Small Number Of Nonzero Entries Research Articles

A note on generalized companion pencils in the monomial basis

In this paper, we introduce a new notion of generalized companion pencils for scalar polynomials over an arbitrary field expressed in the monomial basis. Our definition is quite general and extends the notions of companion pencil in De Teran et al. (Linear Algebra Appl 459:264–333, 2014), generalized companion matrix in Garnett et al. (Linear Algebra Appl 498:360–365, 2016), and Ma–Zhan companion matrices in Ma and Zhan (Linear Algebra Appl 438: 621–625, 2013), as well as the class of quasi-sparse companion pencils introduced in De Teran and Hernando (INdAM Series, Springer, Berlin, pp 157–179, 2019). We analyze some algebraic properties of generalized companion pencils. We determine their Smith canonical form and we prove that they are all nonderogatory. In the last part of the work we will pay attention to the sparsity of these constructions. In particular, by imposing some natural conditions on its entries, we determine the smallest number of nonzero entries of a generalized companion pencil.

Sparse Solutions of Linear Diophantine Equations

We present structural results on solutions to the Diophantine system $A{\boldsymbol y} = {\boldsymbol b}$, ${\boldsymbol y} \in \mathbb Z^t_{\ge 0}$ with the smallest number of non-zero entries. Our tools are algebraic and number theoretic in nature and include Siegel's Lemma, generating functions, and commutative algebra. These results have some interesting consequences in discrete optimization.

An advanced scheme of compressed sensing of acceleration data for telemonintoring of human gait.

BackgroundThe compressed sensing (CS) of acceleration data has been drawing increasing attention in gait telemonitoring application. In such application, there still exist some challenging issues including high energy consumption of body-worn device for acceleration data acquisition and the poor reconstruction performance due to nonsparsity of acceleration data. Thus, the novel scheme of compressive sensing of acceleration data is needed urgently for solutions that are found to these issues.MethodsIn our scheme, the sparse binary matrix is firstly designed as an optimal measurement matrix only containing a smallest number of nonzero entries. And then the block sparse Bayesian learning (BSBL) algorithm is introduced to reconstruct acceleration data with high fidelity by exploiting block sparsity. Finally, some commonly used gait classification models such as multilayer perceptron (MLP), support vector machine (SVM) and KStar are applied to further validate the feasibility of our scheme for gait telemonitoring application.ResultsThe acceleration data were selected from open Human Activity Dataset of Southern California University (USC-HAD). The optimal sparse binary matrix (a smallest number of nonzero entries is 8) is as strong as the full optimal measurement matrix such as Gaussian random matrix. Moreover, BSBL algorithm significantly outperforms existing conventional CS reconstruction algorithms, and reaches the maximal signal-to-noise ratio value (70 dB). In comparison, MLP is best for gait classification, and it can classify upstairs and downstairs patterns with best accuracy of 95 % and seven gait patterns with maximal accuracy of 92 %, respectively.ConclusionsThese results show that sparse binary matrix and BSBL algorithm are feasibly applied in compressive sensing of acceleration data to achieve the perfect compression and reconstruction performance, which has a great potential for gait telemonitoring application.

Local and global strategies for optimal selective mass scaling

The problem of optimal selective mass scaling for linearized elasto-dynamics is discussed. Optimal selective mass scaling should provide solutions for dynamical problems that are close to the ones obtained with a lumped mass matrix, but at much smaller computational costs. It should be equally applicable to all structurally relevant load cases. The three main optimality criteria, namely eigenmode preservation, small number of non-zero entries and good conditioning of the mass matrix are explicitly formulated in the article. An example of optimal mass scaling which relies on redistribution of mass on a global system level is constructed. Alternative local mass scaling strategies are proposed and compared with existing methods using one modal and two transient numerical examples.

Convertible and m-convertible matrices

In this article, we consider the calculation of the permanent of a (0, 1)-matrix using determinants. We investigate when determinants of more than one signing of the original are used to calculate the permanent and we show that non-convertible matrices require at least four different signings. Then, we loosen the restriction that the matrices used to convert the permanent are signings of the original and use this to reduce the number of determinants necessary to convert the permanent of the all 1's matrix by considering a particular partition of the set S n of permutations of {1, … , n}. Finally, we construct a sequence of maximal convertible matrices with a small number of nonzero entries, thus lowering the possible upper bound for the number of nonzero entries of such a matrix, relative to the order.

Linearly scaling direct method for accurately inverting sparse banded matrices

In many problems in computational physics and chemistry, one finds special kinds of sparse matrices, called banded matrices. These matrices, which are defined as having non-zero entries only within a given distance from the main diagonal, often need to be inverted in order to solve the associated linear system of equations. In this work, we introduce a new algorithm for solving such a system, with the size of the matrix being n × n. We derive analytical recursive expressions that allow us to directly obtain the solution. In addition, we describe the extension to deal with matrices that are banded plus a small number of non-zero entries outside the band, and use the same ideas to produce a method for obtaining the full inverse matrix. Finally, we show that our new algorithm is competitive, both in accuracy and in numerical efficiency, when compared with a standard method based on Gaussian elimination. We do this using sets of large random banded matrices, as well as the ones that appear in the calculation of Lagrange multipliers in proteins.

Average Performance of the Sparsest Approximation Using a General Dictionary

Let A be a matrix of size N × M (a dictionary) and let ‖·‖ be a norm on ℝ N . For any data d ∈ ℝ N , we consider the sparsest vector (i.e., the one with the smallest number of nonzero entries) u ∈ ℝ M such that ‖Au − d‖ ≤ τ, for a parameter τ > 0. We say that u is a K-sparse solution if it has less than K ∈ ℕ nonzero entries. In this article, we give a precise geometrical description of the data sets d ∈ ℝ N yielding a K-sparse solution. We parameterize and measure these sets. More precisely, we measure their intersection with a ball defined by any given norm δ and a radius θ. These measures are expressed in terms of the constituents of the optimization problem—namely A, ‖·‖, τ, δ and θ—and they enable these constituents to be rated. This is the core of a new methodology, called Average Performance in Approximation (APA), inaugurated in this work. By way of application, we give the probability of obtaining a K-sparse solution, when d is uniformly distributed in the δ-ball of radius θ. Analyzing the obtained formulas reveals what are the most important features of the dictionary and the norm defining the data fidelity, to obtain sparse solutions. This crucial question is largely discussed. We also provide an example when both ‖·‖ and δ are the Euclidian norm. Some among the wide-ranging perspectives raised by the new APA methodology are described as well.

Minimal density conjugation of binary matrices

We define the notion of minimal density of an n × n binary matrix, which is the smallest number of non-zero entries a matrix can have after conjugation by an element of GL ( n , 2 ) . We give upper bounds on the minimal density for two important cases. We discuss how minimal density behaves with respect to block diagonal matrices and Jordan blocks whose eigenvalues are all one. We also give an algorithm for computing the minimal density which is reasonably fast when the density is low. Finally, we compute the largest minimal density that an n × n matrix can have when n ≠ 6 k + 1 for k ⩾ 2 .