Computing Robust Principal Components by A* Search

Abstract

Principal Component Analysis (PCA) is a classical dimensionality reduction technique that computes a low rank representation of the data. Recent studies have shown how to compute this low rank representation from most of the data, excluding a small amount of outlier data. We show how to convert this problem into graph search, and describe an algorithm that solves this problem optimally by applying a variant of the A* algorithm to search for the outliers. The results obtained by our algorithm are optimal in terms of accuracy, and are shown to be more accurate than results obtained by the current state-of-the- art algorithms which are shown not to be optimal. This comes at the cost of running time, which is typically slower than the current state of the art. We also describe a related variant of the A* algorithm that runs much faster than the optimal variant and produces a solution that is guaranteed to be near the optimal. This variant is shown experimentally to be more accurate than the current state-of-the-art and has a comparable running time.

Accelerated Combinatorial Search for Outlier Detection with Provable Bound on Sub-Optimality

“…The following are special cases that we prove: • If = 0 the algorithm terminates with an optimal outlier selection regardless of the value of the chunk size c. • If c = 1 the algorithm is identical to (Shah et al 2018).…”

“…Still, even in this case, our algorithm has the advantage of producing sub-optimality bounds which are not computed by (Shah et al 2018).…”

“…The authors refer to the algorithm as "iSearch". A different approach was taken in (Shah et al 2018). The authors view outlier detection as a graph search problem, and apply the weighted A * algorithm to solve it.…”

“…The goal of the search is to minimize the PCA error. As reported in (Shah et al 2018) the algorithm comes with an accuracy parameter called . With = 0 the algorithm is guaranteed to terminate with an optimal outlier selection.…”