Hierarchical Tensor Network Structure Search for High-Dimensional Data

Abstract: Tensor network methods provide a scalable solution to represent high-dimensional data. However, their efficacy is often limited by static, expert-defined structures that fail to adapt to evolving data correlations. We address this limitation by formalizing the tensor network structural rounding problem and introducing the hierarchical structure search algorithm HISS, which automatically identifies near-optimal structures and index reshaping for arbitrary tree networks. To navigate the combinatorial explosion of the structural search space, HISS integrates stochastic sub-network sampling with hierarchical refinement. This approach utilizes entropy-guided index clustering to reduce dimensionality and targeted reshaping to expose latent data correlations. Numerical experiments on analytical functions and real-world physics applications, including thermal radiation transport, neutron diffusion, and computational fluid dynamics, demonstrate that HISS exhibits empirical polynomial scaling with dimensionality relative to the sampling budget, bypassing the scalability barriers in prior work. HISS achieves compression ratios $2.5\times$ to $100\times$ higher than standard fixed formats such as Tensor Trains and Hierarchical Tuckers~(peaking at $1000\times$). Furthermore, HISS discovers structures that generalize effectively: applying a structure optimized for one data instance to a related target data typically maintains compression performance within $10\%$ of the result obtained by performing structure search on that target data. These results highlight HISS as a robust, automated tool for adaptive data representation and high-dimensional simulation compression with tensor network methods.