The singularity probability of a random symmetric matrix is exponentially small

Campos, Marcelo; Jenssen, Matthew; Michelen, Marcus; Sahasrabudhe, Julian

doi:10.48550/arxiv.2105.11384

Cited by 6 publications

(8 citation statements)

References 47 publications

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“…Second, the probability changes as a function of the matrix size (Campos, Jenssen, Michelen, & Sahasrabudhe, 2021;Coja-Oghlan, Ergür, Gao, Hetterich, & Rolvien, 2020;Bourgain, Vu, & Wood, 2010;Komlós, 1967Komlós, , 1968Tao & Vu, 2007). Third, certain types of matrices are of interest, such as symmetric ones (Campos, Jenssen, Michelen, & Sahasrabudhe, 2021).…”

Section: Appendixmentioning

confidence: 99%

Theoretical Exploration of Solutions of Feedforward ReLU Networks

Huang¹

2022

Preprint

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This paper aims to interpret the mechanism of feedforward ReLU networks by exploring their solutions for piecewise linear functions through basic rules. The constructed solutions should be universal enough to explain the network architectures of engineering. In order for that, we borrow the methodology of theoretical physics to develop the theories. Some of the consequences of our theories include: Under geometric backgrounds, the solutions of both three-layer networks and deep-layer networks are presented, and the solution universality is ensured by several ways; We give clear and intuitive interpretations of each component of network architectures, such as the parameter-sharing mechanism for multi-output, the function of each layer, the advantage of deep layers, the redundancy of parameters, and so on. We explain three typical network architectures: the subnetwork of last three layers of convolutional networks, multi-layer feedforward networks, and the decoder of autoencoders. This paper is expected to provide a basic foundation of theories of feedforward ReLU networks for further investigations.

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Section: Appendixmentioning

confidence: 99%

Theoretical Exploration of Solutions of Feedforward ReLU Networks

Huang¹

2022

Preprint

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“…We note that estimating the singularity probability for several models of discrete random matrices is a major topic within the combinatorial random matrix theory [49,46,83,10,21,62]. In last few years there has been a significant progress in this research direction (also, as corollaries of quantitative results), in particular, the problem of estimating the sigularity probability of adjacency matrices of random regular (di)graphs [37,61,64], of Bernoulli random matrices [91,56,42] and, more generally, discrete matrices with i.i.d entries [43], of random symmetric matrices [13,29,11,12]. We refer to a recent survey [96] for a discussion and further references.…”

Section: Introductionmentioning

confidence: 99%

Quantitative invertibility of non-Hermitian random matrices

Tikhomirov¹

2022

Preprint

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“…It was noted in [8] that given the techniques in [3,8], Theorem 1.1 (with the weaker probability bound 1 − o n (1)) can be deduced from the following universality statement: the probability that tI n − M n is invertible over F p , for p = n Ω (1) , is essentially the same as for an n × n symmetric matrix whose entries on and above the diagonal are sampled from the uniform distribution on F p . Despite the intensive efforts to study the singularity probability of symmetric Rademacher matrices ([4,6,7, 10,14,20,24] and especially the recent breakthrough [5] which confirms the long-standing conjecture that the singularity probability of symmetric Rademacher matrices is exponentially small), a result of this precision has remained elusive. While for p = o(n 1/8 ), such a result is known due to work of Maples [18], the bound on p is too restrictive to imply Theorem 1.1 (even with the weaker probability 1 − o n (1)).…”

Section: Introductionmentioning

confidence: 99%

Random symmetric matrices: rank distribution and irreducibility of the characteristic polynomial

Ferber¹,

Jain²,

Sah³

et al. 2021

Preprint

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Conditional on the extended Riemann hypothesis, we show that with high probability, the characteristic polynomial of a random symmetric {±1}-matrix is irreducible. This addresses a question raised by Eberhard in recent work. The main innovation in our work is establishing sharp estimates regarding the rank distribution of symmetric random {±1}-matrices over Fp for primes 2 < p ≤ exp(O(n 1/4 )). Previously, such estimates were available only for p = o(n 1/8 ). At the heart of our proof is a way to combine multiple inverse Littlewood-Offord-type results to control the contribution to singularity-type events of vectors in F n p with anticoncentration at least 1/p + Ω(1/p 2 ). Previously, inverse Littlewood-Offord-type results only allowed control over vectors with anticoncentration at least C/p for some large constant C > 1.and applying Theorem B.2 once again finishes.

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The singularity probability of a random symmetric matrix is exponentially small

Abstract: Let A be drawn uniformly at random from the set of all n × n symmetric matrices with entries in {−1, 1}. We show thatwhere c > 0 is an absolute constant, thereby resolving a well-known conjecture.

Cited by 6 publications

References 47 publications

Theoretical Exploration of Solutions of Feedforward ReLU Networks

Theoretical Exploration of Solutions of Feedforward ReLU Networks

Quantitative invertibility of non-Hermitian random matrices

Random symmetric matrices: rank distribution and irreducibility of the characteristic polynomial

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