The RESPECT method. Simple proof of finding estimates from below for minimal singular eigenvalues of random matrices whose entries have zero means and bounded variances

2019 ◽  
Vol 27 (3) ◽  
pp. 167-175
Author(s):  
Vyacheslav L. Girko
Keyword(s):  
Random Matrices ◽  
Lower Bounds ◽  
Simple Proof ◽  
New Method ◽  
The Matrix ◽  
Eigenvalues Of Random Matrices

Abstract The lower bounds for the minimal singular eigenvalue of the matrix whose entries have zero means and bounded variances are obtained. The new method is based on the G-method of perpendiculars and the RESPECT method.

The method of perpendiculars of finding estimates from below for minimal singular eigenvalues of random matrices

2018 ◽  
Vol 26 (2) ◽  
pp. 117-123 ◽  
Author(s):  
Vyacheslav L. Girko
Keyword(s):  
Random Matrices ◽  
Lower Bounds ◽  
New Method ◽  
Martingale Method ◽  
Lindeberg Condition ◽  
The Matrix ◽  
Eigenvalues Of Random Matrices

Abstract The lower bounds for the minimal singular eigenvalue of the matrix are obtained under the G-Lindeberg condition and the G-double stochastic condition for the variances of the matrix entries. The new method is based on the G-method of perpendiculars, the REFORM method, the martingale method, and the theory of canonical spectral equations.

scholarly journals Small-Deviation Inequalities for Sums of Random Matrices

Symmetry ◽  
2019 ◽  
Vol 11 (5) ◽  
pp. 638
Author(s):  
Xianjie Gao ◽  
Chao Zhang ◽  
Hongwei Zhang
Keyword(s):  
Random Matrices ◽  
Large Deviation ◽  
Small Deviation ◽  
High Dimensional ◽  
Main Research ◽  
Infinite Dimensional ◽  
The Matrix ◽  
Largest Eigenvalues ◽  
Deviation Inequalities ◽  
Eigenvalues Of Random Matrices

Random matrices have played an important role in many fields including machine learning, quantum information theory, and optimization. One of the main research focuses is on the deviation inequalities for eigenvalues of random matrices. Although there are intensive studies on the large-deviation inequalities for random matrices, only a few works discuss the small-deviation behavior of random matrices. In this paper, we present the small-deviation inequalities for the largest eigenvalues of sums of random matrices. Since the resulting inequalities are independent of the matrix dimension, they are applicable to high-dimensional and even the infinite-dimensional cases.

scholarly journals LSV-Based Tail Inequalities for Sums of Random Matrices

2017 ◽  
Vol 29 (1) ◽  
pp. 247-262 ◽  
Author(s):  
Chao Zhang ◽  
Lei Du ◽  
Dacheng Tao
Keyword(s):  
Machine Learning ◽  
Laplace Transform ◽  
Random Matrices ◽  
Singular Value ◽  
Diagonal Matrix ◽  
New Method ◽  
Learning Models ◽  
The Matrix ◽  
Diagonalization Method ◽  
Machine Learning Models

The techniques of random matrices have played an important role in many machine learning models. In this letter, we present a new method to study the tail inequalities for sums of random matrices. Different from other work (Ahlswede & Winter, 2002 ; Tropp, 2012 ; Hsu, Kakade, & Zhang, 2012 ), our tail results are based on the largest singular value (LSV) and independent of the matrix dimension. Since the LSV operation and the expectation are noncommutative, we introduce a diagonalization method to convert the LSV operation into the trace operation of an infinitely dimensional diagonal matrix. In this way, we obtain another version of Laplace-transform bounds and then achieve the LSV-based tail inequalities for sums of random matrices.

scholarly journals A new method for identifying the role of marital preferences at shaping marriage patterns

2021 ◽  
pp. 1-27
Author(s):  
Anna Naszodi ◽  
Francisco Mendonca
Keyword(s):  
Contingency Table ◽  
Transformation Method ◽  
New Method ◽  
Marriage Patterns ◽  
The Us ◽  
Analytical Properties ◽  
The Matrix ◽  
Transformation Methods

Abstract We develop a method which assumes that marital preferences are characterized either by the scalar-valued measure proposed by Liu and Lu, or by the matrix-valued generalized Liu–Lu measure. The new method transforms an observed contingency table into a counterfactual table while preserving its (generalized) Liu–Lu value. After exploring some analytical properties of the new method, we illustrate its application by decomposing changes in the prevalence of homogamy in the US between 1980 and 2010. We perform this decomposition with two alternative transformation methods as well where both methods capture preferences differently from Liu and Lu. Finally, we use survey evidence to support our claim that out of the three considered methods, the new transformation method is the most suitable for identifying the role of marital preferences at shaping marriage patterns. These data are also in favor of measuring assortativity in preferences à la Liu and Lu.

