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Alan Edelman

Publications and source records attributed to Alan Edelman.

87 records · Page 5Linked to original sources
arXiv2011

An Efficient Partitioning Oracle for Bounded-Treewidth Graphs

Partitioning oracles were introduced by Hassidim et al. (FOCS 2009) as a generic tool for constant-time algorithms. For any epsilon > 0, a partitioning oracle provides query access to a fixed partition of the input bounded-degree minor-free graph, in which every component has size poly(1/epsilon), and the number of edges removed is at most epsilon*n, where n is the number of vertices in the graph. However, the oracle of Hassidimet al. makes an exponential number of queries to the input graph to answer every query about the partition. In this paper, we construct an efficient partitioning oracle for graphs with constant treewidth. The oracle makes only O(poly(1/epsilon)) queries to the input graph to answer each query about the partition. Examples of bounded-treewidth graph classes include k-outerplanar graphs for fixed k, series-parallel graphs, cactus graphs, and pseudoforests. Our oracle yields poly(1/epsilon)-time property testing algorithms for membership in these classes of graphs. Another application of the oracle is a poly(1/epsilon)-time algorithm that approximates the maximum matching size, the minimum vertex cover size, and the minimum dominating set size up to an additive epsilon*n in graphs with bounded treewidth. Finally, the oracle can be used to test in poly(1/epsilon) time whether the input bounded-treewidth graph is k-colorable or perfect.

cs.DS↗
arXiv2009

Statistical eigen-inference from large Wishart matrices

We consider settings where the observations are drawn from a zero-mean multivariate (real or complex) normal distribution with the population covariance matrix having eigenvalues of arbitrary multiplicity. We assume that the eigenvectors of the population covariance matrix are unknown and focus on inferential procedures that are based on the sample eigenvalues alone (i.e., "eigen-inference"). Results found in the literature establish the asymptotic normality of the fluctuation in the trace of powers of the sample covariance matrix. We develop concrete algorithms for analytically computing the limiting quantities and the covariance of the fluctuations. We exploit the asymptotic normality of the trace of powers of the sample covariance matrix to develop eigenvalue-based procedures for testing and estimation. Specifically, we formulate a simple test of hypotheses for the population eigenvalues and a technique for estimating the population eigenvalues in settings where the cumulative distribution function of the (nonrandom) population eigenvalues has a staircase structure. Monte Carlo simulations are used to demonstrate the superiority of the proposed methodologies over classical techniques and the robustness of the proposed techniques in high-dimensional, (relatively) small sample size settings. The improved performance results from the fact that the proposed inference procedures are "global" (in a sense that we describe) and exploit "global" information thereby overcoming the inherent biases that cripple classical inference procedures which are "local" and rely on "local" information.

math.ST↗
arXiv2007

The polynomial method for random matrices

We define a class of "algebraic" random matrices. These are random matrices for which the Stieltjes transform of the limiting eigenvalue distribution function is algebraic, i.e., it satisfies a (bivariate) polynomial equation. The Wigner and Wishart matrices whose limiting eigenvalue distributions are given by the semi-circle law and the Marcenko-Pastur law are special cases. Algebraicity of a random matrix sequence is shown to act as a certificate of the computability of the limiting eigenvalue density function. The limiting moments of algebraic random matrix sequences, when they exist, are shown to satisfy a finite depth linear recursion so that they may often be efficiently enumerated in closed form. In this article, we develop the mathematics of the polynomial method which allows us to describe the class of algebraic matrices by its generators and map the constructive approach we employ when proving algebraicity into a software implementation that is available for download in the form of the RMTool random matrix "calculator" package. Our characterization of the closure of algebraic probability distributions under free additive and multiplicative convolution operations allows us to simultaneously establish a framework for computational (non-commutative) "free probability" theory. We hope that the tools developed allow researchers to finally harness the power of the infinite random matrix theory.

math.PR↗
arXiv2007

Sample eigenvalue based detection of high dimensional signals in white noise using relatively few samples

We present a mathematically justifiable, computationally simple, sample eigenvalue based procedure for estimating the number of high-dimensional signals in white noise using relatively few samples. The main motivation for considering a sample eigenvalue based scheme is the computational simplicity and the robustness to eigenvector modelling errors which are can adversely impact the performance of estimators that exploit information in the sample eigenvectors. There is, however, a price we pay by discarding the information in the sample eigenvectors; we highlight a fundamental asymptotic limit of sample eigenvalue based detection of weak/closely spaced high-dimensional signals from a limited sample size. This motivates our heuristic definition of the effective number of identifiable signals which is equal to the number of "signal" eigenvalues of the population covariance matrix which exceed the noise variance by a factor strictly greater than 1+sqrt(Dimensionality of the system/Sample size). The fundamental asymptotic limit brings into sharp focus why, when there are too few samples available so that the effective number of signals is less than the actual number of signals, underestimation of the model order is unavoidable (in an asymptotic sense) when using any sample eigenvalue based detection scheme, including the one proposed herein. The analysis reveals why adding more sensors can only exacerbate the situation. Numerical simulations are used to demonstrate that the proposed estimator consistently estimates the true number of signals in the dimension fixed, large sample size limit and the effective number of identifiable signals in the large dimension, large sample size limit.

