Deterministic Matrices Research Articles

A Class of Deterministic Sensing Matrices and Their Application in Harmonic Detection

In this paper, a class of deterministic sensing matrices are constructed by selecting rows from Fourier matrices. These matrices have better performance in sparse recovery than random partial Fourier matrices. The coherence and restricted isometry property of these matrices are given to evaluate their capacity as compressive sensing matrices. In general, compressed sensing requires random sampling in data acquisition, which is difficult to implement in hardware. By using these sensing matrices in harmonic detection, a deterministic sampling method is provided. The frequencies and amplitudes of the harmonic components are estimated from under-sampled data. The simulations show that this under-sampled method is feasible and valid in noisy environments.

Multiple Target Localization in WSNs Based on Compressive Sensing Using Deterministic Sensing Matrices

Accurate and low-cost localization of multiple targets or nodes is one of fundamental and challenging technical issues in wireless sensor networks (WSNs). Furthermore, compressive sensing allows that a sparse signal can be reconstructed from few measurements, and choosing a suitable sensing matrix is also important. For this purpose, random sensing matrices have been studied, while a few researches on deterministic sensing matrices have been considered. In this paper, we use compressive sensing for multiple target localization in WSNs. We formulate multiple target locations as a sparse matrix in the discrete time domain. Then, we exploit received signal strength information to recover noisy measurements, while utilizing deterministic sensing matrices and greedy algorithm to locate each target. The proposal approach reduces the number of measurements in localization process, takes low-cost, and maintains the accuracy as compared to the conventional approach which is noncompressive sensing. Further simulation shows that the proposed approach is practical in use, while being favorably comparable to the existing random sensing matrices in reconstruction performance. This cooperation between the compressive sensing using deterministic sensing matrices and multiple target localization provides a new point of view in WSN localization.

Optimal Dual Certificates for Noise Robustness Bounds in Compressive Sensing

The paper deals with optimizing Lipschitz bounds relating locally the reconstruction error to the measurement error in the RIPless compressive sensing framework. Most recent theoretical papers in the field parametrize such bounds relative to certain families of vectors called dual certificates, which are fundamental to several reconstruction criteria. We show in the paper that such a family of bounds admits a unique minimizer that has a deep geometric meaning and can be explicitly constructed via a convex projection algorithm that we describe. We also give a faster greedy algorithm that provides approximate solutions. The algorithms are numerically illustrated and analyzed for different types of sensing matrices, such as random matrices or deterministic matrices issued from tomography and super-resolution.

Randomization and the Gross–Pitaevskii Hierarchy

We study the Gross–Pitaevskii hierarchy on the spatial domain $${\mathbb{T}^3}$$ . By using an appropriate randomization of the Fourier coefficients in the collision operator, we prove an averaged form of the main estimate which is used in order to contract the Duhamel terms that occur in the study of the hierarchy. In the averaged estimate, we do not need to integrate in the time variable. An averaged spacetime estimate for this range of regularity exponents then follows as a direct corollary. The range of regularity exponents that we obtain is $${\alpha > \frac{3}{4}}$$ . It was shown in our previous joint work with Gressman (J Funct Anal 266(7):4705–4764, 2014) that the range $${\alpha > 1}$$ is sharp in the corresponding deterministic spacetime estimate. This is in contrast to the non-periodic setting, which was studied by Klainerman and Machedon (Commun Math Phys 279(1):169–185, 2008), where the spacetime estimate is known to hold whenever $${\alpha \geq 1}$$ . The goal of our paper is to extend the range of α in this class of estimates in a probabilistic sense. We use the new estimate and the ideas from its proof in order to study randomized forms of the Gross–Pitaevskii hierarchy. More precisely, we consider hierarchies similar to the Gross–Pitaevskii hierarchy, but in which the collision operator has been randomized. For these hierarchies, we show convergence to zero in low regularity Sobolev spaces of Duhamel expansions of fixed deterministic density matrices. We believe that the study of the randomized collision operators could be the first step in the understanding of a nonlinear form of randomization.

Properties of networks with partially structured and partially random connectivity.

