Full-rank Matrices Research Articles

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.

Fixed-time convergence ZNN model for solving rectangular dynamic full-rank matrices inversion

The Moore–Penrose inverse of dynamic matrices has found widespread application and has garnered significant attention. The zeroing neural network (ZNN) has proven to be an effective solution for computing the Moore–Penrose inverse in dynamic matrices. This paper proposes a novel unified fixed-time ZNN (UFTZNN) model designed to achieve fixed-time convergence and solve both left and right inverse problems using a single model. Theoretical analysis of the convergence and robustness of the UFTZNN model is rigorously presented. Numerical simulations comparing the UFTZNN with existing ZNN models confirm its superiority in addressing left and right inverse problems, convergence time, and robustness. The UFTZNN model is applied to the inverse kinematic tracking problem of a six-degree-of-freedom manipulator-based photoelectric tracking system to demonstrate its potential applications and effectiveness.

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.

A closer look at some new lower bounds on the minimum singular value of a matrix

There is an extensive body of literature on estimating the eigenvalues of the sum of two symmetric matrices, P+Q, in relation to the eigenvalues of P and Q. Recently, the authors introduced two novel lower bounds on the minimum eigenvalue, λmin(P+Q), under the conditions that matrices P and Q are symmetric positive semi-definite and their sum P+Q is non-singular. These bounds rely on the Friedrichs angle between the range spaces of matrices P and Q, which are denoted by R(P) and R(Q), respectively. In addition, both results led to the derivation of several new lower bounds on the minimum singular value of full-rank matrices. One significant aspect of the two novel lower bounds on λmin(P+Q) is the distinction of the case where R(P) and R(Q) have no principal angles between 0 and π2. This work offers an explanation for the aforementioned scenario and presents a classification of all matrices that meet the specified criteria. Additionally, we offer insight into the rationale behind selecting the decomposition for the subspace R(Q), which is employed to formulate the lower bounds for λmin(P+Q). At last, an example that showcases the potential for improving these two lower bounds is presented.

Variable selection with the knockoffs: Composite null hypotheses

The fixed-X knockoff filter is a flexible framework for variable selection with false discovery rate (FDR) control in linear models with arbitrary design matrices (of full column rank) and it allows for finite-sample selective inference via the Lasso estimates. In this paper, we extend the theory of the knockoff procedure to tests with composite null hypotheses, which are usually more relevant to real-world problems. The main technical challenge lies in handling composite nulls in tandem with dependent features from arbitrary designs. We develop two methods for composite inference with the knockoffs, namely, shifted ordinary least-squares (S-OLS) and feature-response product perturbation (FRPP), building on new structural properties of test statistics under composite nulls. We also propose two heuristic variants of S-OLS method that outperform the celebrated Benjamini–Hochberg (BH) procedure for composite nulls, which serves as a heuristic baseline under dependent test statistics. Finally, we analyze the loss in FDR when the original knockoff procedure is naively applied on composite tests.

Congruency-Constrained TU Problems Beyond the Bimodular Case

A long-standing open question in integer programming is whether integer programs with constraint matrices with bounded subdeterminants are efficiently solvable. An important special case thereof are congruency-constrained integer programs [Formula: see text] with a totally unimodular constraint matrix T. Such problems are shown to be polynomial-time solvable for m = 2, which led to an efficient algorithm for integer programs with bimodular constraint matrices, that is, full-rank matrices whose n × n subdeterminants are bounded by two in absolute value. Whereas these advances heavily rely on existing results on well-known combinatorial problems with parity constraints, new approaches are needed beyond the bimodular case, that is, for m > 2. We make first progress in this direction through several new techniques. In particular, we show how to efficiently decide feasibility of congruency-constrained integer programs with a totally unimodular constraint matrix for m = 3 using a randomized algorithm. Furthermore, for general m, our techniques also allow for identifying flat directions of infeasible problems and deducing bounds on the proximity between solutions of the problem and its relaxation. Funding: This project received funding from the Swiss National Science Foundation [Grants 200021_184622 and P500PT_206742], the European Research Council under the European Union’s Horizon 2020 research and innovation program [Grant 817750], and the Deutsche Forschungsgemeinschaft (German Research Foundation) under Germany’s Excellence Strategy–GZ 2047/1 [Grant 390685813].

