Biological neural systems often represent information on low-dimensional manifolds that reflect the topology of their encoded variables. This suggests that neural activity can be naturally organized in geometrically meaningful ways, as seen in rodent head direction cells forming circular manifolds. This proposal examines whether artificial neural networks (ANNs) trained on tasks with well-defined topologies—such as planar or spherical coordinates from autonomous driving datasets like Apolloscape, cyclic temporal variables, or graph-structured road networks—develop similar low-dimensional representations aligned with the variables' inherent topology. We consider convolutional and vision transformer models for image data, graph neural networks for road network graphs, and 3D or point-based models for LIDAR point clouds, analyzing their internal activations with dimensionality reduction and topological data analysis. If successful, this approach not only elucidates the nature of internal representations in ANNs but also offers insights into the computational principles that bridge artificial systems and biological cognition.

We investigate the statistical and computational limits of latent **Di**ffusion **T**ransformers (**DiTs**) under the low-dimensional linear latent space assumption. Statistically, we study the universal approximation and sample complexity of the DiTs score function, as well as the distribution recovery property of the initial data. Specifically, under mild data assumptions, we derive an approximation error bound for the score network of latent DiTs, which is sub-linear in the latent space dimension. Additionally, we derive the corresponding sample complexity bound and show that the data distribution generated from the estimated score function converges toward a proximate area of the original one. Computationally, we characterize the hardness of both forward inference and backward computation of latent DiTs, assuming the Strong Exponential Time Hypothesis (SETH). For forward inference, we identify efficient criteria for all possible latent DiTs inference algorithms and showcase our theory by pushing the efficiency toward almost-linear time inference. For backward computation, we leverage the low-rank structure within the gradient computation of DiTs training for possible algorithmic speedup. Specifically, we show that such speedup achieves almost-linear time latent DiTs training by casting the DiTs gradient as a series of chained low-rank approximations with bounded error. Under the low-dimensional assumption, we show that the statistical rates and the computational efficiency are all dominated by the dimension of the subspace, suggesting that latent DiTs have the potential to bypass the challenges associated with the high dimensionality of initial data.