Some Options for L1-Subspace Signal Processing

TL;DR

This paper introduces methods for computing L1-norm signal subspaces that are more robust to outliers than traditional L2 methods, including optimal algorithms for fixed dimension and large sample sizes, with applications in data reduction and direction estimation.

Contribution

It presents the first optimal algorithms for L1 subspace computation in fixed dimension and large sample scenarios, addressing NP-hardness issues and generalizing to multiple components.

Findings

L1 subspace methods are less sensitive to outliers.

Optimal algorithms with complexity O(N^D) are developed.

Applications demonstrated in data reduction and direction-of-arrival estimation.

Abstract

We describe ways to define and calculate $L_1$-norm signal subspaces which are less sensitive to outlying data than $L_2$-calculated subspaces. We focus on the computation of the $L_1$ maximum-projection principal component of a data matrix containing N signal samples of dimension D and conclude that the general problem is formally NP-hard in asymptotically large N, D. We prove, however, that the case of engineering interest of fixed dimension D and asymptotically large sample support N is not and we present an optimal algorithm of complexity $O(N^D)$. We generalize to multiple $L_1$-max-projection components and present an explicit optimal $L_1$ subspace calculation algorithm in the form of matrix nuclear-norm evaluations. We conclude with illustrations of $L_1$-subspace signal processing in the fields of data dimensionality reduction and direction-of-arrival estimation.