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Krishan Lal , Manish

University of British Columbia

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 Fejer 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 Fejer 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.

Text

2023-12-14

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/86955

Mathematics

University of British Columbia

UBCO

http://creativecommons.org/licenses/by-nc-nd/4.0/