• Corpus ID: 88515338

# Subspace Clustering with Missing and Corrupted Data

@article{Charles2017SubspaceCW,
title={Subspace Clustering with Missing and Corrupted Data},
author={Zachary B. Charles and Amin Jalali and Rebecca M. Willett},
journal={arXiv: Machine Learning},
year={2017}
}
• Published 8 July 2017
• Mathematics
• arXiv: Machine Learning
Given full or partial information about a collection of points that lie close to a union of several subspaces, subspace clustering refers to the process of clustering the points according to their subspace and identifying the subspaces. One popular approach, sparse subspace clustering (SSC), represents each sample as a weighted combination of the other samples, with weights of minimal $\ell_1$ norm, and then uses those learned weights to cluster the samples. SSC is stable in settings where each…

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## References

SHOWING 1-10 OF 33 REFERENCES
Group-sparse subspace clustering with missing data
• Computer Science, Mathematics
2016 IEEE Statistical Signal Processing Workshop (SSP)
• 2016
Two novel methods for subspace clustering with missing data are described: (a) group-sparse sub- space clustering (GSSC), which is based on group-sparsity and alternating minimization, and (b) mixture subspace clusters (MSC) which models each data point as a convex combination of its projections onto all subspaces in the union.
Theoretical Analysis of Sparse Subspace Clustering with Missing Entries
• Computer Science, Mathematics
ICML
• 2018
This paper analytically establishes that projecting the zero-filled data onto the observation pattern of the point being expressed leads to a substantial improvement in performance, and gives theoretical guarantees for SSC with incomplete data.
A Geometric Analysis of Subspace Clustering with Outliers
• Mathematics, Computer Science
ArXiv
• 2011
A novel geometric analysis of an algorithm named sparse subspace clustering (SSC) is developed, which signicantly broadens the range of problems where it is provably eective and shows that SSC can recover multiple subspaces, each of dimension comparable to the ambient dimension.
Sparse Subspace Clustering with Missing Entries
• Mathematics, Computer Science
ICML
• 2015
Two new approaches for subspace clustering and completion are proposed and evaluated, which all outperform the natural approach when the data matrix is high-rank or the percentage of missing entries is large.
Noisy Sparse Subspace Clustering
• Mathematics, Computer Science
J. Mach. Learn. Res.
• 2013
It is shown that a modified version of SSC is provably effective in correctly identifying the underlying subspaces, even with noisy data, which extends theoretical guarantee of this algorithm to the practical setting and provides justification to the success of SCC in a class of real applications.
Sparse subspace clustering
• Mathematics, Computer Science
CVPR
• 2009
This work proposes a method based on sparse representation (SR) to cluster data drawn from multiple low-dimensional linear or affine subspaces embedded in a high-dimensional space and applies this method to the problem of segmenting multiple motions in video.
Robust Subspace Clustering
• Computer Science, Mathematics
ArXiv
• 2013
This paper introduces an algorithm inspired by sparse subspace clustering (SSC) to cluster noisy data, and develops some novel theory demonstrating its correctness.
The Information-Theoretic Requirements of Subspace Clustering with Missing Data
• Mathematics, Computer Science
ICML
• 2016
To derive deterministic sampling conditions for SCMD, which give precise information-theoretic requirements and determine sampling regimes, a practical algorithm is given to certify the output of any SCMD method deterministically.
High-Rank Matrix Completion and Clustering under Self-Expressive Models
This work proposes efficient algorithms for simultaneous clustering and completion of incomplete high-dimensional data that lie in a union of low-dimensional subspaces and shows that when the data matrix is low-rank, the algorithm performs on par with or better than low-Rank matrix completion methods, while for high-rank data matrices, the method significantly outperforms existing algorithms.
Generalized principal component analysis (GPCA)
• Mathematics, Medicine
IEEE Transactions on Pattern Analysis and Machine Intelligence
• 2005
An algebro-geometric solution to the problem of segmenting an unknown number of subspaces of unknown and varying dimensions from sample data points and applications of GPCA to computer vision problems such as face clustering, temporal video segmentation, and 3D motion segmentation from point correspondences in multiple affine views are presented.