Intrinsic Grassmann Averages for Online Linear, Robust and Nonlinear Subspace Learning

TL;DR

This paper introduces a geometric framework for computing principal subspaces on the Grassmann manifold, enabling faster PCA, robust PCA, and an online KPCA variant with demonstrated competitive performance.

Contribution

It presents a novel intrinsic Grassmann average approach for subspace learning, applicable to linear, robust, and nonlinear PCA, including an efficient online algorithm.

Findings

The Grassmann average coincides with principal components for Gaussian data.

The proposed methods outperform traditional PCA and KPCA in speed.

The online KPCA achieves competitive results on real and synthetic datasets.

Abstract

Principal Component Analysis (PCA) and Kernel Principal Component Analysis (KPCA) are fundamental methods in machine learning for dimensionality reduction. The former is a technique for finding this approximation in finite dimensions and the latter is often in an infinite dimensional Reproducing Kernel Hilbert-space (RKHS). In this paper, we present a geometric framework for computing the principal linear subspaces in both situations as well as for the robust PCA case, that amounts to computing the intrinsic average on the space of all subspaces: the Grassmann manifold. Points on this manifold are defined as the subspaces spanned by $K$-tuples of observations. The intrinsic Grassmann average of these subspaces are shown to coincide with the principal components of the observations when they are drawn from a Gaussian distribution. We show similar results in the RKHS case and provide an efficient algorithm for computing the projection onto the this average subspace. The result is a method akin to KPCA which is substantially faster. Further, we present a novel online version of the KPCA using our geometric framework. Competitive performance of all our algorithms are demonstrated on a variety of real and synthetic data sets.

Full text

See pages 1-last of tpamiGAforarxiv.pdf