Diffusion Transformer Captures Spatial-Temporal Dependencies: A Theory for Gaussian Process Data

TL;DR

This paper provides a theoretical foundation for diffusion transformers in modeling spatial-temporal dependencies in sequential data, demonstrating their effectiveness in learning Gaussian process data through approximation guarantees and numerical validation.

Contribution

It introduces the first theoretical analysis of diffusion transformers for capturing spatial-temporal dependencies, including score approximation and distribution estimation guarantees.

Findings

Diffusion transformers effectively capture spatial-temporal dependencies.

Theoretical guarantees for learning Gaussian process data are established.

Numerical experiments confirm the theoretical predictions.

Abstract

Diffusion Transformer, the backbone of Sora for video generation, successfully scales the capacity of diffusion models, pioneering new avenues for high-fidelity sequential data generation. Unlike static data such as images, sequential data consists of consecutive data frames indexed by time, exhibiting rich spatial and temporal dependencies. These dependencies represent the underlying dynamic model and are critical to validate the generated data. In this paper, we make the first theoretical step towards bridging diffusion transformers for capturing spatial-temporal dependencies. Specifically, we establish score approximation and distribution estimation guarantees of diffusion transformers for learning Gaussian process data with covariance functions of various decay patterns. We highlight how the spatial-temporal dependencies are captured and affect learning efficiency. Our study proposes a novel transformer approximation theory, where the transformer acts to unroll an algorithm. We support our theoretical results by numerical experiments, providing strong evidence that spatial-temporal dependencies are captured within attention layers, aligning with our approximation theory.