Constructing solutions of the Yang–Baxter-like matrix equation for singular matrices

Let A=UVH be an n×n complex singular matrix of rank r, where U, V are two n×r matrices of full column rank. Some sufficient and necessary conditions are derived for X being a nontrivial (commuting) solution of the quadratic matrix equation AXA=XAX such that we can construct infinitely many (commuting) solutions of the matrix equation. When VHU is nonsingular, all the solutions of the matrix equation are characterized and can be found and constructed by solving smaller matrix equations. In particular, all the commuting solutions can be obtained by finding the projections that commute with VHU. The method of constructing the solutions is free of computations of Jordan canonical forms and matrix inverses.

Toward Complete-View and High-Level Pose-Based Gait Recognition

Model-based gait recognition methods usually adopt the pedestrian walking postures to identify human beings. However, existing methods did not explicitly resolve the large intra-class variance of human pose due to changes in camera view. In this paper, we propose a lower-upper generative adversarial network (LUGAN) to generate multi-view pose sequences for each single-view sample to reduce the cross-view variance. Based on the prior of camera imaging, we prove that the spatial coordinates between cross-view poses satisfy a linear transformation of a full-rank matrix. Hence, LUGAN employs the adversarial training to learn full-rank transformation matrices from the source pose and target views to obtain the target pose sequences. The generator of LUGAN is composed of graph convolutional (GCN) layers, fully connected (FC) layers and two-branch convolutional (CNN) layers: GCN layers and FC layers encode the source pose sequence and target view, then CNN layers take as input the encoded features to learn a lower triangular matrix and an upper one, finally the transformation matrix is formulated by multiplying the lower and upper triangular matrices. For the purpose of adversarial training, we develop a conditional discriminator that distinguishes whether the pose sequence is true or generated. Furthermore, to facilitate the high-level correlation learning, we propose a plug-and-play module, named multi-scale hypergraph convolution (HGC), to replace the spatial graph convolutional layer in baseline, which can simultaneously model the joint-level, part-level and body-level correlations. Extensive experiments on three large gait recognition datasets (i.e., CASIA-B, OUMVLP-Pose and NLPR) demonstrate that our method outperforms the baseline model by a large margin.

Theoretically and Computationally Convenient Geometries on Full-Rank Correlation Matrices

Theoretically and Computationally Convenient Geometries on Full-Rank Correlation Matrices

Almost full rank matrices arising from transitive tournaments

Let F be a field and suppose a := ( a 1 , a 2 , … ) is a sequence of non-zero elements in F . For a tournament T on [ n ] , associate the n × n symmetric matrix M T ( a ) (resp. skew-symmetric matrix M T , skew ( a ) ) with zero diagonal as follows: for i<j, if the edge ij is directed as i → j in T, then set M T ( a ) = a i (resp. M T , skew ( a ) = a i ), else set M T ( a ) = a j (resp. M T , skew ( a ) = a j ). Let M n ( a ) (resp. M n , skew ( a ) ) be the family consisting of all the n × n symmetric matrices M T ( a ) (resp. skew-symmetric matrices M T , skew ( a ) ) as T varies over all tournaments on [ n ] . We show that any matrix in M n ( a ) or M n , skew ( a ) corresponding to a transitive tournament has a rank at least n−1, and this is best possible. This settles in a strong form a conjecture posed in Balachandran et al. [An ensemble of high-rank matrices arising from tournaments; Linear Algebra Appl. 2023;658:310–318]. As a corollary, any matrix in these families has rank at least ⌊ log 2 ⁡ ( n ) ⌋ .

