Sparse Solutions of a Class of Constrained Optimization Problems

In this paper, we consider a well-known sparse optimization problem that aims to find a sparse solution of a possibly noisy underdetermined system of linear equations. Mathematically, it can be modeled in a unified manner by minimizing [Formula: see text] subject to [Formula: see text] for given [Formula: see text] and [Formula: see text]. We then study various properties of the optimal solutions of this problem. Specifically, without any condition on the matrix A, we provide upper bounds in cardinality and infinity norm for the optimal solutions and show that all optimal solutions must be on the boundary of the feasible set when [Formula: see text]. Moreover, for [Formula: see text], we show that the problem with [Formula: see text] has a finite number of optimal solutions and prove that there exists [Formula: see text] such that the solution set of the problem with any [Formula: see text] is contained in the solution set of the problem with p = 0, and there further exists [Formula: see text] such that the solution set of the problem with any [Formula: see text] remains unchanged. An estimation of such [Formula: see text] is also provided. In addition, to solve the constrained nonconvex non-Lipschitz Lp-L1 problem ([Formula: see text] and q = 1), we propose a smoothing penalty method and show that, under some mild conditions, any cluster point of the sequence generated is a stationary point of our problem. Some numerical examples are given to implicitly illustrate the theoretical results and show the efficiency of the proposed algorithm for the constrained Lp-L1 problem under different noises.

A new framework for utilizing side information in sparse representation

An underdetermined system of linear equation has infinitely number of answers. To find a specific solution, regularization method is used. For this propose, we define a cost function based on desired features of the solution and that answer with the best matches to these function is selected as the desired solution. In case of sparse solution, zero-norm function is selected as the cost function. In many engineering cases, there is side information which are omitted because of the zero-norm function. Finding a way to conquer zero-norm function limitation, will help to improve estimation of the desired parameter. In this regard, we utilize maximum a posterior (MAP) estimation and modify the prior information such that both sparsity and side information are utilized. As a consequence, a framework to utilize side information into sparse representation algorithms is proposed. We also test our proposed framework in orthogonal frequency division multiplexing (OFDM) sparse channel estimation problem which indicates, by utilizing our proposed system, the performance of the system improves and fewer resources are required for estimating the channel.

On the number of Hadamard matrices via anti-concentration

Many problems in combinatorial linear algebra require upper bounds on the number of solutions to an underdetermined system of linear equations $Ax = b$ , where the coordinates of the vector x are restricted to take values in some small subset (e.g. $\{\pm 1\}$ ) of the underlying field. The classical ways of bounding this quantity are to use either a rank bound observation due to Odlyzko or a vector anti-concentration inequality due to Halász. The former gives a stronger conclusion except when the number of equations is significantly smaller than the number of variables; even in such situations, the hypotheses of Halász’s inequality are quite hard to verify in practice. In this paper, using a novel approach to the anti-concentration problem for vector sums, we obtain new Halász-type inequalities that beat the Odlyzko bound even in settings where the number of equations is comparable to the number of variables. In addition to being stronger, our inequalities have hypotheses that are considerably easier to verify. We present two applications of our inequalities to combinatorial (random) matrix theory: (i) we obtain the first non-trivial upper bound on the number of $n\times n$ Hadamard matrices and (ii) we improve a recent bound of Deneanu and Vu on the probability of normality of a random $\{\pm 1\}$ matrix.

Sparse Representation for Cardiac Electrical Activity Imaging in Magnetocardiography

Signal sparsity has been widely discussed in communication system, cloud computing, multimedia processing and computational biology. Reconstructing the sparsely distributed current sources of the heart by means of non-invasive magnetocardiography (MCG) measurement and various optimization methods provides a new way to solve the inverse problem of the cardiac magnetic field. The problem of sparse source location of MCG is in the time series of MCG measurement caused by active sparse current source, can the spatiotemporal source be reconstructed accurately and effectively? For the above problem, the scientific contributions of the paper include: (1) A modified focal underdetermined system solver algorithm is proposed for a sparse solution, by combing with dynamic regularization factor and smoothed sparse constraint; (2) Lead field matrix is reduced by prior information of cardiac magnetic field map to reduce under-determination; (3) Spatiotemporal sources are reconstructed for non-invasive cardiac electrical activity imaging. The results of real MCG data demonstrate the effectiveness of this method for cardiac electrical activity imaging. The temporal and spatial changes of the current sources are similar to the depolarization and repolarization process of the ventricle.

