Limiting behavior of random permanents

Rempała, Grzegorz A.; Wesołowski, Jacek

doi:10.1016/s0167-7152(99)00054-1

Cited by 8 publications

(5 citation statements)

References 10 publications

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“…In order to apply Jensen's formula we observe that the left hand side of the formula, for zero mean random matrices A, and g A (z) = Per(J + zA) is essentially bounded from above by the the logarithm of the second moment g A (r). We prove in Lemma 7 (which provides a simplified version of a proof by Rempała and Wesołowski [18]) that this second moment is upper-bounded by a term that scales at most exponentially with the square of r:…”

Section: Roots Of Random Interpolating Polynomialsmentioning

confidence: 90%

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Approximating the Permanent of a Random Matrix with Vanishing Mean

Eldar¹,

Mehraban²

2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

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The permanent is #P -hard to compute exactly on average for natural random matrices including matrices over finite fields or Gaussian ensembles. Should we expect that it remains #P -hard to compute on average if we only care about approximation instead of exact computation?In this work we take a first step towards resolving this question: We present a quasipolynomial time deterministic algorithm for approximating the permanent of a typical n × n random matrix with unit variance and vanishing mean µ = O(ln ln n) −1/8 to within inverse polynomial multiplicative error. Alternatively, one can achieve permanent approximation for matrices with mean µ = 1/polylog(n) in time 2 O(n ε ) , for any ε > 0.The proposed algorithm significantly extends the regime of matrices for which efficient approximation of the permanent is known. This is because unlike previous algorithms which require a stringent correlation between the signs of the entries of the matrix [1, 2] it can tolerate random ensembles in which this correlation is negligible (albeit non-zero). Among important special cases we note:1. Biased Gaussian: each entry is a complex Gaussian with unit variance 1 and mean µ.2. Biased Bernoulli: each entry is −1 + µ with probability 1/2, and 1 with probability 1/2.

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Section: Roots Of Random Interpolating Polynomialsmentioning

confidence: 90%

“…We begin with an upper-bound on the average sensitivity of the permanent of a random matrix that can also be derived from the work of Rempała and Wesołowski [18] (see Proposition 1).…”

Section: Permanent-interpolating-polynomial As a Random Polynomialmentioning

confidence: 99%

Approximating the Permanent of a Random Matrix with Vanishing Mean

Eldar¹,

Mehraban²

2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

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“…Now (12) follows from the estimates in Lemma 7 (for the lower bound in the case k > 1 use k − 1 ≤ n).…”

Section: Proof Of the Upper Bounds In Theoremmentioning

confidence: 98%

“…Say we allow ξ to have a point mass at zero. Then by the known central limit theorem (see [12]) we know that per A n /(n log n) converges to 1 in probability (one needs to truncate ξ at some finite value). In the algorithm from the above proof, one can apply the same argument to the matrix B c , to show that per A n /(n log n) converges in probability to β for β > 1.…”

Section: Lower Bounds and The Proof Of Theoremmentioning

confidence: 99%

“…In the case of i.i.d. elements, relying on earlier results of van Es and Helmers [2] and Borovskikh and Korolyuk [8], they proved central limit theorems for (per A)/E(per A) [12], and later certain strong laws of large numbers [13], when the height of the matrix grows much slower than the width. See also Chapter 3 in [14] for a self-contained discussion of these results.…”

Section: Introductionmentioning

confidence: 99%

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Permanents of heavy-tailed random matrices with positive elements

Antunović¹

2011

Preprint

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We study the asymptotic behavior of permanents of n × n random matrices A with positive entries. We assume that A has either i.i.d. entries or is a symmetric matrix with the i.i.d. upper triangle. Under the assumption that elements have power law decaying tails, we prove a strong law of large numbers for log per A. We calculate the values of the limit limn→∞ log per A n log n in terms of the exponent of the power law distribution decay, and observe a first order phase transition in the limit as the mean becomes infinite. The methods extend to a wide class of rectangular matrices. It is also shown that, in finite mean regime, the limiting behavior holds uniformly over all submatrices of linear size.

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