Block-encoding dense and full-rank kernels using hierarchical matrices: applications in quantum numerical linear algebra

Many quantum algorithms for numerical linear algebra assume black-box access to a block-encoding of the matrix of interest, which is a strong assumption when the matrix is not sparse. Kernel matrices, which arise from discretizing a kernel function k(x,x&amp;#x2032;), have a variety of applications in mathematics and engineering. They are generally dense and full-rank. Classically, the celebrated fast multipole method performs matrix multiplication on kernel matrices of dimension N in time almost linear in N by using the linear algebraic framework of hierarchical matrices. In light of this success, we propose a block-encoding scheme of the hierarchical matrix structure on a quantum computer. When applied to many physical kernel matrices, our method can improve the runtime of solving quantum linear systems of dimension N to O(&amp;#x03BA;polylog&amp;#x2061;(N&amp;#x03B5;)), where &amp;#x03BA; and &amp;#x03B5; are the condition number and error bound of the matrix operation. This runtime is near-optimal and, in terms of N, exponentially improves over prior quantum linear systems algorithms in the case of dense and full-rank kernel matrices. We discuss possible applications of our methodology in solving integral equations and accelerating computations in N-body problems.

Sparse factorization of square matrices with application to neural attention modeling

Square matrices appear in many machine learning problems and models. Optimization over a large square matrix is expensive in memory and in time. Therefore an economic approximation is needed. Conventional approximation approaches factorize the square matrix into a number matrices of much lower ranks. However, the low-rank constraint is a performance bottleneck if the approximated matrix is intrinsically high-rank or close to full rank. In this paper, we propose to approximate a large square matrix with a product of sparse full-rank matrices. In the approximation, our method needs only N(logN)2 non-zero numbers for an N×N full matrix. Our new method is especially useful for scalable neural attention modeling. Different from the conventional scaled dot-product attention methods, we train neural networks to map input data to the non-zero entries of the factorizing matrices. The sparse factorization method is tested for various square matrices, and the experimental results demonstrate that our method gives a better approximation when the approximated matrix is sparse and high-rank. As an attention module, our new method defeats Transformer and its several variants for long sequences in synthetic data sets and in the Long Range Arena benchmarks. Our code is publicly available22https://github.com/RuslanKhalitov/SparseFactorization..

Over-Parametrized Matrix Factorization in the Presence of Spurious Stationary Points

Motivated by the emerging role of interpolating machines in signal processing and machine learning, this work considers the computational aspects of over-parametrized matrix factorization. In this context, the optimization landscape may contain spurious stationary points (SSPs), which are proved to be full-rank matrices. The presence of these SSPs means that it is impossible to hope for any global guarantees in over-parametrized matrix factorization. For example, when initialized at an SSP, the gradient flow will be trapped there forever. Nevertheless, despite these SSPs, we establish in this work that the gradient flow of the corresponding merit function converges to a global minimizer, provided that its initialization is rank-deficient and sufficiently close to the feasible set of the optimization problem. We numerically observe that a heuristic discretization of the proposed gradient flow, inspired by primal-dual algorithms, is successful when initialized randomly. Our result is in sharp contrast with the local refinement methods which require an initialization close to the optimal set of the optimization problem. More specifically, we successfully avoid the traps set by the SSPs because the gradient flow remains rank-deficient at all times, and not because there are no SSPs nearby. The latter is the case for the local refinement methods. Moreover, the widely-used restricted isometry property plays no role in our main result.

