Mathematics Colloquia and Seminars

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Description

L2 regression, also known as the least squares approximation problem, and more generally Lp regression, are fundamental problems that have numerous applications in mathematics and statistical data analysis. Recall that the over-constrained Lp regression problem takes as input an n x d matrix A, where n > d, an n-vector b, and a number p, and it returns as output a d-vector x such that ||Ax-b||_p is minimized. Several randomized algorithms for this problem, each of which provides a relative-error approximation to the exact solution, are described. The first algorithm applies novel ``subspace sampling'' probabilities to randomly sample constraints and thus construct a coreset for the L2 regression problem. The second algorithm applies a random projection and uses a ``fast'' version of the Johnson-Lindenstrauss lemma to obtain an efficient approximation to the L2 regression problem. The third algorithm generalizes the concept of a ``well-conditioned'' basis and of ``subspace-preserving sampling'' to obtain efficiently a coreset for Lp regression, for all $p\in[1,\infty)$. Applications of these random projection and random sampling algorithms to data analysis will be briefly discussed.