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Using Multivariate Statistics

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

This paper considers a natural error correcting problem with real valued input/output. We wish to recover an input vector f/spl isin/R/sup n/ from corrupted measurements y=Af+e. Here, A is an m by n (coding) matrix and e is an arbitrary and unknown vector of errors. Is it possible to recover f exactly from the data y? We prove that under suitable conditions on the coding matrix A, the input f is the unique solution to the /spl lscr//sub 1/-minimization problem (/spl par/x/spl par//sub /spl lscr/1/:=/spl Sigma//sub i/|x/sub i/|) min(g/spl isin/R/sup n/) /spl par/y - Ag/spl par//sub /spl lscr/1/ provided that the support of the vector of errors is not too large, /spl par/e/spl par//sub /spl lscr/0/:=|{i:e/sub i/ /spl ne/ 0}|/spl les//spl rho//spl middot/m for some /spl rho/>0. In short, f can be recovered exactly by solving a simple convex optimization problem (which one can recast as a linear program). In addition, numerical experiments suggest that this recovery procedure works unreasonably well; f is recovered exactly even in situations where a significant fraction of the output is corrupted. This work is related to the problem of finding sparse solutions to vastly underdetermined systems of linear equations. There are also significant connections with the problem of recovering signals from highly incomplete measurements. In fact, the results introduced in this paper improve on our earlier work. Finally, underlying the success of /spl lscr//sub 1/ is a crucial property we call the uniform uncertainty principle that we shall describe in detail.

/pdf/decoding-by-linear-programming-4q5aw83zqx.pdf

Decoding by linear programming

Suppose we are given a vector f in a class FsubeRopfN , e.g., a class of digital signals or digital images. How many linear measurements do we need to make about f to be able to recover f to within precision epsi in the Euclidean (lscr2) metric? This paper shows that if the objects of interest are sparse in a fixed basis or compressible, then it is possible to reconstruct f to within very high accuracy from a small number of random measurements by solving a simple linear program. More precisely, suppose that the nth largest entry of the vector |f| (or of its coefficients in a fixed basis) obeys |f|(n)lesRmiddotn-1p/, where R>0 and p>0. Suppose that we take measurements yk=langf# ,Xkrang,k=1,...,K, where the Xk are N-dimensional Gaussian vectors with independent standard normal entries. Then for each f obeying the decay estimate above for some 0<p<1 and with overwhelming probability, our reconstruction ft, defined as the solution to the constraints yk=langf# ,Xkrang with minimal lscr1 norm, obeys parf-f#parlscr2lesCp middotRmiddot(K/logN)-r, r=1/p-1/2. There is a sense in which this result is optimal; it is generally impossible to obtain a higher accuracy from any set of K measurements whatsoever. The methodology extends to various other random measurement ensembles; for example, we show that similar results hold if one observes a few randomly sampled Fourier coefficients of f. In fact, the results are quite general and require only two hypotheses on the measurement ensemble which are detailed

/pdf/near-optimal-signal-recovery-from-random-projections-2mmu9nt75f.pdf

Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?

This paper considers the classical error correcting problem which is frequently discussed in coding theory. We wish to recover an input vector $f \in \R^n$ from corrupted measurements $y = A f + e$. Here, $A$ is an $m$ by $n$ (coding) matrix and $e$ is an arbitrary and unknown vector of errors. Is it possible to recover $f$ exactly from the data $y$? We prove that under suitable conditions on the coding matrix $A$, the input $f$ is the unique solution to the $\ell_1$-minimization problem ($\|x\|_{\ell_1} := \sum_i |x_i|$) $$ \min_{g \in \R^n} \| y - Ag \|_{\ell_1} $$ provided that the support of the vector of errors is not too large, $\|e\|_{\ell_0} := |\{i : e_i \neq 0\}| \le \rho \cdot m$ for some $\rho > 0$. In short, $f$ can be recovered exactly by solving a simple convex optimization problem (which one can recast as a linear program). In addition, numerical experiments suggest that this recovery procedure works unreasonably well; $f$ is recovered exactly even in situations where a significant fraction of the output is corrupted.

Decoding by Linear Programming

Wigner Matrices and Semicircular Law.- Sample Covariance Matrices and the Mar#x010D enko-Pastur Law.- Product of Two Random Matrices.- Limits of Extreme Eigenvalues.- Spectrum Separation.- Semicircular Law for Hadamard Products.- Convergence Rates of ESD.- CLT for Linear Spectral Statistics.- Eigenvectors of Sample Covariance Matrices.- Circular Law.- Some Applications of RMT.

Spectral Analysis of Large Dimensional Random Matrices

On the empirical distribution of eigenvalues of a class of large dimensional random matrices

In this paper, we give a brief review of the theory of spectral analysis of large dimensional random matrices Most of the existing work in the literature has been stated for real matrices but the corresponding results for the complex case are also of interest, especially for researchers in Electrical and Electronic Engineering Thus, we convert almost all results to the complex case, whenever possible Only the latest results, including some new ones, are stated as theorems here The main purpose of the paper is to show how important methodologies, or mathematical tools, have helped to develop the theory Some unsolved problems are also stated

/pdf/methodologies-in-spectral-analysis-of-large-dimensional-1uksobl2hb.pdf

Methodologies in spectral analysis of large dimensional random matrices, a review

In this paper, the authors show that the smallest (if $p \leq n$) or the $(p - n + 1)$-th smallest (if $p > n$) eigenvalue of a sample covariance matrix of the form $(1/n)XX'$ tends almost surely to the limit $(1 - \sqrt y)^2$ as $n \rightarrow \infty$ and $p/n \rightarrow y \in (0,\infty)$, where $X$ is a $p \times n$ matrix with iid entries with mean zero, variance 1 and fourth moment finite. Also, as a by-product, it is shown that the almost sure limit of the largest eigenvalue is $(1 + \sqrt y)^2$, a known result obtained by Yin, Bai and Krishnaiah. The present approach gives a unified treatment for both the extreme eigenvalues of large sample covariance matrices.

/pdf/limit-of-the-smallest-eigenvalue-of-a-large-dimensional-1d787b2xxq.pdf

Limit of the smallest eigenvalue of a large dimensional sample covariance matrix

Let $B_n = (1/N)T_n^{1/2}X_n X_n^* T_n^{1/2}$, where $X_n$ is $n \times N$ with i.i.d. complex standardized entries having finite fourth moment and $T_n^{1/2}$ is a Hermitian square root of the nonnegative definite Hermitian matrix $T_n$. It is known that, as $n \to \infty$, if $n/N$ converges to a positive number and the empirical distribution of the eigenvalues of $T_n$ converges to a proper probability distribution, then the empirical distribution of the eigenvalues of $B_n$ converges a.s. to a nonrandom limit. In this paper we prove that, under certain conditions on the eigenvalues of $T_n$, for any closed interval outside the support of the limit, with probability 1 there will be no eigenvalues in this interval for all $n$ sufficiently large.

/pdf/no-eigenvalues-outside-the-support-of-the-limiting-spectral-2cpf7510mw.pdf

Zhidong Bai

Papers

Spectral Analysis of Large Dimensional Random Matrices

On the empirical distribution of eigenvalues of a class of large dimensional random matrices

Methodologies in spectral analysis of large dimensional random matrices, a review

Limit of the smallest eigenvalue of a large dimensional sample covariance matrix

No eigenvalues outside the support of the limiting spectral distribution of large-dimensional sample covariance matrices