Rank Of Matrices Research Articles

On the Maximum Probability of Full Rank of Random Matrices over Finite Fields

The problem of determining the conditions under which a random rectangular matrix is of full rank is a fundamental question in random matrix theory, with significant implications for coding theory, cryptography, and combinatorics. In this paper, we study the probability of full rank for a K×N random matrix over the finite field Fq, where q is a prime power, under the assumption that the rows of the matrix are sampled independently from a probability distribution P over FqN. We demonstrate that the probability of full rank attains a local maximum when the distribution P is uniform over FqN∖{0}, for any K⩽N and prime power q. Moreover, we establish that this local maximum is also a global maximum in the special case where K=2. These results highlight the optimality of the uniform distribution in maximizing full rank and represent a significant step toward solving the broader problem of maximizing the probability of full rank for random matrices over finite fields.

Conditional Euclidean distance optimization via relative tangency

We introduce a theory of relative tangency for projective algebraic varieties. The dual variety X Z ∨ X_Z^\vee of a variety X X relative to a subvariety Z Z is the set of hyperplanes tangent to X X at a point of Z Z . We also introduce the concept of polar classes of X X relative to Z Z . We explore the duality of varieties of low rank matrices relative to special linear sections. In this framework, we study the critical points of the Euclidean Distance (ED) function from a data point to X X , lying on Z Z . The locus where the number of such conditional critical points is positive is called the ED data locus of X X given Z Z . The generic number of such critical points defines the conditional ED degree of X X given Z Z . We show the irreducibility of ED data loci, and we compute their dimensions and degrees in terms of relative characteristic classes.

Low-rank decomposition on the antisymmetric product of geminals for strongly correlated electrons

We investigated some variational methods to compute a wavefunction based on antisymmetric product of geminals (APG). The Waring decomposition on the APG wavefunction leads a finite sum of antisymmetrized geminal power (AGP) wavefunctions, each for which the variational principle can be applied. We call this as AGP-CI method which provides a variational solution of the APG wavefunction efficiently. However, number of AGP wavefunctions in the exact AGP-CI formalism become exponentially large in case of many-electron systems. Therefore, we also investigate the low-rank APG wavefunction, in which the coefficient matrices of geminals are factorized by the Schur decomposition. Interestingly, only a few non-zero eigenvalues (up to half number of electrons) were found from the Schur decomposition on the APG wavefunction. We developed some methods to approximate the APG wavefunction by lowering the ranks of coefficient matrices of geminals, and demonstrate their performance. Our new geminal method based on the low-rank decomposition can drastically reduce the number of variational parameters, although no efficient formalism has been elaborated so far, due to the difficulty of optimization caused by the inability to compute analytical derivatives.

Equivariant spaces of matrices of constant rank

Equivariant spaces of matrices of constant rank

The Equivalence Canonical Forms of Two Sets of Five Quaternion Matrices With Applications

ABSTRACTIn this paper, the equivalence canonical forms of two sets of five quaternion matrices are considered. The simultaneous decompositions of these two sets of quaternion matrices are established. Some practical algorithms for computing the decompositions are also presented. Using the equivalence canonical forms, some practical solvability conditions for two systems of generalized Sylvester‐type quaternion matrix equations are derived in terms of the ranks of the coefficient matrices. The general solutions to the systems are also provided. Some numerical examples are given to illustrate the results. The proposed simultaneous decompositions are applied to construct a new framework of color watermarks embedding. In this framework, five color watermarks can be embedded into one host image simultaneously with high efficiency. Moreover, a counterexample for the existence of the equivalence canonical form of the general matrix quaternity with the same row or column numbers is presented. The results extend the existing findings on the equivalence canonical forms of general matrix arrays.

The Rank of Matrices Over Positive Cones of Commutative Semirings

The Rank of Matrices Over Positive Cones of Commutative Semirings

Finite-order method to calculate approximate density matrices in the Fock-space multireference coupled cluster theory

An efficient approach to calculate approximate pure-state and transition reduced density matrices in the framework of the multireference relativistic Fock-space coupled cluster (FS CC) theory is proposed. The method is based on the effective operator formalism and consists of the direct substitution of the FS CC Ansatz for a wave operator into the effective operator expression with the subsequent truncation of expansion at the terms quadratic in cluster amplitudes. The final density matrix is defined by active-space density matrices of different ranks ‘dressed’ with contributions from cluster operators. The method gives a connected expression for pure-state density matrices, provided that the intermediate normalisation condition is fulfilled. Moreover, under some additional assumptions, the connectivity can also be ensured for calculated transition property matrix elements and natural transition spinors. The developed technique allows for fast and accurate calculations of one-particle reduced density matrices for a wide range of electronic states. A pilot application of the new technique to construct averaged atomic natural orbital (ANO) basis sets for fully relativistic electronic structure calculations is presented.

