Self-Representation Based Unsupervised Exemplar Selection in a Union of Subspaces

TL;DR

This paper introduces a novel unsupervised exemplar selection method based on self-representation that effectively handles data from a union of subspaces, improving dataset summarization and clustering, especially for large and imbalanced datasets.

Contribution

It proposes a new exemplar selection model using $ ext{l}_1$ norm reconstruction, along with a farthest first search algorithm, and develops a robust subspace clustering method based on selected exemplars.

Findings

Successfully selects representatives from each subspace in union of subspaces data

Robust to imbalanced data distributions

Enables accurate classification using exemplars as training data

Abstract

Finding a small set of representatives from an unlabeled dataset is a core problem in a broad range of applications such as dataset summarization and information extraction. Classical exemplar selection methods such as $k$-medoids work under the assumption that the data points are close to a few cluster centroids, and cannot handle the case where data lie close to a union of subspaces. This paper proposes a new exemplar selection model that searches for a subset that best reconstructs all data points as measured by the $\ell_1$ norm of the representation coefficients. Geometrically, this subset best covers all the data points as measured by the Minkowski functional of the subset. To solve our model efficiently, we introduce a farthest first search algorithm that iteratively selects the worst represented point as an exemplar. When the dataset is drawn from a union of independent subspaces, our method is able to select sufficiently many representatives from each subspace. We further develop an exemplar based subspace clustering method that is robust to imbalanced data and efficient for large scale data. Moreover, we show that a classifier trained on the selected exemplars (when they are labeled) can correctly classify the rest of the data points.