A Computational Framework for Solving Wasserstein Lagrangian Flows

• URL: http://arxiv.org/abs/2310.10649v3
• Date: Wed, 3 Jul 2024 15:23:42 GMT
• Title: A Computational Framework for Solving Wasserstein Lagrangian Flows
• Authors: Kirill Neklyudov, Rob Brekelmans, Alexander Tong, Lazar Atanackovic, Qiang Liu, Alireza Makhzani,
• Abstract summary: In general, the optimal density path is unknown, and solving these variational problems can be computationally challenging. We propose a novel deep learning based framework approaching all of these problems from a unified perspective. We showcase the versatility of the proposed framework by outperforming previous approaches for the single-cell trajectory inference.
• Score: 48.87656245464521
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The dynamical formulation of the optimal transport can be extended through various choices of the underlying geometry (kinetic energy), and the regularization of density paths (potential energy). These combinations yield different variational problems (Lagrangians), encompassing many variations of the optimal transport problem such as the Schr\"odinger bridge, unbalanced optimal transport, and optimal transport with physical constraints, among others. In general, the optimal density path is unknown, and solving these variational problems can be computationally challenging. We propose a novel deep learning based framework approaching all of these problems from a unified perspective. Leveraging the dual formulation of the Lagrangians, our method does not require simulating or backpropagating through the trajectories of the learned dynamics, and does not need access to optimal couplings. We showcase the versatility of the proposed framework by outperforming previous approaches for the single-cell trajectory inference, where incorporating prior knowledge into the dynamics is crucial for correct predictions.

Related papers

• Optimal Transportation by Orthogonal Coupling Dynamics [0.0]
  We propose a novel framework to address the Monge-Kantorovich problem based on a projection type gradient descent scheme. The micro-dynamics is built on the notion of the conditional expectation, where the connection with the opinion dynamics is explored. We demonstrate that the devised dynamics recovers random maps with favourable computational performance.
  arXiv  Detail & Related papers  (2024-10-10T15:53:48Z)
• Dynamical Measure Transport and Neural PDE Solvers for Sampling [77.38204731939273]
  We tackle the task of sampling from a probability density as transporting a tractable density function to the target. We employ physics-informed neural networks (PINNs) to approximate the respective partial differential equations (PDEs) solutions. PINNs allow for simulation- and discretization-free optimization and can be trained very efficiently.
  arXiv  Detail & Related papers  (2024-07-10T17:39:50Z)
• Neural Optimal Transport with Lagrangian Costs [29.091068250865504]
  We investigate the optimal transport problem between probability measures when the underlying cost function is understood to satisfy a Lagrangian cost. Our contributions are of computational interest, where we demonstrate the ability to efficiently compute geodesics and amortize spline-based paths. Unlike prior work, we also output the resulting Lagrangian optimal transport map without requiring an ODE solver.
  arXiv  Detail & Related papers  (2024-06-01T03:34:00Z)
• Equivariant Deep Weight Space Alignment [54.65847470115314]
  We propose a novel framework aimed at learning to solve the weight alignment problem. We first prove that weight alignment adheres to two fundamental symmetries and then, propose a deep architecture that respects these symmetries.
  arXiv  Detail & Related papers  (2023-10-20T10:12:06Z)
• Backpropagation of Unrolled Solvers with Folded Optimization [55.04219793298687]
  The integration of constrained optimization models as components in deep networks has led to promising advances on many specialized learning tasks. One typical strategy is algorithm unrolling, which relies on automatic differentiation through the operations of an iterative solver. This paper provides theoretical insights into the backward pass of unrolled optimization, leading to a system for generating efficiently solvable analytical models of backpropagation.
  arXiv  Detail & Related papers  (2023-01-28T01:50:42Z)
• Constrained Mass Optimal Transport [0.0]
  This paper introduces the problem of constrained optimal transport. A family of algorithms is introduced to solve a class of constrained saddle point problems. Convergence proofs and numerical results are presented.
  arXiv  Detail & Related papers  (2022-06-05T06:47:25Z)
• Online Learning to Transport via the Minimal Selection Principle [2.3857747529378917]
  We study the Online Learning Transport (OLT) problem where the decision variable is a convex, an-dimensional object. We derive a novel method called the minimal selection or exploration (SoMLT) algorithm to solve OLT problems using mean-field and discretization techniques.
  arXiv  Detail & Related papers  (2022-02-09T21:25:58Z)
• Optimization on manifolds: A symplectic approach [127.54402681305629]
  We propose a dissipative extension of Dirac's theory of constrained Hamiltonian systems as a general framework for solving optimization problems. Our class of (accelerated) algorithms are not only simple and efficient but also applicable to a broad range of contexts.
  arXiv  Detail & Related papers  (2021-07-23T13:43:34Z)
• On the Connection between Dynamical Optimal Transport and Functional Lifting [0.0]
  In this work, we investigate a mathematically rigorous formulation based on embedding into the space over a fixed range $Gamma$ By modifying the continuity equation, the approach can be extended to models with higher-order regularization.
  arXiv  Detail & Related papers  (2020-07-06T08:53:35Z)
• Cross Entropy Hyperparameter Optimization for Constrained Problem Hamiltonians Applied to QAOA [68.11912614360878]
  Hybrid quantum-classical algorithms such as Quantum Approximate Optimization Algorithm (QAOA) are considered as one of the most encouraging approaches for taking advantage of near-term quantum computers in practical applications. Such algorithms are usually implemented in a variational form, combining a classical optimization method with a quantum machine to find good solutions to an optimization problem. In this study we apply a Cross-Entropy method to shape this landscape, which allows the classical parameter to find better parameters more easily and hence results in an improved performance.
  arXiv  Detail & Related papers  (2020-03-11T13:52:41Z)

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.