Nonzero Columns Research Articles

Matrices whose eigenvalues are those of a quadratic matrix polynomial

It is shown how the eigenvalues of sparse matrices, composed of a small number of nonzero block rows and columns, can be obtained from quadratic matrix polynomials with coefficients of a size determined by the number of nonzero rows and columns. This can present advantages, such as revealing eigenvalue properties without the need to compute them, better upper bounds for eigenvalues than those derived directly from the matrix with a smaller computational effort, and lower bounds that cannot otherwise directly be derived from the given matrix. The focus will be on upper and lower eigenvalue bounds, which are useful when eigenvalues are computed iteratively or when an application requires them to be in a particular region of the complex plane.

Joint Multislot Scheduling and Precoding for Unicast and Multicast Scenarios in Multiuser MISO Systems

This paper studies the joint multislot design of user scheduling and precoding to minimize the time needed to serve all the users for unicast and multicast transmission in single-cell multiuser MISO downlink systems. In the literature, the joint design of scheduling and precoding is typically undertaken based on feedback from previous slots. In a system with time-varying channels and QoS requirements, joint multislot designs can achieve better performance since they have the flexibility to schedule users over multiple slots and also can split users across slots efficiently. Further, a joint multislot design can provide a feasible solution even when the sequential design fails. In this paper, scheduling is represented by a binary matrix where the rows represent users, columns represent slots and entries represent scheduling of users in the slots. Noticing that the users may not be permuted across slots for time-varying channels, service time needed for scheduling is rendered as the highest column index corresponding to non-zero columns. With the help of binary scheduling matrix, service time minimization is formulated as a structured mixed-Boolean fractional programming. Further, by exploiting the hidden convex-concave structure in the problem, a convex-concave procedure-based iterative algorithm is proposed. Finally, we vindicate the necessity and illustrate the superiority in performance of joint multislot design over the sequential solution through Monte-Carlo simulations.

Column $\ell_{2,0}$-Norm Regularized Factorization Model of Low-Rank Matrix Recovery and Its Computation

This paper is concerned with the column $\ell_{2,0}$-regularized factorization model of low-rank matrix recovery problems and its computation. The column $\ell_{2,0}$-norm of factor matrices is introduced to promote column sparsity of factors and low-rank solutions. For this nonconvex discontinuous optimization problem, we develop an alternating majorization-minimization (AMM) method with extrapolation, and a hybrid AMM in which a majorized alternating proximal method is proposed to seek an initial factor pair with less nonzero columns and the AMM with extrapolation is then employed to minimize of a smooth nonconvex loss. We provide the global convergence analysis for the proposed AMM methods and apply them to the matrix completion problem with non-uniform sampling schemes. Numerical experiments are conducted with synthetic and real data examples, and comparison results with the nuclear-norm regularized factorization model and the max-norm regularized convex model show that the column $\ell_{2,0}$-regularized factorization model has an advantage in offering solutions of lower error and rank within less time.

Rees algebras of filtrations of covering polyhedra and integral closure of powers of monomial ideals

The aims of this work are to study Rees algebras of filtrations of monomial ideals associated with covering polyhedra of rational matrices with nonnegative entries and nonzero columns using combinatorial optimization and integer programming and to study powers of monomial ideals and their integral closures using irreducible decompositions and polyhedral geometry. We study the Waldschmidt constant and the ic-resurgence of the filtration associated with a covering polyhedron and show how to compute these constants using linear programming. Then, we show a lower bound for the ic-resurgence of the ideal of covers of a graph and prove that the lower bound is attained when the graph is perfect. We also show lower bounds for the ic-resurgence of the edge ideal of a graph and give an algorithm to compute the asymptotic resurgence of squarefree monomial ideals. A classification of when Newton’s polyhedron is the irreducible polyhedron is presented using integral closure.