Networks studied in many disciplines, including neuroscience and mathematical biology, have connectivity that may be stochastic about some underlying mean connectivity represented by a non-normal matrix. Furthermore, the stochasticity may not be independent and identically distributed (iid) across elements of the connectivity matrix. More generally, the problem of understanding the behavior of stochastic matrices with nontrivial mean structure and correlations arises in many settings. We address this by characterizing large random N×N matrices of the form A=M+LJR, where M,L, and R are arbitrary deterministic matrices and J is a random matrix of zero-mean iid elements. M can be non-normal, and L and R allow correlations that have separable dependence on row and column indices. We first provide a general formula for the eigenvalue density of A. For A non-normal, the eigenvalues do not suffice to specify the dynamics induced by A, so we also provide general formulas for the transient evolution of the magnitude of activity and frequency power spectrum in an N-dimensional linear dynamical system with a coupling matrix given by A. These quantities can also be thought of as characterizing the stability and the magnitude of the linear response of a nonlinear network to small perturbations about a fixed point. We derive these formulas and work them out analytically for some examples of M,L, and R motivated by neurobiological models. We also argue that the persistence as N→∞ of a finite number of randomly distributed outlying eigenvalues outside the support of the eigenvalue density of A, as previously observed, arises in regions of the complex plane Ω where there are nonzero singular values of L(-1)(z1-M)R(-1) (for z∈Ω) that vanish as N→∞. When such singular values do not exist and L and R are equal to the identity, there is a correspondence in the normalized Frobenius norm (but not in the operator norm) between the support of the spectrum of A for J of norm σ and the σ pseudospectrum of M.

确定性矩阵可分离压缩成像

确定性矩阵可分离压缩成像

Analysis of the limiting spectral measure of large random matrices of the separable covariance type

Consider the random matrix [Formula: see text] where D and [Formula: see text] are deterministic Hermitian nonnegative matrices with respective dimensions N × N and n × n, and where X is a random matrix with independent and identically distributed centered elements with variance 1/n. Assume that the dimensions N and n grow to infinity at the same pace, and that the spectral measures of D and [Formula: see text] converge as N, n → ∞ towards two probability measures. Then it is known that the spectral measure of ΣΣ* converges towards a probability measure μ characterized by its Stieltjes transform. In this paper, it is shown that μ has a density away from zero, this density is analytical wherever it is positive, and it behaves in most cases as [Formula: see text] near an edge a of its support. In addition, a complete characterization of the support of μ is provided. Aside from its mathematical interest, the analysis underlying these results finds important applications in a certain class of statistical estimation problems.

Limit distributions of random matrices

We study limit distributions of independent random matrices as well as limit joint distributions of their blocks under normalized partial traces composed with classical expectation. In particular, we are concerned with the ensemble of symmetric blocks of independent Hermitian random matrices which are asymptotically free, asymptotically free from diagonal deterministic matrices, and whose norms are uniformly bounded almost surely. This class contains symmetric blocks of unitarily invariant Hermitian random matrices whose asymptotic distributions are compactly supported probability measures on the real line. Our approach is based on the concept of matricial freeness which is a generalization of freeness in free probability. We show that the associated matricially free Gaussian operators provide a unified framework for studying the limit distributions of sums and products of independent rectangular random matrices, including non-Hermitian Gaussian matrices and matrices of Wishart type.

A Reed-Solomon Code Based Measurement Matrix with Small Coherence

In this letter, we construct a class of deterministic measurement matrices which are asymptotically optimal. For this purpose, we first apply the tensor product over the Reed Solomon (R-S) generator matrix to produce a new one; then employing this generator matrix, we construct a measurement matrix. If the R-S code is defined on \BBFq, then the resulting measurement matrices are of dimensions q2 ×q3, where q is an arbitrary prime power and its coherence would be 1/q, which is desirable for compressive sampling. We also illustrate the effectiveness of our proposed matrices in compressed sensing with some simulation examples.

Deterministic Construction of Sparse Sensing Matrices via Finite Geometry

Compressed sensing is a novel sampling technique that provides a fundamentally new approach to data acquisition. Comparing with the traditional method, compressed sensing asserts that a sparse signal can be reconstructed from very few measurements. A central problem in compressed sensing is the construction of sensing matrices. While random sensing matrices have been studied intensively, only a few deterministic constructions are known. As a long-standing subject in combinatorial design theory, the packing design gives rise to deterministic sparse matrices with low coherence. With this framework in mind, we investigate a series of packing designs originated from finite geometry. The connection between the sensing matrix and finite geometry is revealed, using the packing design as a bridge. More specifically, we construct a series of $m \times n$ binary sensing matrices with sparsity order $k=\Theta (m^{1/3})$ or $k=\Theta (m^{1/2})$ . Moreover, we use an embedding operation to merge our binary matrices with matrices having low coherence. Comparing with the original binary matrices, this embedding operation generates modified matrices with better recovery performance. Numerical simulations show that our binary and modified matrices outperform several typical sensing matrices. The sparse property of our matrices helps to accelerate the recovery process, which is very preferable in practice.

