Related papers: Formulating Discrete Probability Flow Through Opti…
The blooming diffusion probabilistic models (DPMs) have garnered significant interest due to their impressive performance and the elegant inspiration they draw from physics. While earlier DPMs relied upon the Markovian assumption, recent…
Despite Flow Matching and diffusion models having emerged as powerful generative paradigms for continuous variables such as images and videos, their application to high-dimensional discrete data, such as language, is still limited. In this…
Optimal transport maps define a one-to-one correspondence between probability distributions, and as such have grown popular for machine learning applications. However, these maps are generally defined on empirical observations and cannot be…
Inspired by the matching of supply to demand in logistical problems, the optimal transport (or Monge--Kantorovich) problem involves the matching of probability distributions defined over a geometric domain such as a surface or manifold. In…
Diffusion models have recently outperformed alternative approaches to model the distribution of natural images, such as GANs. Such diffusion models allow for deterministic sampling via the probability flow ODE, giving rise to a latent space…
Predicting how distributions over discrete variables vary over time is a common task in time series forecasting. But whereas most approaches focus on merely predicting the distribution at subsequent time steps, a crucial piece of…
The design space of discrete-space diffusion or flow generative models are significantly less well-understood than their continuous-space counterparts, with many works focusing only on a simple masked construction. In this work, we aim to…
We present an novel framework for efficiently and effectively extending the powerful continuous diffusion processes to discrete modeling. Previous approaches have suffered from the discrepancy between discrete data and continuous modeling.…
Optimal transportation distances are valuable for comparing and analyzing probability distributions, but larger-scale computational techniques for the theoretically favorable quadratic case are limited to smooth domains or regularized…
We present a deep generative model, named Monge-Amp\`ere flow, which builds on continuous-time gradient flow arising from the Monge-Amp\`ere equation in optimal transport theory. The generative map from the latent space to the data space…
Rectified flow (Liu et al., 2022; Liu, 2022; Wu et al., 2023) is a method for defining a transport map between two distributions, and enjoys popularity in machine learning, although theoretical results supporting the validity of these…
We study the estimation of optimal transport (OT) maps between an arbitrary source probability measure and a log-concave target probability measure. Our contributions are twofold. First, we propose a new evolution equation in the set of…
A partial differential equation governing the global evolution of the joint probability distribution of an arbitrary number of local flow observations, drawn randomly from a control volume, is derived and applied to examples involving…
The study of diffusion in Hamiltonian systems has been a problem of interest for a number of years. In this paper we explore the influence of self-consistency on the diffusion properties of systems described by coupled symplectic maps.…
A discrete model of traffic on a multilane road is considered. The traffic is presented as particles movement with a deterministic component and a stochastic one. Formulas for the traffic characteristics have been found. The model can…
Diffusion models have exhibited excellent performance in various domains. The probability flow ordinary differential equation (ODE) of diffusion models (i.e., diffusion ODEs) is a particular case of continuous normalizing flows (CNFs),…
Representations based on random walks can exploit discrete data distributions for clustering and classification. We extend such representations from discrete to continuous distributions. Transition probabilities are now calculated using a…
Sampling from diffusion probabilistic models (DPMs) can be viewed as a piecewise distribution transformation, which generally requires hundreds or thousands of steps of the inverse diffusion trajectory to get a high-quality image. Recent…
In this work, we propose a novel machine learning approach to compute the optimal transport map between two continuous distributions from their unpaired samples, based on the DeepParticle methods. The proposed method leads to a min-min…
Discrete flow models offer a powerful framework for learning distributions over discrete state spaces and have demonstrated superior performance compared to the discrete diffusion models. However, their convergence properties and error…