Graph Embedding and Extensions: A General Framework for Dimensionality Reduction

TLDR

A large family of supervised and unsupervised algorithms derived from statistics or geometry has been developed to solve dimensionality reduction. This paper proposes a general graph‑embedding formulation that unifies these algorithms and provides a platform for designing new dimensionality‑reduction methods. Graph embedding models each method as an embedding of an intrinsic graph with optional linear, kernel, or tensor extensions, constrained by scale normalization or a penalty graph; the authors propose marginal Fisher analysis, a supervised algorithm that uses an intraclass compactness intrinsic graph and an interclass separability penalty graph. Marginal Fisher analysis surpasses traditional linear discriminant analysis by relaxing distribution assumptions and providing more projection directions, and real face‑recognition experiments demonstrate its superiority over LDA, including its kernel and tensor extensions.

Abstract

A large family of algorithms - supervised or unsupervised; stemming from statistics or geometry theory - has been designed to provide different solutions to the problem of dimensionality reduction. Despite the different motivations of these algorithms, we present in this paper a general formulation known as graph embedding to unify them within a common framework. In graph embedding, each algorithm can be considered as the direct graph embedding or its linear/kernel/tensor extension of a specific intrinsic graph that describes certain desired statistical or geometric properties of a data set, with constraints from scale normalization or a penalty graph that characterizes a statistical or geometric property that should be avoided. Furthermore, the graph embedding framework can be used as a general platform for developing new dimensionality reduction algorithms. By utilizing this framework as a tool, we propose a new supervised dimensionality reduction algorithm called marginal Fisher analysis in which the intrinsic graph characterizes the intraclass compactness and connects each data point with its neighboring points of the same class, while the penalty graph connects the marginal points and characterizes the interclass separability. We show that MFA effectively overcomes the limitations of the traditional linear discriminant analysis algorithm due to data distribution assumptions and available projection directions. Real face recognition experiments show the superiority of our proposed MFA in comparison to LDA, also for corresponding kernel and tensor extensions