Dimension Reduced Channel Feedback for Reconfigurable Intelligent Surface Aided Wireless Communications

Reconfigurable intelligent surface (RIS) has recently received extensive research interest due to its capability to intelligently change the wireless propagation environment. For RIS-aided wireless communications in frequency division duplex (FDD) mode, channel feedback at the user equipment (UE) is essential for the base station (BS) to acquire the downlink channel state information (CSI). In this paper, a dimension reduced channel feedback scheme is proposed to reduce the channel feedback overhead by exploiting the single-structured sparsity of BS-RIS-UE cascaded channel. Since different UEs share the same sparse BS-RIS channel but have their respective RIS-UE channels, there are only limited non-zero column vectors in the BS-RIS-UE cascaded channel matrix, and different UEs share the same indexes of the non-zero columns. Thus, the downlink CSI can be decomposed into <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">user-independent</i> channel information (i.e., the indexes of non-zero columns) and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">user-specific</i> channel information (i.e., the non-zero column vectors), where the former for all UEs can be fed back by only one UE, while the latter can be fed back with a fairly low overhead by different UEs, respectively. Simulation results show that, compared with the conventional method, the proposed scheme can reduce channel feedback overhead by more than 80% for RIS-aided wireless communications.

The Augment Algorithm and its Role in Cognitive Diagnosis

Both the augment algorithm and the reduction algorithm can be used to obtain non-zero knowledge states vector in testing. In particular, the augment algorithm based on the reachability matrix can imply the structure of the Q-matrix and its non-zero columns, thus proving the set of Q-matrix columns forms an algebraic structure (Lattice). Applying the augment algorithm based on the reachability matrix and the test Q-matrix, respectively, we can obtain the theoretical construct validity of the test Q-matrix (i.e., the degree of the test Q-matrix fitting the cognitive model) and use the results to evaluate test quality. We can also use the algorithm when constructing and evaluating cognitive models, as well as when developing cognitive diagnostic models. Moreover, the augment algorithm and its reverse algorithm (reduction algorithm) are suitable for analyzing and evaluating retrofitting data.

Channel Estimation for RIS Assisted Wireless Communications—Part II: An Improved Solution Based on Double-Structured Sparsity

Reconfigurable intelligent surface (RIS) can manipulate the wireless communication environment by controlling the coefficients of RIS elements. However, due to the large number of passive RIS elements without signal processing capability, channel estimation in RIS assisted wireless communication system requires high pilot overhead. In the second part of this invited paper, we propose to exploit the double-structured sparsity of the angular cascaded channels among users to reduce the pilot overhead. Specifically, we first reveal the double-structured sparsity, i.e., different angular cascaded channels for different users enjoy the completely common non-zero rows and the partially common non-zero columns. By exploiting this double-structured sparsity, we further propose the double-structured orthogonal matching pursuit (DS-OMP) algorithm, where the completely common non-zero rows and the partially common non-zero columns are jointly estimated for all users. Simulation results show that the pilot overhead required by the proposed scheme is lower than existing schemes.

Decomposition numbers and global properties

If G is a finite group, p is a prime, and B is a Brauer p-block with decomposition matrix DB, we study non-zero columns of DB and possible interactions with the structure of B.

Fast orthogonal recurrent neural networks employing a novel parametrisation for orthogonal matrices

Training Recurrent Neural Networks (RNNs) is challenging due to the vanishing/exploding gradient problem. Recent progress suggests to solve this problem by constraining the recurrent transition matrix to be unitary/orthogonal during training, but all of which are either limited-capacity, or involve time-consuming operators, e.g., evaluation for the derivation of lengthy matrix chain multiplication, the matrix exponential, or the singular value decomposition. This paper addresses this problem based on the exponentials of sparse antisymmetric matrices with one or more nonzero columns and an equal number of nonzero rows from a geometric view. An analytical expression is presented to simplify the computation of the sparse antisymmetric matrix exponential, which is actually a novel formula for parameterizing orthogonal matrices. The algorithms of this paper are fast, tunable, and full-capacity, where the target variable is updated by optimizing a matrix multiplier, instead of using the explicit gradient descent. Experiments demonstrate the superior performance of our proposed algorithms.

