[Paper Review] Fast algorithm for robust subspace recovery.

Gilad Lerman, Tyler Maunu|arXiv (Cornell University)|Jun 24, 2014

Sparse and Compressive Sensing Techniques19 references3 citations

TL;DR

This paper proposes Fast Median Subspace (FMS), a computationally efficient algorithm for robust subspace recovery in high-dimensional data with outliers. By leveraging median-based optimization, FMS achieves faster computation and competitive accuracy compared to existing methods, especially on large-scale datasets with significant corruption.

ABSTRACT

This paper presents a fast algorithm for robust subspace recovery. The datasets considered include points drawn around a low-dimensional subspace of a higher dimensional ambient space, and a possibly large portion of points that do not lie nearby this subspace. The proposed algorithm, which we refer to as Fast Median Subspace (FMS), is designed to robustly determine the underlying subspace of such datasets, while having lower computational complexity than existing methods. Numerical experiments on synthetic and real data demonstrate its competitive speed and accuracy. 1

Motivation & Objective

• To address the challenge of recovering a low-dimensional subspace from high-dimensional data corrupted by a large fraction of outliers.
• To reduce the computational complexity of existing robust subspace recovery algorithms without sacrificing accuracy.
• To develop a method that scales efficiently to large datasets while maintaining robustness to gross corruptions.
• To provide a practical alternative to existing methods that are computationally expensive for real-time or large-scale applications.

Proposed method

• FMS employs a median-based optimization strategy to minimize the influence of outliers during subspace estimation.
• The algorithm uses a fast iterative reweighting scheme that reduces computational overhead compared to traditional robust regression techniques.
• It leverages the geometric properties of median in high dimensions to identify inliers and estimate the underlying subspace robustly.
• The method alternates between subspace estimation and outlier detection using a robust cost function sensitive to large deviations.
• FMS is designed with computational efficiency in mind, using efficient matrix operations and avoiding expensive SVD decompositions in each iteration.
• The algorithm is initialized with a random subspace and iteratively refines the estimate using a robust objective function.

Experimental results

Research questions

• RQ1 Can a robust subspace recovery algorithm achieve significantly lower computational complexity than existing methods while maintaining high accuracy?
• RQ2 How does FMS perform in terms of speed and accuracy when handling datasets with a high fraction of outliers?
• RQ3 Does the median-based optimization in FMS provide better robustness to gross corruptions compared to traditional L1 or L2-based methods?
• RQ4 To what extent does FMS scale to large-scale datasets compared to state-of-the-art alternatives?

Key findings

• FMS achieves competitive subspace recovery accuracy on both synthetic and real-world datasets, even when up to 50% of the data points are outliers.
• The algorithm runs significantly faster than existing robust subspace recovery methods, particularly on high-dimensional data with large sample sizes.
• Numerical experiments show that FMS maintains stable performance across various noise levels and outlier ratios.
• FMS outperforms baseline methods in terms of computational efficiency, with speedups of up to several times on large-scale datasets.