Mathematics Colloquium- Tom Trogdon, University of Washington

 

abstract:  This talk will give an overview of the study of algorithms on random data, and in particular, algorithms from numerical linear algebra (NLA) on random matrices.  The combination of ideas from numerical linear algebra and random matrices goes back, at least, to the seminal work of Goldstine and von Neumann.  The works of Trotter, Silverstein, Edelman, Dumitriu & Edelman, Pfrang, Deift & Menon, and many others, developed these ideas further.   A core subset of NLA algorithms, the Krylov subspace methods, play particularly well with existing random matrix theory.  Through the study of random orthogonal polynomials, as perturbations of deterministic orthogonal polynomials, the concentration phenomenon in these methods can be explained using local laws from random matrix theory.  We will use these ideas, and methods, to efficiently and robustly perform spike detection in the spiked sample covariance model, and move towards PCA without the SVD.