Anomaly Preserving $\ell _{\scriptscriptstyle 2,\infty }$-Optimal Dimensionality Reduction Over a Grassmann Manifold

In this paper, we address the problem of redundancy reduction of high-dimensional noisy signals that may contain anomaly (rare) vectors, which we wish to preserve. Since anomaly data vectors contribute weakly to the $\ell _{2}$-norm of the signal as compared to the noise, $\ell _{2}$ -based criteria are unsatisfactory for obtaining a good representation of these vectors. As a remedy, a new approach, named Min-Max-SVD (MX-SVD) was recently proposed for signal-subspace estimation by attempting to minimize the maximum of data-residual $\ell _{2}$-norms, denoted as $\ell _{2,\infty }$ and designed to represent well both abundant and anomaly measurements. However, the MX-SVD algorithm is greedy and only approximately minimizes the proposed $\ell _{2,\infty }$-norm of the residuals. In this paper we develop an optimal algorithm for the minization of the $\ell _{2,\infty }$-norm of data misrepresentation residuals, which we call Maximum Orthogonal complements Optimal Subspace Estimation (MOOSE). The optimization is performed via a natural conjugate gradient learning approach carried out on the set of $n$ dimensional subspaces in $ {\rm I\!R}^{m}$, $ m > n$, which is a Grassmann manifold. The results of applying MOOSE, MX-SVD, and $\ell _{2}$– based approaches are demonstrated both on simulated and real hyperspectral data.