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arXiv:2606.23949

Variational inference and density estimation with non-negative tensor of hierarchical tucker format

X W Tang, Haoxuan Chen, Lexing Ying B

Published Jun 22, 2026• 0 citations
arXivHTMLPDF

Abstract

In this work, we present an efficient method to compress a high-dimensional discrete probability function, i.e., a probability tensor, into a non-negative hierarchical Tucker format. The methodology is a two-stage procedure. In the first stage, we take an existing interpolation method to compress the target tensor into a hierarchical Tucker (HT) in a manner similar to the CUR decomposition for low-rank matrix reconstruction. In the second stage, we fit the first-stage output against a non-negative hierarchical Tucker ansatz using a second-order method tailored specifically for this setting. When the tensor is of order \(d\), both stages admit an \(\mathcal{O}(d)\) computational complexity, and therefore the proposed methodology readily extends into high-dimensional settings. Numerical experiments show success in compressing various high-dimensional probability tensors.

DOI
10.48550/arxiv.2606.23949

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In this work, we present an efficient method to compress a high-dimensional discrete probability function, i.e., a probability tensor, into a non-negative hierarchical Tucker format. The methodology is a two-stage procedure. In the first stage, we take an existing interpolation method to compress the target tensor into a hierarchical Tucker (HT) in a manner similar to the CUR decomposition for low-rank matrix reconstruction. In the second stage, we fit the first-stage output against a non-negative hierarchical Tucker ansatz using a second-order method tailored specifically for this setting. When the tensor is of order (d), both stages admit an (\mathcal{O}(d)) computational complexity, and therefore the proposed methodology readily extends into high-dimensional settings. Numerical experiments show success in compressing various high-dimensional probability tensors.
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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.
In the context of big data, high-order tensor decompositions have to face a new challenge in terms of storage and computational costs. The tensor train (TT) decomposition provides a very useful graph-based model reduction, whose storage cost grows linearly with the tensor order DDD. The computation of the TT-core tensors and TT-ranks can be done in a stable sequential (i.e., noniterative) way thanks to the popular TT-SVD algorithm. In this paper, we exploit the ideas developed for the hierarchical/tree Tucker decomposition in the context of the TT decomposition. Specifically, a new efficient estimation scheme, called TT-HSVD (Tensor-Train Hierarchical SVD), is proposed as a solution to compute the TT decomposition of a high-order tensor. The new algorithm simultaneously delivers the TT-core tensors and their TT-ranks in a hierarchical way. It is a stable (i.e., noniterative) and computationally more efficient algorithm than TT-SVD, which is very important when dealing with large-scale data. The TT-HSVD algorithm uses a new reshaping strategy and a tailored partial SVD, which allows us to deal with smaller matrices compared to those of the TT-SVD. In addition, TT-HSVD is well suited for a parallel processing architecture. An algebraic analysis of the two algorithms is carried out, showing that TT-SVD and TT-HSVD compute the same TT-ranks and TT-core tensors up to specific bases. Simulation results for different tensor orders and dimensions corroborate the effectiveness of the proposed algorithm.
We introduce a new SVD-based tensor decomposition method for tensor networks with arbitrary graph topologies, extending classical hierarchical SVD-based techniques to networks with cycles and general connectivity. We also introduce addition and rounding procedures for traceable tensor graphs, enabling step-rounding time integration of high-dimensional PDEs directly in graph format, with rank truncation controlled to a prescribed tolerance at every time step. We demonstrate the new method on the decomposition of multivariate functions and on the numerical solution of the Fokker-Planck equation, and find that the graph-format representation attains comparable or better accuracy than the classical tensor train and hierarchical Tucker tensor formats, while using substantially fewer degrees of freedom at lower computational cost.
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.