Mathematical Morphology in Machine Learning

We present \textit{CluProp}, a novel framework that reimagines varied-density clustering in high-dimensional spaces as a label propagation process over neighborhood graphs. Our approach formally bridges the gap between density-based clustering and graph connectivity, leveraging efficient propagation mechanisms from network science to mitigate the parameter sensitivity inherent in traditional density-based methods. Specifically, we introduce a deterministic density-based propagation strategy to ensure scalable neighborhood identification. The framework is agnostic to the choice of distance metric and exhibits superior performance on large-scale data, processing millions of points in minutes while consistently outperforming existing baselines in accuracy.

The silhouette is one of the most widely used measures to assess the quality of a $k$-clustering of a dataset of $n$ elements. Its evaluation requires no information beyond the clustering assignment. In addition, the silhouette is extremely easy to interpret, providing a score to measure the quality of a clustering as a whole or for each element. The exact computation of the: (i) silhouette of each element of a dataset; and (ii) the global silhouette of the clustering; require $Θ(n^2)$ distance calculations, under general metrics. The quadratic complexity $Θ(n^2)$ is extremely prohibitive, especially on massive modern datasets. Surprisingly, existing approximate methods using $O(n^2)$ distance calculations are heuristics not offering provable and controllable guarantees on the quality of their results. We introduce the first rigorous and efficient algorithms to estimate: (i) the (local) silhouette of each element of a dataset; and (ii) the (global) silhouette; of any metric $k$-clustering. Our methods, based on sampling, perform $O(nk\varepsilon^{-2}\ln (nk/δ))$ distance computations, and provide estimates with additive error $O(\varepsilon)$ with probability at least $1-δ$. That is, parameters $\varepsilon$ and $δ$ in $(0,1)$ control the trade-off between accuracy and efficiency. We also introduce a scalable and distributed design of our methods for the MapReduce and Massively Parallel Computing (MPC) frameworks. Our distributed algorithms use a constant number of rounds and sublinear local memory. Finally, we perform extensive experiments against state-of-the-art approaches. The results show that our new techniques yield the best trade-off between accuracy and efficiency for both local and global silhouette estimation. In addition, our methods scale efficiently to massive datasets for which an exact computation of the silhouette is not practical.

K-Means clustering algorithm is one of the most commonly used clustering algorithms because of its simplicity and efficiency. K-Means clustering algorithm based on Euclidean distance only pays attention to the linear distance between Euclidean distance is an efficient and interpretable similarity measurement, but its effectiveness may deteriorate in sample spaces with anisotropic structures, redundant features, or complex feature interactions. In this paper, we propose a novel distance metric called View distance. Inspired by orthographic projection, the proposed metric projects the sample space onto $n(n-1)/2$ two-dimensional planes and defines the final distance as the sum of the Euclidean distances across the projected planes. Theoretical derivations verify that the View distance strictly satisfies the metric axioms and norm constraints. Beyond that, the View distance achieves feature coupling through projection and not only enables constant features to indirectly participate in distance calculation, but also suppresses interference from redundant features while exhibiting anisotropic geometric properties. Furthermore, to address the high computational complexity and poor scalability of full-projection View distance, we propose a two-dimensional projection plane selection strategy based on iterative Maximum Weight Matching, which reduces the computational complexity of distance calculation from $\mathcal{O}(n^2)$ to $\mathcal{O}(k)$. Extensive experiments on 12 diverse datasets demonstrate that View distance and the deterministic selection strategy provide competitive or superior performance compared with Euclidean distance and other $L_{p}$ metrics while maintaining strong interpretability and computational efficiency. The View distance provides a new perspective and option for similarity measurements.