arXiv subjects

Daizhan Cheng

Publications and source records attributed to Daizhan Cheng.

At least 19 recordsLinked to original sources

arXiv2026

From Eigenvalues/Eigenvectors of Hypermatrices to Canonical Form of Tensors

The rings of non-square matrices based on dimension-keeping (DK-) semi-tensor product (STP) are considered. Using the ring structure to hypermatrices, four kinds of eigenvalues/eigenvectors (EEs), namely, ordinary EE (OEE), universal EE (UEE), diagonal EE (DEE), horizontal DEE (HDEE) of hypermatrices are proposed with respect to preassigned special matricizings of hypermatrices respectively. Kronecker canonical form (KCF) of non-square pencils is used to calculate OEE. Then the monic decomposition algorithm (MDA) is used to pick out UEE, DEE, and MEE respectively. Finally, the KCF of non-square pencils is used to construct the KCF of a tensor, which reveals all the EEs of the tensor. KCF of tensors not only has similar properties of the Jordan canonical form of matrices but also includes Jordan canonical form as its special case. All the EEs of a hypermatrix are straightforward computable via its KCF.

math.RA↗

arXiv2026

A Numerical Solution to KPD

A stationary value based algorithm (SVA) is provided to solve the nearest Kronecker product decomposition (KPD) problem of vector form hypermatrices. Using the algorithm successively, the finite sum KPD is also solved. Then the permutation matrix is introduced. Using it, the KPD of matrix form hypermatrices is converted to its equivalent KPD of vector forms, and then the SVA is also applicable to solve the same problems for vector form hypermatrix. Some numerical examples are presented to demonstrate the new algorithm and to compare it with existing methods.

math.NA↗

arXiv2026

On Dimension-Varying Control Systems: A Universal State Space Approach

This paper develops a unified framework for the analysis and design of dimension-varying control systems by constructing an intrinsic quotient state space, $\Omega$. A significant challenge in non-fixed-dimensional systems is the lack of a common metric space that enables comparison of states across dimensions without relying on arbitrary external embeddings. To address this, we propose a cross-dimensional pseudo-metric $d_{\mathcal{V}}$ on $\mathbb{R}^{\infty}$ and derive $\Omega$ by identifying zero-distance representatives. We demonstrate that $\Omega$ preserves the essential topological and metric geometry of Euclidean space, providing the necessary foundation to extend fundamental control notions to the dimension-varying case. Specifically, we establish conditions for controllability, observability and stabilizability, and we address the complexities of Lipschitz switching and disturbance decoupling within this common space. The framework is further extended to hierarchical dimension-varying networks. The practical utility of the results is illustrated through a generator-removal-and-reconnection scenario in a three-machine power system. This case study demonstrates the use of translated representatives and projection/lift benchmarks, quantifies event-wise $d_{\mathcal{V}}$-gaps, and provides a finite-schedule dwell-time consistency check to validate the system's structural transitions.

math.OC↗

arXiv2025

A Dimension-Keeping Semi-Tensor Product Framework for Compressed Sensing

In compressed sensing (CS), sparse signals can be reconstructed from significantly fewer samples than required by the Nyquist-Shannon sampling theorem. While non-sparse signals can be sparsely represented in appropriate transformation domains, conventional CS frameworks rely on the incoherence of the measurement matrix columns to guarantee reconstruction performance. This paper proposes a novel method termed Dimension-Keeping Semi-Tensor Product Compressed Sensing (DK-STP-CS), which leverages intra-group correlations while maintaining inter-group incoherence to enhance the measurement matrix design. Specifically, the DK-STP algorithm is integrated into the design of the sensing matrix, enabling dimensionality reduction while preserving signal recovery capability. For image compression and reconstruction tasks, the proposed method achieves notable noise suppression and improves visual fidelity. Experimental results demonstrate that DK-STP-CS significantly outperforms traditional CS and STP-CS approaches, as evidenced by higher Peak Signal-to-Noise Ratio (PSNR) values between the reconstructed and original images. The robustness of DK-STP-CS is further validated under noisy conditions and varying sampling rates, highlighting its potential for practical applications in resource-constrained environments.

