Go With the Flow: Fast Diffusion for Gaussian Mixture Models

• URL: http://arxiv.org/abs/2412.09059v5
• Date: Fri, 30 May 2025 19:11:37 GMT
• Title: Go With the Flow: Fast Diffusion for Gaussian Mixture Models
• Authors: George Rapakoulias, Ali Reza Pedram, Fengjiao Liu, Lingjiong Zhu, Panagiotis Tsiotras,
• Abstract summary: Schrodinger Bridges (SBs) are diffusion processes that steer in finite time, a given initial distribution to another final one while minimizing a suitable cost functional.<n>We propose an analytic parametrization of a set of feasible policies for solving low dimensional problems.<n>We showcase the potential of this approach in low-to-image problems such as image-to-image translation in the latent space of an autoencoder, learning of cellular dynamics using multi-marginal momentum SB problems and various other examples.
• Score: 16.07896640031724
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Schrodinger Bridges (SBs) are diffusion processes that steer, in finite time, a given initial distribution to another final one while minimizing a suitable cost functional. Although various methods for computing SBs have recently been proposed in the literature, most of these approaches require computationally expensive training schemes, even for solving low-dimensional problems. In this work, we propose an analytic parametrization of a set of feasible policies for steering the distribution of a dynamical system from one Gaussian Mixture Model (GMM) to another. Instead of relying on standard non-convex optimization techniques, the optimal policy within the set can be approximated as the solution of a low-dimensional linear program whose dimension scales linearly with the number of components in each mixture. The proposed method generalizes naturally to more general classes of dynamical systems, such as controllable linear time-varying systems, enabling efficient solutions to multi-marginal momentum SB between GMMs, a challenging distribution interpolation problem. We showcase the potential of this approach in low-to-moderate dimensional problems such as image-to-image translation in the latent space of an autoencoder, learning of cellular dynamics using multi-marginal momentum SB problems, and various other examples. We also test our approach on an Entropic Optimal Transport (EOT) benchmark problem and show that it outperforms state-of-the-art methods in cases where the boundary distributions are mixture models while requiring virtually no training.

