An efficient algorithm for designing projection matrix in compressive sensing based on alternating optimization

Sparsity and incoherence are the two key ingredients in compressive sensing (CS). Given a sparsifying dictionary D, the projection matrix P must be as incoherent with D as possible for the CS system to be efficient. Thus the design of projection matrix is naturally a problem of minimizing the coherence between P andD. Unfortunately, this turns out to be a nonconvex, nonsmooth, large-scale problem even for a CS system of moderate size. In this paper, the above-mentioned problem is investigated in a formulation where the problem is converted into a sequence of nonsmooth but convex subproblems. A subgradient projection algorithm is proposed to solve the nonsmooth subproblems that converges to a projection matrix with improved performance. The performance of the proposed algorithm is evaluated by simulations and comparisons with several existing techniques.

Compressive sensing (CS) is mainly concerned with low-coherence pairs, since the number of samples needed to recover the signal is proportional to the mutual coherence between projection matrix and sparsifying matrix. Until now, papers on CS always assume the projection matrix to be a random matrix. In this paper, aiming at minimizing the mutual coherence, a method is proposed to optimize the projection matrix. This method is based on equiangular tight frame (ETF) design because an ETF has minimum coherence. It is impossible to solve the problem exactly because of the complexity. Therefore, an alternating minimization type method is used to find a feasible solution. The optimally designed projection matrix can further reduce the necessary number of samples for recovery or improve the recovery accuracy. The proposed method demonstrates better performance than conventional optimization methods, which brings benefits to both basis pursuit and orthogonal matching pursuit.

Finite tight frames play an important role in miscellaneous areas, including quantum information theory. Here we apply a class of tight frames, equiangular tight frames, to address the problem of detecting the entanglement of bipartite states. Here we derive some entanglement criteria based on positive operator-valued measurements built from equiangular tight frames. We also present a class of entanglement witnesses based on the equiangular tight frames. At last, we generalize the entanglement criterion for bipartite systems to multipartite systems.

A projection matrix design method for MSE reduction in adaptive compressive sensing

In this article, we propose a new projection test for linear hypotheses on regression coefficient matrices in linear models with high-dimensional responses. We systematically study the theoretical properties of the proposed test. We first derive the optimal projection matrix for any given projection dimension to achieve the best power and provide an upper bound for the optimal dimension of projection matrix. We further provide insights into how to construct the optimal projection matrix. One- and two-sample mean problems can be formulated as special cases of linear hypotheses studied in this article. We both theoretically and empirically demonstrate that the proposed test can outperform the existing ones for one- and two-sample mean problems. We conduct Monte Carlo simulation to examine the finite sample performance and illustrate the proposed test by a real data example.

We present the theory of equiangular tight frames (ETFs) from association schemes as a generalization of harmonic ETFs. This theory has the pleasing consequence of including not only harmonic ETFs from abelian groups but also real ETFs with centroidal symmetry and the ETFs discovered by Renes and Strohmer coming from skew-symmetric conference matrices. Finally we present a theory of nonabelian harmonic ETFs from the perspective of association schemes. An infinite class of ETFs arising as group frames from nonabelian groups is presented.

An equiangular tight frame (ETF) is an M N matrix which has orthogonal equal norm rows, equal normcolumns, and the inner products of all pairs of columns have the same modulus. In this paper we studyETFs in which all of the entries are unimodular, and in particular pth roots of unity. A new constructionof unimodular ETFs based on resolvable Steiner systems is presented. This construction gives many newexamples of unimodular ETFs. In particular, an new in nite class of ETFs with entries in f1; 1g ispresented.Keywords: tight frames, equiangular, Hadamard matrices, Steiner systems, resolvable 1. INTRODUCTION Let F be an M N matrix with real or complex entries. If there exists A > 0 such that FF = AI , where Iis the identity matrix, then F is called a tight frame . If F F has constant diagonal and all of the o -diagonalentries have the same modulus, then F is called equiangular. If all of the entries of F have modulus 1 thenwe call F unimodular . If all of the entries of F are pth roots of unity, then we call F unimodular of degree p.This article is concerned with equiangular tight frames (ETFs) which are unimodular of degree p. Inparticular, we are concerned with nding unimodular ETFs of degree 2, which we shall also refer to asHadamard ETFs. Note that in the usual de nition of ETFs the columns are taken to have norm 1. However,since we are concerned with unimodular ETFs we will avoid normalizing, thus the columns of an M Nunimodular ETF have norm M .The paper is organized as follows. In Section 2 we present some necessary conditions on the existence ofunimodular ETFs. In Section 3 we review the construction of harmonic ETFs, which is an important classof ETFs that also happen to be unimodular. In Section 4 we review the construction of the Steiner ETFs ofFickus, Mixon, and Tremain,