Compressed Sampling of Spectrally Sparse Signals Using Sparse Circulant Matrices

AbstractCirculant measurement matrices constructed by partial cyclically shifts of one generating sequence, are easier to be implemented in hardware than widely used random measurement matrices; however, the diminishment of randomness makes it more sensitive to signal noise. Selecting a deterministic sequence with optimal periodic autocorrelation property (PACP) as generating sequence, would enhance the noise robustness of circulant measurement matrix, but this kind of deterministic circulant matrices only exists in the fixed periodic length. Actually, the selection of generating sequence doesn't affect the compressive performance of circulant measurement matrix but the subspace energy in spectrally sparse signals. Sparse circulant matrices, whose generating sequence is a sparse sequence, could keep the energy balance of subspaces and have similar noise robustness to deterministic circulant matrices. In addition, sparse circulant matrices have no restriction on length and are more suitable for the compressed sampling of spectrally sparse signals at arbitrary dimensionality.

Spectral Properties of Random and Deterministic CMV Matrices

The CMV matrices are unitary analogues of the discrete one-dimensional Schrodinger operators. We review spectral properties of a few classes of CMV matrices and describe families of random and deterministic CMV matrices which exhibit a transition in the distribution of their eigenvalues.

The strong asymptotic freeness of Haar and deterministic matrices

In this paper, we are interested in sequences of q-tuple of N-by-N random matrices having a strong limiting distribution (i.e. given any non-commutative polynomial in the matrices and their conjugate transpose, its normalized trace and its norm converge). We start with such a sequence having this property, and we show that this property pertains if the q-tuple is enlarged with independent unitary Haar distributed random matrices. Besides, the limit of norms and traces in non-commutative polynomials in the enlarged family can be computed with reduced free product construction. This extends results of one author (C. M.) and of Haagerup and Thorbjornsen. We also show that a p-tuple of independent orthogonal and symplectic Haar matrices have a strong limiting distribution, extending a recent result of Schultz.

The Road to Deterministic Matrices with the Restricted Isometry Property

The restricted isometry property (RIP) is a well-known matrix condition that provides state-of-the-art reconstruction guarantees for compressed sensing. While random matrices are known to satisfy this property with high probability, deterministic constructions have found less success. In this paper, we consider various techniques for demonstrating RIP deterministically, some popular and some novel, and we evaluate their performance. In evaluating some techniques, we apply random matrix theory and inadvertently find a simple alternative proof that certain random matrices are RIP. Later, we propose a particular class of matrices as candidates for being RIP, namely, equiangular tight frames (ETFs). Using the known correspondence between real ETFs and strongly regular graphs, we investigate certain combinatorial implications of a real ETF being RIP. Specifically, we give probabilistic intuition for a new bound on the clique number of Paley graphs of prime order, and we conjecture that the corresponding ETFs are RIP in a manner similar to random matrices.

An Evaluation of the Sparsity Degree for Sparse Recovery with Deterministic Measurement Matrices

The paper deals with the estimation of the maximal sparsity degree for which a given measurement matrix allows sparse reconstruction through ℓ 1-minimization. This problem is a key issue in different applications featuring particular types of measurement matrices, as for instance in the framework of tomography with low number of views. In this framework, while the exact bound is NP hard to compute, most classical criteria guarantee lower bounds that are numerically too pessimistic. In order to achieve an accurate estimation, we propose an efficient greedy algorithm that provides an upper bound for this maximal sparsity. Based on polytope theory, the algorithm consists in finding sparse vectors that cannot be recovered by ℓ 1-minimization. Moreover, in order to deal with noisy measurements, theoretical conditions leading to a more restrictive but reasonable bounds are investigated. Numerical results are presented for discrete versions of tomography measurement matrices, which are stacked Radon transforms corresponding to different tomograph views.

Generalized Helmert–Ledermann orthogonal matrices and rom simulation

Deterministic rectangular orthonormal matrices satisfying a hyper-plane constraint plays a central role in random orthogonal matrix (ROM) simulation. The multivariate skewness and kurtosis sampling properties are encrypted in a given orthonormal matrix. We consider a subclass of generalized Helmert–Ledermann (GHL) orthogonal matrices that have fixed last column, are generated by the Cayley transform, and satisfy the required hyper-plane constraint. The algebraic structure of GHL orthogonal matrices is determined. Simple and convenient skewness and kurtosis formulas are obtained. We exhibit an asymptotically growing range of variation for skewness and kurtosis, which points to an increased flexibility in ROM simulation.

