Self-Dictionary Sparse Regression for Hyperspectral Unmixing: Greedy Pursuit and Pure Pixel Search are Related

This paper considers a recently emerged hyperspectral unmixing formulation based on sparse regression of a self-dictionary multiple measurement vector (SD-MMV) model, wherein the measured hyperspectral pixels are used as the dictionary. Operating under the pure pixel assumption, this SD-MMV formalism is special in that it allows simultaneous identification of the endmember spectral signatures and the number of endmembers. Previous SD-MMV studies mainly focus on convex relaxations. In this study, we explore the alternative of greedy pursuit, which generally provides efficient and simple algorithms. In particular, we design a greedy SD-MMV algorithm using simultaneous orthogonal matching pursuit. Intriguingly, the proposed greedy algorithm is shown to be closely related to some existing pure pixel search algorithms, especially, the successive projection algorithm (SPA). Thus, a link between SD-MMV and pure pixel search is revealed. We then perform exact recovery analyses, and prove that the proposed greedy algorithm is robust to noise---including its identification of the (unknown) number of endmembers---under a sufficiently low noise level. The identification performance of the proposed greedy algorithm is demonstrated through both synthetic and real-data experiments

Robust Linear Regression Analysis - A Greedy Approach

The task of robust linear estimation in the presence of outliers is of particular importance in signal processing, statistics and machine learning. Although the problem has been stated a few decades ago and solved using classical (considered nowadays) methods, recently it has attracted more attention in the context of sparse modeling, where several notable contributions have been made. In the present manuscript, a new approach is considered in the framework of greedy algorithms. The noise is split into two components: a) the inlier bounded noise and b) the outliers, which are explicitly modeled by employing sparsity arguments. Based on this scheme, a novel efficient algorithm (Greedy Algorithm for Robust Denoising - GARD), is derived. GARD alternates between a least square optimization criterion and an Orthogonal Matching Pursuit (OMP) selection step that identifies the outliers. The case where only outliers are present has been studied separately, where bounds on the \textit{Restricted Isometry Property} guarantee that the recovery of the signal via GARD is exact. Moreover, theoretical results concerning convergence as well as the derivation of error bounds in the case of additional bounded noise are discussed. Finally, we provide extensive simulations, which demonstrate the comparative advantages of the new technique

A Contextual Bandit Bake-off

Contextual bandit algorithms are essential for solving many real-world interactive machine learning problems. Despite multiple recent successes on statistically and computationally efficient methods, the practical behavior of these algorithms is still poorly understood. We leverage the availability of large numbers of supervised learning datasets to empirically evaluate contextual bandit algorithms, focusing on practical methods that learn by relying on optimization oracles from supervised learning. We find that a recent method (Foster et al., 2018) using optimism under uncertainty works the best overall. A surprisingly close second is a simple greedy baseline that only explores implicitly through the diversity of contexts, followed by a variant of Online Cover (Agarwal et al., 2014) which tends to be more conservative but robust to problem specification by design. Along the way, we also evaluate various components of contextual bandit algorithm design such as loss estimators. Overall, this is a thorough study and review of contextual bandit methodology