RANSAC Revisited: An Improved Algorithm for Robust Subspace Recovery under Adversarial and Noisy Corruptions

Abstract: In this paper, we study the problem of robust subspace recovery (RSR) in the presence of both strong adversarial corruptions and Gaussian noise. Specifically, given a limited number of noisy samples -- some of which are tampered by an adaptive and strong adversary -- we aim to recover a low-dimensional subspace that approximately contains a significant fraction of the uncorrupted samples, up to an error that scales with the Gaussian noise. Existing approaches to this problem often suffer from high computational costs or rely on restrictive distributional assumptions, limiting their applicability in truly adversarial settings. To address these challenges, we revisit the classical random sample consensus (RANSAC) algorithm, which offers strong robustness to adversarial outliers, but sacrifices efficiency and robustness against Gaussian noise and model misspecification in the process. We propose a two-stage algorithm, RANSAC+, that precisely pinpoints and remedies the failure modes of standard RANSAC. Our method is provably robust to both Gaussian and adversarial corruptions, achieves near-optimal sample complexity without requiring prior knowledge of the subspace dimension, and is more efficient than existing RANSAC-type methods.

An Improved Algorithm for Robust Subspace Recovery in Adversarial Settings

The paper "RANSAC Revisited: An Improved Algorithm for Robust Subspace Recovery under Adversarial and Noisy Corruptions" tackles the problem of robust subspace recovery (RSR) in contexts where both adversarially corrupted data and Gaussian noise are prevalent. Traditional methodologies, while robust to outliers, typically lack efficiency or necessitate stringent distributional assumptions that render them unsuitable in adversarial scenarios. This research revisits the classical random sample consensus (RANSAC) algorithm, a staple in the field due to its strong resistance to adversarial outliers, but often criticized for its inefficiency and susceptibility to noise and model misspecification.

Objective and Methodology

The paper focuses on recovering a low-dimensional subspace from a set of high-dimensional data points, where some points are corrupted by adversarial contamination and all are perturbed by Gaussian noise. The primary objective is to identify a dimension-$r^\star$ subspace that captures the core structure of uncorrupted data up to noise-determined precision. Traditional RANSAC struggles to handle pervasive adversarial settings efficiently, largely due to its sensitivity to noise and reliance on prior knowledge of subspace dimensions.

To address these limitations, the authors introduce a refined two-stage approach, termed RANSAC+. This algorithm aims to combine the robustness of the original RANSAC against adversarial corruption with enhanced efficiency and reduced sensitivity to Gaussian noise and rank misspecification. The RANSAC+ operates effectively without needing the exact subspace dimension beforehand, a notable challenge in practical applications.

Results and Contributions

Theoretical Insights: The authors provide robust theoretical analyses demonstrating that RANSAC+ achieves subspace recovery with an error margin that scales with the noise level, while also reducing computational costs significantly as compared to classical RANSAC. The proposed methodology achieves this by:

1. Coarse-Grained Estimation: This novel stage estimates an initial subspace using a smaller data subset, ensuring high sample efficiency and resilience to noise. The derived subspace dimension is approximately $O(r^\star)$, requiring sample sizes proportional to $\rstar \log(\rstar)$, which is near-optimal.
2. Fine-Grained Estimation: The second stage refines the subspace estimation by projecting the original dataset onto the initially estimated subspace. Here, the authors apply a robustified variant of RANSAC to a reduced dimensionality, enabling finer accuracy and dimension determination.

Computational Efficiency: The proposed algorithm exhibits an overall complexity of $O\left(nd r^\star \log(r^\star) + \rstar^3 e^{\rstar}\right)$, marking a significant improvement over traditional approaches. This improvement stems from operating predominantly in lower-dimensional subspaces and ensuring convergence through rigorous characterization of the sample complexity and runtime.

Numerical Results: Empirical evaluations highlight the superiority of RANSAC+ over existing RSR methods, particularly under strong adversarial corruption and various noise levels. It stabilizes performance across diverse corruption rates and showcases reduced sensitivity to dimension overestimation and Gaussian noise, typical pitfalls of classical RANSAC.

Implications and Future Directions

This study provides a strong foundational advancement for robust subspace recovery in adversarial environments. It establishes significant strides towards computational feasibility and robustness to noise. The methodology could be pivotal in applications ranging from computer vision to machine learning, where high-dimensional data is subject to noise and systematic contamination.

Future investigations could explore extending these methodologies to address other complex noise and corruption models while ensuring scalability for real-time applications. Additionally, there's potential in adapting similar methodologies to more generalized manifold learning contexts, thereby broadening the application scope of the presented techniques. The paper does not explicitly cover the implications of model misspecification beyond subspace dimension, which could be an intriguing direction for subsequent research endeavors.