On the Hopf-Cole Transform for Control-affine Schrödinger Bridge

• URL: http://arxiv.org/abs/2503.17640v1
• Date: Sat, 22 Mar 2025 04:08:10 GMT
• Title: On the Hopf-Cole Transform for Control-affine Schrödinger Bridge
• Authors: Alexis Teter, Abhishek Halder,
• Abstract summary: We show that the Hopf-Cole transform applied to the conditions of optimality for generic control-affine Schr"odinger bridge problems.<n>We explain how the resulting PDEs can be interpreted as nonlinear forward-backward advection-diffusion-reaction equations.
• Score: 0.4972323953932129
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The purpose of this note is to clarify the importance of the relation $\boldsymbol{gg}^{\top}\propto \boldsymbol{\sigma\sigma}^{\top}$ in solving control-affine Schr\"{o}dinger bridge problems via the Hopf-Cole transform, where $\boldsymbol{g},\boldsymbol{\sigma}$ are the control and noise coefficients, respectively. We show that the Hopf-Cole transform applied to the conditions of optimality for generic control-affine Schr\"{o}dinger bridge problems, i.e., without the assumption $\boldsymbol{gg}^{\top}\propto\boldsymbol{\sigma\sigma}^{\top}$, gives a pair of forward-backward PDEs that are neither linear nor equation-level decoupled. We explain how the resulting PDEs can be interpreted as nonlinear forward-backward advection-diffusion-reaction equations, where the nonlinearity stem from additional drift and reaction terms involving the gradient of the log-likelihood a.k.a. the score. These additional drift and reaction vanish when $\boldsymbol{gg}^{\top}\propto\boldsymbol{\sigma\sigma}^{\top}$, and the resulting boundary-coupled system of linear PDEs can then be solved by dynamic Sinkhorn recursions. A key takeaway of our work is that the numerical solution of the generic control-affine Schr\"{o}dinger bridge requires further algorithmic development, possibly generalizing the dynamic Sinkhorn recursion or otherwise.

