STOR Colloquium: Jong-Shi Pang, U. of Southern California

Jong-Shi Pang

University of Southern California

Title:     Structural Properties of Affine Sparsity Constraints

Abstract:   We introduce a new constraint system for sparse variable selection in statistical learning.   Such a

system arises when there are logical conditions on the sparsity of certain unknown model parameters that need

to be incorporated into their selection process.  Formally, extending a cardinality constraint, an affine sparsity

constraint (ASC) is defined by a linear inequality with two sets of variables: one set of continuous variables

and the other set represented by their nonzero patterns.  This paper studies an ASC system consisting

of finitely many affine sparsity constraints.  We investigate a number of fundamental structural properties

of the solution set of such a non-standard system of inequalities, including its closedness and the description

of its closure, continuous approximations and their set convergence, and characterizations of its tangent

cones for use in optimization.  Based on the obtained structural properties of an ASC system, we establish

the convergence of B(ouligand) stationary solutions when the ASC is approximated by surrogates of the

ell0 step function commonly employed in sparsity representation.  Our study lays a solid mathematical

foundation for solving optimization problems involving these affine sparsity constraints through their

continuous approximations.

This is joint work with Professor Hongbo Dong (Washington State University) and graduate student

Miju Ahn (University of Southern California).

Refreshments will be served in the lounge area of Hanes Hall at 3:00 pm