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Semi-discrete quadratic Wasserstein energy and state-dependent Langevin exploration — arXiv:2609.03405

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.03405
License (as archived): CC0-1.0
Archived: 2026-10-17T02:26:54.708Z
SHA-256 of the archived text: 06ee795804a1beaf6b6965f5042530ddecc2438e5be7c9d1b6c1dfd67acc1c4e
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How to cite

Semi-discrete quadratic Wasserstein energy and state-dependent Langevin exploration — arXiv:2609.03405 — arXiv (en), https://arxiv.org/abs/2609.03405 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_f59ca3ce99ba (Metraipolis archive dc_f59ca3ce99ba, sha256 06ee795804a1beaf6b6965f5042530ddecc2438e5be7c9d1b6c1dfd67acc1c4e)

BibTeX

@misc{dcf59ca3ce99ba,
  title = {Semi-discrete quadratic Wasserstein energy and state-dependent Langevin exploration — arXiv:2609.03405},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.03405},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_f59ca3ce99ba; sha256 06ee795804a1beaf6b6965f5042530ddecc2438e5be7c9d1b6c1dfd67acc1c4e},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.03405. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We study the semi-discrete quadratic Wasserstein energy. The energy is nonsmooth at collisions of sites. We prove local Lipschitz continuity on the full configuration space, together with global semiconcavity, coercivity, and dissipativity; show that every global minimizer is interior and collision free; and establish $C^2$ regularity on the collision-free configuration space. The gradient is expressed through the barycenters of the balanced Laguerre cells, while the Hessian is given by an explicit facet formula and satisfies a global one-sided bound. We also solve the one-dimensional problem explicitly in each ordering chamber and give a two-site example on the unit square with non-minimizing Lloyd fixed points. For $d\ge 2$, we then formulate an entropy-regularized relaxed control of the Langevin temperature. The controlled dynamics is strongly well posed, nonexplosive, and collision free. Its value function is a classical interior solution of the exploratory Hamilton-Jacobi-Bellman equation; the Laplacian of the value function is locally $C^1$, which yields a locally Lipschitz optimal temperature feedback. Independently of this optimal-control result, for every fixed Borel temperature rule bounded away from zero, and every sufficiently small step size, the associated Gaussian Euler chain is geometrically ergodic with a full-support invariant law. The raw iterates do not converge, whereas the best-so-far energy converges almost surely to the global minimum and the running record approaches the set of global minimizers.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-04 in math.OC).
Categorie arXiv: Numerical Analysis (math.NA) ; Optimization and Control (math.OC); Probability (math.PR).
Pagina del lavoro: https://arxiv.org/abs/2609.03405 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-17T02:07:43.357Z dalla pagina https://arxiv.org/catchup/math.OC/2026-09-04?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).