Related papers

• Markov Kernels, Distances and Optimal Control: A Parable of Linear Quadratic Non-Gaussian Distribution Steering [3.497013356387396]
  kernel for the Ito Markov diffusion $mathrmdboldsymbolx_t=boldsymbolA_tboldsymbolx_t=boldsymbolA_tboldsymbolx_t=boldsymbolA_tboldsymbolx_t=boldsymbolA_tboldsymbolx_t=boldsymbolA_tboldsymbolx_t=boldsymbolA
  arXiv  Detail & Related papers  (2025-04-22T10:07:43Z)
• Conditional Mutual Information Based Diffusion Posterior Sampling for Solving Inverse Problems [3.866047645663101]
  In computer vision, tasks such as inpainting, deblurring, and super-resolution are commonly formulated as inverse problems.<n>Recently, diffusion models (DMs) have emerged as a promising approach for addressing noisy linear inverse problems.<n>We propose an information-theoretic approach to improve the effectiveness of DMs in solving inverse problems.
  arXiv  Detail & Related papers  (2025-01-06T09:45:26Z)
• Learning with Norm Constrained, Over-parameterized, Two-layer Neural Networks [54.177130905659155]
  Recent studies show that a reproducing kernel Hilbert space (RKHS) is not a suitable space to model functions by neural networks. In this paper, we study a suitable function space for over- parameterized two-layer neural networks with bounded norms.
  arXiv  Detail & Related papers  (2024-04-29T15:04:07Z)
• Resilience of Rademacher chaos of low degree [5.3390449198644445]
  resilience of Rademacher chaos is the maximum number of adversarial sign-flips that the chaos can sustain.<n>We provide probabilistic lower-bound guarantees for the resilience of Rademacher chaos of arbitrary yet sufficiently low degree.<n>Our results for decoupled Rademacher chaos of order two and that of higher order whilst are established through the same conceptual framework, differ substantially.
  arXiv  Detail & Related papers  (2024-02-16T08:27:55Z)
• On the Contraction Coefficient of the Schr\"odinger Bridge for Stochastic Linear Systems [15.022863946000495]
  A popular method to numerically solve the Schr"odinger bridge problems is via contractive fixed point recursions. We study a priori estimates for the contraction coefficients associated with the convergence of respective Schr"odinger systems.
  arXiv  Detail & Related papers  (2023-09-12T22:24:05Z)
• Transformers as Support Vector Machines [54.642793677472724]
  We establish a formal equivalence between the optimization geometry of self-attention and a hard-margin SVM problem. We characterize the implicit bias of 1-layer transformers optimized with gradient descent. We believe these findings inspire the interpretation of transformers as a hierarchy of SVMs that separates and selects optimal tokens.
  arXiv  Detail & Related papers  (2023-08-31T17:57:50Z)
• Planar Curve Registration using Bayesian Inversion [0.0]
  We study parameterisation-independent closed curve matching as a Bayesian inverse problem. The motion of the curve is modelled via a curve on the diffeomorphism group acting on the ambient space. We adopt ensemble Kalman inversion using a negative Sobolev mismatch penalty to measure the discrepancy between the target and the ensemble mean shape.
  arXiv  Detail & Related papers  (2023-07-10T21:26:43Z)
• Near-optimal fitting of ellipsoids to random points [68.12685213894112]
  A basic problem of fitting an ellipsoid to random points has connections to low-rank matrix decompositions, independent component analysis, and principal component analysis. We resolve this conjecture up to logarithmic factors by constructing a fitting ellipsoid for some $n = Omega(, d2/mathrmpolylog(d),)$. Our proof demonstrates feasibility of the least squares construction of Saunderson et al. using a convenient decomposition of a certain non-standard random matrix.
  arXiv  Detail & Related papers  (2022-08-19T18:00:34Z)
• The Franke-Gorini-Kossakowski-Lindblad-Sudarshan (FGKLS) Equation for Two-Dimensional Systems [62.997667081978825]
  Open quantum systems can obey the Franke-Gorini-Kossakowski-Lindblad-Sudarshan (FGKLS) equation. We exhaustively study the case of a Hilbert space dimension of $2$.
  arXiv  Detail & Related papers  (2022-04-16T07:03:54Z)
• The Schr\"odinger Bridge between Gaussian Measures has a Closed Form [101.79851806388699]
  We focus on the dynamic formulation of OT, also known as the Schr"odinger bridge (SB) problem. In this paper, we provide closed-form expressions for SBs between Gaussian measures.
  arXiv  Detail & Related papers  (2022-02-11T15:59:01Z)
• Projection-free Graph-based Classifier Learning using Gershgorin Disc Perfect Alignment [59.87663954467815]
  In graph-based binary learning, a subset of known labels $hatx_i$ are used to infer unknown labels. When restricting labels $x_i$ to binary values, the problem is NP-hard. We propose a fast projection-free method by solving a sequence of linear programs (LP) instead.
  arXiv  Detail & Related papers  (2021-06-03T07:22:48Z)
• GO Hessian for Expectation-Based Objectives [73.06986780804269]
  GO gradient was proposed recently for expectation-based objectives $mathbbE_q_boldsymbolboldsymbolgamma(boldsymboly) [f(boldsymboly)]$. Based on the GO gradient, we present for $mathbbE_q_boldsymbolboldsymbolgamma(boldsymboly) [f(boldsymboly)]$ an unbiased low-variance Hessian estimator, named GO Hessian.
  arXiv  Detail & Related papers  (2020-06-16T02:20:41Z)
• The generalization error of max-margin linear classifiers: Benign overfitting and high dimensional asymptotics in the overparametrized regime [11.252856459394854]
  Modern machine learning classifiers often exhibit vanishing classification error on the training set. Motivated by these phenomena, we revisit high-dimensional maximum margin classification for linearly separable data.
  arXiv  Detail & Related papers  (2019-11-05T00:15:27Z)

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.