Georgia Institute of Technology College of Sciences

In recent years, significant progress has been made in our understanding of the quantitative behavior of random matrices. Such results include delocalization properties of eigenvectors and tail estimates for the smallest singular value. A key ingredient in their proofs is a 'distance theorem', which is a small ball estimate for the distance between a random vector and subspace.  Building on work of Livshyts and Livshyts, Tikhomirov and Vershynin, we introduce a new distance theorem for inhomogeneous vectors and subspaces spanned by the columns of an inhomogeneous matrix. Such a result has a number of applications for generalizing results about the quantitative behavior of i.i.d. matrices to matrices without any identical distribution assumptions. To highlight this, we show that the smallest singular value estimate of Rudelson and Vershynin, proven for i.i.d. subgaussian rectangular matrices, holds true for inhomogeneous and heavy-tailed matrices.
This talk is partially based on joint work with Max Dabagia.