On Singular Values of Random Matrices

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Bounding the smallest singular value of a random matrix without concentration (2013, background): “ isotropic (that is, for every t ∈ Rn, E X,t 2 = ktk2 ℓn 2 ) and its distribution is a log-concave measure [1, 2]; that X is isotropic, kXkℓn 2 ≤ c 1 √ n and supt∈Sn−1 k X,t kLp ≤ L for some p > 8 [9]; and that X is isotropic and satisﬁes the assumption that, for every orthogonal projection P, kP”
Sample covariance matrices of heavy-tailed distributions (2016, methodology): “Unlike the above computations, the next lemma is a new addition to the arguments employed in [20, 10].”
The smallest singular value of random rectangular matrices with no moment assumptions on entries (2014, background): “Later, the log-concavity was replaced by weaker assumptions (see, in particular, [2], [17], [9], [4]).”
The smallest singular value of inhomogeneous square random matrices (2019, background): “The behavior of the smallest singular values of random matrices have been extensively studied [2, 4, 5, 11, 15, 16, 17, 20, 22, 24, 25, 26, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43].”
Learning without Concentration (2014, background): “stimate is one-sided. A two-sided bound, obtained by an upper estimate on centred quadratic process sup f∈F∩sD f∗ 1 N XN i=1 (f−f∗)2(Xi) −E(f−f∗)2 , requires F to be highly regular in some sense (see [20, 21, 22, 23] for twosided estimates on the quadratic process). And, as noted earlier, a two-side”