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Sampling, Diffusions, and Stochastic Localization
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Sampling, Diffusions, and Stochastic Localization
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Diffusions are a successful technique to sample from high-dimensional distributions. The target distribution can be either explicitly given or learnt from a collection of samples. They implement a diffusion process whose endpoint is a sample from the target distribution. The drift of the diffusion process is typically represented as a neural network. Stochastic localization is a successful technique to prove mixing of Markov Chains and other functional inequalities in high dimension. An algorithmic version of stochastic localization was recently proposed in order to sample from certain statistical mechanics models. This expository article has three objectives: $(i)$~Generalize the algorithmic construction to other stochastic localization processes. This construction is both simple and broadly applicable; $(ii)$~Clarify the connection between diffusions and stochastic localization. This allows to derive several known sampling schemes in a unified fashion; $(iii)$~Describe the insights that follow from this unified viewpoint.
Forward citations
Cited by 9 Pith papers
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Weak Poincar\'e Inequalities via Approximate Stochastic Localization: Application to Sampling the Sherrington-Kirkpatrick Model
Approximate stochastic localization plus conductance transfers yield a weak Poincaré inequality for the SK model at β < 1/2, enabling efficient Glauber sampling from a warm start.
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On McDiarmid's Inequality under Dependence via Approximate Tensorization of Entropy
Derives McDiarmid-type inequalities for dependent variables via approximate tensorization of entropy, with applications improving DKW rates to 1/sqrt(n) under weak dependence for log-concave measures.
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Potential Hessian Ascent III: Sampling the Sherrington--Kirkpatrick Model at Beta < 1/2
Polynomial-time algorithm samples the Sherrington-Kirkpatrick Gibbs measure at beta < 1/2 with o(1) TVD error by combining potential Hessian ascent, stochastic localization, covariance estimates, and Jarzynski equalit...
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Potential Hessian Ascent III: Sampling the Sherrington--Kirkpatrick Model at Beta < 1/2
A polynomial-time algorithm samples the SK model Gibbs measure with o(1) TVD error for β < 1/2 by combining potential Hessian ascent, stochastic localization, Jarzynski equality, and Glauber dynamics.
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Binomial flows: Denoising and flow matching for discrete ordinal data
Binomial flows close the gap between continuous flow matching and discrete ordinal data by using binomial distributions to enable unified denoising, sampling, and exact likelihoods in diffusion models.
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Discrete Stochastic Localization for Non-autoregressive Generation
Discrete Stochastic Localization lets a single trained network support an entire family of per-token SNR paths for discrete sequence generation, with masked diffusion as a special case, and improves MAUVE scores when ...
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Stochastic Interpolants: A Unifying Framework for Flows and Diffusions
Stochastic interpolants unify flow-based and diffusion-based generative models by bridging target densities exactly via latent-variable processes whose drifts minimize quadratic objectives.
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A note on connections between the F\"ollmer process and the denoising diffusion probabilistic model
Discretized Föllmer processes supply hyper-parameter settings for DDPM samplers that recover state-of-the-art sampling error bounds with slight improvements.
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A Mathematical Introduction to Diffusion Models
An educational exposition that layers core definitions, simplified estimates, and research-level theorems on diffusion sampling for probability-background graduate students.
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