Abstract Algebraic Subspace Clustering

Subspace clustering is the problem of clustering data that lie close to a union of linear subspaces. In this paper, we consider an abstract version of this problem, where one is given a set of points lying in general position inside the algebraic variety of a union of subspaces, and the objective is to decompose the underlying variety into its constituent subspaces. Prior algebraic-geometric approaches to this problem require the subspaces to be of equal dimension, or the number of subspaces to be known. While an algorithm addressing the general case of an unknown number of subspaces of possibly different dimensions had been proposed, a proof for its correctness had not been given. In this paper, we propose a provably correct algorithm for addressing the general case. Our algorithm uses the gradient of a vanishing polynomial at a point in the variety to find a hyperplane containing the subspace passing through that point. By intersecting the variety with this hyperplane and recursively applying the procedure until no non-zero vanishing polynomial exists, our algorithm identifies the subspace containing that point. By repeating this procedure for other points, our algorithm eventually identifies all the subspaces by returning a basis for their orthogonal complement. 1. Introduction. Given a set of points drawn from a union of linear subspaces, subspace clustering refers to the problem of identifying the number of subspaces, their dimensions, a basis for each subspace, and the clustering of the data points according to their membership to the subspaces. This is an important problem with widespread applications in computer vision [32], systems theory [20] and genomics [15].

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