Bridging discrete and continuous state spaces: Exploring the Ehrenfest process in time-continuous diffusion models

• URL: http://arxiv.org/abs/2405.03549v1
• Date: Mon, 6 May 2024 15:12:51 GMT
• Title: Bridging discrete and continuous state spaces: Exploring the Ehrenfest process in time-continuous diffusion models
• Authors: Ludwig Winkler, Lorenz Richter, Manfred Opper,
• Abstract summary: We study time-continuous Markov jump processes on discrete state spaces. We show that the time-reversal of the Ehrenfest process converges to the time-reversed Ornstein-Uhlenbeck process.
• Score: 4.186575888568896
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: Generative modeling via stochastic processes has led to remarkable empirical results as well as to recent advances in their theoretical understanding. In principle, both space and time of the processes can be discrete or continuous. In this work, we study time-continuous Markov jump processes on discrete state spaces and investigate their correspondence to state-continuous diffusion processes given by SDEs. In particular, we revisit the $\textit{Ehrenfest process}$, which converges to an Ornstein-Uhlenbeck process in the infinite state space limit. Likewise, we can show that the time-reversal of the Ehrenfest process converges to the time-reversed Ornstein-Uhlenbeck process. This observation bridges discrete and continuous state spaces and allows to carry over methods from one to the respective other setting. Additionally, we suggest an algorithm for training the time-reversal of Markov jump processes which relies on conditional expectations and can thus be directly related to denoising score matching. We demonstrate our methods in multiple convincing numerical experiments.

Related papers

• Adversarial Schrödinger Bridge Matching [66.39774923893103]
  Iterative Markovian Fitting (IMF) procedure alternates between Markovian and reciprocal projections of continuous-time processes. We propose a novel Discrete-time IMF (D-IMF) procedure in which learning of processes is replaced by learning just a few transition probabilities in discrete time. We show that our D-IMF procedure can provide the same quality of unpaired domain translation as the IMF, using only several generation steps instead of hundreds.
  arXiv  Detail & Related papers  (2024-05-23T11:29:33Z)
• Using Ornstein-Uhlenbeck Process to understand Denoising Diffusion Probabilistic Model and its Noise Schedules [3.4235611902516263]
  We show that Denoising Diffusion Probabilistic Model DDPM can be represented by a time-homogeneous continuous-time Markov process. Surprisingly, this continuous-time Markov process is the well-known and well-studied Ornstein-Ohlenbeck (OU) process.
  arXiv  Detail & Related papers  (2023-11-29T14:36:33Z)
• DiffuSeq-v2: Bridging Discrete and Continuous Text Spaces for Accelerated Seq2Seq Diffusion Models [58.450152413700586]
  We introduce a soft absorbing state that facilitates the diffusion model in learning to reconstruct discrete mutations based on the underlying Gaussian space. We employ state-of-the-art ODE solvers within the continuous space to expedite the sampling process. Our proposed method effectively accelerates the training convergence by 4x and generates samples of similar quality 800x faster.
  arXiv  Detail & Related papers  (2023-10-09T15:29:10Z)
• On the path integral simulation of space-time fractional Schroedinger equation with time independent potentials [0.0]
  A Feynman-Kac path integral method has been proposed for solving the Cauchy problems associated with the space-time fractional Schroedinger equations. We have been able to simulate the space-time fractional diffusion process with comparable simplicity and convergence rate as in the case of a standard diffusion.
  arXiv  Detail & Related papers  (2023-06-25T20:14:40Z)
• Inference and Sampling of Point Processes from Diffusion Excursions [26.111388335046197]
  We propose a point process construction that describes arrival time observations in terms of the state of a latent diffusion process. Based on the developments in Ito's excursion theory, we propose methods for inferring and sampling from the point process derived from the latent diffusion process.
  arXiv  Detail & Related papers  (2023-06-01T14:56:23Z)
• Blackout Diffusion: Generative Diffusion Models in Discrete-State Spaces [0.0]
  We develop a theoretical formulation for arbitrary discrete-state Markov processes in the forward diffusion process. As an example, we introduce Blackout Diffusion'', which learns to produce samples from an empty image instead of from noise.
  arXiv  Detail & Related papers  (2023-05-18T16:24:12Z)
• ShiftDDPMs: Exploring Conditional Diffusion Models by Shifting Diffusion Trajectories [144.03939123870416]
  We propose a novel conditional diffusion model by introducing conditions into the forward process. We use extra latent space to allocate an exclusive diffusion trajectory for each condition based on some shifting rules. We formulate our method, which we call textbfShiftDDPMs, and provide a unified point of view on existing related methods.
  arXiv  Detail & Related papers  (2023-02-05T12:48:21Z)
• Score-based Continuous-time Discrete Diffusion Models [102.65769839899315]
  We extend diffusion models to discrete variables by introducing a Markov jump process where the reverse process denoises via a continuous-time Markov chain. We show that an unbiased estimator can be obtained via simple matching the conditional marginal distributions. We demonstrate the effectiveness of the proposed method on a set of synthetic and real-world music and image benchmarks.
  arXiv  Detail & Related papers  (2022-11-30T05:33:29Z)
• Contrastive learning of strong-mixing continuous-time stochastic processes [53.82893653745542]
  Contrastive learning is a family of self-supervised methods where a model is trained to solve a classification task constructed from unlabeled data. We show that a properly constructed contrastive learning task can be used to estimate the transition kernel for small-to-mid-range intervals in the diffusion case.
  arXiv  Detail & Related papers  (2021-03-03T23:06:47Z)
• The Connection between Discrete- and Continuous-Time Descriptions of Gaussian Continuous Processes [60.35125735474386]
  We show that discretizations yielding consistent estimators have the property of invariance under coarse-graining' This result explains why combining differencing schemes for derivatives reconstruction and local-in-time inference approaches does not work for time series analysis of second or higher order differential equations.
  arXiv  Detail & Related papers  (2021-01-16T17:11:02Z)
• Discovering Causal Structure with Reproducing-Kernel Hilbert Space $\epsilon$-Machines [0.0]
  We present a method that infers causal structure directly from observations of a system's behaviors. The method robustly estimates causal structure in the presence of varying external and measurement noise levels.
  arXiv  Detail & Related papers  (2020-11-23T23:41:16Z)

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.