Bayesian mmWave Channel Estimation via Exploiting Joint Sparse and Low-Rank Structures

We consider the problem of channel estimation for millimeter wave (mmWave) systems, where both the base station and the mobile station employ a single radio frequency (RF) chain to reduce the hardware cost and power consumption. Recent real-world channel measurements reveal that the mmWave channels incur a certain amount of spread over the angular domains due to the scattering clusters. The angular spreads give rise to a joint sparse and low-rank channel matrix in the angular domain. To utilize this joint sparse and low-rank structure, we address the channel estimation problem within a Bayesian framework. Specifically, we adopt a matrix factorization formulation and translate the problem of channel estimation into one of searching for two-factor matrices. To encourage a joint sparse and low-rank solution, independent sparsity-promoting priors are placed on entries of the two-factor matrices, which aims to promote sparse factor matrices with only a few non-zero columns. Based on the proposed prior model, we develop a variational Bayesian inference method for the mmWave channel estimation. The simulation results show that our proposed method presents a considerable performance improvement over the state-of-the-art compressed sensing-based channel estimation methods.

Quadratic polynomial maps with Jacobian rank two

Let K be any field and x=(x1,x2,…,xn). We classify all matrices M∈Matm,n(K[x]) whose entries are polynomials of degree at most 1, for which rkM≤2. As a special case, we describe all such matrices M, which are the Jacobian matrix JH (the matrix of partial derivatives) of a polynomial map H from Kn to Km.Among other things, we show that up to composition with linear maps over K, M=JH has only two nonzero columns or only three nonzero rows in this case. In addition, we show that trdegKK(H)=rkJH for quadratic polynomial maps H over K such that 12∈K and rkJH≤2.Furthermore, we prove that up to conjugation with linear maps over K, nilpotent Jacobian matrices N of quadratic polynomial maps, for which rkN≤2, are triangular (with zeroes on the diagonal), regardless of the characteristic of K. This generalizes several results by others.In addition, we prove the same result for Jacobian matrices N of quadratic polynomial maps, for which N2=0. This generalizes a result by others, namely the case where 12∈K and N(0)=0.

Independent component analysis employing exponentials of sparse antisymmetric matrices

Abstract Independent component analysis (ICA) is a standard method for separating a multivariate signal into additive components that are non-Gaussian and independent from each other. This paper introduced a novel algorithm to perform ICA employing matrix exponentials, which performs similarly to geodesic based methods but based on a different insight. First, we showed that the ICA problem can be formulated as an optimization problem in the space of orthogonal matrices whose determinants are one, which can be further transformed into an equivalent problem in the space of antisymmetric matrices. Then, an efficient approach was presented for iteratively solving this problem using the antisymmetric matrices with one or more nonzero columns and rows. Especially, we proved that in the sense of local optimization it is sufficient to employ antisymmetric matrices with only one nonzero column and row. The analytical expressions of exponentials of such special antisymmetric matrices were also explicitly established in this paper. Compared to other competing algorithms, experimental results indicated that the proposed method can achieve separation with superior performance in term of the precision and running speed.

A Randomized Subspace Learning Based Anomaly Detector for Hyperspectral Imagery

This paper proposes a randomized subspace learning based anomaly detector (RSLAD) for hyperspectral imagery (HSI). Improved from robust principal component analysis, the RSLAD assumes that the background matrix is low-rank, and the anomaly matrix is sparse with a small portion of nonzero columns (i.e., column-wise). It also assumes the anomalies do not lie in the column subspace of the background and aims to find a randomized subspace of the background to detect the anomalies. First, random techniques including random sampling and random Hadamard projections are implemented to construct a coarse randomized columns subspace of the background with reduced computational cost. Second, anomaly columns are searched and removed from the coarse randomized column subspace by solving a series of least squares problems, resulting in a purified randomized column subspace. Third, the nonzero columns in the anomaly matrix are located by projecting all the pixels on the orthogonal subspace of the purified subspace, and the anomalies are finally detected based on the L2 norm of the columns in the anomaly matrix. The detection performance of RSLAD is compared with four state-of-the-art methods, including global Reed-Xiaoli (GRX), local RX (LRX), collaborative-representation based detector (CRD), and low-rank and sparse matrix decomposition base anomaly detector (LRaSMD). Experimental results show good detection performance of RSLAD with lower computational cost. Therefore, the proposed RSLAD offers an alternative option for hyperspectral anomaly detection.

