# CUR Decompositions, Similarity Matrices, and Subspace Clustering

arXiv: https://arxiv.org/abs/1711.04178 · HTML: https://mlsanity.com/paper/1711.04178/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Nov 2017.

## Built on by (most cited first, 4 of 4)

- [Robust CUR Decomposition: Theory and Imaging Applications](https://mlsanity.com/paper/2101.05231/) (2021)
- [Mode-wise Tensor Decompositions: Multi-dimensional Generalizations of CUR Decompositions](https://mlsanity.com/paper/2103.11037/) (2021)
- [Error Analysis of Tensor-Train Cross Approximation](https://mlsanity.com/paper/2207.04327/) (2022)
- [On Matrix Factorizations in Subspace Clustering](https://mlsanity.com/paper/2106.12016/) (2021)

## Abstract (arXiv)

A general framework for solving the subspace clustering problem using the CUR decomposition is presented. The CUR decomposition provides a natural way to construct similarity matrices for data that come from a union of unknown subspaces mathscrU=underseti=1oversetMbigcupSi. The similarity matrices thus constructed give the exact clustering in the noise-free case. Additionally, this decomposition gives rise to many distinct similarity matrices from a given set of data, which allow enough flexibility to perform accurate clustering of noisy data. We also show that two known methods for subspace clustering can be derived from the CUR decomposition. An algorithm based on the theoretical construction of similarity matrices is presented, and experiments on synthetic and real data are presented to test the method. Additionally, an adaptation of our CUR based similarity matrices is utilized to provide a heuristic algorithm for subspace clustering; this algorithm yields the best overall performance to date for clustering the Hopkins155 motion segmentation dataset.