Scalable Iterative Algorithm for Robust Subspace Clustering

TL;DR

This paper introduces a fast, scalable iterative algorithm for robust subspace clustering that improves convergence speed and accuracy on large high-dimensional datasets by using a simplified update procedure and robust objectives.

Contribution

It presents a novel, efficient iterative algorithm for robust subspace clustering that guarantees convergence and outperforms existing methods in speed and accuracy.

Findings

Converges an order of magnitude faster than prior algorithms.

Achieves 2-approximation for the robust PCA objective.

Outperforms previous methods in accuracy on MNIST dataset.

Abstract

Subspace clustering (SC) is a popular method for dimensionality reduction of high-dimensional data, where it generalizes Principal Component Analysis (PCA). Recently, several methods have been proposed to enhance the robustness of PCA and SC, while most of them are computationally very expensive, in particular, for high dimensional large-scale data. In this paper, we develop much faster iterative algorithms for SC, incorporating robustness using a {\em non-squared} $\ell_2$-norm objective. The known implementations for optimizing the objective would be costly due to the alternative optimization of two separate objectives: optimal cluster-membership assignment and robust subspace selection, while the substitution of one process to a faster surrogate can cause failure in convergence. To address the issue, we use a simplified procedure requiring efficient matrix-vector multiplications for subspace update instead of solving an expensive eigenvector problem at each iteration, in addition to release nested robust PCA loops. We prove that the proposed algorithm monotonically converges to a local minimum with approximation guarantees, e.g., it achieves 2-approximation for the robust PCA objective. In our experiments, the proposed algorithm is shown to converge at an order of magnitude faster than known algorithms optimizing the same objective, and have outperforms prior subspace clustering methods in accuracy and running time for MNIST dataset.