Some Applications of the Poincare Separation Theorem in Data Analysis

We examine in this paper various consequences of the Poincari separation theorem. First, Poincari's theorem is used to prove results establishing a constrainted principal component analysis, both at the sample and at the population level. Introduction of constraints in principal component analysis paves the way for promising applications; some of them are pointed out in this article. Incidentally, the main result also allows to elegantly unify formulas in analytic geometry, by readily providing parametric as well as cartesian systems of equations characterising affine subspaces in IRp passing through specified affine subspaces of lower dimension. Then, we show that a direct consequence of the Poincari separation theorem implies (but is not implied by) the principal component analysis procedure (and therefore that this theorem is stronger than principal component analysis). Showing the intimate link between Poincari's theorem and principal component analysis allows to bring to light various results in linear algebra that are interesting per se. Finally, investigation of the geometrical meaning of Poincari's theorem underlines the fact that projected clouds of points formed by the rows of matrices of data lose their dispersion in two distincts ways when projected onto linear subspaces.