Neural Sampling from Boltzmann Densities: Fisher-Rao Curves in the Wasserstein Geometry

• URL: http://arxiv.org/abs/2410.03282v1
• Date: Fri, 4 Oct 2024 09:54:11 GMT
• Title: Neural Sampling from Boltzmann Densities: Fisher-Rao Curves in the Wasserstein Geometry
• Authors: Jannis Chemseddine, Christian Wald, Richard Duong, Gabriele Steidl,
• Abstract summary: We deal with the task of sampling from an unnormalized Boltzmann density $rho_D$ by learning a Boltzmann curve given by $f_t$. Inspired by M'at'e and Fleuret, we propose an which parametrizes only $f_t$ and fixes an appropriate $v_t$. This corresponds to the Wasserstein flow of the Kullback-Leibler divergence related to Langevin dynamics.
• Score: 1.609940380983903
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We deal with the task of sampling from an unnormalized Boltzmann density $\rho_D$ by learning a Boltzmann curve given by energies $f_t$ starting in a simple density $\rho_Z$. First, we examine conditions under which Fisher-Rao flows are absolutely continuous in the Wasserstein geometry. Second, we address specific interpolations $f_t$ and the learning of the related density/velocity pairs $(\rho_t,v_t)$. It was numerically observed that the linear interpolation, which requires only a parametrization of the velocity field $v_t$, suffers from a "teleportation-of-mass" issue. Using tools from the Wasserstein geometry, we give an analytical example, where we can precisely measure the explosion of the velocity field. Inspired by M\'at\'e and Fleuret, who parametrize both $f_t$ and $v_t$, we propose an interpolation which parametrizes only $f_t$ and fixes an appropriate $v_t$. This corresponds to the Wasserstein gradient flow of the Kullback-Leibler divergence related to Langevin dynamics. We demonstrate by numerical examples that our model provides a well-behaved flow field which successfully solves the above sampling task.

Related papers

• Relative-Translation Invariant Wasserstein Distance [82.6068808353647]
  We introduce a new family of distances, relative-translation invariant Wasserstein distances ($RW_p$) We show that $RW_p distances are also real distance metrics defined on the quotient set $mathcalP_p(mathbbRn)/sim$ invariant to distribution translations.
  arXiv  Detail & Related papers  (2024-09-04T03:41:44Z)
• Gradient Flows and Riemannian Structure in the Gromov-Wasserstein Geometry [29.650065650233223]
  We study gradient flows in the Gromov-Wasserstein (GW) geometry. We focus on the inner product GW (IGW) distance between distributions on $mathbbRd. We identify the intrinsic IGW geometry that gives rise to the intrinsic IGW geometry, using which we establish a Benamou-Brenier-like formula for IGW.
  arXiv  Detail & Related papers  (2024-07-16T14:53:23Z)
• Efficient Sampling on Riemannian Manifolds via Langevin MCMC [51.825900634131486]
  We study the task efficiently sampling from a Gibbs distribution $d pi* = eh d vol_g$ over aian manifold $M$ via (geometric) Langevin MCMC. Our results apply to general settings where $pi*$ can be non exponential and $Mh$ can have negative Ricci curvature.
  arXiv  Detail & Related papers  (2024-02-15T22:59:14Z)
• Projected Langevin dynamics and a gradient flow for entropic optimal transport [0.8057006406834466]
  We introduce analogous diffusion dynamics that sample from an entropy-regularized optimal transport. By studying the induced Wasserstein geometry of the submanifold $Pi(mu,nu)$, we argue that the SDE can be viewed as a Wasserstein gradient flow on this space of couplings.
  arXiv  Detail & Related papers  (2023-09-15T17:55:56Z)
• An Explicit Expansion of the Kullback-Leibler Divergence along its Fisher-Rao Gradient Flow [8.052709336750823]
  We show that when $pirhollback$ exhibits multiple modes, $pirhollback$ is that textitindependent of the potential function. We provide an explicit expansion of $textKL. KL. KL. KL. KL. KL. KL. KL. KL. KL. KL. KL. KL.
  arXiv  Detail & Related papers  (2023-02-23T18:47:54Z)
• Learning Interpolations between Boltzmann Densities [0.0]
  We introduce a training objective for continuous normalizing flows in the absence of samples. We find a time-dependent vector field $V_t$ that transports samples along the family $p_t$ of densities.
  arXiv  Detail & Related papers  (2023-01-18T09:32:33Z)
• A Law of Robustness beyond Isoperimetry [84.33752026418045]
  We prove a Lipschitzness lower bound $Omega(sqrtn/p)$ of robustness of interpolating neural network parameters on arbitrary distributions. We then show the potential benefit of overparametrization for smooth data when $n=mathrmpoly(d)$. We disprove the potential existence of an $O(1)$-Lipschitz robust interpolating function when $n=exp(omega(d))$.
  arXiv  Detail & Related papers  (2022-02-23T16:10:23Z)
• Quantum information measures of the Dirichlet and Neumann hyperspherical dots [0.0]
  $mathttd$-dimensional hyperspherical quantum dot with either Dirichlet or Neumann boundary conditions (BCs) This paves the way to an efficient computation in either space of Shannon, R'enyi and Tsallis entropies, Onicescu energies and Fisher informations.
  arXiv  Detail & Related papers  (2021-03-24T11:08:31Z)
• Optimal Robust Linear Regression in Nearly Linear Time [97.11565882347772]
  We study the problem of high-dimensional robust linear regression where a learner is given access to $n$ samples from the generative model $Y = langle X,w* rangle + epsilon$ We propose estimators for this problem under two settings: (i) $X$ is L4-L2 hypercontractive, $mathbbE [XXtop]$ has bounded condition number and $epsilon$ has bounded variance and (ii) $X$ is sub-Gaussian with identity second moment and $epsilon$ is
  arXiv  Detail & Related papers  (2020-07-16T06:44:44Z)
• A diffusion approach to Stein's method on Riemannian manifolds [65.36007959755302]
  We exploit the relationship between the generator of a diffusion on $mathbf M$ with target invariant measure and its characterising Stein operator. We derive Stein factors, which bound the solution to the Stein equation and its derivatives. We imply that the bounds for $mathbb Rm$ remain valid when $mathbf M$ is a flat manifold.
  arXiv  Detail & Related papers  (2020-03-25T17:03:58Z)
• Quantum Algorithms for Simulating the Lattice Schwinger Model [63.18141027763459]
  We give scalable, explicit digital quantum algorithms to simulate the lattice Schwinger model in both NISQ and fault-tolerant settings. In lattice units, we find a Schwinger model on $N/2$ physical sites with coupling constant $x-1/2$ and electric field cutoff $x-1/2Lambda$. We estimate observables which we cost in both the NISQ and fault-tolerant settings by assuming a simple target observable---the mean pair density.
  arXiv  Detail & Related papers  (2020-02-25T19:18:36Z)

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.