math.ST↗
arXiv2007

Sample size cognizant detection of signals in white noise

The detection and estimation of signals in noisy, limited data is a problem of interest to many scientific and engineering communities. We present a computationally simple, sample eigenvalue based procedure for estimating the number of high-dimensional signals in white noise when there are relatively few samples. We highlight a fundamental asymptotic limit of sample eigenvalue based detection of weak high-dimensional signals from a limited sample size and discuss its implication for the detection of two closely spaced signals. This motivates our heuristic definition of the 'effective number of identifiable signals.' Numerical simulations are used to demonstrate the consistency of the algorithm with respect to the effective number of signals and the superior performance of the algorithm with respect to Wax and Kailath's "asymptotically consistent" MDL based estimator.

cs.IT↗
arXiv2006

From Random Matrices to Stochastic Operators

We propose that classical random matrix models are properly viewed as finite difference schemes for stochastic differential operators. Three particular stochastic operators commonly arise, each associated with a familiar class of local eigenvalue behavior. The stochastic Airy operator displays soft edge behavior, associated with the Airy kernel. The stochastic Bessel operator displays hard edge behavior, associated with the Bessel kernel. The article concludes with suggestions for a stochastic sine operator, which would display bulk behavior, associated with the sine kernel.

math-ph↗
arXiv2005

Numerical Methods for Eigenvalue Distributions of Random Matrices

We present efficient numerical techniques for calculation of eigenvalue distributions of random matrices in the beta-ensembles. We compute histograms using direct simulations on very large matrices, by using tridiagonal matrices with appropriate simplifications. The distributions are also obtained by numerical solution of the Painleve II and V equations with high accuracy. For the spacings we show a technique based on the Prolate matrix and Richardson extrapolation, and we compare the distributions with the zeros of the Riemann zeta function.

math-ph↗
arXiv2004

MOPS: Multivariate Orthogonal Polynomials (symbolically)

In this paper we present a Maple library (MOPs) for computing Jack, Hermite, Laguerre, and Jacobi multivariate polynomials, as well as eigenvalue statistics for the Hermite, Laguerre, and Jacobi ensembles of Random Matrix theory. We also compute multivariate hypergeometric functions, and offer both symbolic and numerical evaluations for all these quantities. We prove that all algorithms are well-defined, analyze their complexity, and illustrate their performance in practice. Finally, we also present a few of the possible applications of this library.

math-ph↗
arXiv2004

Eigenvalues of Hermite and Laguerre ensembles: Large Beta Asymptotics

In this paper we examine the zero and first order eigenvalue fluctuations for the $β$-Hermite and $β$-Laguerre ensembles, using the matrix models we described in \cite{dumitriu02}, in the limit as $β\to \infty$. We find that the fluctuations are described by Gaussians of variance $O(1/β)$, centered at the roots of a corresponding Hermite (Laguerre) polynomial. We also show that the approximation is very good, even for small values of $β$, by plotting exact level densities versus sum of Gaussians approximations.

math-ph↗
arXiv2002

Matrix Models for Beta Ensembles

This paper constructs tridiagonal random matrix models for general ($β>0$) $β$-Hermite (Gaussian) and $β$-Laguerre (Wishart) ensembles. These generalize the well-known Gaussian and Wishart models for $β= 1,2,4$. Furthermore, in the cases of the $β$-Laguerre ensembles, we eliminate the exponent quantization present in the previously known models. We further discuss applications for the new matrix models, and present some open problems.

math-ph↗
arXiv1998

The Geometry of Algorithms with Orthogonality Constraints

In this paper we develop new Newton and conjugate gradient algorithms on the Grassmann and Stiefel manifolds. These manifolds represent the constraints that arise in such areas as the symmetric eigenvalue problem, nonlinear eigenvalue problems, electronic structures computations, and signal processing. In addition to the new algorithms, we show how the geometrical framework gives penetrating new insights allowing us to create, understand, and compare algorithms. The theory proposed here provides a taxonomy for numerical linear algebra algorithms that provide a top level mathematical view of previously unrelated algorithms. It is our hope that developers of new algorithms and perturbation theories will benefit from the theory, methods, and examples in this paper.

physics.comp-ph↗
arXiv1998

Multiscale Computation with Interpolating Wavelets

Multiresolution analyses based upon interpolets, interpolating scaling functions introduced by Deslauriers and Dubuc, are particularly well-suited to physical applications because they allow exact recovery of the multiresolution representation of a function from its sample values on a finite set of points in space. We present a detailed study of the application of wavelet concepts to physical problems expressed in such bases. The manuscript describes algorithms for the associated transforms which, for properly constructed grids of variable resolution, compute correctly without having to introduce extra grid points. We demonstrate that for the application of local homogeneous operators in such bases, the non-standard multiply of Beylkin, Coifman and Rokhlin also proceeds exactly for inhomogeneous grids of appropriate form. To obtain less stringent conditions on the grids, we generalize the non-standard multiply so that communication may proceed between non-adjacent levels. The manuscript concludes with timing comparisons against naive algorithms and an illustration of the scale-independence of the convergence rate of the conjugate gradient solution of Poisson's equation using a simple preconditioning, suggesting that this approach leads to an O(n) solution of this equation.

cond-mat.mtrl-sci↗
arXiv1995

How many zeros of a random polynomial are real?

We provide an elementary geometric derivation of the Kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients. We show that the expected number of real zeros is simply the length of the moment curve $(1,t,\ldots,t^n)$ projected onto the surface of the unit sphere, divided by $π$. The probability density of the real zeros is proportional to how fast this curve is traced out. We then relax Kac's assumptions by considering a variety of random sums, series, and distributions, and we also illustrate such ideas as integral geometry and the Fubini-Study metric.

math.CA↗