Phase Transition in Convex Optimization Problems with Random Data

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Faghih Mirzaei, Delbar | 2021

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 54229 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Alishahi, Kasra
Abstract:
In the behavior of many convex optimization problems with random constraints in high dimensions, sudden changes or phase transitions have been observed in terms of the number of constraints. A well-known example of this is the problem of reconstructing a thin vector or a low-order matrix based on a number of random linear observations. In both cases, methods based on convex optimization have been developed, observed, and proved that when the number of observations from a certain threshold becomes more (less), the answer to the problem with a probability of close to one (zero) is correct and the original matrix is reconstructed. Recently, results have been obtained that explain why this phenomenon of phase transition is pervasive in convex optimization problems with stochastic data. These results rely on the concepts of high-dimensional stochastic geometry and theorems about the accumulation of intrinsic volumes, which have led to a statistical definition of convex angles and convex optimization problems and have established connections between already known topics. The purpose of this dissertation is to introduce and review these new results
Keywords:
Convex Optimization ; Sparse Vector Method ; Stochastic Geometry ; Phase Transition ; Low-Rank Matrix