Journal
IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING
Volume 12, Issue 6, Pages 1520-1533Publisher
IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
DOI: 10.1109/JSTSP.2018.2867446
Keywords
Affine subspace clustering; affine sparse subspace clustering; subspace-preserving property; nonnegative solution; subspace-dense solution
Categories
Funding
- National Natural Science Foundation of China [61876022]
- Key Laboratory of Machine Perception (MOE), Peking University
- National Science Foundation [1447822]
- Direct For Computer & Info Scie & Enginr
- Div Of Information & Intelligent Systems [1447822] Funding Source: National Science Foundation
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Sparse subspace clustering (SSC) is a state-of-the-art method for segmenting a set of data points drawn from a union of subspaces into their respective subspaces. It is now well understood that SSC produces subspace-preserving data affinity under broad geometric conditions but suffers from a connectivity issue. In this paper, we develop a novel geometric analysis for a variant of SSC, named affine SSC (ASSC), for the problem of clustering data from a union of affine subspaces. Our contributions include a new concept called affine independence for capturing the arrangement of a collection of affine subspaces. Under the affine independence assumption, we show that ASSC is guaranteed to produce subspace-preserving affinity. Moreover, inspired by the phenomenon that the l(1) regularization no longer induces sparsity when the solution is nonnegative, we further show that subspace-preserving recovery can be achieved under much weaker conditions for all data points other than the extreme points of samples from each subspace. In addition, we confirm a curious observation that the affinity produced by ASSC may be subspace-dense-which could guarantee the subspace-preserving affinity of ASSC to produce correct clustering under rather weak conditions. We validate the theoretical findings on carefully designed synthetic data and evaluate the performance of ASSC on several real datasets.
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