Maximizing throughput in RLNC-based multi-source multi-relay with guaranteed decoding

In this paper, an efficient and practical solution to cope with random linear network coding (RLNC) drawbacks and maximize the system throughput is proposed. No header overheads and no retransmissions are required as a result of the introduction of a unique codebook at the nodes in charge of RLNC coding and decoding. The codebook contains the seeds that allow for the pseudo-random generation of full-rank matrices of network coding elements over the finite Galois field GF(q), where q is the field size, for N generations. Each relay generates the same matrix and uses its corresponding row of N coding GF(q) coefficients. The destination, whose seeds codebook is synchronized on the same index value, generates the same full-rank matrix to decode the source nodes packets. Results show that the number of successfully transmitted packets is not generally affected by the choice of the generation size or the underlying finite field, which demonstrates the outperformance of the proposed technique over the traditional RLNC procedure.

Principal Bundle Structure of Matrix Manifolds

In this paper, we introduce a new geometric description of the manifolds of matrices of fixed rank. The starting point is a geometric description of the Grassmann manifold Gr(Rk) of linear subspaces of dimension r&lt;k in Rk, which avoids the use of equivalence classes. The set Gr(Rk) is equipped with an atlas, which provides it with the structure of an analytic manifold modeled on R(k−r)×r. Then, we define an atlas for the set Mr(Rk×r) of full rank matrices and prove that the resulting manifold is an analytic principal bundle with base Gr(Rk) and typical fibre GLr, the general linear group of invertible matrices in Rk×k. Finally, we define an atlas for the set Mr(Rn×m) of non-full rank matrices and prove that the resulting manifold is an analytic principal bundle with base Gr(Rn)×Gr(Rm) and typical fibre GLr. The atlas of Mr(Rn×m) is indexed on the manifold itself, which allows a natural definition of a neighbourhood for a given matrix, this neighbourhood being proved to possess the structure of a Lie group. Moreover, the set Mr(Rn×m) equipped with the topology induced by the atlas is proven to be an embedded submanifold of the matrix space Rn×m equipped with the subspace topology. The proposed geometric description then results in a description of the matrix space Rn×m, seen as the union of manifolds Mr(Rn×m), as an analytic manifold equipped with a topology for which the matrix rank is a continuous map.

Joint-diagonalizability-constrained multichannel nonnegative matrix factorization based on time-variant multivariate complex sub-Gaussian distribution

Multichannel nonnegative matrix factorization (MNMF) is a common blind source separation technique that employs full-rank spatial covariance matrices (SCMs). The full-rank SCMs can simulate reverberant mixing systems where the sources are spatially spread. In conventional MNMF, spectrograms of observed signals are modeled by some types of distribution, e.g., the Gaussian distribution and Student’s t distribution. However, MNMF based on the sub-Gaussian distribution has not been proposed because its cost function is difficult to minimize. In this paper, we address the statistical model extension of MNMF to the sub-Gaussian distribution to improve the source separation accuracy. In the proposed method, the generalized Gaussian distribution is utilized as the sub-Gaussian model. Moreover, to design an auxiliary function for the proposed cost function, we introduce the joint-diagonalizability constraint to SCMs similarly to FastMNMF. Two types of update rule for the proposed MNMF are derived on the basis of the majorization-minimization (MM) and majorization-equalization (ME) algorithms. Since the optimization speed of each parameter affects the source separation performance, we experimentally analyze the best combination of MM- and ME-algorithm-based update rules in the proposed method. Experiments of blind source separation reveal that the proposed MNMF based on the sub-Gaussian model can outperform conventional methods.

PhysEx: A novel procedure to estimate full-rank physical matrices of a structure from an incomplete modal model

