Sparse Random Matrices have Simple Spectrum

Luh, Kyle; Vu, Van

doi:10.48550/arxiv.1802.03662

Cited by 8 publications

(27 citation statements)

References 20 publications

(36 reference statements)

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“…We note that since its first appearance in [25], the RLCD has been used in many works (see, e.g., [13,14,16,19,26]); the MRLCD (and median threshold, for discrete distributions) can replace these applications in a black-box manner, and likely lead to improved quantitative estimates. We also note that a related use of combinatorially incorporating arithmetic unstructure of different projections of a vector appeared in recent work of the authors [8]; however, the interaction with both the net and anticoncentration estimates is more delicate here.…”

Section: Introductionmentioning

confidence: 99%

On the smallest singular value of symmetric random matrices

Jain¹,

Sah²,

Sawhney³

2020

Preprint

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We show that for an n × n random symmetric matrix An, whose entries on and above the diagonal are independent copies of a sub-Gaussian random variable ξ with mean 0 and variance 1,) for all ǫ ≥ 0. This improves a result of Vershynin, who obtained such a bound with n 1/2 replaced by n c for a small constant c, and 1/8 replaced by (1/8) + η (with implicit constants also depending on η > 0). Furthermore, when ξ is a Rademacher random variable, we prove thatThe special case ǫ = 0 improves a recent result of Campos, Mattos, Morris, and Morrison, which showed that P[sn(AnThe main innovation in our work are new notions of arithmetic structure -the Median Regularized Least Common Denominator and the Median Threshold, which we believe should be more generally useful in contexts where one needs to combine anticoncentration information of different parts of a vector.

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Section: Introductionmentioning

confidence: 99%

On the smallest singular value of symmetric random matrices

Jain¹,

Sah²,

Sawhney³

2020

Preprint

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“…However, this does not rule out the possibility of having gaps ∆ and as such, to extract bounds on eigenvalue gaps one needs to look at the local spectral statistics of ĀG(n,p) . It has been recently proven that ĀG(n,p) has no degenerate eigenvalues (simple spectrum), almost surely as long as C log 6 (n)/n ≤ p ≤ 1 − C log 6 (n)/n for some constant C > 0 [37,42,43]. Note that these bounds are quite tight: for p = 1, we know that ĀG(n,p) has repeated eigenvalues while on the other hand for p = o(log(n)/n), the underlying random graph is disconnected, implying again that ĀG(n,p) has repeated eigenvalues.…”

Section: Mixing Of Quantum Walksmentioning

confidence: 99%

“…there is a non-zero gap between any two eigenvalues of A G(n,p) . Recently in [42], this was generalized to show that even sparse random graphs have simple spectrum. We recall the result here.…”

Section: Spectral Properties Of Erd öS-renyi Random Networkmentioning

confidence: 99%

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How Fast Do Quantum Walks Mix?

2020

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The fundamental problem of sampling from the limiting distribution of quantum walks on networks, known as mixing, finds widespread applications in several areas of quantum information and computation. Of particular interest in most of these applications, is the minimum time beyond which the instantaneous probability distribution of the quantum walk remains close to this limiting distribution, known as the quantum mixing time. However this quantity is only known for a handful of specific networks. In this letter, we prove an upper bound on the quantum mixing time for almost all networks, i.e. the fraction of networks for which our bound holds, goes to one in the asymptotic limit. To this end, using several results in random matrix theory, we find the quantum mixing time of Erdös-Renyi random networks: networks of n nodes where each edge exists with probability p independently. For example for dense random networks, where p is a constant, we show that the quantum mixing time is O(n 3/2+o(1) ). Besides opening avenues for the analytical study of quantum dynamics on random networks, our work could find applications beyond quantum information processing. Owing to the universality of Wigner random matrices, our results on the spectral properties of random graphs hold for general classes of random matrices that are ubiquitous in several areas of physics. In particular, our results could lead to novel insights into the equilibration times of isolated quantum systems defined by random Hamiltonians, a foundational problem in quantum statistical mechanics.The quantum dynamics of any discrete system can be captured by a quantum walk on a network, which is a universal model for quantum computation [1]. Besides being a useful primitive to design quantum algorithms [2][3][4][5][6], quantum walks are a powerful tool to model transport in quantum systems such as the transfer of excitations in light-harvesting systems [7][8][9]. Studying the long-time dynamics of quantum walks on networks is crucial to the understanding of these diverse problems. As quantum evolutions are unitary and hence distancepreserving, quantum walks never converge to a limiting distribution, unlike their classical counterpart. However, given a network of n nodes, one can define the limiting distribution of quantum walk on the network as the long-time average probability distribution of finding the walker in each node [10]. Of particular interest is the quantum mixing time: starting from some initial state, the minimum time after which the underlying quantum walk remains close to its limiting distribution.The importance of the problem of mixing for quantum walks cannot be overstated: this is at the heart of quantum speedups for a number of quantum algorithms [2,11] and is also key to demonstrating the equivalence between the standard (circuit) and Hamiltonian-based models of quantum computation [12,13]. Unfortunately, no general result exists for quantum mixing time on networks: it has been estimated for a handful of specific graphs (graphs and networks are ...

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“…Nevertheless, we found empirical evidence that A − → w does not have repeated eigenvalues (in magnitude) for a wide range of − → w . In fact, it has been proved that certain types of random matrices have simple spectrum with high probability (Tao & Vu, 2017;Luh & Vu, 2018). For completeness, to handle the rare scenario when λ…”

Section: Spectral Clustering With Maximal Eigenratiomentioning

confidence: 99%

Spectral clustering via adaptive layer aggregation for multi-layer networks

Huang¹,

Weng²,

Feng³

2020

Preprint

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One of the fundamental problems in network analysis is detecting community structure in multi-layer networks, of which each layer represents one type of edge information among the nodes. We propose integrative spectral clustering approaches based on effective convex layer aggregations. Our aggregation methods are strongly motivated by a delicate asymptotic analysis of the spectral embedding of weighted adjacency matrices and the downstream k-means clustering, in a challenging regime where community detection consistency is impossible. In fact, the methods are shown to estimate the optimal convex aggregation, which minimizes the mis-clustering error under some specialized multi-layer network models. Our analysis further suggests that clustering using Gaussian mixture models is generally superior to the commonly used k-means in spectral clustering. Extensive numerical studies demonstrate that our adaptive aggregation techniques, together with Gaussian mixture model clustering, make the new spectral clustering remarkably competitive compared to several popularly used methods.

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Sparse Random Matrices have Simple Spectrum

Cited by 8 publications

References 20 publications

On the smallest singular value of symmetric random matrices

On the smallest singular value of symmetric random matrices

How Fast Do Quantum Walks Mix?

Spectral clustering via adaptive layer aggregation for multi-layer networks

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