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arXiv:2603.27856 · Journal of Computational Physics

Hierarchical search of tree tensor networks for high-dimensional data

Zheng Kui Guo, Aditya Deshpande, Xinyu Wang, Brian C. Kiedrowski, Alex Arkady Gorodetsky

Published Jul 22, 2026• 0 citations
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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 structural rounding problem for tree tensor networks and introducing a hierarchical search algorithm HiST , which automatically identifies optimized structures with index reshaping for input tree tensor networks. To navigate the combinatorial explosion of the structural search space, HiST 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 HiST exhibits empirical polynomial scaling with dimensionality relative to the sampling budget, bypassing the scalability barriers in prior work. HiST achieves compression ratios 2.5 × to 100 × higher than standard fixed formats such as Tensor Trains and Hierarchical Tuckers (peaking at 1000 × ). Furthermore, HiST 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 HiST as a robust, automated tool for adaptive data representation and high-dimensional simulation compression with tensor network methods.

DOI
10.1016/j.jcp.2026.115230

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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 structural rounding problem for tree tensor networks and introducing a hierarchical search algorithm HiST , which automatically identifies optimized structures with index reshaping for input tree tensor networks. To navigate the combinatorial explosion of the structural search space, HiST 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 HiST exhibits empirical polynomial scaling with dimensionality relative to the sampling budget, bypassing the scalability barriers in prior work. HiST achieves compression ratios 2.5 × to 100 × higher than standard fixed formats such as Tensor Trains and Hierarchical Tuckers (peaking at 1000 × ). Furthermore, HiST 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 HiST as a robust, automated tool for adaptive data representation and high-dimensional simulation compression with tensor network methods.
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Tensor networks are a compressed format for multi-dimensional data. One dimensional tensor networks—often referred to as tensor trains (TT) or matrix product states (MPS)—are increasingly being used as a numerical Ansatz for continuum functions by ``quantizing’’ the inputs into discrete binary digits. Here we demonstrate the power of more general tree tensor networks (TTNs) for this purpose. We provide direct constructions of a number of elementary functions as generic tree tensor networks and interpolative constructions for more complicated functions via a generalization of the tensor cross interpolation algorithm. For a range of multi-dimensional functions we show how more structured tree tensor networks offer a significantly more efficient Ansatz than the commonly used tensor train. Finally, we demonstrate how the methods introduced in this work can be used to realize a TTN-based solver for multi-dimensional, non-linear Fredholm equations.
High performance computers produce extreme-scale data sets that require sampling or compression if they are to be used to their full potential. Existing data compression techniques typically exploit features such as sparsity in the data, homogeneity in the data, or {\it a priori} knowledge of what subsets of data are of most interest. Fluid dynamics data in general do not exhibit these features and so are attractive test beds for generic compression techniques that are objective, robust, and tuneable with respect to information lost due to compression. Presented here is a method based on tensor networks, specifically matrix product states or tensor trains, that meets these requirements. The method is demonstrated for compression in one-dimension and is extensible to higher dimensionality. Lossless compression is demonstrated for random Fourier series for sufficiently high bond dimension of the tensor network, with the memory required to store the tensor network scaling directly proportional to the bond dimension. The lossy compression exhibited at lower bond dimension can be well within the relative error of many fluid simulations. The compression algorithm is tested for the time evolution of Burger's equation with excellent results. We additionally demonstrate the capability to perform computations in the compressed form through a tensor network periodic convolution that can be orders of magnitude faster than using fast Fourier transforms and the convolution theorem. In addition to being an attractive method for working with data sets generated by existing computers, the tensor network methods utilised are directly translatable to the emerging paradigm of quantum computing.
We present two novel algorithms for approximating and incrementally updating the hierarchical Tucker (HT) decomposition of tensor streams. The first algorithm, Batch Hierarchical Tucker - leaf to root ( BHT-l2r ), introduces a restructured HT topology that anchors the batch dimension at the root node, enabling efficient compression and downstream latent space extraction for batches of similar tensors. The second algorithm, Hierarchical Tucker - Rapid Incremental Subspace Expansion ( HT-RISE ), provides a deterministic subspace tracking approach to continuously update this representation as new data batches arrive. Designed for the online setting, HT-RISE never requires full storage or reconstruction of the historical data. Unlike randomized sketching methods that defer network construction, HT-RISE explicitly maintains a fully orthogonal tensor network state at every time step, dynamically adapting tensor ranks to satisfy certified, user-defined error bounds. Empirical evaluations on several datasets demonstrate the superior performance of both methods. BHT-l2r achieves up to 6.2 × greater compression and a 3.7 × speedup over standard HT formats, while HT-RISE delivers up to 3.1 × higher compression and a 3.2 × speedup compared to state-of-the-art incremental Tensor Train algorithms.
The quantum state preparation of probability distributions is an important subroutine for many quantum algorithms. When embedding D -dimensional multivariate probability distributions by discretizing each dimension into 2n points, we need a state preparation circuit comprising a total of nD qubits, which is often difficult to compile. In this study, we propose a scalable method to generate state preparation circuits for D -dimensional multivariate normal distributions, utilizing tree tensor networks (TTN). We establish theoretical guarantees that multivariate normal distributions with 1D correlation structures can be efficiently represented using TTN. Based on these analyses, we propose a compilation method that uses automatic structural optimization to find the most efficient network structure and compact circuit. We apply our method to state preparation circuits for various high-dimensional random multivariate normal distributions. The numerical results suggest that our method can dramatically reduce the circuit depth and CNOT count while maintaining fidelity compared to existing approaches.
Tensor decomposition techniques have shown great successes in machine learning and data science by extending classical algorithms based on matrix factorization to multi-modal and multi-way data. However, there exist many tensor decomposition models~(CP, Tucker, Tensor Train, etc.) and the rank of such a decomposition is typically a collection of integers rather than a unique number, making model and hyper-parameter selection a tedious and costly task. At the same time, tensor network methods are powerful tools developed in the physics community which have recently shown their potential for machine learning applications and offer a unifying view of the various tensor decomposition models. In this paper, we leverage the tensor network formalism to develop a generic and efficient adaptive algorithm for tensor learning. Our method is based on a simple greedy approach optimizing a differentiable loss function starting from a rank one tensor and successively identifying the most promising tensor network edges for small rank increments. Our algorithm can adaptively identify tensor network structures with small number of parameters that effectively optimize the objective function from data. The framework we introduce is very broad and encompasses many common tensor optimization problems. Experiments on tensor decomposition and tensor completion tasks with both synthetic and real world data demonstrate the effectiveness of the proposed algorithm.