Nonzero Entries Research Articles

Quadratic Spline Wavelets for Sparse Discretization of Jump–Diffusion Models

This paper is concerned with a construction of new quadratic spline wavelets on a bounded interval satisfying homogeneous Dirichlet boundary conditions. The inner wavelets are translations and dilations of four generators. Two of them are symmetrical and two anti-symmetrical. The wavelets have three vanishing moments and the basis is well-conditioned. Furthermore, wavelets at levels i and j where i - j > 2 are orthogonal. Thus, matrices arising from discretization by the Galerkin method with this basis have O 1 nonzero entries in each column for various types of differential equations, which is not the case for most other wavelet bases. To illustrate applicability, the constructed bases are used for option pricing under jump–diffusion models, which are represented by partial integro-differential equations. Due to the orthogonality property and decay of entries of matrices corresponding to the integral term, the Crank–Nicolson method with Richardson extrapolation combined with the wavelet–Galerkin method also leads to matrices that can be approximated by matrices with O 1 nonzero entries in each column. Numerical experiments are provided for European options under the Merton model.

New two‐stage approach to ECG denoising

A new algorithm for denoising electrocardiogram (ECG) signals contaminated by additive white Gaussian noise is proposed in this study. In the proposed algorithm, a clean ECG signal is modelled as a combination of a smooth signal representing the P-wave and the T-wave, and a group-sparse (GS) signal representing the QRS-complex, where a GS signal is a sparse signal in which its non-zero entries tend to concentrate in groups. Accordingly, the proposed approach consists of two stages. In the first stage, an algorithm previously developed by the author is adapted to extract the GS signal representing the QRS-complex, while in the second stage a new algorithm is developed to smooth the remaining signal. Each one of these two algorithms depends on a regularisation parameter, which is selected automatically in the proposed algorithms. Simulation results on real and simulated ECG data show that the proposed algorithm can be successfully utilised to denoise ECG data. In addition, the proposed algorithm is also shown to produce significantly improved results compared to existing techniques used for performing similar tasks.

A sparser matrix representation of the Mertens function

Redheffer introduced a sparse (0,1) matrix whose determinant provides an unorthodox representation of the Mertens function. The growth of the Mertens function is intimately tied to the locations of the nontrivial zeros of the Riemann zeta function.We introduce a (0,1) matrix whose determinant also provides a nontrivial representation of the Mertens function, but it is sparser than the matrix of Redheffer. It is sparser in two ways: the matrix has fewer nonzero entries (O(n) compared to O(nlog⁡n)), and it has substantially fewer eigenvalues different from 1. This construction is a corollary of a more general result that holds for multiplicative sequences.

Minimization of Fraction Function Penalty in Compressed Sensing.

In this paper, we study the minimization problem of a non-convex sparsity-promoting penalty function, i.e., fraction function, in compressed sensing. First, we discuss the equivalence of l0 minimization and fraction function minimization. It is proved that the optimal solution to fraction function minimization solves l0 minimization and the optimal solution to the regularization problem also solves fraction function minimization if the certain conditions are satisfied, which is similar to the regularization problem in a convex optimization theory. Second, we study the properties of the optimal solution to the regularization problem, including the first-order and second-order optimality conditions and the lower and upper bounds of the absolute value for its nonzero entries. Finally, we derive the closed-form representation of the optimal solution to the regularization problem and propose an iterative FP thresholding algorithm to solve the regularization problem. We also provide a series of experiments to assess the performance of the FP algorithm, and the experimental results show that the FP algorithm performs well in sparse signal recovery with and without measurement noise.

The Spectrum and Fine Spectrum of q-Cesàro Matrices with 0 < q < 1 on c0

One of the q-analogs of the Cesáro matrix of order one is the triangular matrix with nonzero entries where In this paper, we determine the spectrum and various spectral separations of the spectrum of which is q -analog of the Cesáro matrix regarded as a bounded linear operator on the sequence space c0.

Fault Classification for Transmission Lines Based on Group Sparse Representation

Fault classification is an important aspect of the protective relaying system for transmission lines. This paper proposes a new method based on group sparse representation for fault classification in transmission lines in which half-cycle superimposed current signals are measured for the classification task. When compared to conventional feature extraction methods, the proposed method in this paper alleviates the requirement to manually design feature. Signals are factorized over an over-complete basis in which elements are the fault signals themselves. The algorithm of classification is based on the idea that the training samples of a particular fault type approximately form a linear basis for any test sample belonging to that class. Solved by ${l}^{{2,1}}$ -minimization, the coefficient should be group sparse, and its non-zero entries correspond to particular group of correlated training samples. It is illustrated that the proposed classification method can be properly modified to deal with noise-containing signals. Moreover, dimension reduction is performed using random mapping technique. The results of several simulations are carried out by PSCAD/EMTDC and field data in real system indicate that the proposed method is accurate and fast for fault classification, and has a high robustness to noise.