cs.IT↗

arXiv2025

Universal Solution to Kronecker Product Decomposition

This paper provides a general solution for the Kronecker product decomposition (KPD) of vectors, matrices, and hypermatrices. First, an algorithm, namely, monic decomposition algorithm (MDA), is reviewed. It consists of a set of projections from a higher dimension Euclidian space to its factor-dimension subspaces. It is then proved that the KPD of vectors is solvable, if and only if, the project mappings provide the required decomposed vectors. Hence it provides an easily verifiable necessary and sufficient condition for the KPD of vectors. Then an algorithm is proposed to calculate the least square error approximated decomposition. Using it finite times a finite sum (precise) KPD of any vectors can be obtained. Then the swap matrix is used to make the elements of a matrix re-arranging, and then provides a method to convert the KPD of matrices to its corresponding KPD of vectors. It is proved that the KPD of a matrix is solvable, if and only if, the KPD of its corresponding vector is solvable. In this way, the necessary and sufficient condition, and the followed algorithms for approximate and finite sum KPDs for matrices are also obtained. Finally, the permutation matrix is introduced and used to convert the KPD of any hypermatrix to KPD of its corresponding vector. Similarly to matrix case, the necessary and sufficient conditions for solvability and the techniques for vectors and matrices are also applicable for hypermatrices, though some additional algorithms are necessary. Several numerical examples are included to demonstrate this universal KPD solving method.

math.NA↗

arXiv2025

Semi-Tensor-Product Based Convolutional Neural Networks

The semi-tensor product of vectors generalizes the conventional inner product, enabling algebraic operations between vectors of different dimensions. Building upon this foundation, we introduce a domain-based convolutional product and integrate it with the STP to formulate a padding-free convolutional operation. This new operation inherently avoids zero or other artificial padding, thereby eliminating redundant information and boundary artifacts commonly present in conventional convolutional neural networks. Based on this operation, we further develop an STP-based CNN framework that extends convolutional computation to irregular and cross-dimensional data domains. Applications to image processing and third-order signal identification demonstrate the proposed method's effectiveness in handling irregular, incomplete, and high-dimensional data without the distortions caused by padding.

eess.SY↗

arXiv2025

On Dimension-Free Transformer: An Application of STP to AI

The matrix expressions for every parts of a transformer are firstly described. Based on semi-tensor product (STP) of matrices the hypervectors are reconsidered and the linear transformation over hypervectors is constructed by using projection. Its properties and calculating formulas are obtained. Using projection-based transformation of hypervector (PBTH), the framework of dimension-free transformer (DFT) is proposed by verifying each linear transformation in a transformer and replacing it by a proper PBTH, which allows the inputs and outputs being of arbitrary dimensions. Using balanced information about all entries, DFT must be more efficient in dealing with signals.

cs.LG↗

arXiv2024

t-Product and t-STP of Cubic Matrices With Application to Hyper-Networked Systems

Motivated by the study of dynamic control systems, this paper proposes novel algebraic operations on cubic matrices to construct both linear and nonlinear controlled dynamics. The standard t-product of cubic matrices imposes strict dimensional constraints; to resolve this, we first introduce the dimension-keeping semi-tensor product (DK-STP), which generalizes the matrix product for arbitrary dimensions. However, the DK-STP yields decoupled subsystem dynamics because it fails to capture interactions across subsystems corresponding to frontal slices. To overcome this limitation, we propose the t-semi-tensor product (t-STP), an integration of the t-product and the DK-STP that allows for coupled subsystems and greater modeling flexibility. We systematically study the algebraic structures derived from the t-STP over cubic matrices, including groups, rings, modules, and Lie groups. Finally, we obtain t-STP-based dynamic control systems over cubic matrices and demonstrate the utility of this framework by applying it to a hyper-networked evolutionary game modeling supply chain interactions.

