Lp準ノルム(0＜p＜1)に基づく2つの最小2乗問題(Lp正則化最小2乗問題とLp制約付き最小2乗問題)の構造を研究した。まず, 過剰決定系の場合を考察し, 2つの問題の本質的な相違点を明らかにするとともに, スパース最適化に対する貪欲アルゴリズムとの関係を明らかにした。次に, 圧縮センシングなどで注目されている劣決定系の場合を考察し, ある仮定の下, 原点とスパース解(多数のゼロ成分を持つ解)を結ぶ連続な臨界点パスの存在性を理論的に証明した。最後に, 得られた知見をスパース適応フィルタと再生核適応フィルタに応用し, その有効性を示した。
The structures of two least square problems (Lp-regularized least squares and Lp-constrained least squares for 0＜p＜1) have been elucidated. First, for the over-determined linear system, the essential difference between the two problems has been clarified, and the relation between the least squares based on the Lp quasi-norm and the greedy algorithm for sparse optimization has been established. Second, for the under-determined system, the existence of a continuous path that connects the origin and the sparsest least square solution has been proven mathematically under a certain condition. Third, the obtained results have been applied to the sparse/kernel adaptive filters.
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