On deterministic conditions for subspace clustering under missing data

In this paper we present deterministic analysis of sufficient conditions for sparse subspace clustering under missing data, when data is assumed to come from a Union of Subspaces (UoS) model. In this context we consider two cases, namely Case I when all the points are sampled at the same co-ordinates, and Case II when points are sampled at different locations. We show that results for Case I directly follow from several existing results in the literature, while results for Case II are not as straightforward and we provide a set of dual conditions under which, perfect clustering holds true. We provide extensive set of simulation results for clustering as well as completion of data under missing entries, under the UoS model. Our experimental results indicate that in contrast to the full data case, accurate clustering does not imply accurate subspace identification and completion, indicating the natural order of relative hardness of these problems.

[1]  Joel A. Tropp,et al.  Robust computation of linear models, or How to find a needle in a haystack , 2012, ArXiv.

[2]  Robert D. Nowak,et al.  On the sample complexity of subspace clustering with missing data , 2014, 2014 IEEE Workshop on Statistical Signal Processing (SSP).

[3]  P. Wedin.  On angles between subspaces of a finite dimensional inner product space , 1983 .

[4]  Emmanuel J. Candès,et al.  A Geometric Analysis of Subspace Clustering with Outliers , 2011, ArXiv.

[5]  Helmut Bölcskei,et al.  Dimensionality-reduced subspace clustering , 2015, ArXiv.

[6]  Tetsunao Matsuta,et al.  国際会議開催報告:2013 IEEE International Symposium on Information Theory , 2013 .

[7]  Ulrike von Luxburg,et al.  A tutorial on spectral clustering , 2007, Stat. Comput..

[8]  Nigel Boston,et al.  Deterministic conditions for subspace identifiability from incomplete sampling , 2014, 2015 IEEE International Symposium on Information Theory (ISIT).

[9]  René Vidal,et al.  Sparse Subspace Clustering: Algorithm, Theory, and Applications , 2012, IEEE transactions on pattern analysis and machine intelligence.

[10]  Constantine Caramanis,et al.  Greedy Subspace Clustering , 2014, NIPS.

[11]  Lek-Heng Lim,et al.  Schubert Varieties and Distances between Subspaces of Different Dimensions , 2014, SIAM J. Matrix Anal. Appl..

[12]  Allen Y. Yang,et al.  Estimation of Subspace Arrangements with Applications in Modeling and Segmenting Mixed Data , 2008, SIAM Rev..

[13]  Gene H. Golub,et al.  Numerical methods for computing angles between linear subspaces , 1971, Milestones in Matrix Computation.

[14]  Peter Gritzmann,et al.  Inner and outerj-radii of convex bodies in finite-dimensional normed spaces , 1992, Discret. Comput. Geom..

[15]  René Vidal,et al.  Abstract Algebraic Subspace Clustering , 2015, ArXiv.

[16]  Daniel P. Robinson,et al.  Sparse Subspace Clustering with Missing Entries , 2015, ICML.

[17]  Nigel Boston,et al.  A characterization of deterministic sampling patterns for low-rank matrix completion , 2015, 2015 53rd Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[18]  Robert D. Nowak,et al.  K-subspaces with missing data , 2012, 2012 IEEE Statistical Signal Processing Workshop (SSP).

[19]  Robert D. Nowak,et al.  High-Rank Matrix Completion and Subspace Clustering with Missing Data , 2011, ArXiv.

[20]  Emmanuel J. Candès,et al.  Exact Matrix Completion via Convex Optimization , 2008, Found. Comput. Math..

[21]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[22]  Michael I. Jordan,et al.  On Spectral Clustering: Analysis and an algorithm , 2001, NIPS.

[23]  Helmut Bölcskei,et al.  Robust Subspace Clustering via Thresholding , 2013, IEEE Transactions on Information Theory.

[24]  Robert D. Nowak,et al.  Group-sparse subspace clustering with missing data , 2016, 2016 IEEE Statistical Signal Processing Workshop (SSP).