Overconstrained Cable-Driven Parallel Manipulators Statics Analysis based on Simplified Static Cable Model

In recent years, cable-driven parallel manipulators (CDPM) become more and more interesting topics of robot researchers due to its outstanding advantages. Unlike traditional parallel robots, CDPMs use many flexible cables in order to connect the robot fixed frame and the moving platform instead of using conventional rigid links. Since cables used in CDPM is very light compared to rigid links, its workspace can be very large. Besides, CDPMs are often enhanced load capacity by adding redundant actuators. They also help to widen the singularity-free workspace of CDPM. On the other hand, the redundant actuators produce the underdetermined system i.e. the system has non-unique solutions. Moreover, the elasticity and bendability of flexible cable caused by self-weight and external forces act on it, resulting in the kinematic problem of CDPMs are no longer related to the geometric problem. Therefore, the system of CDPM become non-linear when the deformation of cable is considered. In this study, we introduce the simplified static cable model and use it to linearize the static model of redundantly actuated CDPM. The algorithm to solve the force distribution problem is proposed in Sect. 4. The static-workspace and the performance of those are analyzed in a numerical test.

Estimation and Mitigation of Epidemic Risk on a Public Transit Route using Automatic Passenger Count Data

This paper studies the potential spread of infectious disease through passenger encounters in a public transit system using automatic passenger count (APC) data. An algorithmic procedure is proposed to evaluate three different measures to quantify these encounters. The first two measures quantify the increased possibility of disease spread from passenger interaction when traveling between different origin–destination pairs. The third measure evaluates an aggregate measure quantifying the relative risk of boarding at a particular stop of the transit route. For calculating these measures, compressed sensing is employed to estimate a sparse passenger flow matrix planted in the underdetermined system of equations obtained from the APC data. Using the APC data of Route 5 in Minneapolis/St. Paul region during the COVID-19 pandemic, it was found that all three measures grow abruptly with the number of passengers on board. The passenger contact network is densely connected, which further increases the potential risk of disease transmission. To reduce the relative risk, it is proposed to restrict the number of passengers on-board and analyze the effect of this using a simulation framework. It was found that a considerable reduction in the relative risk can be achieved when the maximum number of passengers on-board is restricted below 15. To account for the reduced capacity and still maintain reasonable passenger wait times, it would then be necessary to increase the frequency of the route.

Scalable signal reconstruction for a broad range of applications

Signal reconstruction problem (SRP) is an important optimization problem where the objective is to identify a solution to an underdetermined system of linear equations that is closest to a given prior. It has a substantial number of applications in diverse areas, such as network traffic engineering, medical image reconstruction, acoustics, astronomy, and many more. Unfortunately, most of the common approaches for solving SRP do not scale to large problem sizes. We propose a novel and scalable algorithm for solving this critical problem. Specifically, we make four major contributions. First, we propose a dual formulation of the problem and develop the DIRECT algorithm that is significantly more efficient than the state of the art. Second, we show how adapting database techniques developed for scalable similarity joins provides a substantial speedup over DIRECT. Third, we describe several practical techniques that allow our algorithm to scale---on a single machine---to settings that are orders of magnitude larger than previously studied. Finally, we use the database techniques of materialization and reuse to extend our result to dynamic settings where the input to the SRP changes. Extensive experiments on real-world and synthetic data confirm the efficiency, effectiveness, and scalability of our proposal.