math.RA↗

arXiv2024

From Signal Space To STP-CS

Under the assumption that a finite signal with different sampling lengths or different sampling frequencies is considered as equivalent, the signal space is considered as the quotient space of $\mathbb{R}^{\infty}$ over equivalence. The topological structure and the properties of signal space are investigated. Using them some characteristics of semi-tensor product based compressed sensing (STP-CS) are revealed. Finally, a systematic analysis of the construction of sensing matrix based on balanced incomplete block design (BIBD) is presented.

eess.SY↗

arXiv2024

Signal Processing via Cross-Dimensional Projection

Using projection between Euclidian spaces of different dimensions, the signal compression and decompression become straightforward. This encoding/decoding technique requires no preassigned measuring matrix as in compressed sensing. Moreover, in application there is no dimension or size restrictions. General formulas for encoding/decoding of any finite dimensional signals are provided. Their main properties are revealed. Particularly, it is shown that under the equivalence assumption the technique provides the best approximation with least square error.

eess.SY↗

arXiv2024

Cross-Dimensional Mathematics: A Foundation For STP/STA

A new mathematical structure, called the cross-dimensional mathematics (CDM), is proposed. The CDM considered in this paper consists of three parts: hyper algebra, hyper geometry, and hyper Lie group/Lie algebra. Hyper algebra proposes some new algebraic structures such as hyper group, hyper ring, and hyper module over matrices and vectors with mixed dimensions (MVMDs). They have sets of classical groups, rings, and modules as their components and cross-dimensional connections among their components. Their basic properties are investigated. Hyper geometry starts from mixed dimensional Euclidian space, and hyper vector space. Then the hyper topological vector space, hyper inner product space, and hyper manifold are constructed. They have a joined cross-dimensional geometric structure. Finally, hyper metric space, topological hyper group and hyper Lie algebra are built gradually, and finally, the corresponding hyper Lie group is introduced. All these concepts are built over MVMDs, and to reach our purpose in addition to existing semi-tensor products (STPs) and semi-tensor additions (STAs), a couple of most general STP and STA are introduced. Some existing structures/results about STPs/STAs have also been resumed and integrated into this CDM.

math.RA↗

arXiv2024

Observer-Based Realization of Control Systems

A novel model reduction framework for large-scale complex systems is proposed by introducing function-type dynamic control systems via the dimension-keeping semi-tensor product (DK-STP) of matrices. Utilizing bridge matrices, the DK-STP facilitates the construction of an approximate observer-based realization (OR) of a linear control system in the form of a function-type control system, where the functions serve as observers. A necessary and sufficient condition is established for the OR-system to admit exact observer dynamics. When an exact OR-system does not exist, an extended OR-system is developed by incorporating the original system's observers into its state. Furthermore, a minimal feedback extended OR-system is constructed, and its relationship to Kalman's minimal realization is analyzed. Finally, the proposed approach is extended to nonlinear control-affine systems.

math.OC↗

arXiv2023

Design of zero-determinant strategies and its application to networked repeated games

Using semi-tensor product (STP) of matrices, the profile evolutionary equation (PEE) for repeated finite games is obtained. By virtue of PEE, the zero-determinant (ZD) strategies are developed for general finite games. A formula is then obtained to design ZD strategies for general finite games with multi-player and asymmetric strategies. A necessary and sufficient condition is obtained to ensure the availability of the designed ZD strategies. It follows that player $i$ is able to unilaterally design $k_i-1$ (one less than the number of her strategies) dominating linear relations about the expected payoffs of all players. Finally, the fictitious opponent player is proposed for networked repeated games (NRGs). A technique is proposed to simplify the model by reducing the number of frontier strategies.

