A Robust Spectral Clustering Algorithm for Sub-Gaussian Mixture Models with Outliers
This addresses the issue of outlier sensitivity in clustering algorithms for data analysis applications, offering a provably robust method with competitive performance, though it is incremental as it builds on existing spectral clustering techniques.
The paper tackles the problem of clustering datasets with arbitrary outliers by developing a robust spectral clustering algorithm that applies a denoising rounding scheme to a Gaussian kernel matrix, showing that the misclassification error decays exponentially with the signal-to-noise ratio when outliers are a small fraction of inliers, matching the best-known bound for SDPs without outliers.
We consider the problem of clustering datasets in the presence of arbitrary outliers. Traditional clustering algorithms such as k-means and spectral clustering are known to perform poorly for datasets contaminated with even a small number of outliers. In this paper, we develop a provably robust spectral clustering algorithm that applies a simple rounding scheme to denoise a Gaussian kernel matrix built from the data points and uses vanilla spectral clustering to recover the cluster labels of data points. We analyze the performance of our algorithm under the assumption that the "good" data points are generated from a mixture of sub-gaussians (we term these "inliers"), while the outlier points can come from any arbitrary probability distribution. For this general class of models, we show that the misclassification error decays at an exponential rate in the signal-to-noise ratio, provided the number of outliers is a small fraction of the inlier points. Surprisingly, this derived error bound matches with the best-known bound for semidefinite programs (SDPs) under the same setting without outliers. We conduct extensive experiments on a variety of simulated and real-world datasets to demonstrate that our algorithm is less sensitive to outliers compared to other state-of-the-art algorithms proposed in the literature.