Randomized subspace-based robust principal component analysis for hyperspectral anomaly detection

A randomized subspace-based robust principal component analysis (RSRPCA) method for anomaly detection in hyperspectral imagery (HSI) is proposed. The RSRPCA combines advantages of randomized column subspace and robust principal component analysis (RPCA). It assumes that the background has low-rank properties, and the anomalies are sparse and do not lie in the column subspace of the background. First, RSRPCA implements random sampling to sketch the original HSI dataset from columns and to construct a randomized column subspace of the background. Structured random projections are also adopted to sketch the HSI dataset from rows. Sketching from columns and rows could greatly reduce the computational requirements of RSRPCA. Second, the RSRPCA adopts the columnwise RPCA (CWRPCA) to eliminate negative effects of sampled anomaly pixels and that purifies the previous randomized column subspace by removing sampled anomaly columns. The CWRPCA decomposes the submatrix of the HSI data into a low-rank matrix (i.e., background component), a noisy matrix (i.e., noise component), and a sparse anomaly matrix (i.e., anomaly component) with only a small proportion of nonzero columns. The algorithm of inexact augmented Lagrange multiplier is utilized to optimize the CWRPCA problem and estimate the sparse matrix. Nonzero columns of the sparse anomaly matrix point to sampled anomaly columns in the submatrix. Third, all the pixels are projected onto the complemental subspace of the purified randomized column subspace of the background and the anomaly pixels in the original HSI data are finally exactly located. Several experiments on three real hyperspectral images are carefully designed to investigate the detection performance of RSRPCA, and the results are compared with four state-of-the-art methods. Experimental results show that the proposed RSRPCA outperforms four comparison methods both in detection performance and in computational time.

Extreme points of the local differential privacy polytope

We study the convex polytope of n×n stochastic matrices that define locally ϵ-differentially private mechanisms. We first present invariance properties of the polytope and results reducing the number of constraints needed to define it. Our main results concern the extreme points of the polytope. In particular, we completely characterise these for matrices with 1, 2 or n non-zero columns.

Markov Chain Model for the Decoding Probability of Sparse Network Coding

Random Linear Network Coding (RLNC) has been proved to offer an efficient communication scheme, leveraging an interesting robustness against packet losses. However, it suffers from a high computational complexity and some novel approaches, which follow the same idea, have been recently proposed. One of such solutions is Tunable Sparse Network Coding (TSNC), where only few packets are combined in each transmissions. The amount of data packets to be combined in each transmissions can be set from a density parameter/distribution, which could be eventually adapted. In this work we present an analytical model that captures the performance of SNC on an accurate way. We exploit an absorbing Markov process where the states are defined by the number of useful packets received by the decoder, i.e the decoding matrix rank, and the number of non-zero columns at such matrix. The model is validated by means of a thorough simulation campaign, and the difference between model and simulation is negligible. A mean square error less than $4 \cdot 10^{-4}$ in the worst cases. We also include in the comparison some of more general bounds that have been recently used, showing that their accuracy is rather poor. The proposed model would enable a more precise assessment of the behavior of sparse network coding techniques. The last results show that the proposed analytical model can be exploited by the TSNC techniques in order to select by the encoder the best density as the transmission evolves.

Low-Rank Matrix Recovery With Simultaneous Presence of Outliers and Sparse Corruption

We study a data model in which the data matrix D can be expressed as D = L + S + C, where L is a low rank matrix, S an element-wise sparse matrix and C a matrix whose non-zero columns are outlying data points. To date, robust PCA algorithms have solely considered models with either S or C, but not both. As such, existing algorithms cannot account for simultaneous element-wise and column-wise corruptions. In this paper, a new robust PCA algorithm that is robust to simultaneous types of corruption is proposed. Our approach hinges on the sparse approximation of a sparsely corrupted column so that the sparse expansion of a column with respect to the other data points is used to distinguish a sparsely corrupted inlier column from an outlying data point. We also develop a randomized design which provides a scalable implementation of the proposed approach. The core idea of sparse approximation is analyzed analytically where we show that the underlying ell_1-norm minimization can obtain the representation of an inlier in presence of sparse corruptions.