math.OC↗

arXiv2023

On Universal Eigenvalues and Eigenvectors of Hypermatrices

A generalized eigenvector of a hypermatrix, called the universal (U-) eigenvector, is proposed, which extended the notion of diagonal (D-) eigenvectors in the literature. Using the semi-tensor product, the homogeneous U-eigenequation can be converted into a general eigenequation of matrix $(A-\lambda B)x=0$. A general technique for solving this equation is proposed, which leads to two kinds of eigenvalues: essential and quasi eigenvalues. The technique to convert nonhomogeneous eigenequation to homogeneous ones is also revealed. Then a hypervector decomposing method, called the monic decomposition algorithm (MDA), is developed. Using the MDA, the U-eigenproblem (including the D-eigenproblem) can be converted into general matrix eigenproblems. Some examples are presented, demonstrating the geometric meaning and potential applications of the U-eigenvalue/eigenvector.

math.NA↗

arXiv2023

From DK-STP to Non-square General Linear Algebras and General Linear Groups

A new matrix product, called dimension-keeping semi-tensor product (DK-STP), is proposed. Under DK-STP, the set of $m\times n$ matrices becomes a semi-group $G({m\times n},\mathbb{F})$, and a ring, denoted by $R(m\times n,\mathbb{F})$. Moreover, the Lie bracket can also be defined, which turns the ring into a Lie algebra, called non-square (or STP) general linear algebra, denoted by $\mathrm{gl}(m\times n, \mathbb{F})$. Then the action of semi-group $G(m\times n,\mathbb{F})$ on dimension-free Euclidian space, denoted by $\mathbb{R}^{\infty}$, is discussed. This action leads to discrete-time and continuous time S-systems. Their trajectories are calculated, and their invariant subspaces are revealed. As byproduct of this study, some important concepts for square matrices, such as eigenvalue, eigenvector, determinant, invertibility, etc., have been extended to non-square matrices. Particularly, it is surprising that the famous Cayley-Hamilton theory can also been extended to non-square matrices. Finally, a Lie group, called the non-square (or STP) general Lie group and denoted by $\mathrm{GL}(m\times n,\mathbb{F})$, is constructed, which has $\mathrm{GL}(m\times n,\mathbb{F})$ as its Lie algebra. Their relations with classical Lie group $\mathrm{GL}(m,\mathbb{F})$ and Lie algebra $\mathrm{gl}(m,\mathbb{F})$ are revealed.

math.RA↗

arXiv2023

Contracted Product of Hypermatrices via STP of Matrices

An equivalent definition of hypermatrices is introduced. The matrix expression of hypermatrices is proposed. Using permutation matrices, the conversion of different matrix expressions is revealed. The various contracted products of hypermatrices are realized by semi-tensor products (STP) of matrices via matrix expressions of hypermatrices.

math.NA↗

arXiv2023

Transition System Representation of Boolean Control Networks

First, the topological structure of a transition system is studied. Then, two types of transition system (TS) representations of Boolean networks (BNs) and Boolean control networks (BCNs) are investigated. The first kind of representation is state-based, which converts a BCN into a TS with either distinct control or non-distinct control. The second representation is output-based, which is also called the simulation of the original BCN. Some applications are also studied.

eess.SY↗

arXiv2023

Aggregated (Bi-)Simulation of Finite Valued Networks

The paper provides a method to approximate a large-scale finite-valued network by a smaller model called the aggregated simulation, which is a combination of aggregation and (bi-)simulation. First, the algebraic state space representation (ASSR) of a transition system is presented. Under output equivalence, the quotient system is obtained, which is called the simulation of the original transition system. The ASSR of the quotient system is obtained. The aggregated (bi-)simulation is execueted in several steps: a large scale finite-valued network is firstly aggregated into several blocks, each of which is considered as a network where the in-degree nodes and out-degree nodes are considered as the block inputs and block outputs respectively. Then the dynamics of each block is converted into its quotient system, called its simulation. Then the overall network can be approximated by the quotient systems of each blocks, which is called the aggregated simulation. If the simulation of a block is a bi-simulation, the approximation becomes a lossless transformation. Otherwise, the quotient system is only a (non-deterministic) transition system, and it can be replaced by a probabilistic networks. Aggregated simulation can reduce the dimension of the original network, while a tradeoff between computation complexity and approximation error need to be decided.

eess.SY↗