SOME REMARKS ON THE DOZIER–SILVERSTEIN THEOREM FOR RANDOM MATRICES WITH DEPENDENT ENTRIES

The Dozier–Silverstein theorem asserts the almost sure convergence of the empirical spectral distribution of information plus noise matrices, i.e. perturbations of deterministic matrices whose spectral distribution converges (information matrices) by random matrices with i.i.d. entries (noise matrices). We show that a modification of the original proof given by Dozier and Silverstein allows to extend this result to more general noise matrices, in particular matrices with independent columns satisfying a natural concentration inequality for quadratic forms, matrices with independent entries, satisfying a Lindeberg-type condition (recovering a recent result by Xie), matrices with heavy-tailed entries in the domain of attraction of the Gaussian distribution and certain classes of matrices with dependencies among columns (generalizing those investigated recently by O'Rourke and containing matrices with exchangeable entries). As a corollary we obtain the circular law for random matrices with independent log-concave isotropic rows.

Verified stochastic methods

Markov chains provide quite attractive features for simulating a system's behavior under consideration of uncertainties. However, their use is somewhat limited because of their deterministic transition matrices. Vague probabilistic information and imprecision appear in the modeling of real-life systems, thus causing difficulties in the pure probabilistic model set-up. Moreover, their accuracy suffers due to implementations on computers with floating point arithmetics. Our goal is to address these problems by extending the Dempster-Shafer with Intervals toolbox for MATLAB with novel verified algorithms for modeling that work with Markov chains with imprecise transition matrices, known as Markov set-chains. Additionally, in order to provide a statistical estimation tool that can handle imprecision to set up Markov chain models, we develop a new verified algorithm for computing relations between the mean and the standard deviation of fuzzy sets.

Minimal representation of matrix valued white stochastic processes and U–D factorisation of algorithms for optimal control

Two different descriptions are used in the literature to formulate the optimal dynamic output feedback control problem for linear dynamical systems with white stochastic parameters and quadratic criteria, called the optimal compensation problem. One describes the matrix valued white stochastic processes involved, using a sum of deterministic matrices each one multiplied by a scalar stochastic process that is independent of the others. Another, that is more general and concise, uses Kronecker products instead. This article relates the statistics of both descriptions and shows their advantages and disadvantages. As to the first description, an important result that comes out is the minimum number of matrices multiplied by scalar, independent, stochastic processes needed to represent a certain matrix valued white stochastic process, together with an associated minimal representation. As to the second description, an important result concerns the generation of all Kronecker products that represent relevant statistics. Both results facilitate the specification of statistics of systems with white stochastic parameters. The second part of this article further exploits these results to perform an U–D factorisation of an algorithm to compute optimal dynamic output feedback controllers (optimal compensators) for linear discrete-time systems with white stochastic parameters and quadratic sum criteria. U–D factorisation of this type of algorithm is new. By solving several numerical examples, the U–D factored algorithm is compared with a conventional algorithm.

Toward deterministic compressed sensing

During the past decade, compressed sensing has delivered significant advances in the theory and application of measuring and compressing data. Consider capturing a 10-megapixel image with a digital camera. Emailing an image of this size requires an unnecessary amount of storage space and bandwidth. Instead, users employ a standard digital compression scheme, such as JPEG, to represent the image as a 64-kb file. The compressed image is completely recognizable even though the dimension of the compressed version is a tiny fraction of the original 10 million dimensions. Compressed sensing takes this mathematical phenomenon one step further. Is it possible to capture the pertinent information, such as the 64-kb image, without first measuring the full 10 million pixel values? If so, how should we perform the measurements? If we capture the important information, can we still reconstruct the image from this limited number of observations? Compressed sensing exploded in 2004 when Donoho (1, 2) and Candes and Tao (3) definitively answered these questions by incorporating randomness in the measurement process. Because engineering a truly random process is impossible, a major open problem in compressed sensing is the search for deterministic methods for sparse signal measurement that capture the relevant information in the signal and permit accurate reconstruction. In PNAS, Monajemi et al. (4) provide a major step forward in understanding the potential for deterministic measurement matrices in compressed sensing.