Detecting or classifying already known sparse signals contaminated by Gaussian noise from compressive measurements is different from reconstructing sparse signals, as its objective is to minimize the error probability which describes performance of the detectors or classifiers. This paper is concerned about the performance improvement of a commonly used Compressive Classifier. We prove that when the arbitrary sensing matrices used to get the Compressive Measurements are transformed into Equi-Norm Tight Frames, i.e. the matrices that are row-orthogonal, The Compressive Classifier achieves better performance. Although there are other proofs that among all Equi-Norm Tight Frames the Equiangular tight Frames (ETFs) bring best worst-case performance, the existence and construction of ETFs on some dimensions is still an open problem. As the construction of Equi-Norm Tight Frames from any arbitrary matrices is very easy and practical compared with ETF matrices, the result of this paper can also provide a practical method to design an improved sensing matrix for Compressive Classification. We can conclude that: Tight is Better!

We introduce the study of frames and equiangular lines in classical geometries over finite fields. After developing the basic theory, we give several examples and demonstrate finite field analogs of equiangular tight frames (ETFs) produced by modular difference sets, and by translation and modulation operators. Using the latter, we prove that Gerzon's bound is attained in each unitary geometry of dimension d=22l+1 over the field F32. We also investigate interactions between complex ETFs and those in finite unitary geometries, and we show that every complex ETF implies the existence of ETFs with the same size over infinitely many finite fields.

The relationship between equiangular tight frames and strongly regular graphs has been known for several years. This relationship has been exploited to construct many of the latest examples of new strongly regular graphs. Recently it was shown that there is a similar relationship between two-distance tight frames and strongly regular graphs. In this paper we present a new tensor like construction of two-distance tight frames, and hence a family of strongly regular graphs. While graphs with these parameters were known to exist, this new construction is very simple, requiring only the existence of an affine plane, whereas the original constructions often require more complicated objects such as generalized quadrangles.

Due to sensor drift, instrumental variation, or change of measurement object, the distributions of the datasets (domains) acquired by the sensors are often different. A domain adaptive framework called MCSP-SSS is proposed to reduce the distribution discrepancy across domains for high-accuracy sensor data classification. The framework consists of two key parts: Multi-Constraint Subspace Projection (MCSP) and Source Sample Selection (SSS). MCSP is a subspace-projection-based approach, which introduces four constraints to get an optimized projection matrix: Principal Component Analysis (PCA) is used to reduce the redundancy of features; Mean Distribution Discrepancy (MDD) is applied to minimize the difference between source and target data; Hilbert-Schmidt Independence Criterion (HSIC) is adopted to maximizing the dependence between features and labels; A so-called weighted-within-class scatter matrix is introduced to minimize the within-class variance to avoid samples with different labels to overlap in the subspace. SSS is designed to remove the outliers in projected source samples so as to reduce distribution discrepancy between the source and target samples projected by MCSP. Experimental results show that our framework can achieve the best accuracies in all classification tasks in comparison with other state-of-the-art approaches. The accuracy of MCSP-SSS is 3.57 percentage points (pps) higher on average than that of the second best approach for single-source domain adaptation, and 7.00 pps for multi-source domain adaptation. Source code is available at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/threedteam/tsc_subspace_projection</uri> .

An efficient method is employed to determine the scattering characteristics of a beam wave from inhomogeneous dielectric fibers. The fibers are approximated by a series of stratified dielectric layers and only 2 × 2 matrix operations are used. The results apply to an inhomogeneous fiber positioned with its axis of symmetry perpendicular to the direction of propagation of the beam wave. The fiber may have arbitrary, radially symmetric variations of its refractive-index profile. Scattering patterns for several functional dependences of the index of refraction of the fiber are presented.

A SIC is a maximal equiangular tight frame in a finite dimensional Hilbert space. Given a SIC in dimension d, there is good evidence that there always exists an aligned SIC in dimension d(d − 2), having predictable symmetries and smaller equiangular tight frames embedded in them. We provide a recipe for how to calculate sets of vectors in dimension d(d − 2) that share these properties. They consist of maximally entangled vectors in certain subspaces defined by the numbers entering the d dimensional SIC. However, the construction contains free parameters and we have not proven that they can always be chosen so that one of these sets of vectors is a SIC. We give some worked examples that, we hope, may suggest to the reader how our construction can be improved. For simplicity we restrict ourselves to the case of odd dimensions.

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