  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.