New Lower Bounds on Circuit Size of Multi-output Functions

Let B n, m be the set of all Boolean functions from {0, 1} n to {0, 1} m , B n = B n, 1 and U 2 = B 2∖{⊕, ≡}. In this paper, we prove the following two new lower bounds on the circuit size over U 2. 1. A lower bound \(C_{U_{2}}(f) \ge 5n-o(n)\) for a linear function f ∈ B n − 1,logn . The lower bound follows from the following more general result: for any matrix A ∈ {0, 1} m × n with n pairwise different non-zero columns and b ∈ {0, 1} m , $$C_{U_{2}}(Ax \oplus b)\ge 5(n-m).$$ 2. A lower bound \(C_{U_{2}}(f) \ge 7n-o(n)\) for f ∈ B n, n . Again, this is a consequence of the following result: for any f ∈ B n satisfying a certain simple property, $$C_{U_{2}}(g(f)) \ge \min \{C_{U_{2}}(f|_{x_{i} = a, x_{j} = b}) \colon x_{i} \neq x_{j}, a,b, \in \{0,1\}\} +2n-\varTheta (1)$$ where g(f) ∈ B n, n is defined as follows: g(f) = (f ⊕ x 1, … , f ⊕ x n ) (to get a 7n − o(n) lower bound it remains to plug in a known function f ∈ B n, 1 with \(C_{U_{2}}(f) \ge 5n-o(n)\)).

Optimal Periodic Sensor Scheduling in Networks of Dynamical Systems

We consider the problem of finding optimal time-periodic sensor schedules for estimating the state of discrete-time dynamical systems. We assume that multiple sensors have been deployed and that the sensors are subject to resource constraints, which limits the number of times each can be activated over one period of the periodic schedule. We seek an algorithm that strikes a balance between estimation accuracy and total sensor activations over one period. We make a correspondence between active sensors and the nonzero columns of the estimator gain. We formulate an optimization problem in which we minimize the trace of the error covariance with respect to the estimator gain while simultaneously penalizing the number of nonzero columns of the estimator gain. This optimization problem is combinatorial in nature, and we employ the alternating direction method of multipliers (ADMM) to find its locally optimal solutions. Numerical results and comparisons with other sensor scheduling algorithms in the literature are provided to illustrate the effectiveness of our proposed method.

Speaker adaptation based on regularized speaker-dependent eigenphone matrix estimation

Eigenphone-based speaker adaptation outperforms conventional maximum likelihood linear regression (MLLR) and eigenvoice methods when there is sufficient adaptation data. However, it suffers from severe over-fitting when only a few seconds of adaptation data are provided. In this paper, various regularization methods are investigated to obtain a more robust speaker-dependent eigenphone matrix estimation. Element-wise l1 norm regularization (known as lasso) encourages the eigenphone matrix to be sparse, which reduces the number of effective free parameters and improves generalization. Squared l2 norm regularization promotes an element-wise shrinkage of the estimated matrix towards zero, thus alleviating over-fitting. Column-wise unsquared l2 norm regularization (known as group lasso) acts like the lasso at the column level, encouraging column sparsity in the eigenphone matrix, i.e., preferring an eigenphone matrix with many zero columns as solution. Each column corresponds to an eigenphone, which is a basis vector of the phone variation subspace. Thus, group lasso tries to prevent the dimensionality of the subspace from growing beyond what is necessary. For nonzero columns, group lasso acts like a squared l2 norm regularization with an adaptive weighting factor at the column level. Two combinations of these methods are also investigated, namely elastic net (applying l1 and squared l2 norms simultaneously) and sparse group lasso (applying l1 and column-wise unsquared l2 norms simultaneously). Furthermore, a simplified method for estimating the eigenphone matrix in case of diagonal covariance matrices is derived, and a unified framework for solving various regularized matrix estimation problems is presented. Experimental results show that these methods improve the adaptation performance substantially, especially when the amount of adaptation data is limited. The best results are obtained when using the sparse group lasso method, which combines the advantages of both the lasso and group lasso methods. Using speaker-adaptive training, performance can be further improved.