Transform-Based Multilinear Algebra via Tensor Decomposition

Tensor-Based Reduced-Order Modeling for Optimization-Based Inverse Problems

We develop a tensor reduced-order modeling (TROM) framework for optimization-based inverse problems governed by parameter-dependent dynamical systems. The approach approximates the parameter-to-observation map directly in tensor-train format, using either TT-SVD or TT-Cross compression, and integrates the resulting representation into a regularized nonlinear least-squares formulation. Beyond accelerating forward evaluations, the low-rank tensor structure is used to reformulate the inverse problem in reduced coordinates, assemble the Gauss--Newton quantities without forming the full observation-space Jacobian, and perform TROM-based objective minimization over the discrete parameter grid. This tensor optimization step can be used either as a stand-alone approximate minimization procedure or as a data-informed initialization for a subsequent Gauss--Newton solve. The method is studied for two inverse problems: an inverse heat-transfer problem in a heterogeneous medium, where the unknown parameters describe the locations of multiple low-conductivity inclusions, and a FitzHugh--Nagumo parameter-estimation problem with a highly nonconvex optimization landscape. Numerical experiments assess the effects of ROM approximation error, measurement noise, regularization, initialization, spatial discretization, and increasing parameter dimension. The results show that TROM can reproduce the behavior of full-order inversion at a substantially reduced online cost. The experiments also demonstrate that reduced-coordinate inversion, tensor-based optimization, and appropriate regularization improve robustness in higher-dimensional, noisy, and strongly nonconvex regimes.

Iterative tensor network transformations for element-wise evaluation of elementary and filtering functions

Tensor networks are powerful formats for compressing large-scale data. However, their application to general data processing has been limited by the difficulty of performing nonlinear operations. Here, we introduce iterative tensor network transformations (ITNTs), a general algorithmic framework for the element-wise evaluation of elementary and nonlinear filtering functions on data encoded as tensor trains (TTs), a class of tensor networks. Our approach operates entirely in the compressed domain, enabling efficient computation on exponentially large datasets while maintaining a controlled computational cost. We demonstrate its power in two key areas: (I) evaluating highly nonlinear elementary and filtering functions on a 3D reactive flow field, enabling high-fidelity reaction rate computation and region filtering, and (II) finding extrema in complex optimization problems, such as solving Max-SAT instances on spaces up to $2^{70}$ configurations. These results establish ITNT as a foundational tool that provides tensor network methods with the capability for general-purpose data science and large-scale optimization.

Randomized QLP Decomposition for Third-Order Tensors with Unitary Transform

Nonlocal and Nonconvex Robust Tensor Completion with Tensor Dictionary Learning for Multi-Dimensional Image Recovery

Robust tensor completion aims to recover a clean tensor from noisy and incomplete observations, where the observed tensor is corrupted by Gaussian noise and sparse noise simultaneously. Existing methods only exploit one or two priors out of global tensor low-rankness, local properties, and nonlocal self-similarity, leading to suboptimal recovery performance. In this paper, we propose a nonconvex model combined with nonlocal self-similarity and tensor dictionary learning for robust tensor completion. Specifically, by partitioning the tensor into several overlapping cubes, the similar cubes are grouped together. Then, we unfold the cubes into matrices and stack these matrices into a third-order tensor. Subsequently, the minimax concave penalty (MCP) is employed on the singular values of all frontal slices of the sub-tensors in the transformed domain to explore the low-rankness of the underlying sub-tensor. The tensor dictionary learning based on Tucker decomposition is used to explore the local patterns of the underlying sub-tensor. Moreover, the MCP is employed onto each entry of the sparse noise tensor to explore the sparsity. A proximal alternating linearized minimization algorithm is adopted to solve the resulting model. Extensive numerical experiments demonstrate that the proposed method outperforms the competing state-of-the-art methods in both visual quality and quantitative metrics.

Computing with traceable tensor networks

A new SVD-based tensor decomposition method for tensor networks with arbitrary graph topologies is introduced, and it is found 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-Train Methods for 3D Linear Elasticity: Block and Global Operator Representations with Solver Performance Analysis

This work develops tensor-train (TT) formulations for solving large-scale three-dimensional linear elasticity problems discretized by isogeometric analysis. By exploiting the tensor-product structure of the basis functions and the low-rank structure of geometry-dependent coefficient fields, the stiffness operator, mass operator, force vector, and displacement solution are represented in TT format. Two solution strategies are investigated: a block-operator formulation, in which the coupled elasticity operator is stored as separated TT blocks, and a single-operator formulation, in which the full coupled system is stored as one monolithic TT operator. A matrix-free three-field TT conjugate-gradient solver is introduced for the block formulation, while AMEn is used for the single-operator formulation. Numerical examples demonstrate substantial compression of both operators and solutions compared with conventional sparse full-grid representations, showing that TT-based formulations provide an efficient and scalable approach for large-scale three-dimensional elasticity simulations.