aiqbench analysis
This paper presents a novel trust-region optimization framework for minimizing potential-interaction energies within the Wasserstein space of probability measures. By leveraging second-order information and explicit self-adjoint operators, the method ensures convergence to stationary points even in nonconvex landscapes defined by smooth particle interactions. The approach offers rigorous complexity guarantees independent of particle number and dimension, making it a significant advancement for high-dimensional distribution approximation and optimal transport problems.
Trust-Region Optimization for Smooth Potential-Interaction Energies in Wasserstein Space
arXiv:2610.08883v1 Announce Type: cross Abstract: Finding low-energy configurations of interacting particles and approximating probability distributions lead to the minimization of potential-interaction energies in Wasserstein space. These energies can be nonconvex, making it important to exploit second-order information while controlling the reliability of local approximations. We study trust-region optimization of smooth potential-interaction energies on the Wasserstein space of probability measures with finite second moment. The method uses a quadratic model along pushforward curves, an $L^2(\rho)$ step radius, and a Steihaug-Toint subsolver with an explicit self-adjoint second-variation operator. A ratio test determines acceptance and guides the radius update. Under a lower energy bound and globally bounded Hessians of the potential and interaction kernel, we prove that the objective is nonincreasing, the Wasserstein-gradient norms converge to zero, and an $\varepsilon$-stationary iterate is reached within $O(\varepsilon^{-2})$ total outer trials, including rejected trials. If the potential is quadratically coercive, every weak accumulation point is stationary. The analysis applies to arbitrary initial measures with finite second moment. For empirical measures, the iteration is a finite-dimensional trust-region method in the $L^2(\rho_N)$ inner product, with complexity constants independent of particle number and dimension when the initial objective gaps are uniformly bounded. Numerical experiments include a smooth soft-particle energy, maximum-mean-discrepancy minimization for non-Gaussian targets, component ablations, and scaling studies in particle number and dimension.
Abstract:Finding low-energy configurations of interacting particles and approximating probability distributions lead to the minimization of potential-interaction energies in Wasserstein space. These energies can be nonconvex, making it important to exploit second-order information while controlling the reliability of local approximations. We study trust-region optimization of smooth potential-interaction energies on the Wasserstein space of probability measures with finite second moment. The method uses a quadratic model along pushforward curves, an $L^2(\rho)$ step radius, and a Steihaug-Toint subsolver with an explicit self-adjoint second-variation operator. A ratio test determines acceptance and guides the radius update. Under a lower energy bound and globally bounded Hessians of the potential and interaction kernel, we prove that the objective is nonincreasing, the Wasserstein-gradient norms converge to zero, and an $\varepsilon$-stationary iterate is reached within $O(\varepsilon^{-2})$ total outer trials, including rejected trials. If the potential is quadratically coercive, every weak accumulation point is stationary. The analysis applies to arbitrary initial measures with finite second moment. For empirical measures, the iteration is a finite-dimensional trust-region method in the $L^2(\rho_N)$ inner product, with complexity constants independent of particle number and dimension when the initial objective gaps are uniformly bounded. Numerical experiments include a smooth soft-particle energy, maximum-mean-discrepancy minimization for non-Gaussian targets, component ablations, and scaling studies in particle number and dimension.
Machine Learning (stat.ML); Machine Learning (cs.LG); Optimization and Control (math.OC); Probability (math.PR)
arXiv:2610.08883 [stat.ML]
(or arXiv:2610.08883v1 [stat.ML] for this version)
https://doi.org/10.48550/arXiv.2610.08883 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
gpu-package/README.md
gpu-package/payload/README-inputs.md
gpu-package/payload/cluster_run_gpu.sh
gpu-package/payload/numerics/figures/benchmark-results.json
