### Abstract

ℓ^{1}-Graph, which learns a sparse graph over the data by sparse representation, has been demonstrated to be effective in clustering especially for high dimensional data. Although it achieves compelling performance, the sparse graph generated by ℓ^{1}-Graph ignores the geometric information of the data by sparse representation for each datum separately. To obtain a sparse graph that is aligned to the underlying manifold structure of the data, we propose the novel Neighborhood Regularized ℓ^{1}-Graph (NRℓ^{1}-Graph). NRℓ^{1}-Graph learns sparse graph with locally consistent neighborhood by encouraging nearby data to have similar neighbors in the constructed sparse graph. We present the optimization algorithm of NRℓ^{1}-Graph with theoretical guarantee on the convergence and the gap between the suboptimal solution and the globally optimal solution in each step of the coordinate descent, which is essential for the overall optimization of NRℓ^{1}-Graph. Its provable accelerated version, NRℓ^{1}-Graph by Random Projection (NRℓ^{1}-Graph-RP) that employs randomized data matrix decomposition, is also presented to improve the efficiency of the optimization of NRℓ^{1}-Graph. Experimental results on various real data sets demonstrate the effectiveness of both NRℓ^{1}-Graph and NRℓ^{1}- Graph-RP.-Graph, which learns a sparse graph over the data by sparse representation, has been demonstrated to be effective in clustering especially for high dimensional data. Although it achieves compelling performance, the sparse graph generated by ℓ^{1}-Graph ignores the geometric information of the data by sparse representation for each datum separately. To obtain a sparse graph that is aligned to the underlying manifold structure of the data, we propose the novel Neighborhood Regularized ℓ^{1}-Graph (NRℓ^{1}-Graph). NRℓ^{1}-Graph learns sparse graph with locally consistent neighborhood by encouraging nearby data to have similar neighbors in the constructed sparse graph. We present the optimization algorithm of NRℓ^{1}-Graph with theoretical guarantee on the convergence and the gap between the suboptimal solution and the globally optimal solution in each step of the coordinate descent, which is essential for the overall optimization of NRℓ^{1}-Graph. Its provable accelerated version, NRℓ^{1}-Graph by Random Projection (NRℓ^{1}-Graph-RP) that employs randomized data matrix decomposition, is also presented to improve the efficiency of the optimization of NRℓ^{1}-Graph. Experimental results on various real data sets demonstrate the effectiveness of both NRℓ^{1}-Graph and NRℓ^{1}- Graph-RP.

Original language | English (US) |
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State | Published - Jan 1 2017 |

Event | 33rd Conference on Uncertainty in Artificial Intelligence, UAI 2017 - Sydney, Australia Duration: Aug 11 2017 → Aug 15 2017 |

### Other

Other | 33rd Conference on Uncertainty in Artificial Intelligence, UAI 2017 |
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Country | Australia |

City | Sydney |

Period | 8/11/17 → 8/15/17 |

### ASJC Scopus subject areas

- Artificial Intelligence

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## Cite this

*Neighborhood regularized ℓ*. Paper presented at 33rd Conference on Uncertainty in Artificial Intelligence, UAI 2017, Sydney, Australia.

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