Robust Fuzzy local k-plane clustering with mixture distance of hinge loss and L1 norm

Being robust to the presence of outliers is crucial for applying clustering algorithms in practice. In the \textit{robust k-Means} problem (i.e., $k$-Means with outliers), the goal is to remove $z$ outliers and minimize the $k$-Means cost on the remaining points. Despite the close connection between robust $k$-Means and outlier detection, both theoretical and empirical understanding of the effectiveness of $\textit{classic outlier detection heuristics}$ for robust $k$-Means remains limited. In this paper, we prove that under a practical assumption on the optimal cluster sizes, simply removing points with large $K$-Nearest-Neighbor distances achieves performance comparable to prior work in terms of approximation guarantees: it yields a constant-factor reduction from robust $k$-Means to standard $k$-Means, without introducing additional centers or discarding extra outliers, as is commonly required by existing approaches. Empirically, experiments on real-world datasets show that our method outperforms or matches several more sophisticated algorithms in terms of clustering cost and runtime. These results demonstrate that simple KNN-based heuristics can be surprisingly effective for robust clustering, highlighting new opportunities to bridge techniques from outlier detection and clustering.

Lloyd's $K$-means algorithm, also known as naïve $K$-means, is a widely used ad hoc optimization heuristic, designed to minimize the sum of squared errors (SSE) across all $K$-partitions of a dataset via iterative cluster refinement. In this work, we establish a novel connection between Lloyd's algorithm and the Frank-Wolfe (FW) algorithm, a prominent first-order method for projection-free optimization. We demonstrate that Lloyd's algorithm is a special case of FW. Leveraging recent advances in FW methods for concave objectives, we derive a non-asymptotic $\mathcal{O}(1/t)$ convergence rate to a local minimum of the SSE objective. To account for empty clusters, an outcome possible under Lloyd's greedy assignment, we develop an FW variant for semismooth objectives while retaining the same convergence rate that is solely controlled by the initial SSE value. We illustrate our findings with a simulation study for spherical Gaussian mixtures and a real-world image segmentation dataset.

Functional data present unique challenges for clustering due to their infinite-dimensional nature and potential sensitivity to outliers. An extension of the OCLUST algorithm to the functional setting is proposed to address these issues. The approach leverages the OCLUST framework, creating a robust method to cluster curves and trim outliers. The methodology is evaluated on both simulated and real-world functional datasets, demonstrating strong performance in clustering and outlier identification.