61 research outputs found
Tree-Based Diffusion Schr\"odinger Bridge with Applications to Wasserstein Barycenters
Multi-marginal Optimal Transport (mOT), a generalization of OT, aims at
minimizing the integral of a cost function with respect to a distribution with
some prescribed marginals. In this paper, we consider an entropic version of
mOT with a tree-structured quadratic cost, i.e., a function that can be written
as a sum of pairwise cost functions between the nodes of a tree. To address
this problem, we develop Tree-based Diffusion Schr\"odinger Bridge (TreeDSB),
an extension of the Diffusion Schr\"odinger Bridge (DSB) algorithm. TreeDSB
corresponds to a dynamic and continuous state-space counterpart of the
multimarginal Sinkhorn algorithm. A notable use case of our methodology is to
compute Wasserstein barycenters which can be recast as the solution of a mOT
problem on a star-shaped tree. We demonstrate that our methodology can be
applied in high-dimensional settings such as image interpolation and Bayesian
fusion
Diffusion Schr\"odinger Bridge Matching
Solving transport problems, i.e. finding a map transporting one given
distribution to another, has numerous applications in machine learning. Novel
mass transport methods motivated by generative modeling have recently been
proposed, e.g. Denoising Diffusion Models (DDMs) and Flow Matching Models
(FMMs) implement such a transport through a Stochastic Differential Equation
(SDE) or an Ordinary Differential Equation (ODE). However, while it is
desirable in many applications to approximate the deterministic dynamic Optimal
Transport (OT) map which admits attractive properties, DDMs and FMMs are not
guaranteed to provide transports close to the OT map. In contrast,
Schr\"odinger bridges (SBs) compute stochastic dynamic mappings which recover
entropy-regularized versions of OT. Unfortunately, existing numerical methods
approximating SBs either scale poorly with dimension or accumulate errors
across iterations. In this work, we introduce Iterative Markovian Fitting, a
new methodology for solving SB problems, and Diffusion Schr\"odinger Bridge
Matching (DSBM), a novel numerical algorithm for computing IMF iterates. DSBM
significantly improves over previous SB numerics and recovers as
special/limiting cases various recent transport methods. We demonstrate the
performance of DSBM on a variety of problems
Diffusion Schr\"odinger Bridge with Applications to Score-Based Generative Modeling
Progressively applying Gaussian noise transforms complex data distributions
to approximately Gaussian. Reversing this dynamic defines a generative model.
When the forward noising process is given by a Stochastic Differential Equation
(SDE), Song et al. (2021) demonstrate how the time inhomogeneous drift of the
associated reverse-time SDE may be estimated using score-matching. A limitation
of this approach is that the forward-time SDE must be run for a sufficiently
long time for the final distribution to be approximately Gaussian. In contrast,
solving the Schr\"odinger Bridge problem (SB), i.e. an entropy-regularized
optimal transport problem on path spaces, yields diffusions which generate
samples from the data distribution in finite time. We present Diffusion SB
(DSB), an original approximation of the Iterative Proportional Fitting (IPF)
procedure to solve the SB problem, and provide theoretical analysis along with
generative modeling experiments. The first DSB iteration recovers the
methodology proposed by Song et al. (2021), with the flexibility of using
shorter time intervals, as subsequent DSB iterations reduce the discrepancy
between the final-time marginal of the forward (resp. backward) SDE with
respect to the prior (resp. data) distribution. Beyond generative modeling, DSB
offers a widely applicable computational optimal transport tool as the
continuous state-space analogue of the popular Sinkhorn algorithm (Cuturi,
2013).Comment: 58 pages, 18 figures (correction of Proposition 5
Diffusion Models for Constrained Domains
Denoising diffusion models are a recent class of generative models which
achieve state-of-the-art results in many domains such as unconditional image
generation and text-to-speech tasks. They consist of a noising process
destroying the data and a backward stage defined as the time-reversal of the
noising diffusion. Building on their success, diffusion models have recently
been extended to the Riemannian manifold setting. Yet, these Riemannian
diffusion models require geodesics to be defined for all times. While this
setting encompasses many important applications, it does not include manifolds
defined via a set of inequality constraints, which are ubiquitous in many
scientific domains such as robotics and protein design. In this work, we
introduce two methods to bridge this gap. First, we design a noising process
based on the logarithmic barrier metric induced by the inequality constraints.
Second, we introduce a noising process based on the reflected Brownian motion.
As existing diffusion model techniques cannot be applied in this setting, we
derive new tools to define such models in our framework. We empirically
demonstrate the applicability of our methods to a number of synthetic and
real-world tasks, including the constrained conformational modelling of protein
backbones and robotic arms
From Denoising Diffusions to Denoising Markov Models
Denoising diffusions are state-of-the-art generative models which exhibit
remarkable empirical performance and come with theoretical guarantees. The core
idea of these models is to progressively transform the empirical data
distribution into a simple Gaussian distribution by adding noise using a
diffusion. We obtain new samples whose distribution is close to the data
distribution by simulating a "denoising" diffusion approximating the time
reversal of this "noising" diffusion. This denoising diffusion relies on
approximations of the logarithmic derivatives of the noised data densities,
known as scores, obtained using score matching. Such models can be easily
extended to perform approximate posterior simulation in high-dimensional
scenarios where one can only sample from the prior and simulate synthetic
observations from the likelihood. These methods have been primarily developed
for data on while extensions to more general spaces have been
developed on a case-by-case basis. We propose here a general framework which
not only unifies and generalizes this approach to a wide class of spaces but
also leads to an original extension of score matching. We illustrate the
resulting class of denoising Markov models on various applications
- …