Riesz-Kernel Stein Variational Gradient Descent: Renormalized Entropy and Long-Time Particle Limits

• URL: http://arxiv.org/abs/2607.14527v1
• Date: Thu, 16 Jul 2026 03:30:20 GMT
• Title: Riesz-Kernel Stein Variational Gradient Descent: Renormalized Entropy and Long-Time Particle Limits
• Abstract summary: Stein variational gradient descent (SVGD) transports interacting particles toward a target distribution through deterministic kernelized dynamics.<n>We study periodic Riesz SVGD with self-interaction removed and prove a many-particle, long-time sampling singularity theorem.
• Score: 9.465013037575494
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Stein variational gradient descent (SVGD) transports interacting particles toward a target distribution through deterministic kernelized dynamics. Singular Riesz kernels are attractive because they can provide quantitative population-level convergence, but at the finite-particle level the corresponding Stein energy has infinite self-interaction. We study periodic Riesz SVGD with self-interaction removed and prove a many-particle, long-time sampling theorem. Throughout the range in which the singular Stein energy is locally integrable, under a uniform bound on the initial relative entropy per particle, the time-averaged empirical-measure law converges weakly to the point mass \(δ_π\) at the target as the particle number and any diverging averaging horizon tend to infinity. We also show that the empirical-measure laws induced by invariant particle laws of finite relative entropy converge weakly to \(δ_π\), without a uniform entropy bound. Below the logarithmic singularity threshold, we obtain an explicit algebraic finite-particle error bound. These results extend the joint-entropy approach for smooth-kernel SVGD to singular interactions.

Related papers

• Quantitative Target Convergence and Uniform-in-Time Propagation of Chaos for Langevin-Regularized SVGD [2.457924087844968]
  We show convergence to the target and uniform-in-time propagation of chaos for Langevin-regularized Stein descent.<n>For propagation of chaos, we develop two complementary finite-time approaches.
  arXiv  Detail & Related papers  (2026-08-28T19:56:49Z)
• Uniform-in-time Propagation-of-Chaos for Stein Variational Gradient Descent [16.01291417072387]
  We study uniform-in-time propagation-of-chaos for continuous-time Stein Variational Gradient Descent flows.<n>For broad distributional metrics, we introduce a cutoff strategy which combines finite-time propagation-of-chaos estimates up to an $N$-dependent horizon.<n>We also develop a finite-dimensional theory for matrix-valued finite-rank kernels.
  arXiv  Detail & Related papers  (2026-06-30T20:27:39Z)
• Magnetic long-range order at finite temperature in two-dimensional hyperbolic lattices [45.88028371034407]
  Infrared singularities of gapless Goldstone modes preclude magnetic long-range order at finite temperature in conventional two-dimensional systems.<n>We find that the zero-energy collective modes required by symmetry carry vanishing local spectral weight.<n>We estimate the ordering temperature from the thermal spin-wave correction to the ordered moment.
  arXiv  Detail & Related papers  (2026-06-25T18:00:01Z)
• Quantitative Local Convergence of Mean-Field Stein Variational Gradient Flow [10.511277414974613]
  In the mean-field and continuous-time limit, it is known that the flow converges weakly toward the target, but no quantitative rate is known for the last singularity.<n>We establish quantitative local convergence in strong norms for this dynamics, when the interaction kernel is of Riesz type on the $d$-dimensional torus.<n>We show that these rates are sharp in certain regimes, and support the theory with numerical experiments.
  arXiv  Detail & Related papers  (2026-05-10T10:15:25Z)
• Effective Dynamics for Weakly Interacting Bosons in an Iterated High-Density Thermodynamic Limit [0.0]
  We study the time evolution of weakly interacting Bose gases on a three-dimensional torus of arbitrary volume.<n>We take into account a class of initial states exhibiting quasi-complete Bose-Einstein condensation.
  arXiv  Detail & Related papers  (2025-10-01T12:48:53Z)
• Symmetries, Conservation Laws and Entanglement in Non-Hermitian Fermionic Lattices [37.69303106863453]
  Non-Hermitian quantum many-body systems feature steady-state entanglement transitions driven by unitary dynamics and dissipation.<n>We show that the steady state is obtained by filling single-particle right eigenstates with the largest imaginary part of the eigenvalue.<n>We illustrate these principles in the Hatano-Nelson model with periodic boundary conditions and the non-Hermitian Su-Schrieffer-Heeger model.
  arXiv  Detail & Related papers  (2025-04-11T14:06:05Z)
• Closed-form solutions for the Salpeter equation [41.94295877935867]
  We study the propagator of the $1+1$ dimensional Salpeter Hamiltonian, describing a relativistic quantum particle with no spin. The analytical extension of the Hamiltonian in the complex plane allows us to formulate the equivalent problem, namely the B"aumer equation. This B"aumera corresponds to the Green function of a relativistic diffusion process that interpolates between Cauchy for small times and Gaussian diffusion for large times.
  arXiv  Detail & Related papers  (2024-06-26T15:52:39Z)
• Measurement phase transitions in the no-click limit as quantum phase transitions of a non-hermitean vacuum [77.34726150561087]
  We study phase transitions occurring in the stationary state of the dynamics of integrable many-body non-Hermitian Hamiltonians. We observe that the entanglement phase transitions occurring in the stationary state have the same nature as that occurring in the vacuum of the non-hermitian Hamiltonian.
  arXiv  Detail & Related papers  (2023-01-18T09:26:02Z)
• A Finite-Particle Convergence Rate for Stein Variational Gradient Descent [47.6818454221125]
  We provide the first finite-particle convergence rate for Stein variational descent gradient (SVGD) Our explicit, non-asymptotic proof strategy will serve as a template for future refinements.
  arXiv  Detail & Related papers  (2022-11-17T17:50:39Z)
• Utilising the CLT Structure in Stochastic Gradient based Sampling : Improved Analysis and Faster Algorithms [14.174806471635403]
  We consider approximations of sampling algorithms, such as Gradient Langevin Dynamics (SGLD) and the Random Batch Method (RBM) for Interacting Particle Dynamcs (IPD) We observe that the noise introduced by the approximation is nearly Gaussian due to the Central Limit Theorem (CLT) while the driving Brownian motion is exactly Gaussian. We harness this structure to absorb the approximation error inside the diffusion process, and obtain improved convergence guarantees for these algorithms.
  arXiv  Detail & Related papers  (2022-06-08T10:17:40Z)
• A Non-Asymptotic Analysis for Stein Variational Gradient Descent [44.30569261307296]
  We provide a novel finite time analysis for the Stein Variational Gradient Descent algorithm. We provide a descent lemma establishing that the algorithm decreases the objective at each iteration. We also provide a convergence result of the finite particle system corresponding to the practical implementation of SVGD to its population version.
  arXiv  Detail & Related papers  (2020-06-17T12:01:33Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.