Stability of two direct methods for bidiagonalization and partial least squares (Q2877089)

The paper studies two methods for solving partial least squares (PLS) problems, namely the Householder reflexions based Golub-Kahan bidiagonalization and a nonlinear iterative partial least squares (NIPALS PLS) algorithm. In PLS, the aim is to construct a sequence of least squares solutions to the problem \(A x = b\) with the restriction to a prescribed Krylov subspace of increasing dimension. The bidiagonalization based approach is known to be stable, since it is based on applying a sequence of orthogonal Householder matrices. In the NIPALS PLS algorithm, the basis of the Krylov subspace is constracted explicitely using orthogonal projections similarly as in modified Gram-Schmidt method. The widely considered implementation of the NIPALS PLS algorithm uses the simplification that a possible numerical deflation in \(b\) is ommited. The paper shows on numerical examples, that without this simplification the computational time increases only slightly while the stability of the algorithm becomes comparable to the bidiagonalization based approach. The presented results motivate a conjecture that a correctly implemented NIPALS PLS is also stable. However, the proof is not given.