Fast and explainable clustering in the Manhattan and Tanimoto distance

TL;DR

This work extends the CLASSIX clustering framework from Euclidean distance to Manhattan and Tanimoto distances by sorting data points on a norm-based score and applying a reverse-triangle bound to prune neighbor searches. It introduces Manhattan-specific orthant shifting and a Baldi intersection inequality to aggressively prune binary fingerprint neighbors under the Tanimoto metric, along with probabilistic analyses and extensive experiments. The results show massive speedups over Taylor--Butina and DBSCAN on real chemical fingerprint datasets while delivering higher-quality clusters, and the Manhattan variant demonstrates strong performance on standard benchmarks and in conjunction with dimensionality reduction. The approach preserves deterministic aggregation/merging steps, enabling transparent and explainable clustering and scalable memory usage suitable for large datasets and potential GPU acceleration.

Abstract

The CLASSIX algorithm is a fast and explainable approach to data clustering. In its original form, this algorithm exploits the sorting of the data points by their first principal component to truncate the search for nearby data points, with nearness being defined in terms of the Euclidean distance. Here we extend CLASSIX to other distance metrics, including the Manhattan distance and the Tanimoto distance. Instead of principal components, we use an appropriate norm of the data vectors as the sorting criterion, combined with the triangle inequality for search termination. In the case of Tanimoto distance, a provably sharper intersection inequality is used to further boost the performance of the new algorithm. On a real-world chemical fingerprint benchmark, CLASSIX Tanimoto is about 30 times faster than the Taylor--Butina algorithm, and about 80 times faster than DBSCAN, while computing higher-quality clusters in both cases.

Figures (9)

• Figure 1: Comparison of search area vs possible location of vectors satisfying the radius condition.
• Figure 2: Efficiency of the search termination criterion based on different flip probabilities and similarity thresholds for different values of $\alpha_i$, as a result of direct computation of probabilities.
• Figure 3: Efficiency of the search termination criterion based on different flip probabilities and similarity thresholds for different values of $\alpha_i$, as a result of simulations on synthetic data.
• Figure 4: Runtime of CLASSIX_T (red crosses), DBSCAN (blue dots) and OPTICS (green triangles) as the number of data points, $n$, varies. The data dimension $d=1,000$ and number of clusters $\texttt{num\_clusters} = 10$ are kept fixed.
• Figure 5: Scaling of CLASSIX_T (red crosses) and DBSCAN (blue dots) with the separability of the data with respect to the pop-count scores. Total number of data points is kept constant $n=50,000$.