Smallest singular value of random matrices and geometry of random polytopes

By Alexander E. Litvak, Alain Pajor, Mark Rudelson, Nicole Tomczak-Jaegermann. 2005

297 citations recorded by the source corpus.

Subfields: Applied Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/e7d40ff5a87b8ba08d3bbb020576ed3bd85f7673

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In Applied Mathematics

In Statistics and Probability

Citations in the corpus

Around the circular law (2011, methodology): “The idea, originated from [102], is now to use the rate of convergence given by the Berry-Esseen theorem to upper bound this last expression.”
Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies? (2004, unknown): “Fortunately, Litvak, Pajor, Rudelson and Tomczak-Jaegermann [41][Theorem 3.”
The smallest singular value of a random rectangular matrix (2008, background): “Indeed, this was proved in [12] for arbitrary aspect ratios λ < 1 − c/ log n and for general random matrices with independent subgaussian entries.”
Subgaussian random variables : An expository note (2012, background): “For instance, subgaussian random variables have been studied in connection with random series in [2]; in connection with the geometry of Banach Spaces in [9]; with the spectral properties of random matrices in [7], [12].”
Invertibility of Sparse non-Hermitian matrices (2015, background): “The idea of splitting the sphere into compressible (close to sparse) and incompressible ones originated in [16] and was further developed in [19, 21].”
Smallest singular value of a random rectangular matrix (2009, background): “Indeed, this was proved in [18] for arbitrary aspect ratios λ < 1− c/ log n and for general random matrices with independent subgaussian entries.”
The LASSO Risk for Gaussian Matrices (2010, background): “Large deviation estimates [21] imply that the probability that the minimum singular value is smaller than a constant is exponentially small.”
The Littlewood-Offord problem and invertibility of random matrices (2007, methodology): “Estimates on the smallest singular value are known for rectangular random matrices [18].”
Stability of matrix factorization for collaborative filtering (2012, methodology): “We remark that the bound on singular values we used has been generalized to random matrices following subgaussian (Rudelson & Vershynin, 2009) and log-concave distributions (Litvak et al., 2005).”
From the Littlewood-Offord problem to the Circular Law: universality of the spectral distribution of random matrices (2008, background): “Results of this type first appear in [37] (see also [33] for some earlier related work for the least singualar value of rectangular matrices).”
On the Marchenko-Pastur and Circular Laws for some Classes of Random Matrices with Dependent Entries (2011, unknown): “in [31, 35, 25].”