9 research outputs found
Stein Variational Gradient Descent:many-particle and long-time asymptotics
Stein variational gradient descent (SVGD) refers to a class of methods for Bayesian inference based on in-teracting particle systems. In this paper, we consider the originally proposed deterministic dynamics as well asa stochastic variant, each of which represent one of the two main paradigms in Bayesian computational statis-tics:variational inferenceandMarkov chain Monte Carlo. As it turns out, these are tightly linked througha correspondence between gradient flow structures and large-deviation principles rooted in statistical physics.To expose this relationship, we develop the cotangent space construction for the Stein geometry, prove its ba-sic properties, and determine the large-deviation functional governing the many-particle limit for the empiricalmeasure. Moreover, we identify theStein-Fisher information(orkernelised Stein discrepancy) as its leadingorder contribution in the long-time and many-particle regime in the sense ofΓ-convergence, shedding some lighton the finite-particle properties of SVGD. Finally, we establish a comparison principle between the Stein-Fisherinformation and RKHS-norms that might be of independent interes
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