@techreport{oai:ipsj.ixsq.nii.ac.jp:00211738,
 author = {舛田, 昂生 and 右田, 剛史 and 高橋, 規一 and Takao, Masuda and Tsuyoshi, Migita and Norikazu, Takahashi},
 issue = {4},
 month = {Jun},
 note = {非負値行列因子分解 (Nonnegative Matrix Factorization: NMF) は，与えられた非負値行列を二つの非負値因子行列に分解する処理である．最近，大規模非負値行列に対する NMF を高速化するためのアプローチとして，非負値行列にランダム行列を掛けて次元を削減してから NMF を実行するランダム化 NMF が提案された．ランダム化 NMF は元の NMF とは異なる制約付き最適化問題に定式化されるため，それに適したアルゴリズムの開発が必要である．しかし，従来のアルゴリズムには最適化問題の制約条件が満たされないという重大な欠点がある．そのため実行可能解が得られる保証がない．また，アルゴリズムの収束性に関する議論も行われていない．本報告では，これらの欠点を解消するために最適化問題にわずかな修正を加え，修正された最適化問題を解くための，階層的交互最小二乗法に基づくアルゴリズムを提案する．また，提案アルゴリズムの大域収束性を証明する．, Nonnegative Matrix Factorization (NMF) is the process of decomposing a given nonnegative matrix into two nonnegative factor matrices. Recently, randomized NMF has been proposed as an approach to fast NMF of large nonnegative matrices. The main idea of this approach is to perform NMF after reducing the dimensionality of the given nonnegative matrix by multiplying it by a random matrix. Since randomized NMF is formulated as a constrained optimization problem which is slightly different from the one for original NMF, it is necessary to develop suitable algorithms for solving it. However, the conventional algorithm has a serious drawback that the constraints of the optimization problem are not satisfied. Hence there is no guarantee that a feasible solution can be obtained. In addition, the convergence of the algorithm has not been analyzed. In this report, in order to overcome the drawback, we propose to modify the optimization problem slightly and design an algorithm based on the hierarchical alternating least squares method to solve the modified optimization problem. We also prove the global convergence of the designed algorithm.},
 title = {ランダム化NMFにおける最適化問題の修正とHALS法に基づく解法の提案},
 year = {2021}
}