Related papers

• Self-Supervised Coarsening of Unstructured Grid with Automatic Differentiation [55.88862563823878]
  In this work, we present an original algorithm to coarsen an unstructured grid based on the concepts of differentiable physics.<n>We demonstrate performance of the algorithm on two PDEs: a linear equation which governs slightly compressible fluid flow in porous media and the wave equation.<n>Our results show that in the considered scenarios, we reduced the number of grid points up to 10 times while preserving the modeled variable dynamics in the points of interest.
  arXiv  Detail & Related papers  (2025-07-24T11:02:13Z)
• Steering Large Agent Populations using Mean-Field Schrodinger Bridges with Gaussian Mixture Models [13.03355083378673]
  Mean-Field Schrodinger Bridge (MFSB) problem is an optimization problem aiming to find the minimum effort control policy. In the context of multiagent control, the objective is to control the configuration of a swarm of identical, interacting cooperative agents.
  arXiv  Detail & Related papers  (2025-03-31T04:01:04Z)
• Training Deep Learning Models with Norm-Constrained LMOs [56.00317694850397]
  We study optimization methods that leverage the linear minimization oracle (LMO) over a norm-ball. We propose a new family of algorithms that uses the LMO to adapt to the geometry of the problem and, perhaps surprisingly, show that they can be applied to unconstrained problems.
  arXiv  Detail & Related papers  (2025-02-11T13:10:34Z)
• Multi-Agent Path Finding in Continuous Spaces with Projected Diffusion Models [57.45019514036948]
  Multi-Agent Path Finding (MAPF) is a fundamental problem in robotics.<n>This work proposes a novel approach that integrates constrained optimization with diffusion models for MAPF in continuous spaces.
  arXiv  Detail & Related papers  (2024-12-23T21:27:19Z)
• A Stochastic Approach to Bi-Level Optimization for Hyperparameter Optimization and Meta Learning [74.80956524812714]
  We tackle the general differentiable meta learning problem that is ubiquitous in modern deep learning. These problems are often formalized as Bi-Level optimizations (BLO) We introduce a novel perspective by turning a given BLO problem into a ii optimization, where the inner loss function becomes a smooth distribution, and the outer loss becomes an expected loss over the inner distribution.
  arXiv  Detail & Related papers  (2024-10-14T12:10:06Z)
• A Simulation-Free Deep Learning Approach to Stochastic Optimal Control [12.699529713351287]
  We propose a simulation-free algorithm for the solution of generic problems in optimal control (SOC) Unlike existing methods, our approach does not require the solution of an adjoint problem.
  arXiv  Detail & Related papers  (2024-10-07T16:16:53Z)
• Momentum Particle Maximum Likelihood [2.4561590439700076]
  We propose an analogous dynamical-systems-inspired approach to minimizing the free energy functional. By discretizing the system, we obtain a practical algorithm for Maximum likelihood estimation in latent variable models. The algorithm outperforms existing particle methods in numerical experiments and compares favourably with other MLE algorithms.
  arXiv  Detail & Related papers  (2023-12-12T14:53:18Z)
• Ensemble Kalman Filtering Meets Gaussian Process SSM for Non-Mean-Field and Online Inference [47.460898983429374]
  We introduce an ensemble Kalman filter (EnKF) into the non-mean-field (NMF) variational inference framework to approximate the posterior distribution of the latent states. This novel marriage between EnKF and GPSSM not only eliminates the need for extensive parameterization in learning variational distributions, but also enables an interpretable, closed-form approximation of the evidence lower bound (ELBO) We demonstrate that the resulting EnKF-aided online algorithm embodies a principled objective function by ensuring data-fitting accuracy while incorporating model regularizations to mitigate overfitting.
  arXiv  Detail & Related papers  (2023-12-10T15:22:30Z)
• Gaussian Mixture Solvers for Diffusion Models [84.83349474361204]
  We introduce a novel class of SDE-based solvers called GMS for diffusion models. Our solver outperforms numerous SDE-based solvers in terms of sample quality in image generation and stroke-based synthesis.
  arXiv  Detail & Related papers  (2023-11-02T02:05:38Z)
• Energy-Guided Continuous Entropic Barycenter Estimation for General Costs [95.33926437521046]
  We propose a novel algorithm for approximating the continuous Entropic OT (EOT) barycenter for arbitrary OT cost functions. Our approach is built upon the dual reformulation of the EOT problem based on weak OT.
  arXiv  Detail & Related papers  (2023-10-02T11:24:36Z)
• An Optimization-based Deep Equilibrium Model for Hyperspectral Image Deconvolution with Convergence Guarantees [71.57324258813675]
  We propose a novel methodology for addressing the hyperspectral image deconvolution problem. A new optimization problem is formulated, leveraging a learnable regularizer in the form of a neural network. The derived iterative solver is then expressed as a fixed-point calculation problem within the Deep Equilibrium framework.
  arXiv  Detail & Related papers  (2023-06-10T08:25:16Z)
• Interfacing Finite Elements with Deep Neural Operators for Fast Multiscale Modeling of Mechanics Problems [4.280301926296439]
  In this work, we explore the idea of multiscale modeling with machine learning and employ DeepONet, a neural operator, as an efficient surrogate of the expensive solver. DeepONet is trained offline using data acquired from the fine solver for learning the underlying and possibly unknown fine-scale dynamics. We present various benchmarks to assess accuracy and speedup, and in particular we develop a coupling algorithm for a time-dependent problem.
  arXiv  Detail & Related papers  (2022-02-25T20:46:08Z)
• A Deterministic Sampling Method via Maximum Mean Discrepancy Flow with Adaptive Kernel [5.618322163107168]
  We propose a novel deterministic sampling method to approximate a target distribution $rho*$ by minimizing the kernel discrepancy.<n>We use the EVI-MMD algorithm to solve two types of sampling problems.
  arXiv  Detail & Related papers  (2021-11-21T03:09:07Z)
• Semi-Implicit Neural Solver for Time-dependent Partial Differential Equations [4.246966726709308]
  We propose a neural solver to learn an optimal iterative scheme in a data-driven fashion for any class of PDEs. We provide theoretical guarantees for the correctness and convergence of neural solvers analogous to conventional iterative solvers.
  arXiv  Detail & Related papers  (2021-09-03T12:03:10Z)
• 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)
• A Bayesian Multiscale Deep Learning Framework for Flows in Random Media [0.0]
  Fine-scale simulation of complex systems governed by multiscale partial differential equations (PDEs) is computationally expensive and various multiscale methods have been developed for addressing such problems. In this work, we introduce a novel hybrid deep-learning and multiscale approach for multiscale PDEs with limited training data. For demonstration purposes, we focus on a porous media flow problem. We use an image-to-image supervised deep learning model to learn the mapping between the input permeability field and the multiscale basis functions.
  arXiv  Detail & Related papers  (2021-03-08T23:11:46Z)
• Pushing the Envelope of Rotation Averaging for Visual SLAM [69.7375052440794]
  We propose a novel optimization backbone for visual SLAM systems. We leverage averaging to improve the accuracy, efficiency and robustness of conventional monocular SLAM systems. Our approach can exhibit up to 10x faster with comparable accuracy against the state-art on public benchmarks.
  arXiv  Detail & Related papers  (2020-11-02T18:02:26Z)
• GACEM: Generalized Autoregressive Cross Entropy Method for Multi-Modal Black Box Constraint Satisfaction [69.94831587339539]
  We present a modified Cross-Entropy Method (CEM) that uses a masked auto-regressive neural network for modeling uniform distributions over the solution space. Our algorithm is able to express complicated solution spaces, thus allowing it to track a variety of different solution regions.
  arXiv  Detail & Related papers  (2020-02-17T20:21:20Z)
• A Near-Optimal Gradient Flow for Learning Neural Energy-Based Models [93.24030378630175]
  We propose a novel numerical scheme to optimize the gradient flows for learning energy-based models (EBMs) We derive a second-order Wasserstein gradient flow of the global relative entropy from Fokker-Planck equation. Compared with existing schemes, Wasserstein gradient flow is a smoother and near-optimal numerical scheme to approximate real data densities.
  arXiv  Detail & Related papers  (2019-10-31T02:26:20Z)

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.