- Analytical formulas for alternating projection sequences of positive semidefinite cones and their application to convergence analysis (arXiv)
Author: Hiroyuki Ochiai, Yoshiyuki Sekiguchi, Hayato Waki
Summary: Find an analytic formula for the alternating projection method for a positive semidefinite matrix and an affine subspace cone Sn+. More precisely, we find a recursive relation of parameters representing a sequence constructed by alternating projections. We analyze the alternating projection method in detail by applying Eq. and show that the upper limit given by the singularity degree is indeed strict when we apply the alternating projection method to S3+ and 3 planes whose intersection is a singleton and has singularity. Masu. 2. △ Few
2. Differential geometry with extreme eigenvalues in positive semidefinite cones (arXiv)
Authors: Silas Mostageran, Nasael da Costa, Graham Van Gofflier, Rodolphe Sepulcher
Abstract: Differential geometry approaches to analyze and process data in the form of symmetric positive definite (SPD) matrices have achieved remarkable success in many fields such as computer vision, medical image processing, and machine learning. The dominant geometrical paradigm for such applications has consisted of some Riemannian geometries associated with large-scale, high-dimensional, and costly spectral computations. We present a path to a scalable geometric framework for the analysis and processing of SPD value data, based on the efficient computation of extreme generalized eigenvalues by Hilbert and Thompson geometry of semidefinite cones. We investigate in detail a particular geodesic spatial structure based on Thompson geometry and establish some properties associated with this structure. Furthermore, we define a new iterative mean of the SPD matrix based on this geometry and prove its existence and uniqueness for a given finite set of points. Finally, we state and prove some desirable properties fulfilled by this measure.
