Non-asymptotic bounds for forward processes in denoising diffusions: Ornstein-Uhlenbeck is hard to beat

• URL: http://arxiv.org/abs/2408.13799v1
• Date: Sun, 25 Aug 2024 10:28:31 GMT
• Title: Non-asymptotic bounds for forward processes in denoising diffusions: Ornstein-Uhlenbeck is hard to beat
• Authors: Miha Brešar, Aleksandar Mijatović,
• Abstract summary: This paper presents explicit non-asymptotic bounds on the forward diffusion error in total variation (TV) We parametrise multi-modal data distributions in terms of the distance $R$ to their furthest modes and consider forward diffusions with additive and multiplicative noise.
• Score: 49.1574468325115
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Denoising diffusion probabilistic models (DDPMs) represent a recent advance in generative modelling that has delivered state-of-the-art results across many domains of applications. Despite their success, a rigorous theoretical understanding of the error within DDPMs, particularly the non-asymptotic bounds required for the comparison of their efficiency, remain scarce. Making minimal assumptions on the initial data distribution, allowing for example the manifold hypothesis, this paper presents explicit non-asymptotic bounds on the forward diffusion error in total variation (TV), expressed as a function of the terminal time $T$. We parametrise multi-modal data distributions in terms of the distance $R$ to their furthest modes and consider forward diffusions with additive and multiplicative noise. Our analysis rigorously proves that, under mild assumptions, the canonical choice of the Ornstein-Uhlenbeck (OU) process cannot be significantly improved in terms of reducing the terminal time $T$ as a function of $R$ and error tolerance $\varepsilon>0$. Motivated by data distributions arising in generative modelling, we also establish a cut-off like phenomenon (as $R\to\infty$) for the convergence to its invariant measure in TV of an OU process, initialized at a multi-modal distribution with maximal mode distance $R$.

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