arXiv subjects

Ziqin He

Publications and source records attributed to Ziqin He.

7 recordsLinked to original sources

arXiv2026

Transform-Based Multilinear Algebra via Tensor Decomposition

Transform-based tensor products, including the T-product and its more general form, namely the higher-order tensor-tensor product, have become fundamental tools for multilinear data analysis in applications such as image processing, signal reconstruction, and robotics. While invertible transforms enable tensor computations to be carried out via matrix operations in the transform domain, the resulting storage and computational costs remain prohibitive for high-dimensional, higher-order tensors. To address this challenge, we integrate low-rank tensor decomposition techniques, specifically tensor train decomposition (TTD) and hierarchical Tucker decomposition (HTD), into transform-based multilinear algebra to improve computational and memory efficiency. In particular, we develop TTD- and HTD-based formulations for the T-product and its associated key algebra, such as block diagonalization and tensor singular value decomposition, by operating directly on the factor matrices or tensors of the decompositions. The framework is further generalized to the higher-order tensor-tensor product and applied to multilinear model order reduction problems. We demonstrate the effectiveness and efficiency of our framework with numerical examples.

math.NA↗

arXiv2025

Quadratic Formula-based Nonlinear Approximation

This paper presents a quadratic formula-based nonlinear representation for a given single-variable function f(x), $-1 \leq x \leq 1$. First, we construct the explicit polynomial coefficient functions a(x), b(x), and c(x) using a least-squares approach. Then, f is reconstructed by solving the degree-2 polynomial equation a(x) f^2 - b(x) f - c(x)=0 for any $x \in [-1,1]$, where an index function is used to select the correct sign in the quadratic formula. The quadratic formula-based nonlinear approximation (degree-2 in f) outperforms classical orthogonal polynomial-based least-squares approximation (degree-0 in f) and rational approximation (degree-1 in f) for functions with sharp transitions or discontinuities. As a potential application, we apply the degree-2 representation to data denoising. Instead of relying on more complex "edge-preserving" metric-based optimization techniques, the smooth coefficient functions a(x), b(x), and c(x) enable effective least-squares-based denoising on the low-dimensional manifold described by the algebraic variety a(x) f^2 - b(x) f - c(x)=0. Denoising the index function, which determines the appropriate root to select, can be achieved using classical statistical or modern classification/clustering techniques. Numerical results and data denoising examples are provided to demonstrate the effectiveness of the degree-2 nonlinear approximation technique. The new nonlinear, quadratic formula-based representation also raises theoretical and numerical questions, including strategies for identifying numerically stable representations, developing optimal algorithms to construct the polynomial coefficient functions a(x), b(x), and c(x), and achieving economical representation and denoising of the index function.

math.NA↗

arXiv2025

Tensor Dynamic Mode Decomposition

Dynamic mode decomposition (DMD) has become a powerful data-driven method for analyzing the spatiotemporal dynamics of complex, high-dimensional systems. However, conventional DMD methods are limited to matrix-based formulations, which might be inefficient or inadequate for modeling inherently multidimensional data including images, videos, and higher-order networks. In this letter, we propose tensor dynamic mode decomposition (TDMD), a novel extension of DMD to third-order tensors based on the recently developed T-product framework. By incorporating tensor factorization techniques, TDMD achieves more efficient computation and better preservation of spatial and temporal structures in multiway data for tasks such as state reconstruction and dynamic component separation, compared to standard DMD with data flattening. We demonstrate the effectiveness of TDMD on both synthetic and real-world datasets.

eess.SY↗

arXiv2025

Data-driven model order reduction for T-Product-Based dynamical systems

Model order reduction plays a crucial role in simplifying complex systems while preserving their essential dynamic characteristics, making it an invaluable tool in a wide range of applications, including robotic systems, signal processing, and fluid dynamics. However, traditional model order reduction techniques like balanced truncation are not designed to handle tensor data directly and instead require unfolding the data, which may lead to the loss of important higher-order structural information. In this article, we introduce a novel framework for data-driven model order reduction of T-product-based dynamical systems (TPDSs), which are often used to capture the evolution of third-order tensor data such as images and videos through the T-product. Specifically, we develop advanced T-product-based techniques, including T-balanced truncation, T-balanced proper orthogonal decomposition, and the T-eigensystem realization algorithm for input-output TPDSs by leveraging the unique properties of T-singular value decomposition. We demonstrate that these techniques offer significant memory and computational savings while achieving reduction errors that are comparable to those of conventional methods. The effectiveness of the proposed framework is further validated through synthetic and real-world examples.

eess.SY↗

arXiv2025

Tensor-based homogeneous polynomial dynamical system analysis from data

Numerous complex real-world systems, such as those in biological, ecological, and social networks, exhibit higher-order interactions that are often modeled using polynomial dynamical systems or homogeneous polynomial dynamical systems (HPDSs). However, identifying system parameters and analyzing key system-theoretic properties remain challenging due to their inherent nonlinearity and complexity, particularly for large-scale systems. To address these challenges, we develop an innovative computational framework in this article that leverages advanced tensor decomposition techniques, namely tensor train and hierarchical Tucker decompositions, to facilitate efficient identification and analysis of HPDSs that can be equivalently represented by tensors. Specifically, we introduce memory-efficient system identification techniques for directly estimating system parameters represented through tensor decompositions from time-series data. Additionally, we develop necessary and sufficient conditions for determining controllability and observability using the tensor decomposition-based representations of HPDSs, accompanied by detailed complexity analyses that demonstrate significant reductions in computational demands. The effectiveness and efficiency of our framework are validated through numerical examples.

math.DS↗

arXiv2025

Data-driven Control of T-Product-based Dynamical Systems

Data-driven control is a powerful tool that enables the design and implementation of control strategies directly from data without explicitly identifying the underlying system dynamics. While various data-driven control techniques, such as stabilization, linear quadratic regulation, and model predictive control, have been extensively developed, these methods are not inherently suited for multi-linear dynamical systems, where the states are represented as higher-order tensors. In this article, we propose a novel framework for data-driven control of T-product-based dynamical systems (TPDSs), where the system evolution is governed by the T-product between a third-order dynamic tensor and a third-order state tensor. In particular, we offer necessary and sufficient conditions to determine the data informativity for system identification, stabilization by state feedback, and T-product quadratic regulation of TPDSs with detailed complexity analyses. Finally, we validate our framework through numerical examples.

eess.SY↗

arXiv2024

Data-driven Analysis of T-Product-based Dynamical Systems

A wide variety of data can be represented using third-order tensors, spanning applications in chemometrics, psychometrics, and image processing. However, traditional data-driven frameworks are not naturally equipped to process tensors without first unfolding or flattening the data, which can result in a loss of crucial higher-order structural information. In this article, we introduce a novel framework for the data-driven analysis of T-product-based dynamical systems (TPDSs), where the system evolution is governed by the T-product between a third-order dynamic tensor and a third-order state tensor. In particular, we examine the data informativity of TPDSs concerning system identification, stability, controllability, and stabilizability and illustrate significant computational improvements over traditional approaches by leveraging the unique properties of the T-product. The effectiveness of our framework is demonstrated through numerical examples.

eess.SY↗