# Robust Subspace Clustering via Thresholding

@article{Heckel2015RobustSC, title={Robust Subspace Clustering via Thresholding}, author={Reinhard Heckel and Helmut B{\"o}lcskei}, journal={IEEE Transactions on Information Theory}, year={2015}, volume={61}, pages={6320-6342} }

The problem of clustering noisy and incompletely observed high-dimensional data points into a union of low-dimensional subspaces and a set of outliers is considered. The number of subspaces, their dimensions, and their orientations are assumed unknown. We propose a simple low-complexity subspace clustering algorithm, which applies spectral clustering to an adjacency matrix obtained by thresholding the correlations between data points. In other words, the adjacency matrix is constructed from the…

## 140 Citations

Subspace clustering via thresholding and spectral clustering

- Computer Science2013 IEEE International Conference on Acoustics, Speech and Signal Processing
- 2013

A simple and low-complexity clustering algorithm based on thresholding the correlations between the data points followed by spectral clustering is proposed, which proves that this algorithm succeeds even when the subspaces intersect, and when the dimensions of the subSpaces scale linearly in the ambient dimension.

Noisy subspace clustering via thresholding

- Computer Science2013 IEEE International Symposium on Information Theory
- 2013

This work considers the problem of clustering noisy high-dimensional data points into a union of low-dimensional subspaces and a set of outliers and finds that the simple outlier detection scheme introduced in [1] provably succeeds in the noisy case.

Greedy Subspace Clustering

- Computer ScienceNIPS
- 2014

The statistical analysis shows that the algorithms are guaranteed exact (perfect) clustering performance under certain conditions on the number of points and the affinity between subspaces, which are weaker than those considered in the standard statistical literature.

Subspace Clustering using Ensembles of $K$-Subspaces

- Computer ScienceArXiv
- 2017

This work presents a novel geometric approach to the subspace clustering problem that leverages ensembles of the K-subspaces algorithm via the evidence accumulation clustering framework and proves general recovery guarantees for any algorithm that forms an affinity matrix with entries close to a monotonic transformation of pairwise absolute inner products.

Dimensionality-reduced subspace clustering

- Computer ScienceArXiv
- 2015

It is found that dimensionality reduction down to the order of the subspace dimensions is possible without incurring significant performance degradation, and these results are order-wise optimal in the sense that reducing the dimensionality further leads to a fundamentally ill-posed clustering problem.

Dimensionality-reduced subspace clustering

- Computer Science
- 2016

It is found that dimensionality reduction down to the order of the subspace dimensions is possible without incurring significant performance degradation, and these results are order-wise optimal in the sense that reducing the dimensionality further leads to a fundamentally ill-posed clustering problem.

A Theoretical Analysis of Noisy Sparse Subspace Clustering on Dimensionality-Reduced Data

- Computer ScienceIEEE Transactions on Information Theory
- 2019

This paper study's analysis applies to the most general fully deterministic model, where both underlying subspaces and data points within each subspace are deterministically positioned, and also a wide range of dimensionality reduction techniques that fall into a subspace embedding framework.

Dimensionality Reduction for Sparse Subspace Clustering

- Computer Science
- 2014

Analysis of performance guarantees for sparse subspace clustering (SSC) applied to data whose dimensionality was reduced using random projections shows that the dimensionality of the data can be reduced to the order of the dimensions of the subspaces without compromising the clustering performance, and reveals a tradeoff between the amount of dimensionality reduction tolerated and the affinities between the subSpaces.

Neighborhood selection for thresholding-based subspace clustering

- Computer Science2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)
- 2014

This paper proposes a variation of the recently introduced thresholding-based subspace clustering algorithm, which applies spectral clustering to an adjacency matrix constructed from the nearest neighbors of each data point with respect to the spherical distance measure.

Graph Connectivity in Noisy Sparse Subspace Clustering

- Computer ScienceAISTATS
- 2016

These results provide the first exact clustering guarantee of noisy SSC for subspaces of dimension greater then 3 and show that a simple post-processing procedure is capable of delivering consistent clustering under certain "general position" or "restricted eigenvalue" assumptions.

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