Comb Tensor Networks vs. Matrix Product States: Enhanced Efficiency in High-Dimensional Spaces

TL;DR

It is demonstrated that beyond a certain threshold in data and bond dimensions, a comb-shaped tensor network architecture can yield more efficient contractions than a standard MPS.

Abstract

Modern approaches to generative modeling of continuous data using tensor networks incorporate compression layers to capture the most meaningful features of high-dimensional inputs. These methods, however, rely on traditional Matrix Product States (MPS) architectures. Here, we demonstrate that beyond a certain threshold in data and bond dimensions, a comb-shaped tensor network architecture can yield more efficient contractions than a standard MPS. This finding suggests that for continuous and high-dimensional data distributions, transitioning from MPS to a comb tensor network representation can substantially reduce computational overhead while maintaining accuracy.

Figures (1)

• Figure 1: Threshold roots $x_+$ and $x_-$ as a function of $d$ for $M = 50$. The curves depict the conditions under which a comb tensor network outperforms a regular MPS. As $d$ grows, the beneficial regime for the comb structure becomes apparent, guiding the selection of parameters for efficient generative modeling of continuous data.