This work is devoted to presenting and applying a novel methodology that estimates a linear physical model, represented by full-rank mass, stiffness and damping matrices, capable of replicating the dynamic behaviour of an incomplete modal model. No hypothesis is made with respect to the damping matrix, so the procedure is applicable assuming any linear viscous damping model. It is shown that, if the number of considered modes satisfies a certain restriction with respect to the number of degrees of freedom, the problem has infinite solutions that differ in their exogenous eigenvalues. By controlling their effect on the dynamic behaviour of the model in the frequency range of interest, a proper final solution is obtained. The methodology is applied to a three-degrees-of-freedom discrete model and a discretized cantilever beam. Comparisons are made in both frequency and time domains which reveal that the methodology provides physical models that accurately replicate the dynamic behaviour of the incomplete modal model. Even for complete models, where the solvability condition is not met, the methodology manages to provide meaningful results in some situations. To prove the usefulness of the estimated models, a mass is added on one DOF of both examples to show that they reproduce the same effects as the original modified ones. Finally, in the interests of reproducibility, all models and algorithms presented in this article have been made publicly available.

Inverse-Free Discrete ZNN Models Solving for Future Matrix Pseudoinverse via Combination of Extrapolation and ZeaD Formulas.

Time-varying matrix pseudoinverse (TVMP) problem has been investigated by many researchers in recent years, but a new class of matrix termed Zhang matrix has been found and not been handled by some conventional models, e.g., Getz-Marsden dynamic model. On the other way, future matrix pseudoinverse (FMP), as a more challenging and intractable discrete-time problem, deserves more attention due to its significant role-playing on some engineering applications, such as redundant manipulator. Based on the zeroing neural network (ZNN), this article concentrates on designing new discrete ZNN models appropriately for computing the FMPs of all matrices of full rank, including the Zhang matrix. First, an inverse-free continuous ZNN model for computing TVMP is derived. Subsequently, Zhang et al. discretization (ZeaD) formulas and equidistant extrapolation formulas are used to discretize the continuous ZNN model to two discrete ZNN models for computing FMPs with different truncation errors. The numerical experiments are conducted for the five conventional discrete models and two new discrete ZNN models. Distinct numerical results substantiate the effectiveness and choiceness of newly proposed models. Finally, one of the newly proposed models is implemented on simulating and physical instances of robot manipulators, respectively, to show its practicability.

BLAS IV: A BLAS for Rk Matrix Algebra

Basic Linear Algebra Subroutines (BLAS) are well-known low-level workhorse subroutines for linear algebra vector-vector, matrixvector and matrix-matrix operations for full rank matrices. The advent of block low rank (Rk) full wave direct solvers, where most blocks of the system matrix are Rk, an extension to the BLAS III matrix-matrix work horse routine is needed due to the agony of Rk addition. This note outlines the problem of BLAS III for Rk LU and solve operations and then outlines an alternative approach, which we will call BLAS IV. This approach utilizes the thrill of Rk matrix-matrix multiply and uses the Adaptive Cross Approximation (ACA) as a methodology to evaluate sums of Rk terms to circumvent the agony of low rank addition.

On the Detection of the Correct Number of Factors in Two-Facet Models by Means of Parallel Analysis.

Methods for optimal factor rotation of two-facet loading matrices have recently been proposed. However, the problem of the correct number of factors to retain for rotation of two-facet loading matrices has rarely been addressed in the context of exploratory factor analysis. Most previous studies were based on the observation that two-facet loading matrices may be rank deficient when the salient loadings of each factor have the same sign. It was shown here that full-rank two-facet loading matrices are, in principle, possible, when some factors have positive and negative salient loadings. Accordingly, the current simulation study on the number of factors to extract for two-facet models was based on rank-deficient and full-rank two-facet population models. The number of factors to extract was estimated from traditional parallel analysis based on the mean of the unreduced eigenvalues as well as from nine other rather traditional versions of parallel analysis (based on the 95th percentile of eigenvalues, based on reduced eigenvalues, based on eigenvalue differences). Parallel analysis based on the mean eigenvalues of the correlation matrix with the squared multiple correlations of each variable with the remaining variables inserted in the main diagonal had the highest detection rates for most of the two-facet factor models. Recommendations for the identification of the correct number of factors are based on the simulation results, on the results of an empirical example data set, and on the conditions for approximately rank-deficient and full-rank two-facet models.