Dissertation Defence: Nonconvex Projections Arising in Bilinear Mathematical Models

Manish Krishan Lal, supervised by Dr. Heinz Bauschke & Dr. Shawn Wang, will defend their dissertation titled “Nonconvex Projections Arising in Bilinear Mathematical Models” in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics.

An abstract for Manish Krishan Lal’s dissertation is included below.

Examinations are open to all members of the campus community as well as the general public. Please email heinz.bauschke@ubc.ca or shawn.wang@ubc.ca to receive the Zoom link for this defence.

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ABSTRACT

This thesis contributes to the study of projection operators associated with bilinear sets. Bilinear sets are not convex and appear in many applications such as deep learning, inverse problems, and other bilinear models in control and optimization.

The closed-form projection formulas for some of these bilinear sets namely crosses, hyperbolas, and hyperbolic paraboloids are provided. Along the way, a convenient presentation for the roots of cubic polynomials is highlighted and utilized further to develop more projection formulas and proximal mappings which are essential tools used in projection algorithms and proximal splitting algorithms.

The notion of Fej´er monotonicity is instrumental in unifying the convergence proofs of many iterative methods, such as the Krasnoselskii-Mann iteration, the proximal point method, and the projection algorithms. In a finite-dimensional Hilbert space, the sequences generated by the proximal point algorithm enjoy directionally asymptotic properties. A comprehensive study of directionally asymptotical results of strongly convergent subsequences of Fej´er monotone sequences in general Hilbert spaces is provided along with some detailed examples.

We also provide a foundational mathematical model of Elser’s framework for matrix factorization.