Dynamical Systems for Joint Principal and Minor Component Analysis

Abstract

Most known principal or a minor subspace (or component)

Joint computation of principal and minor components using gradient dynamical systems over stiefel manifolds

“…This result has been stated and analyzed in [11] using constrained optimization techniques. In this section, we examine the stability of a gradient dynamical system that is based on the gradient of F 2 on the Stiefel's manifold S defined in (4):…”

“…The eigen-spread of a symmetric matrix A may be charaterized by Mirsky result [12] (Theorem 3). Another charaterization of the eigen-spread of the matrix A is given in [11] and is shown to be equivalent to Mirsky result. In this paper, several variations will be derived by generalizing the methods presented in [11].…”

“…Another charaterization of the eigen-spread of the matrix A is given in [11] and is shown to be equivalent to Mirsky result. In this paper, several variations will be derived by generalizing the methods presented in [11]. It will be assumed that λ 1 > λn for otherwise A is an identity matrix.…”

“…In most known methods, either a principal or a minor subspace (or component) but not both are computed. Methods for joint computing eigenspaces corresponding to both maximum and minimum eigenvalues are given in [11]. In this paper, additional methods that expand those of [11] are proposed.…”

“…Methods for joint computing eigenspaces corresponding to both maximum and minimum eigenvalues are given in [11]. In this paper, additional methods that expand those of [11] are proposed. These involve algorithms for joint computation of both (PSA and MSA) or (PCA and MCA).…”