On the cardinality of matrices with prescribed rank and partial trace over a finite field

On the cardinality of matrices with prescribed rank and partial trace over a finite field

A fast method to estimate the Moore-Penrose inverse for well-determined numerical rank matrices based on the Tikhonov regularization

A fast method to estimate the Moore-Penrose inverse for well-determined numerical rank matrices based on the Tikhonov regularization

Oriented posets, rank matrices and q-deformed Markov numbers

Oriented posets, rank matrices and q-deformed Markov numbers

On the rank structure of the Moore-Penrose inverse of singular k-banded matrices

On the rank structure of the Moore-Penrose inverse of singular k-banded matrices

Rank Matrix Approach for Endometriosis: Integrating Data and Constructing Diagnostic Models

Background: Endometriosis is a debilitating gynecological disorder characterized by chronic pain, infertility, and the growth of endometrial tissue outside the uterus. Accurate and early detection of this condition is crucial for effective management and treatment. Methods: We developed a gene rank matrix-based model to integrate endometriosis cohorts across multiple platforms. After removing batch effects, we identified 83 genes associated with endometriosis and further refined a diagnostic model using 11 of these genes. The model was trained on two platforms and validated on two others using SVM, Random Forest, Logistic Regression, and gradient-boosting machine learning algorithms. Results: The integration via the gene rank matrix effectively mitigated batch effects. Utilizing a gradient boosting classifier with a subset of 11 genes, the model demonstrated commendable diagnostic efficacy, achieving an Area Under the Curve (AUC) of 0.77, an accuracy of 0.72, and an F1 score of 0.72 for the training dataset. When subjected to validation, the model maintained its performance, yielding an AUC of 0.769, an accuracy of 0.719, and an F1 score of 0.732. These 11 genes were found to be associated with immunosuppression. Conclusion: Our approach to integrating gene rank matrices effectively consolidates endometriosis data across diverse platforms. The diagnostic model, harnessing the predictive power of 11 specific genes, surpasses alternative models, thereby offering promising prospects for aiding clinical diagnosis of endometriosis. Further validation is imperative to elucidate the functional significance of these 11 genes. Our study underscores the potential of data integration coupled with machine learning techniques in advancing the diagnosis of intricate diseases, such as endometriosis.

Four-point condition matrices of edge-weighted trees

Abstract Formulas for the determinant of distance matrix D T {D}_{T} of tree T T are known in the unweighted case and in the case when the edges of T T have commuting variable weights. Associated with the four-point condition (4PC) and a tree T T are two matrices, the Max4PC T {{\rm{Max4PC}}}_{T} and the Min4PC T {{\rm{Min4PC}}}_{T} . These are not full rank matrices and their rank, a basis B B , and formulas for the determinant when restricted to the rows and columns of B B are known. In this work, we generalize both these matrices to the case when the edges of T T have commuting variable weights and determine edge-weighted counterparts of known results.

Kronecker product of matrices and solutions of Sylvestertype matrix polynomial equations

We investigate the solutions of the Sylvester-type matrix polynomial equation $$A(\lambda)X(\lambda)+Y(\lambda)B(\lambda)=C(\lambda),$$ where\ $A(\lambda),$ \ $ B(\lambda),$\ and \ $C(\lambda)$ are the polynomial matrices with elements in a ring of polynomials \ $\mathcal{F}[\lambda],$ \ $\mathcal{F}$ is a field,\ $X(\lambda)$\ and \ $Y(\lambda)$ \ are unknown polynomial matrices. Solving such a matrix equation is reduced to the solving a system of linear equations $$G \left\|\begin{array}{c}\mathbf{x} \\ \mathbf{y} \end{array} \right\|=\mathbf{c}$$ over a field $\mathcal{F}.$ In this case, the Kronecker product of matrices is applied. In terms of the ranks of matrices over a field $\mathcal{F},$ which are constructed by the coefficients of the Sylvester-type matrix polynomial equation,the necessary and sufficient conditions for the existence of solutions \ $X_0(\lambda)$\ and \ $Y_0(\lambda)$ \ of given degrees to the Sylvester-type matrix polynomial equation are established. The solutions of this matrix polynomial equation are constructed from the solutions of the linear equations system.As a consequence of the obtained results, we give the necessary and sufficient conditions for the existence of the scalar solutions \ $X_0$\ and \ $Y_0,$ \ whose entries are elements in a field $\mathcal{F},$ to the Sylvester-type matrix polynomial equation.

