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We present a randomized approximation scheme for the permanent of a matrix with nonnegative entries. Our scheme extends a recursive rejection sampling method of Huber and Law (SODA 2008) by replacing the upper bound for the permanent with a…
Concentration of measure is a phenomenon in which a random variable that depends in a smooth way on a large number of independent random variables is essentially constant. The random variable will "concentrate" around its median or…
We revisit the topic of probability measures on CAT(0) cube complexes and prove that an amenable group acting on a CAT(0) cube complex, regardless of dimension, necessarily preserves an interval in the Roller compactification. In the finite…
Binary embedding of high-dimensional data aims to produce low-dimensional binary codes while preserving discriminative power. State-of-the-art methods often suffer from high computation and storage costs. We present a simple and fast…
The salient properties of large empirical covariance and correlation matrices are studied for three datasets of size 54, 55 and 330. The covariance is defined as a simple cross product of the returns, with weights that decay logarithmically…
We study random points on the real line generated by the eigenvalues in unitary invariant random matrix ensembles or by more general repulsive particle systems. As the number of points tends to infinity, we prove convergence of the…
Data analysis in cosmology requires reliable covariance matrices. Covariance matrices derived from numerical simulations often require a very large number of realizations to be accurate. When a theoretical model for the covariance matrix…
Fourier matrices naturally appear in many applications and their stability is closely tied to performance guarantees of algorithms. The starting point of this article is a result that characterizes properties of an exponential system on a…
We consider estimation of the covariance matrix of a multivariate random vector under the constraint that certain covariances are zero. We first present an algorithm, which we call Iterative Conditional Fitting, for computing the maximum…
In this paper, we show that the largest and smallest eigenvalues of a sample correlation matrix stemming from $n$ independent observations of a $p$-dimensional time series with iid components converge almost surely to $(1+\sqrt{\gamma})^2$…
It is shown that if the wave function of a quantum system undergoes an arbitrary random transformation such that the diagonal elements of the density matrix in the decoherence basis associated with a preferred observable remain constant,…
Let $M_n$ be a random matrix of size $n\times n$ and let $\lambda_1,...,\lambda_n$ be the eigenvalues of $M_n$. The empirical spectral distribution $\mu_{M_n}$ of $M_n$ is defined as $$\mu_{M_n}(s,t)=\frac{1}{n}# \{k\le n, \Re(\lambda_k)\le…
We present a fast algorithm for global rigid symmetry detection with approximation guarantees. The algorithm is guaranteed to find the best approximate symmetry of a given shape, to within a user-specified threshold, with very high…
We study decoherence of two non-interacting qubits. The environment and its interaction with the qubits are modelled by random matrices. Decoherence, measured in terms of purity, is calculated in linear response approximation. Monte Carlo…
How many samples are sufficient to guarantee that the eigenvectors and eigenvalues of the sample covariance matrix are close to those of the actual covariance matrix? For a wide family of distributions, including distributions with finite…
Continuous phase estimation is known to be superior in accuracy as compared to static estimation. The estimation process is, however, desired to be made robust to uncertainties in the underlying parameters. Here, homodyne phase estimation…
We present fixed domain asymptotic results that establish consistent estimates of the variance and scale parameters for a Gaussian random field with a geometric anisotropic Mat\'ern autocovariance in dimension $d>4$. When $d<4$ this is…
We study the complexity of classically sampling from the output distribution of an Ising spin model, which can be implemented naturally in a variety of atomic, molecular, and optical systems. In particular, we construct a specific example…
We propose a numerical method, based on the shift-and-invert power iteration, that answers whether a symmetric matrix is positive definite ("yes") or not ("no"). Our method uses randomization. But, it returns the correct answer with high…
This study presents an innovative direct numerical simulation approach for complex particle systems with irregular shapes and large numbers. Using partially saturated methods, it accurately models arbitrary shapes, albeit at considerable…