scholarly journals Lower bounds for the smallest singular value of structured random matrices

2018 ◽  
Vol 46 (6) ◽  
pp. 3442-3500 ◽  
Author(s):  
Nicholas Cook
Keyword(s):  
Random Matrices ◽  
Lower Bounds ◽  
Singular Value ◽  
Smallest Singular Value

scholarly journals The smallest singular value of inhomogeneous square random matrices

2021 ◽  
Vol 49 (3) ◽  
Author(s):  
Galyna V. Livshyts ◽  
Konstantin Tikhomirov ◽  
Roman Vershynin
Keyword(s):  
Random Matrices ◽  
Singular Value ◽  
Smallest Singular Value

scholarly journals A new lower bound for the smallest singular value

Filomat ◽  
2019 ◽  
Vol 33 (9) ◽  
pp. 2711-2723
Author(s):  
Ksenija Doroslovacki ◽  
Ljiljana Cvetkovic ◽  
Ernest Sanca
Keyword(s):  
Lower Bound ◽  
Boundary Value Problems ◽  
Lower Bounds ◽  
Large Scale ◽  
Block Structure ◽  
Upper Bounds ◽  
Singular Value ◽  
Ecological Modelling ◽  
Infinity Norm ◽  
Smallest Singular Value

The aim of this paper is to obtain new lower bounds for the smallest singular value for some special subclasses of nonsingularH-matrices. This is done in two steps: first, unifying principle for deriving new upper bounds for the norm 1 of the inverse of an arbitrary nonsingular H-matrix is presented, and then, it is combined with some well-known upper bounds for the infinity norm of the inverse. The importance and efficiency of the results are illustrated by an example from ecological modelling, as well as on a type of large-scale matrices posessing a block structure, arising in boundary value problems.

scholarly journals Lower Bounds of the Smallest Singular Value of Matrices

2021 ◽  
Vol 13 (5) ◽  
pp. 1
Author(s):  
Liao Ping
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Singular Value ◽  
Euclidean Norm ◽  
Numerical Examples ◽  
Smallest Singular Value

In this paper, we get a lower bound of the smallest singular value of an arbitrarily matrix A by the trace of H(A) and the Euclidean norm of H(A), where H(A) is Hermitian part of A, numerical examples show the e ectiveness of our results.

scholarly journals Random polytopes obtained by matrices with heavy-tailed entries

2019 ◽  
Vol 22 (04) ◽  
pp. 1950027
Author(s):  
O. Guédon ◽  
A. E. Litvak ◽  
K. Tatarko
Keyword(s):  
Compressed Sensing ◽  
Convex Hull ◽  
Random Matrices ◽  
Random Matrix ◽  
Singular Value ◽  
Small Ball ◽  
Random Polytopes ◽  
Heavy Tailed ◽  
Smallest Singular Value ◽  
Quotient Property

Let [Formula: see text] be an [Formula: see text] random matrix with independent entries and such that in each row entries are i.i.d. Assume also that the entries are symmetric, have unit variances, and satisfy a small ball probabilistic estimate uniformly. We investigate properties of the corresponding random polytope [Formula: see text] in [Formula: see text] (the absolute convex hull of rows of [Formula: see text]). In particular, we show that [Formula: see text] where [Formula: see text] depends only on parameters in small ball inequality. This extends results of [A. E. Litvak, A. Pajor, M. Rudelson and N. Tomczak-Jaegermann, Smallest singular value of random matrices and geometry of random polytopes, Adv. Math. 195 (2005) 491–523] and recent results of [F. Krahmer, C. Kummerle and H. Rauhut, A quotient property for matrices with heavy-tailed entries and its application to noise-blind compressed sensing, preprint (2018); arXiv:1806.04261]. This inclusion is equivalent to so-called [Formula: see text]-quotient property and plays an important role in compressed sensing (see [F. Krahmer, C. Kummerle and H. Rauhut, A quotient property for matrices with heavy-tailed entries and its application to noise-blind compressed sensing, preprint (2018); arXiv:1806.04261] and references therein).

scholarly journals Further lower bounds for the smallest singular value

1998 ◽  
Vol 272 (1-3) ◽  
pp. 169-179 ◽  
Author(s):  
Charles R. Johnson ◽  
Tomasz Szulc
Keyword(s):  
Lower Bounds ◽  
Singular Value ◽  
Smallest Singular Value

scholarly journals Smallest singular value of random matrices and geometry of random polytopes

2005 ◽  
Vol 195 (2) ◽  
pp. 491-523 ◽  
Author(s):  
A.E. Litvak ◽  
A. Pajor ◽  
M. Rudelson ◽  
N. Tomczak-Jaegermann
Keyword(s):  
Random Matrices ◽  
Singular Value ◽  
Random Polytopes ◽  
Smallest Singular Value
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