The smallest singular value of tensor matrices
Colin Matthias Kleinschmidt · Multimedialen Archiv und Publikationsserver der Christian-Albrechts-Universität zu Kiel (Christian-Albrechts-Universität zu Kiel) · 2016
In this thesis we investigate the smallest singular value of random square matrices whose entries are independent subgaussian random variables. Statements about the behaviour of the singular values of a random matrix are expressed in stochastic terms as singular values of a random matrix are random variables themselves. We estimate the order of the expectation of the smallest singular value of a Gaussian square matrix with disturbed last column, i.e. all entries of the random matrix are independent centered Gaussian random variables, but the variance of the entries of the last column is very small, whereas the other entries are standard normal distributed random variables. This example of a random square matrix with not necessarily identically distributed entries is crucial to understand the behaviour of the smallest singular value. In the more general case of tensor square matrices whose entries are independent subgaussian random variables, we estimate the probability that the smallest singular value is smaller/ greater than some bound. To do so we use a technique called decomposition of the sphere due to M. Rudelson and R. Vershynin, where the Euclidean sphere is decomposed in compressible vectors, i.e. unit vectors that carry most of their mass only in a few coordinates, and incompressible vectors, which are unit vectors that are not compressible. This represents a new approach in this context, as we use the properties of this decomposition to get in a position, where we can use Hölder’s inequality and get rid of the factors. We also generalize a result of M. Rudelson and R. Vershynin to estimate the tail probability of the smallest singular value explicitly and we show that if we lower the assumptions one can generalize the tail estimate to almost square matrices.