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Finding a small set of representatives from an unlabeled dataset is a coreproblem in a broad range of applications such as dataset summarization andinformation extraction. Classical exemplar selection methods such as$k$-medoids work under the assumption that the data points are close to a fewcluster centroids, and cannot handle the case where data lie close to a unionof subspaces. This paper proposes a new exemplar selection model that searchesfor a subset that best reconstructs all data points as measured by the $\ell_1$norm of the representation coefficients. Geometrically, this subset best coversall the data points as measured by the Minkowski functional of the subset. Tosolve our model efficiently, we introduce a farthest first search algorithmthat iteratively selects the worst represented point as an exemplar. When thedataset is drawn from a union of independent subspaces, our method is able toselect sufficiently many representatives from each subspace. We further developan exemplar based subspace clustering method that is robust to imbalanced dataand efficient for large scale data. Moreover, we show that a classifier trainedon the selected exemplars (when they are labeled) can correctly classify therest of the data points.

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