2014/06/03 by Shukai Du, Du, Shukai, Nailin Du +1 · 1 citation
Computer Science · Engineering · Mathematics · #15A09 #47A52 #65J20 #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Sparse and Compressive Sensing Techniques
paper · pdf · doi:10.48550/arxiv.1406.0578
openalex publication_date 2014/06/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper investigates the least-squares projection method for bounded linear operators, which provides a natural regularization scheme by projection for many ill-posed problems. Yet, without additional assumptions, the convergence of this approximation scheme cannot be guaranteed. We reveal that the convergence of least-squares projection method is determined by two independent factors -- the kernel approximability and the offset angle. The kernel approximability is a necessary condition of convergence described with kernel N(T) and its subspaces N(T)∩Xn, and we give several equivalent characterizations for it (Theorem 1). The offset angle of Xn is defined as the largest canonical angle between space T^*T(Xn) and T†T(Xn) (which are subspaces of N(T)^\bot), and it geometrically reflects the rate of convergence (Theorem 2). The paper also presents new observations for the unconvergence examples of Seidman [10, Example 3.1] and Du [2, Example 2.10] under the notions of kernel approximability and offset angle.