Study of Matrix Factorization Problem Based on Bayesian Inference Method

2023 Fiscal Year Final Research Report

• Project/Area Number: 18K11175
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 60020:Mathematical informatics-related
• Research Institution: Ibaraki University
• Principal Investigator: Takeda Koujin 茨城大学, 理工学研究科(工学野), 准教授 (70397040)
• Project Period (FY): 2018-04-01 – 2024-03-31
• Keywords: 行列分解 / ベイズ推定 / 統計物理学 / 数理神経科学 / スパース性

Outline of Final Research Achievements

The main results of this research project are summarized as follows. (A) We derived a variational Bayesian solution to the sparse matrix factorization problem analytically, and confirmed that the matrix factorization algorithm based on the analytical solution exhibits good performance in seeking a sparse factorized matrix solution. Furthermore, we proposed an automatic tuning method for the sparsity hyperparameter. (B) Based on mathematical neuroscience and thermodynamics, we developed a method to analyze the properties of the solution and the dynamical behavior for the matrix factorization algorithm. (C) We pointed out the effectiveness of sparse matrix factorization for feature extraction from fMRI data of neuronal activity in the brain. In addition, we proposed a method of independent component analysis incorporating sparsity.

Free Research Field

統計物理学

Academic Significance and Societal Importance of the Research Achievements

行列分解問題は与えられた行列を2つの行列の積に分解する問題であり、データ解析や信号処理等の分野で応用される情報科学の基本的な問題である。しかし行列分解問題には様々な問題設定が存在するが、それらの問題の数理構造を統一的に理解する試みは十分に行われていなかった。本研究の成果により、ベイズ推定の枠組みで様々な設定の行列分解問題の数理構造が系統的に解析できることが示された。この成果は行列分解問題の研究に新たな知見を与えるとともに、行列分解の実問題への応用にも今後大いに役立つと考えられる。