Roweis Discriminant Analysis: A Generalized Subspace Learning Method

Abstract

We present a new method which generalizes subspace learning based on eigenvalue and generalized eigenvalue problems. This method, Roweis Discriminant Analysis (RDA), is named after Sam Roweis to whom the field of subspace learning owes significantly. RDA is a family of infinite number of algorithms where Principal Component Analysis (PCA), Supervised PCA (SPCA), and Fisher Discriminant Analysis (FDA) are special cases. One of the extreme special cases, which we name Double Supervised Discriminant Analysis (DSDA), uses the labels twice; it is novel and has not appeared elsewhere. We propose a dual for RDA for some special cases. We also propose kernel RDA, generalizing kernel PCA, kernel SPCA, and kernel FDA, using both dual RDA and representation theory. Our theoretical analysis explains previously known facts such as why SPCA can use regression but FDA cannot, why PCA and SPCA have duals but FDA does not, why kernel PCA and kernel SPCA use kernel trick but kernel FDA does not, and why PCA is the best linear method for reconstruction. Roweisfaces and kernel Roweisfaces are also proposed generalizing eigenfaces, Fisherfaces, supervised eigenfaces, and their kernel variants. We also report experiments showing the effectiveness of RDA and kernel RDA on some benchmark datasets.

Multidimensional Scaling, Sammon Mapping, and Isomap: Tutorial and Survey

“…Sammon mapping (Sammon, 1969) is a special case of metric MDS; hence, it is a nonlinear method. It is probably correct to call this method the first proposed nonlinear method for manifold learning (Ghojogh et al, 2019b). This method has different names in the literature such as Sammon's nonlinear mapping, Sammon mapping, and Nonlinear Mapping (NLM) (Lee & Verleysen, 2007).…”

“…In later approaches, Sammon mapping (Sammon, 1969) was proposed which is a special case of the distance-based metric MDS. One can consider Sammon mapping as the first proposed nonlinear manifold learning method (Ghojogh et al, 2019b). The disadvantage of Sammon mapping is its iterative solution of optimization, which makes this method a little slow.…”

Weighted Fisher Discriminant Analysis in the Input and Feature Spaces

“…1). For example, the non-zero pairs (2,20), (4,14), (13,6), (19,20), (17,6) in AW-FDA and the pairs (2,20), (4,14), (19,20), (17,14) in AW-KFDA make sense visually because of having glasses so their classes are close to one another.…”

“…Treating closer classes need more attention because classifiers may more easily confuse them whereas classes far from each other are generally easier to separate. The same problem exists in Kernel FDA (KFDA) [3] and in most of subspace learning methods that are based on generalized eigenvalue problem such as FDA and KFDA [4]; hence, a weighting procedure might be more appropriate.…”