Hierarchical Agglomerative Clustering (HAC) is likely the earliest and most flexible clustering method, because it can be used with many distances, similarities, and various linkage strategies. It is often used when the number of clusters the data set forms is unknown and some sort of hierarchy in the data is plausible. Most algorithms for HAC operate on a full distance matrix, and therefore require quadratic memory. The standard algorithm also has cubic runtime to produce a full hierarchy. Both memory and runtime are especially problematic in the context of embedded or otherwise very resource-constrained systems. In this section, we present how data aggregation with BETULA, a numerically stable version of the well known BIRCH data aggregation algorithm, can be used to make HAC viable on systems with constrained resources with only small losses on clustering quality, and hence allow exploratory data analysis of very large data sets.
Partitioning trees are efficient data structures for $k$-nearest neighbor search. Machine learning libraries commonly use a special type of partitioning trees called $k$d-trees to perform $k$-nn search. Unfortunately, $k$d-trees can be ineffective in high dimensions because they need more tree levels to decrease the vector quantization (VQ) error. Random projection trees rpTrees solve this scalability problem by using random directions to split the data. A collection of rpTrees is called rpForest. $k$-nn search in an rpForest is influenced by two factors: 1) the dispersion of points along the random direction and 2) the number of rpTrees in the rpForest. In this study, we investigate how these two factors affect the $k$-nn search with varying $k$ values and different datasets. We found that with larger number of trees, the dispersion of points has a very limited effect on the $k$-nn search. One should use the original rpTree algorithm by picking a random direction regardless of the dispersion of points.
We introduce the breathing k-means algorithm, which on average significantly improves solutions obtained by the widely-known greedy k-means++ algorithm, the default method for k-means clustering in the scikit-learn package. The improvements are achieved through a novel ``breathing'' technique, that cyclically increases and decreases the number of centroids based on local error and utility measures. We conducted experiments using greedy k-means++ as a baseline, comparing it with breathing k-means and five other k-means algorithms. Among the methods investigated, only breathing k-means and better k-means++ consistently outperformed the baseline, with breathing k-means demonstrating a substantial lead. This superior performance was maintained even when comparing the best result of ten runs for all other algorithms to a single run of breathing k-means, highlighting its effectiveness and speed. Our findings indicate that the breathing k-means algorithm outperforms the other k-means techniques, especially greedy k-means++ with ten repetitions, which it dominates in both solution quality and speed. This positions breathing k-means (with the built-in initialization by a single run of greedy k-means++) as a superior alternative to running greedy k-means++ on its own.
We propose a novel accelerated exact k-means algorithm, which performs better than the current state-of-the-art low-dimensional algorithm in 18 of 22 experiments, running up to 3 times faster. We also propose a general improvement of existing state-of-the-art accelerated exact k-means algorithms through better estimates of the distance bounds used to reduce the number of distance calculations, and get a speedup in 36 of 44 experiments, up to 1.8 times faster. We have conducted experiments with our own implementations of existing methods to ensure homogeneous evaluation of performance, and we show that our implementations perform as well or better than existing available implementations. Finally, we propose simplified variants of standard approaches and show that they are faster than their fully-fledged counterparts in 59 of 62 experiments.