Feature Selection by Canonical Correlation Search in High-Dimensional Multiresponse Models With Complex Group Structures

High-dimensional multiresponse models with complex group structures in both the response variables and the covariates arise from current researches in important fields such as genetics and medicine. However, no enough research has been done on such models. One of a few researches, if not the only one, is the article by Li, Nan, and Zhu where the sparse group Lasso approach is extended to such models. In this article, we propose a novel approach named the sequential canonical correlation search (SCCS) procedure. In the SCCS procedure, the nonzero group by group blocks of regression coefficients are searched stepwise using a canonical correlation measure. Each step of the procedure consists of a block selection and a sparsity identification. The model selection criterion, EBIC, is used as the stopping rule of the procedure. We establish the selection consistency of the SCCS procedure and conduct simulation studies for the comparison of existing methods. The SCCS procedure has two advantages over the sparse grouped Lasso method: (i) it is more accurate in the identification of nonzero coefficient blocks and their nonzero entries, and (ii) its implementation is not limited by the dimensionality of the models and requires much less computation. A real example in genetic studies is also considered. Supplementary materials for this article are available online.

Determinant formulas of some Toeplitz–Hessenberg matrices with Catalan entries

In this paper, we consider determinants of some families of Toeplitz–Hessenberg matrices having various translates of the Catalan numbers for the nonzero entries. These determinant formulas may also be rewritten as identities involving sums of products of Catalan numbers and multinomial coefficients. Combinatorial proofs may be given for several of the identities that are obtained.

The High Order Block RIP Condition for Signal Recovery

In this paper, we consider the recovery of block sparse signals, whose nonzero entries appear in blocks (or clusters) rather than spread arbitrarily throughout the signal, from incomplete linear measurement. A high order sufficient condition based on block RIP is obtained to guarantee the stable recovery of all block sparse signals in the presence of noise, and robust recovery when signals are not exactly block sparse via mixed $l_{2}/l_{1}$ minimization. Moreover, a concrete example is established to ensure the condition is sharp. The significance of the results presented in this paper lies in the fact that recovery may be possible under more general conditions by exploiting the block structure of the sparsity pattern instead of the conventional sparsity pattern.

On the C⁎-algebra of matrix-finite bounded operators

Let H be a separable Hilbert space with a fixed orthonormal basis. Let B(k)(H) denote the set of operators, whose matrices have no more than k non-zero entries in each line and in each column. The closure of the union (over k∈N) of B(k)(H) is a C⁎-algebra. We study some properties of this C⁎-algebra. We show that this C⁎-algebra is not an AW⁎-algebra, its group of invertibles is contractible. and it gives rise to an example of a C⁎-algebra with a dense ⁎-subalgebra and a maximal closed ideal with zero intersection.

Sparse power factorization: balancing peakiness and sample complexity

In many applications, one is faced with an inverse problem, where the known signal depends in a bilinear way on two unknown input vectors. Often at least one of the input vectors is assumed to be sparse, i.e., to have only few non-zero entries. Sparse power factorization (SPF), proposed by Lee, Wu, and Bresler, aims to tackle this problem. They have established recovery guarantees for a somewhat restrictive class of signals under the assumption that the measurements are random. We generalize these recovery guarantees to a significantly enlarged and more realistic signal class at the expense of a moderately increased number of measurements.

Exactly Solvable Record Model for Rainfall.

Daily precipitation time series are composed of null entries corresponding to dry days and nonzero entries that describe the rainfall amounts on wet days. Assuming that wet days follow a Bernoulli process with success probability p, we show that the presence of dry days induces negative correlations between record-breaking precipitation events. The resulting nonmonotonic behavior of the Fano factor of the record counting process is recovered in empirical data. We derive the full probability distribution P(R,n) of the number of records R_{n} up to time n, and show that for large n, it converges to a Poisson distribution with parameter ln(pn). We also study in detail the joint limit p→0, n→∞, which yields a random record model in continuous time t=pn.

Analysis of hard-thresholding for distributed compressed sensing with one-bit measurements

Abstract A simple hard-thresholding operation is shown to be able to uniformly recover $L$ signals $\textbf{x}_1,...,\textbf{x}_L \in{\mathbb{R}}^n$ that share a common support of size $s$ from $m = \mathscr{O}(s)$ one-bit measurements per signal if $L \geqslant \ln (en/s)$. This result improves the single signal recovery bounds with $m = \mathscr{O}(s\ln (en/s))$ measurements in the sense that asymptotically fewer measurements per non-zero entry are needed. Numerical evidence supports the theoretical considerations.

