Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the smallest singular value of symmetric random matrices
260
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We show that for an $ntimes n$ random symmetric matrix $A_n$, whose entries on and above the diagonal are independent copies of a sub-Gaussian random variable $xi$ with mean $0$ and variance $1$, [mathbb{P}[s_n(A_n) le epsilon/sqrt{n}] le O_{xi}(epsilon^{1/8} + exp(-Omega_{xi}(n^{1/2}))) quad text{for all } epsilon ge 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 $(1/8) + eta$ (with implicit constants also depending on $eta > 0$). Furthermore, when $xi$ is a Rademacher random variable, we prove that [mathbb{P}[s_n(A_n) le epsilon/sqrt{n}] le O(epsilon^{1/8} + exp(-Omega((log{n})^{1/4}n^{1/2}))) quad text{for all } epsilon ge 0.] The special case $epsilon = 0$ improves a recent result of Campos, Mattos, Morris, and Morrison, which showed that $mathbb{P}[s_n(A_n) = 0] le O(exp(-Omega(n^{1/2}))).$ The 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.
rate research
Read More
In this paper we introduce the algorithm and the fixed point hardware to calculate the normalized singular value decomposition of a non-symmetric matrices using Givens fast (approximate) rotations. This algorithm only uses the basic combinational logic modules such as adders, multiplexers, encoders, Barrel shifters (B-shifters), and comparators and does not use any lookup table. This method in fact combines the iterative properties of singular value decomposition method and CORDIC method in one single iteration. The introduced architecture is a systolic architecture that uses two different types of processors, diagonal and non-diagonal processors. The diagonal processor calculates, transmits and applies the horizontal and vertical rotations, while the non-diagonal processor uses a fully combinational architecture to receive, and apply the rotations. The diagonal processor uses priority encoders, Barrel shifters, and comparators to calculate the rotation angles. Both processors use a series of adders to apply the rotation angles. The design presented in this work provides $2.83sim649$ times better energy per matrix performance compared to the state of the art designs. This performance achieved without the employment of pipelining; a better performance advantage is expected to be achieved employing pipelining.
Conditional on the extended Riemann hypothesis, we show that with high probability, the characteristic polynomial of a random symmetric ${pm 1}$-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 ${pm 1}$-matrices over $mathbb{F}_p$ for primes $2 < p leq exp(O(n^{1/4}))$. 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 $mathbb{F}_p^{n}$ with anticoncentration at least $1/p + Omega(1/p^2)$. Previously, inverse Littlewood--Offord-type results only allowed control over vectors with anticoncentration at least $C/p$ for some large constant $C > 1$.
This short note presents upper bounds of the expectations of the largest singular values/eigenvalues of various types of random tensors in the non-asymptotic sense. For a standard Gaussian tensor of size $n_1timescdotstimes n_d$, it is shown that the expectation of its largest singular value is upper bounded by $sqrt {n_1}+cdots+sqrt {n_d}$. For the expectation of the largest $ell^d$-singular value, it is upper bounded by $2^{frac{d-1}{2}}prod_{j=1}^{d}n_j^{frac{d-2}{2d}}sum^d_{j=1}n_j^{frac{1}{2}}$. We also derive the upper bounds of the expectations of the largest Z-/H-($ell^d$)/M-/C-eigenvalues of symmetric, partially symmetric, and piezoelectric-type Gaussian tensors, which are respectively upper bounded by $dsqrt n$, $dcdot 2^{frac{d-1}{2}}n^{frac{d-1}{2}}$, $2sqrt m+2sqrt n$, and $3sqrt n$.
We study the behaviour of the inverse participation ratio and the localization transition in infinitely large random matrices through the cavity method. Results are shown for two ensembles of random matrices: Laplacian matrices on sparse random graphs and fully-connected Levy matrices. We derive a critical line separating localized from extended states in the case of Levy matrices. Comparison between theoretical results and diagonalization of finite random matrices is shown.
We consider the empirical eigenvalue distribution of an $mtimes m$ principle submatrix of an $ntimes n$ random unitary matrix distributed according to Haar measure. Earlier work of Petz and Reffy identified the limiting spectral measure if $frac{m}{n}toalpha$, as $ntoinfty$; under suitable scaling, the family ${mu_alpha}_{alphain(0,1)}$ of limiting measures interpolates between uniform measure on the unit disc (for small $alpha$) and uniform measure on the unit circle (as $alphato1$). In this note, we prove an explicit concentration inequality which shows that for fixed $n$ and $m$, the bounded-Lipschitz distance between the empirical spectral measure and the corresponding $mu_alpha$ is typically of order $sqrt{frac{log(m)}{m}}$ or smaller. The approach is via the theory of two-dimensional Coulomb gases and makes use of a new Coulomb transport inequality due to Chafai, Hardy, and Maida.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
