Distributions of Demmel and Related Condition Numbers

Dharmawansa, K.D.P.; McKay, Matthew R.; Chen, Yang

doi:10.1137/110843423

Cited by 10 publications

(12 citation statements)

References 48 publications

(88 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Note that the extreme value statistics of such random variables (the largest, ω max = y max / N i=1 y i , and the smallest, ω min = y min / N i=1 y i , which is the reciprocal of the so called Demmel condition number, see, e.g., [40] and more recent [41,42]) plays a key role in various scale independent hypothesis testing procedures, both in classical statistics as well as in signal processing. Classical examples (see, e.g., [13,14]) include testing for the presence of interactions in multi-way data and testing for equality of the population covariance to a scaled identity matrix.…”

Section: B β-Ft Vs Scaled-variables Ensemblementioning

confidence: 99%

Random pure states: Quantifying bipartite entanglement beyond the linear statistics

2016

View full text Add to dashboard Cite

We analyze the properties of entangled random pure states of a quantum system partitioned into two smaller subsystems of dimensions N and M. Framing the problem in terms of random matrices with a fixed-trace constraint, we establish, for arbitrary N≤M, a general relation between the n-point densities and the cross moments of the eigenvalues of the reduced density matrix, i.e., the so-called Schmidt eigenvalues, and the analogous functionals of the eigenvalues of the Wishart-Laguerre ensemble of the random matrix theory. This allows us to derive explicit expressions for two-level densities, and also an exact expression for the variance of von Neumann entropy at finite N,M. Then, we focus on the moments E{K^{a}} of the Schmidt number K, the reciprocal of the purity. This is a random variable supported on [1,N], which quantifies the number of degrees of freedom effectively contributing to the entanglement. We derive a wealth of analytical results for E{K^{a}} for N=2 and 3 and arbitrary M, and also for square N=M systems by spotting for the latter a connection with the probability P(x_{min}^{GUE}≥sqrt[2N]ξ) that the smallest eigenvalue x_{min}^{GUE} of an N×N matrix belonging to the Gaussian unitary ensemble is larger than sqrt[2N]ξ. As a by-product, we present an exact asymptotic expansion for P(x_{min}^{GUE}≥sqrt[2N]ξ) for finite N as ξ→∞. Our results are corroborated by numerical simulations whenever possible, with excellent agreement.

show abstract

Section: B β-Ft Vs Scaled-variables Ensemblementioning

confidence: 99%

Random pure states: Quantifying bipartite entanglement beyond the linear statistics

2016

View full text Add to dashboard Cite

show abstract

“…7 Specific steps pertaining to this evaluation are not given here as the detailed steps of solving an analogous integral have been given in [19].…”

Section: Cumulative Distribution Of the Minimum Eigenvaluementioning

confidence: 99%

Some new results on the eigenvalues of complex non-central Wishart matrices with a rank-1 mean

Dharmawansa

2016

Journal of Multivariate Analysis

Self Cite

View full text Add to dashboard Cite

Recently, D. Wang [44] has devised a new contour integral based method to simplify certain matrix integrals. Capitalizing on that approach, we derive a new expression for the probability density function (p.d.f.) of the joint eigenvalues of a complex noncentral Wishart matrix with a rank-1 mean. The resulting functional form in turn enables us to use powerful classical orthogonal polynomial techniques in solving three problems related to the non-central Wishart matrix. To be specific, for an n × n complex non-central Wishart matrix W with m degrees of freedom (m ≥ n) and a rank-1 mean, we derive a new expression for the cumulative distribution function (c.d.f.) of the minimum eigenvalue (λ min ). The c.d.f. is expressed as the determinant of a square matrix, the size of which depends only on the difference m − n. This further facilitates the analysis of the microscopic limit for the minimum eigenvalue which takes the form of the determinant of a square matrix of size m − n with the Bessel kernel. We also develop a moment generating function based approach to derive the p.d.f. of the random variable tr(W) λmin , where tr(·) denotes the trace of a square matrix. This random quantity is of great importance in the so-called smoothed analysis of Demmel condition number. Finally, we find the average of the reciprocal of the characteristic polynomial det[zI n + W], | arg z| < π, where I n and det[·] denote the identity matrix of size n and the determinant, respectively.Let us first define the p.d.f. of a complex non-central Wishart matrix.

show abstract

“…Another fundamental form introduced by Demmel in his seminal work on the probabilistic analysis of the degree of difficulty associated with numerical analysis problems [5] is defined as κ D (A) = ||A|| F ||A −1 || 2 , where || • || F denotes the Frobenius norm. This definition naturally extends to rectangular matrices [9], [14] giving…”

Section: Introductionmentioning

confidence: 99%

“…≤ λ r are the non-zero eigenvalues of X * X (or XX * ) with (•) * denoting the conjugate transpose operator. Since the statistical characteristics of κ SC (X) are of paramount importance in many scientific disciplines, it is common to assume X to be real/complex Gaussian distributed with m ≥ n [13], [14], [16]- [18] which in turn gives…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Distribution of the Scaled Condition Number of Single-spiked Complex Wishart Matrices

Pasan¹,

Dharmawansa²,

Chen³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

Let X ∈ C m×n (m ≥ n) be a random matrix with independent rows each distributed as complex multivariate Gaussian with zero mean and single-spiked covariance matrix I n + ηuu * , where I n is the n × n identity matrix, u ∈ C n×n is an arbitrary vector with a unit Euclidean norm, η ≥ 0 is a non-random parameter, and (•) * represents conjugate-transpose. This paper investigates the distribution of the random quantity κ 2 SC (X) = n k=1 λ k /λ 1 , where 0 < λ 1 < λ 2 < . . . < λ n < ∞ are the ordered eigenvalues of X * X (i.e., single-spiked Wishart matrix). This random quantity is intimately related to the so called scaled condition number or the Demmel condition number (i.e., κ SC (X)) and the minimum eigenvalue of the fixed trace Wishart-Laguerre ensemble (i.e., κ −2 SC (X)). In particular, we use an orthogonal polynomial approach to derive an exact expression for the probability density function of κ 2 SC (X) which is amenable to asymptotic analysis as matrix dimensions grow large. Our asymptotic results reveal that, as m, n → ∞ such that m − n is fixed and when η scales on the order of 1/n, κ 2 SC (X) scales on the order of n 3 . In this respect we establish simple closed-form expressions for the limiting distributions.

show abstract

Distributions of Demmel and Related Condition Numbers

Cited by 10 publications

References 48 publications

Random pure states: Quantifying bipartite entanglement beyond the linear statistics

Random pure states: Quantifying bipartite entanglement beyond the linear statistics

Some new results on the eigenvalues of complex non-central Wishart matrices with a rank-1 mean

Distribution of the Scaled Condition Number of Single-spiked Complex Wishart Matrices

Contact Info

Product

Resources

About