gpu-package/payload/numerics/numerics_baselines.py
gpu-package/payload/requirements-gpu-cu126.txt
gpu-package/payload/requirements.txt
gpu-package/payload/runs/gaussian-interaction-runtime-2026-09-23/0.npy
gpu-package/payload/runs/gaussian-interaction-runtime-2026-09-23/1.npy
gpu-package/payload/runs/gaussian-interaction-runtime-2026-09-23/10.npy
gpu-package/payload/runs/gaussian-interaction-runtime-2026-09-23/11.npy
gpu-package/payload/runs/gaussian-interaction-runtime-2026-09-23/2.npy
gpu-package/payload/runs/gaussian-interaction-runtime-2026-09-23/3.npy
gpu-package/payload/runs/gaussian-interaction-runtime-2026-09-23/4.npy
gpu-package/payload/runs/gaussian-interaction-runtime-2026-09-23/5.npy
gpu-package/payload/runs/gaussian-interaction-runtime-2026-09-23/6.npy
gpu-package/payload/runs/gaussian-interaction-runtime-2026-09-23/7.npy
gpu-package/payload/runs/gaussian-interaction-runtime-2026-09-23/8.npy
gpu-package/payload/runs/gaussian-interaction-runtime-2026-09-23/9.npy
gpu-package/payload/runs/gaussian-interaction-runtime-2026-09-23/protocol.md
gpu-package/payload/runs/gaussian-interaction-runtime-2026-09-23/run_experiment_gpu.py
gpu-package/slurm/run_gaussian_gpu.sh
numerics/figure-inputs/fig-mmd-targets-before-column-removal.pdf
numerics/figures/application-results.json
numerics/figures/dendrimer-g4-ppi-phenylurea.xyz
numerics/figures/make_dendrimer_structure.py
numerics/figures/make_mmd_targets_proportioned.py
numerics/figures/scaling-part-00-of-04.json
numerics/figures/scaling-part-01-of-04.json
numerics/figures/scaling-part-02-of-04.json
numerics/figures/scaling-part-03-of-04.json
numerics/figures/scaling-results.json
numerics/figures/table-gaussian-runtime.tex
numerics/numerics_applications.py
numerics/numerics_scaling.py
numerics/requirements.txt
runs/dendrimer-gaussian-2026-10-04/README.md
runs/dendrimer-gaussian-2026-10-04/config.json
runs/dendrimer-gaussian-2026-10-04/dendrimer_gaussian_experiment.py
runs/dendrimer-gaussian-2026-10-04/results/initial_configuration.png
runs/dendrimer-gaussian-2026-10-04/results/objective_traces.png
runs/dendrimer-gaussian-2026-10-04/results/raw_trials.csv
runs/dendrimer-gaussian-2026-10-04/results/summary.json
runs/gaussian-interaction-runtime-2026-09-24-gpu-cluster/0.npy
runs/gaussian-interaction-runtime-2026-09-24-gpu-cluster/1.npy
runs/gaussian-interaction-runtime-2026-09-24-gpu-cluster/10.npy
runs/gaussian-interaction-runtime-2026-09-24-gpu-cluster/11.npy
runs/gaussian-interaction-runtime-2026-09-24-gpu-cluster/2.npy
runs/gaussian-interaction-runtime-2026-09-24-gpu-cluster/3.npy
runs/gaussian-interaction-runtime-2026-09-24-gpu-cluster/4.npy
runs/gaussian-interaction-runtime-2026-09-24-gpu-cluster/5.npy
runs/gaussian-interaction-runtime-2026-09-24-gpu-cluster/6.npy
runs/gaussian-interaction-runtime-2026-09-24-gpu-cluster/7.npy
runs/gaussian-interaction-runtime-2026-09-24-gpu-cluster/8.npy
runs/gaussian-interaction-runtime-2026-09-24-gpu-cluster/9.npy
runs/gaussian-interaction-runtime-2026-09-24-gpu-cluster/README.md
runs/gaussian-interaction-runtime-2026-09-24-gpu-cluster/checks.json
runs/gaussian-interaction-runtime-2026-09-24-gpu-cluster/configuration.json
runs/gaussian-interaction-runtime-2026-09-24-gpu-cluster/display-grid.json
runs/gaussian-interaction-runtime-2026-09-24-gpu-cluster/environment-prepare.json
runs/gaussian-interaction-runtime-2026-09-24-gpu-cluster/extract_display_curves.py
runs/gaussian-interaction-runtime-2026-09-24-gpu-cluster/extract_sampled_histories.sh
runs/gaussian-interaction-runtime-2026-09-24-gpu-cluster/fig-gaussian-runtime-display.pdf
runs/gaussian-interaction-runtime-2026-09-24-gpu-cluster/fig-gaussian-runtime-display.png
runs/gaussian-interaction-runtime-2026-09-24-gpu-cluster/fig-gaussian-runtime.pdf
runs/gaussian-interaction-runtime-2026-09-24-gpu-cluster/fig-gaussian-runtime.png
runs/gaussian-interaction-runtime-2026-09-24-gpu-cluster/make_display_figure.py
runs/gaussian-interaction-runtime-2026-09-24-gpu-cluster/per-initialization.csv
runs/gaussian-interaction-runtime-2026-09-24-gpu-cluster/raw-result-manifest.txt
runs/gaussian-interaction-runtime-2026-09-24-gpu-cluster/sampled-histories.csv
runs/gaussian-interaction-runtime-2026-09-24-gpu-cluster/summary.json
runs/gaussian-interaction-runtime-2026-09-24-gpu-cluster/table-gaussian-runtime.tex
(69 additional files not shown) You must enabled JavaScript to view entire file list.
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.
Full text under the publisher's open licence · 399 words