Minimum number of non-zero-entries in a 7 × 7 stable matrix

We prove that if a 7×7 sign pattern matrix is potentially stable, then it has at least 11 non-zero entries. The results for n×n matrix with n up to 6 are known previously. We prove the result by making a list of possible associated digraphs with at most 10 edges, and then use algebraic conditions to show all of these digraphs or matrices cannot be potentially stable. In relation to this, we also determine the minimum number of edges in a strongly connected digraph depending on its circumference.

Adaptive JSPA in distributed colocated MIMO radar network for multiple targets tracking

In this study, the authors present a distributed resource-awareness colocated multiple-input-multiple-output radar network for multiple targets tracking. Specifically, taking posterior Cramer-Rao lower bound as the performance metric, they propose a joint scheduling and power allocation (JSPA) scheme as a sparse optimisation problem. The sparsity of the solution enables the JSPA scheme to jointly optimise the schedule and corresponding transmit power of radars by the non-zero entries with associated amplitudes. The JSPA scheme has the robustness of feasibility even if there is not enough resource to achieve the prescribed performance requirements of targets. In this case, the JSPA scheme will automatically relax the performance bounds for individual targets at the lowest price and decide the number of targets to be tracked according to the priorities. They also propose a global optimal fusion rule that builds the connection between resource allocation and distributed tracking. Simulation results demonstrate the effectiveness of the proposed methods. The JSPA scheme can dynamically adjust the resource utilisation of the tracking of individual targets according to the accuracy demands. The performance of important targets is favoured to be satisfied, whereas a target with low priority may be dropped by the JSPA scheme if necessary.

Ray pattern matrices requiring Perron–Frobenius properties

A ray pattern matrix is a matrix whose nonzero entries are all unimodular. An n×n ray pattern matrix A is said to require the weak Perron–Frobenius property if for any n×n entrywise positive matrix K, the spectral radius ρ(K∘A) is one of the eigenvalues of K∘A; to require the strong Perron–Frobenius property if ρ(K∘A) is an algebraically simple eigenvalue of K∘A and the left and right eigenvectors associated to ρ(K∘A) are strictly nonzero. In this paper, necessary and sufficient conditions for ray pattern matrices requiring the weak/strong Perron–Frobenius property are given.

Matrix permutation meets block compressed sensing

Traditional block compressed sensing (CS) schemes encode nature images via a fixed sampling rate without taking the sparsity level differences among the blocks into consideration. In order to improve sampling efficiency, permutation-based block CS (BCS) schemes are proposed. In these schemes, the crux is to find a good permutation strategy which can make the nonzero entries evenly distributed among the blocks. In order to make the nonzero entries distributed among the blocks as evenly as possible, a novel matrix permutation strategy is proposed in this paper. Then, a BCS scheme with matrix permutation (BCS-MP) is proposed, which can be utilized to encode nature images effectively. Simulation results show that the proposed approach gets a significant gain of peak signal-to-noise ratio (PSNR) of the reconstructed-images compared with the state-of-the-art permutation-based ones and the traditional non-permutation one at the cost of slightly increasing the encoding time.

Adaptive Estimation in Two-way Sparse Reduced-rank Regression

This paper studies the problem of estimating a large coefficient matrix in a multiple response linear regression model when the coefficient matrix could be both of low rank and sparse in the sense that most nonzero entries concentrate on a few rows and columns. We are especially interested in the high dimensional settings where the number of predictors and/or response variables can be much larger than the number of observations. We propose a new estimation scheme, which achieves competitive numerical performance and at the same time allows fast computation. Moreover, we show that (a slight variant of) the proposed estimator achieves near optimal non-asymptotic minimax rates of estimation under a collection of squared Schatten norm losses simultaneously by providing both the error bounds for the estimator and minimax lower bounds. The effectiveness of the proposed algorithm is also demonstrated on an \textit{in vivo} calcium imaging dataset.

Optimized structured sparse sensing matrices for compressive sensing

Abstract We consider designing a robust structured sparse sensing matrix consisting of a sparse matrix with a few non-zero entries per row and a dense base matrix for capturing signals efficiently. We design the robust structured sparse sensing matrix through minimizing the distance between the Gram matrix of the equivalent dictionary and the target Gram of matrix holding small mutual coherence. Moreover, a regularization is added to enforce the robustness of the optimized structured sparse sensing matrix to the sparse representation error (SRE) of signals of interests. An alternating minimization algorithm with global sequence convergence is proposed for solving the corresponding optimization problem. Numerical experiments on synthetic data and natural images show that the obtained structured sensing matrix results in a higher signal reconstruction than a random dense sensing matrix.

The sequence of non k-th powers of a finite group G

We examine the connection between the sequence of not k-th powers of a finite group G and structural information about G. It is known that a single nonzero entry in the sequence bounds the order of G. We show that information about the number of specific not k-th powers can determine whether G is nilpotent, has a normal Hall subgroup, and other structural information about G. Furthermore, we show that the set of not k-th powers uniquely determines finite abelian groups up to isomorphism.