Dimensionality-reduced subspace clustering


Reinhard Heckel, Michael Tschannen, and Helmut Bölcskei


Information and Inference: A Journal of the IMA, Vol. 6, No. 3, pp. 246-283, Sept. 2017.

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Subspace clustering refers to the problem of clustering unlabeled high-dimensional data points into a union of low-dimensional linear subspaces, whose number, orientations, and dimensions are all unknown. In practice one may have access to dimensionality-reduced observations of the data only, resulting, e.g., from undersampling due to complexity and speed constraints on the acquisition device or mechanism. More pertinently, even if the high-dimensional data set is available it is often desirable to first project the data points into a lower-dimensional space and to perform clustering there; this reduces storage requirements and computational cost. The purpose of this paper is to quantify the impact of dimensionality reduction through random projection on the performance of three subspace clustering algorithms, all of which are based on principles from sparse signal recovery. Specifi cally, we analyze the thresholding based subspace clustering (TSC) algorithm, the sparse subspace clustering (SSC) algorithm, and an orthogonal matching pursuit variant thereof (SSC-OMP). We fi nd, for all three algorithms, that dimensionality reduction down to the order of the subspace dimensions is possible without incurring signifi cant performance degradation. Moreover, these results are order-wise optimal in the sense that reducing the dimensionality further leads to a fundamentally ill-posed clustering problem. Our fi ndings carry over to the noisy case as illustrated through analytical results for TSC and simulations for SSC and SSC-OMP. Extensive experiments on synthetic and real data complement our theoretical findings.


Subspace clustering, dimensionality reduction, random projection, sparse signal recovery

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Copyright Notice: © 2017 R. Heckel, M. Tschannen, and H. Bölcskei.

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