Spectral analysis of random sparse matrices

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Structured Random Matrices

Random matrix theory is a well-developed area of probability theory that has numerous connections with other areas of mathematics and its applications. Much of the literature in this area is concerned with matrices that possess many exact or approximate symmetries, such as matrices with i.i.d. entries, for which precise analytic results and limit theorems are available. Much less well understood are matrices that are endowed with an arbitrary structure, such as sparse Wigner matrices or matrices whose entries possess a given variance pattern. The challenge in investigating such structured random matrices is to understand how the given structure of the matrix is reflected in its spectral properties. This chapter reviews a number of recent results, methods, and open problems in this direction, with a particular emphasis on sharp spectral norm inequalities for Gaussian random matrices.Comment: 46 pages; to appear in IMA Volume "Discrete Structures: Analysis and Applications" (Springer

The power of sum-of-squares for detecting hidden structures

We study planted problems---finding hidden structures in random noisy inputs---through the lens of the sum-of-squares semidefinite programming hierarchy (SoS). This family of powerful semidefinite programs has recently yielded many new algorithms for planted problems, often achieving the best known polynomial-time guarantees in terms of accuracy of recovered solutions and robustness to noise. One theme in recent work is the design of spectral algorithms which match the guarantees of SoS algorithms for planted problems. Classical spectral algorithms are often unable to accomplish this: the twist in these new spectral algorithms is the use of spectral structure of matrices whose entries are low-degree polynomials of the input variables. We prove that for a wide class of planted problems, including refuting random constraint satisfaction problems, tensor and sparse PCA, densest-k-subgraph, community detection in stochastic block models, planted clique, and others, eigenvalues of degree-d matrix polynomials are as powerful as SoS semidefinite programs of roughly degree d. For such problems it is therefore always possible to match the guarantees of SoS without solving a large semidefinite program. Using related ideas on SoS algorithms and low-degree matrix polynomials (and inspired by recent work on SoS and the planted clique problem by Barak et al.), we prove new nearly-tight SoS lower bounds for the tensor and sparse principal component analysis problems. Our lower bounds for sparse principal component analysis are the first to suggest that going beyond existing algorithms for this problem may require sub-exponential time

Scalable Parallel Factorizations of SDD Matrices and Efficient Sampling for Gaussian Graphical Models

Motivated by a sampling problem basic to computational statistical inference, we develop a nearly optimal algorithm for a fundamental problem in spectral graph theory and numerical analysis. Given an $n\times n$ SDDM matrix ${\bf \mathbf{M}}$, and a constant $-1 \leq p \leq 1$, our algorithm gives efficient access to a sparse $n\times n$ linear operator $\tilde{\mathbf{C}}$ such that $${\mathbf{M}}^{p} \approx \tilde{\mathbf{C}} \tilde{\mathbf{C}}^\top.$$ The solution is based on factoring ${\bf \mathbf{M}}$ into a product of simple and sparse matrices using squaring and spectral sparsification. For ${\mathbf{M}}$ with $m$ non-zero entries, our algorithm takes work nearly-linear in $m$, and polylogarithmic depth on a parallel machine with $m$ processors. This gives the first sampling algorithm that only requires nearly linear work and $n$ i.i.d. random univariate Gaussian samples to generate i.i.d. random samples for $n$-dimensional Gaussian random fields with SDDM precision matrices. For sampling this natural subclass of Gaussian random fields, it is optimal in the randomness and nearly optimal in the work and parallel complexity. In addition, our sampling algorithm can be directly extended to Gaussian random fields with SDD precision matrices

Spectra of random matrices

The recent interest of the scientific community about the properties of networks is based on the possibility to study complex real world systems by renouncing the exact knowledge of the nature of system itself. This approach allows to model the system, for example, as a large collection of agents linked together in pairs to form a network. The networks are very studied in different scientific fields and, particularly, in ecological one, in order to understand the dynamics of the evolution related to a community composed by different species interacting with each other. A random matrix can incorporate many information according to the type of the system. By using the graph’s theory, it is possible to extrapolate information about the matrix and, therefore, about the system considered. The statistical features of the eigenvalues of large random matrices have been the focus of wide interest in mathematics and physics. This thesis is mainly focused on the study of the spectral density of sparse random matrices. Symmetric random matrices and non-Hermitian matrices have been considered in this work, paying attention to both the analytical and numerical approach of the eigenvalues distribution calculation. There are different mathematical methods used to analyze ensembles of random matrices with a particular underlying symmetry. It is well-known that the spectral density of random matrices ensembles will converge, as the matrix dimensions grows, to a precise limit. One example is Girko elliptic law. The introduction of the sparsity is one of the factors that complicate enormously the mathematical analysis and new techniques for the calculation of the spectral density are welcome. The cavity method is a new approach presented to extend our knowledge about large-scale statistical behavior of eigenvalues of random sparse Hermitian and non-Hermitian matrices. Therefore, the cavity method provides a specific analysis related to the study about how the modularity structure influences the stability in the ecological communities

Spectral Theory of Sparse Non-Hermitian Random Matrices

Sparse non-Hermitian random matrices arise in the study of disordered physical systems with asymmetric local interactions, and have applications ranging from neural networks to ecosystem dynamics. The spectral characteristics of these matrices provide crucial information on system stability and susceptibility, however, their study is greatly complicated by the twin challenges of a lack of symmetry and a sparse interaction structure. In this review we provide a concise and systematic introduction to the main tools and results in this field. We show how the spectra of sparse non-Hermitian matrices can be computed via an analogy with infinite dimensional operators obeying certain recursion relations. With reference to three illustrative examples --- adjacency matrices of regular oriented graphs, adjacency matrices of oriented Erd\H{o}s-R\'{e}nyi graphs, and adjacency matrices of weighted oriented Erd\H{o}s-R\'{e}nyi graphs --- we demonstrate the use of these methods to obtain both analytic and numerical results for the spectrum, the spectral distribution, the location of outlier eigenvalues, and the statistical properties of eigenvectors.Comment: 60 pages, 10 figure

Spectra of Sparse Random Matrices

We compute the spectral density for ensembles of of sparse symmetric random matrices using replica, managing to circumvent difficulties that have been encountered in earlier approaches along the lines first suggested in a seminal paper by Rodgers and Bray. Due attention is payed to the issue of localization. Our approach is not restricted to matrices defined on graphs with Poissonian degree distribution. Matrices defined on regular random graphs or on scale-free graphs, are easily handled. We also look at matrices with row constraints such as discrete graph Laplacians. Our approach naturally allows to unfold the total density of states into contributions coming from vertices of different local coordination.Comment: 22 papges, 8 figures (one on graph-Laplacians added), one reference added, some typos eliminate