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arXiv:2502.20655 · Multiscale Modeling and Simulation

Wavelet-Based Density Sketching with Functional Hierarchical Tensor

Xun Tang, Lexing Ying B

Published May 27, 2026• 1 citation
arXivHTMLPDF

Abstract

Abstract. We introduce the functional hierarchical tensor under a wavelet basis (FHT-W) ansatz for high-dimensional density estimation in lattice models. Recently, the functional tensor network has emerged as a suitable candidate for density estimation due to its ability to calculate the normalization constant exactly, a defining feature not enjoyed by neural network alternatives such as energy-based models or diffusion models. While current functional tensor network models show good performance for lattice models with weak or moderate couplings, we show that they face significant model capacity constraints when applied to lattice models with strong couplings. To address this issue, this work proposes to perform density estimation on the lattice model under a wavelet transformation. Motivated by the literature on scale separation, we perform iterative wavelet coarsening to separate the lattice model into different scales. Based on this multiscale structure, we design a new functional hierarchical tensor ansatz using a hierarchical tree topology, whereby information on the finer scale is further away from the root node of the tree. Our experiments show that the numerical rank of typical lattice models is significantly lower under an appropriate wavelet transformation. Furthermore, we show that our proposed model allows one to model challenging Gaussian field models and Ginzburg–Landau models.

DOI
10.1137/25m1737912

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Abstract. We introduce the functional hierarchical tensor under a wavelet basis (FHT-W) ansatz for high-dimensional density estimation in lattice models. Recently, the functional tensor network has emerged as a suitable candidate for density estimation due to its ability to calculate the normalization constant exactly, a defining feature not enjoyed by neural network alternatives such as energy-based models or diffusion models. While current functional tensor network models show good performance for lattice models with weak or moderate couplings, we show that they face significant model capacity constraints when applied to lattice models with strong couplings. To address this issue, this work proposes to perform density estimation on the lattice model under a wavelet transformation. Motivated by the literature on scale separation, we perform iterative wavelet coarsening to separate the lattice model into different scales. Based on this multiscale structure, we design a new functional hierarchical tensor ansatz using a hierarchical tree topology, whereby information on the finer scale is further away from the root node of the tree. Our experiments show that the numerical rank of typical lattice models is significantly lower under an appropriate wavelet transformation. Furthermore, we show that our proposed model allows one to model challenging Gaussian field models and Ginzburg–Landau models.
Abstract. Solving high-dimensional partial differential equations necessitates methods free of exponential scaling in the dimension of the problem. This work introduces a tensor network approach for the Kolmogorov backward equation via approximating directly the Markov operator. We show that the high-dimensional Markov operator can be obtained under a functional hierarchical tensor (FHT) ansatz with a hierarchical sketching algorithm. When the terminal condition admits an FHT ansatz, the proposed operator outputs an FHT ansatz for the PDE solution through an efficient functional tensor network contraction procedure. In addition, the proposed operator-based approach also provides an efficient way to solve the Kolmogorov forward equation when the initial distribution is in an FHT ansatz. We apply the proposed approach successfully to two challenging time-dependent Ginzburg–Landau models with hundreds of variables.
Abstract. We propose a particle-based workflow for approximating the time-dependent law of finite-volume discretizations of the Dean–Kawasaki model. After discretization, the state is a nonnegative vector whose total mass is conserved by the finite-volume scheme, and it is therefore supported on a probability simplex. To enable tensor-network density estimation, we map the simplex to an unconstrained Euclidean space using a centered logarithmic transform and then apply a wavelet transform that organizes degrees of freedom by spatial scale. On the transformed variables, we fit the probability density with a functional hierarchical tensor over a wavelet basis, i.e., a hierarchical-Tucker/tree-tensor-network representation of the coefficient tensor of a fixed univariate basis expansion. We illustrate the method on 1D and 2D examples with [Formula: see text] degrees of freedom, including cases with external potentials and pairwise interactions. The method accurately captures the site-wise correlations and other observables of the true model. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/Xun-Tang123/FHT_for_deans_equation and in the supplementary materials ( FHT_for_deans_equation-main.zip [27.5MB]). [Formula: see text]
The numerical representation of high-dimensional Gibbs distributions is challenging due to the curse of dimensionality manifesting through the intractable normalization constant calculations. This work addresses this challenge by performing a particle-based high-dimensional parametric density estimation subroutine, and the input to the subroutine is Gibbs samples generated by leveraging advanced sampling techniques. Specifically, to generate Gibbs samples, we employ ensemble-based annealed importance sampling, a population-based approach for sampling multimodal distributions. These samples are then processed using functional hierarchical tensor sketching, a tensor-network-based density estimation method for high-dimensional distributions, to obtain the numerical representation of the Gibbs distribution. We successfully apply the proposed approach to complex Ginzburg-Landau models with hundreds of variables. In particular, we show that the approach proposed is successful at addressing the metastability issue under difficult numerical cases.
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.
We present a multiscale treatment of electron correlations based on hyperbolic wavelet expansions of Jastrow-type correlation functions. Wavelets provide hierarchical basis sets that can be locally adapted to the length- and energy-scales of physical phenomena. Combined with hyperbolic tensor products and local adaptive refinement near the interelectron cusp, these wavelet bases enable sparse representations of Jastrow factors. The computational efficiency of wavelets in electronic structure calculations is demonstrated within the coupled electron-pair approximation (local ansatz). Based on a diagrammatic multiresolution analysis, we discuss various kinds of sparsity features for matrix elements required by the local ansatz. Sparsity originates from the hierarchical structure and vanishing moments property of wavelet bases. This led us to a recurrence scheme for the evaluation of matrix elements with almost linear computational complexity with respect to the size of the underlying isotropic 3d-wavelet basis. Numerical studies for selected diagrams are presented for a homogeneous electron gas.
Tensor network methods are powerful and efficient tools for studying the properties and dynamics of statistical and quantum systems, in particular in one and two dimensions. In recent years, these methods have been applied to lattice gauge theories, yet these theories remain a challenge in ( 2 + 1 ) dimensions. In this article, we present a new (decorated) tensor network algorithm, in which the tensors encode the lattice gauge amplitude expressed in the fusion basis. This has several advantages—firstly, the fusion basis does diagonalize operators measuring the magnetic fluxes and electric charges associated to a hierarchical set of regions. The algorithm allows therefore a direct access to these observables. Secondly the fusion basis is, as opposed to the previously employed spin network basis, stable under coarse-graining. Thirdly, due to the hierarchical structure of the fusion basis, the algorithm does implement predefined disentanglers. We apply this new algorithm to lattice gauge theories defined for the quantum group SU ( 2 ) k and identify a weak and a strong coupling phase for various levels k . As we increase the level k , the critical coupling g c decreases linearly, suggesting the absence of a deconfining phase for the continuous group SU ( 2 ) . Moreover, we illustrate the scaling behaviour of the Wilson loops in the two phases.