Computational methods for periodic systems: Recent developments

In the past few decades, there has been a significant focus on computational methods for periodic systems. In some of these works, various considerations were outlined to establish and categorize algorithms for designing periodic discrete‐time state‐space systems. In a recent work, a direct algorithm was proposed to obtain low‐dimensional realizations from a proper lifted system in state‐space form. This algorithm, which lies in a category often called “fast” algorithms, takes advantage on the structure of lifted representations being highly efficient. In a 1993 work by Ching‐An Lin and Chwan‐Wen King, another direct algorithm was proposed that appears to be quite similar to the algorithm outlined recently. These two algorithms generate state‐space periodic realizations that are both uniform, meaning that the dimension of the state remains constant at each time instant. The algorithms proposed in these two works require both a rank test applied to two different types of matrices, and , respectively (in chronologic order). Although the former algorithm establishes an upper bound for the state dimension à priori, by determining the maximum rank of the matrices , the latter obtains an upper bound à posteriori, based on the rank of matrices . In this work, we proved that these matrices and have the same rank for all , and we use this information to revise the latter algorithm and assess whether it represents an improvement. The algorithm presented as well its modified versions that were coded using the Python programming language. Surprisingly, the revised algorithm did not demonstrate any significant improvement in terms of computational efficiency.

Low-rank matrices, tournaments, and symmetric designs

Low-rank matrices, tournaments, and symmetric designs

New Attacks on LowMC Using Partial Sets in the Single-Data Setting

The LowMC family of block ciphers was proposed by Albrecht et al. in Eurocrypt 2015, specifically targeting adoption in FHE and MPC applications due to its low multiplicative complexity. The construction operates a 3-bit quadratic S-box as the sole non-linear transformation in the algorithm. In contrast, both the linear layer and round key generation are achieved through multiplications of full rank matrices over GF(2). The cipher is instantiable using a diverse set of default configurations, some of which have partial non-linear layers i.e., in which the S-boxes are not applied over the entire internal state of the cipher. The significance of cryptanalysing LowMC was elevated by its inclusion into the NIST PQC digital signature scheme PICNIC in which a successful key recovery using a single plaintext/ciphertext pair is akin to retrieving the secret signing key. The current state-of-the-art attack in this setting is due to Dinur at Eurocrypt 2021, in which a novel way of enumerating roots of a Boolean system of equation is morphed into a key-recovery procedure that undercuts an ordinary exhaustive search in terms of time complexity for the variants of the cipher up to five rounds. In this work, we demonstrate that this technique can efficiently be enriched with a specific linearization strategy that reduces the algebraic degree of the non-linear layer as put forward by Banik et al. at IACR ToSC 2020(4). This amalgamation yields new attacks on certain instances of LowMC up to seven rounds.

On the Hankel matrices of finite rank and generalized Fibonacci sequences

On the Hankel matrices of finite rank and generalized Fibonacci sequences

Numerical solution of the boundary value problems for the biharmonic equations via quasiseparable representations

The paper incorporates new methods of numerical linear algebra for the approximation of the biharmonic equation with potential, namely, numerical solution of the Dirichlet problem for ddx4u(x)+c(x)u(x)=ϕ(x),0<x<1.High-order discrete finite difference operators are presented, constructed on the basis of discrete Hermitian derivatives, and the associated Discrete Biharmonic Operator (DBO). It is shown that the matrices associated with the discrete operator belong to a class of quasiseparable matrices of low rank matrices. The application of quasiseparable representation of rank structured matrices yields fast and stable algorithm for variable potentials c(x). Numerical examples corroborate the claim of high order accuracy of the algorithm, with optimal complexity O(N).

Delegation-Relegation for Boolean Matrix Factorization

The Boolean Matrix Factorization (BMF) problem aims to represent a n×m Boolean matrix as the Boolean product of two matrices of small rank k, where the product is computed using Boolean algebra operations. However, finding a BMF of minimum rank is known to be NP-hard, posing challenges for heuristic algorithms and exact approaches in terms of rank found and computation time, particularly as matrix size or the number of entries equal to 1 grows. In this paper, we present a new approach to simplifying the matrix to be factorized by reducing the number of 1-entries, which allows to directly recover a Boolean factorization of the original matrix from its simplified version. We introduce two types of simplification: one that performs numerous simplifications without preserving the original rank and another that performs fewer simplifications but guarantees that an optimal BMF on the simplified matrix yields an optimal BMF on the original matrix. Furthermore, our experiments show that our approach outperforms existing exact BMF algorithms.