-rigid Surface Registration; Left ventricle; ASSD (mm): 0.407.

- [Affine Surface Registration on Liver (ASSD, HD90)](<https://www.sota2.com/research/sota/affine-surface-registration-on-liver-assd-hd90.md>): Affine Surface Registration; Liver; ASSD (mm): 7.555.

- [Affine Surface Registration on Pancreas (ASSD, HD90)](<https://www.sota2.com/research/sota/affine-surface-registration-on-pancreas-assd-hd90.md>): Affine Surface Registration; Pancreas; ASSD (mm): 5.511.

- [Affine Surface Registration on Left ventricle (ASSD, HD90)](<https://www.sota2.com/research/sota/affine-surface-registration-on-left-ventricle-assd-hd90.md>): Affine Surface Registration; Left ventricle; ASSD (mm): 2.172.

- [Affine Surface Registration on Liver](<https://www.sota2.com/research/sota/affine-surface-registration-on-liver.md>): Affine Surface Registration; Liver; Runtime (s): 2.53.

- [Affine Surface Registration on Pancreas](<https://www.sota2.com/research/sota/affine-surface-registration-on-pancreas.md>): Affine Surface Registration; Pancreas; Runtime (s): 2.82.

- [Affine Surface Registration on Left ventricle](<https://www.sota2.com/research/sota/affine-surface-registration-on-left-ventricle.md>): Affine Surface Registration; Left ventricle; Runtime (second): 1.6.

- [Non-rigid Surface Registration on Liver](<https://www.sota2.com/research/sota/non-rigid-surface-registration-on-liver.md>): Non-rigid Surface Registration; Liver; Runtime (s): 2.53.

- [Non-rigid Surface Registration on Pancreas](<https://www.sota2.com/research/sota/non-rigid-surface-registration-on-pancreas.md>): Non-rigid Surface Registration; Pancreas; Runtime (s): 1.43.

- [Non-rigid Surface Registration on Left ventricle](<https://www.sota2.com/research/sota/non-rigid-surface-registration-on-left-ventricle.md>): Non-rigid Surface Registration; Left ventricle; Runtime (s): 0.41.

## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2310.04006>)

[PDF](<https://arxiv.org/pdf/2310.04006.pdf>)