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Solving a more general matrix completion problem with Tikhonov regularization and APGL method

Title
Solving a more general matrix completion problem with Tikhonov regularization and APGL method
Authors
강주연
Issue Date
2020
Department/Major
대학원 수학과
Publisher
이화여자대학교 대학원
Degree
Master
Advisors
김선영
Abstract
Matrix completion problems have been studied and used in various fields including machine learning and recommendation systems. This problem is usually solved as a norm minimization problem, which is the convex relaxation of the original problem. Various algorithms have been presented so far, and among them, we will focus on APGL suggested by K. Toh, S. Yun. The APGL is a state of art algorithm, using accelerated proximal gradient method(APG). In this thesis it is reconstructed and presented, to be used for the Tikhonov regularized problem with a strongly convex term added. The efficiency and stability of the algorithm can be confirmed by the numerical experiment results on large random matrices and well-known real data matrices;행렬 완성 문제(Matrix completion problem)는 머신러닝, 추천시스템을 포함한 다양한 분야에서 연구되며 쓰여지고 있다. 이 문제는 보편적으로 행렬의 계수를 최소화하는 문제로 쓰여지는데, NP-난해(NP-hard)하다고 알려져있어 볼록 완화(convex relaxation)를 이용하여 행렬의 노름 최소화(norm minimization)문제로 대신 풀이된다. 이 논문에서는 가속된 경사 하강법(Accelerated Proximal Gradient)을 이용한 APGL 알고리즘을 강한 볼록 항(strongly convex term)을 추가한 문제(Tikhonov regularized problem)에 사용할 수 있게 만든 알고리즘을 제시한다. 이를 크기가 큰 임의행렬과 잘 알려진 실 데이터 행렬에 적용하여, 효율성(efficieny)와 안정성(stability)의 향상을 확인할 수 있다.
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