An Analysis of Weighted Least Squares Method and Layered Least Squares Method with the Basis Block Lower Triangular Matrix Form∗

In this paper, we analyze the limiting behavior of the weighted least squares problem minx∈<n Pp i=1 kDi(Aix − bi)k2, where each Di is a positive definite diagonal matrix. We consider the situation where the magnitude of the weights are drastically different block-wisely so that max(D1) ≥ min(D1) À max(D2) ≥ min(D2) À max(D3) ≥ . . . À max(Dp−1) ≥ min(Dp−1)À max(Dp). Here max(·) and min(·) represents the maximum and minimum entries of diagonal elements, respectively. Specifically, we consider the case when the gap g ≡ mini 1/(kD−1 i kkDi+1k) is very large or tends to infinity. Vavasis and Ye proved that the limiting solution exists (when the proportion of diagonal elements within each block Di is unchanged and only the gap g tends to∞), and showed that the limit is characterized as the solution of a variant of the least squares problem called the layered least squares (LLS) problem. We analyze the difference between the solutions of WLS and LLS quantitatively and show that the norm of the difference of the two solutions is bounded above by O(χAχ̄ 2(p+1) A g −2kbk) and O(χ̄ A g −2kbk) in the variable and the residual spaces, respectively, using the two condition numbers χA ≡ maxB∈B kB−1k and χ̄A ≡ maxB∈B kB−1Ak of A, where B is the set of all nonsingular n×n submatrix of A, A = [A1; . . . ;Ap] and b = [b1; . . . ; bp]. A remarkable feature of this result is the error bound is represented in terms of A, g (and b) and independent of the weights Di, i = 1, . . . , p. The analysis is carried out by making the change of variables to convert the matrix A into a basis lower-triangular form and then by applying the Sharmann-Morrison-Woodbury formula. ∗A part of this research was supported with Grant-in-Aid for Scientific Research (B), 2008, 20340024 from the Japan Society for the Promotion of Science. †Graduate School of Decision Science and Technology, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8552 Japan ([email protected]). ‡The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, Tokyo 106-8569 Japan ([email protected]).