TL;DR
This paper establishes the existence and uniqueness of solutions for a modified SPDE derived from the Fokker-Planck equation related to Score-based Generative Models, using a variational approach and novel functional spaces.
Contribution
It introduces a modified SPDE for SGMs, proves solution existence and uniqueness, and develops a new functional space inspired by Ornstein-Uhlenbeck operators.
Findings
Proved existence and uniqueness of solutions for the SPDE.
Developed a novel functional space for analysis.
Enhanced understanding of the mathematical foundation of SGMs.
Abstract
This paper investigates a Stochastic Partial Differential Equation (SPDE) derived from the Fokker-Planck equation associated with Score-based Generative Models. We modify the standard Fokker-Planck equation to better represent practical SGMs and introduce noise to mitigate potential discretization issues. The primary goal is to prove the existence and uniqueness of solutions for this SPDE. This aspect requires careful consideration due to the time-dependent operator and unbounded domain. To overcome these hurdles, we employ a variational approach and introduce a novel space inspired by Ornstein-Uhlenbeck operators. By demonstrating that this space and its subspace satisfy the necessary assumptions, they establish the existence of a solution for the given SPDE.
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Taxonomy
TopicsGenerative Adversarial Networks and Image Synthesis · Stochastic Gradient Optimization Techniques · Statistical Mechanics and Entropy