scholarly journals Matrix-Based RSA Encryption of Streaming Data

2021 ◽  
Author(s):  
Michael Prendergast
Keyword(s):  
Streaming Data ◽  
New Method ◽  
Encryption Algorithm ◽  
Public Key ◽  
Invertible Matrix ◽  
Integer Coefficients ◽  
Private Key ◽  
Matrix Generalization ◽  
Euler Totient Function ◽  
The Matrix

This paper describes a new method for performing secure encryption of blocks of streaming data. This algorithm is an extension of the RSA encryption algorithm. Instead of using a public key (e,n) where n is the product of two large primes and e is relatively prime to the Euler Totient function, φ(n), one uses a public key (n,m,E), where m is the rank of the matrix E and E is an invertible matrix in GL(m,φ(n)). When m is 1, this last condition is equivalent to saying that E is relatively prime to φ(n), which is a requirement for standard RSA encryption. Rather than a secret private key (d,φ(n)) where d is the inverse of e (mod φ(n)), the private key is (D,φ(n)), where D is the inverse of E (mod (φ(n)). The key to making this generalization work is a matrix generalization of the scalar exponentiation operator that maps the set of m-dimensional vectors with integer coefficients modulo n, onto itself.

scholarly journals An asymptotic property of large matrices with identically distributed Boolean independent entries

2019 ◽  
Vol 22 (04) ◽  
pp. 1950024
Author(s):  
Mihai Popa ◽  
Zhiwei Hao
Keyword(s):  
Recent Result ◽  
Recent Work ◽  
Random Matrices ◽  
Asymptotic Property ◽  
Limit Distribution ◽  
Asymptotic Independence ◽  
Moment Conditions ◽  
The Matrix ◽  
Combinatorial Condition ◽  
Independence Relations

Motivated by the recent work on asymptotic independence relations for random matrices with non-commutative entries, we investigate the limit distribution and independence relations for large matrices with identically distributed and Boolean independent entries. More precisely, we show that, under some moment conditions, such random matrices are asymptotically [Formula: see text]-diagonal and Boolean independent from each other. This paper also gives a combinatorial condition under which such matrices are asymptotically Boolean independent from the matrix obtained by permuting the entries (thus extending a recent result in Boolean probability). In particular, we show that the random matrices considered are asymptotically Boolean independent from some of their partial transposes. The main results of the paper are based on combinatorial techniques.

From random matrices to random tensors

2021 ◽  
pp. 166-177
Author(s):  
Adrian Tanasa
Keyword(s):  
Matrix Models ◽  
Random Matrices ◽  
Mathematical Physics ◽  
Natural Generalization ◽  
Random Surface ◽  
The Third ◽  
Planar Surfaces ◽  
Tensor Models ◽  
Feynman Graphs ◽  
The Matrix

After a brief presentation of random matrices as a random surface QFT approach to 2D quantum gravity, we focus on two crucial mathematical physics results: the implementation of the large N limit (N being here the size of the matrix) and of the double-scaling mechanism for matrix models. It is worth emphasizing that, in the large N limit, it is the planar surfaces which dominate. In the third section of the chapter we introduce tensor models, seen as a natural generalization, in dimension higher than two, of matrix models. The last section of the chapter presents a potential generalisation of the Bollobás–Riordan polynomial for tensor graphs (which are the Feynman graphs of the perturbative expansion of QFT tensor models).

DETECTION OF NEW COPIES OF SINE IN THE RICE GENOME

2020 ◽  
pp. 267-269
Author(s):  
A. Kamionskaya ◽  
E. Korotkov
Keyword(s):  
Rice Genome ◽  
False Positives ◽  
New Method ◽  
The Matrix

We represent here a new method for the detection of new copies of SINE elements. The method is based on the correlation of pairs of symbols. The correlation is used for the construction of a position-specific matrix as well as for the search of new repeat copies using the matrix. This allows us to enlarge the alphabet and to increase the sensitivity of the method. The method was used to study the rice genome. As a result, new copies of SINE repeats that were not included in the standard annotation were found. The number of false positives was evaluated.

Sign in / Sign up

Export Citation Format

Share Document