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Hyperoctant Search Clustering: A Method for Clustering Data in High-Dimensional Hyperspheres

Mauricio Toledo-Acosta, Luis Ángel Ramos-García, Jorge Hermosillo-Valadez

TL;DR

This work introduces Hyperoctant Search Clustering (HOS-Clustering), a density-based clustering method designed for high-dimensional data under the angular metric. By partitioning space into sign-defined hyperoctants and modeling proximity with a Levenshtein-distance graph over labels, the method detects dense regions and provides topological insights into the data distribution. A key contribution is the rotation-based centering step that enhances hyperoctant alignment, enabling a stable clustering process with a tunable density parameter and a BFS-based exploration of the reduced graph. Experiments on text embeddings for topic detection demonstrate competitive performance to DBSCAN and reveal the method’s capacity to reveal the topology of the embedding space, suggesting broad applicability to high-dimensional, non-Euclidean representations.

Abstract

Clustering of high-dimensional data sets is a growing need in artificial intelligence, machine learning and pattern recognition. In this paper, we propose a new clustering method based on a combinatorial-topological approach applied to regions of space defined by signs of coordinates (hyperoctants). In high-dimensional spaces, this approach often reduces the size of the dataset while preserving sufficient topological features. According to a density criterion, the method builds clusters of data points based on the partitioning of a graph, whose vertices represent hyperoctants, and whose edges connect neighboring hyperoctants under the Levenshtein distance. We call this method HyperOctant Search Clustering. We prove some mathematical properties of the method. In order to as assess its performance, we choose the application of topic detection, which is an important task in text mining. Our results suggest that our method is more stable under variations of the main hyperparameter, and remarkably, it is not only a clustering method, but also a tool to explore the dataset from a topological perspective, as it directly provides information about the number of hyperoctants where there are data points. We also discuss the possible connections between our clustering method and other research fields.

Hyperoctant Search Clustering: A Method for Clustering Data in High-Dimensional Hyperspheres

TL;DR

This work introduces Hyperoctant Search Clustering (HOS-Clustering), a density-based clustering method designed for high-dimensional data under the angular metric. By partitioning space into sign-defined hyperoctants and modeling proximity with a Levenshtein-distance graph over labels, the method detects dense regions and provides topological insights into the data distribution. A key contribution is the rotation-based centering step that enhances hyperoctant alignment, enabling a stable clustering process with a tunable density parameter and a BFS-based exploration of the reduced graph. Experiments on text embeddings for topic detection demonstrate competitive performance to DBSCAN and reveal the method’s capacity to reveal the topology of the embedding space, suggesting broad applicability to high-dimensional, non-Euclidean representations.

Abstract

Clustering of high-dimensional data sets is a growing need in artificial intelligence, machine learning and pattern recognition. In this paper, we propose a new clustering method based on a combinatorial-topological approach applied to regions of space defined by signs of coordinates (hyperoctants). In high-dimensional spaces, this approach often reduces the size of the dataset while preserving sufficient topological features. According to a density criterion, the method builds clusters of data points based on the partitioning of a graph, whose vertices represent hyperoctants, and whose edges connect neighboring hyperoctants under the Levenshtein distance. We call this method HyperOctant Search Clustering. We prove some mathematical properties of the method. In order to as assess its performance, we choose the application of topic detection, which is an important task in text mining. Our results suggest that our method is more stable under variations of the main hyperparameter, and remarkably, it is not only a clustering method, but also a tool to explore the dataset from a topological perspective, as it directly provides information about the number of hyperoctants where there are data points. We also discuss the possible connections between our clustering method and other research fields.

Paper Structure

This paper contains 15 sections, 2 theorems, 28 equations, 10 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Methods
  4. Notation and Background
  5. Hyperoctants
  6. Hyperoctants as strings of signs.
  7. Distance between hyperoctants.
  8. Encoding hyperoctants' proximity using graphs.
  9. Centering Value for the data set
  10. Rotations in RD
  11. The Clustering Method
  12. Experimental design to evaluate the method
  13. Experiments and Results
  14. Discussion
  15. Conclusions

Key Result

Proposition 1

Any minimal length path in $P_D$ between $v$ and $w$ has length $d_L(v,w)$.

Figures (10)

  • Figure 1: The Levenshtein distance can be a good substitute for the angular distance (left), but not so much for the Euclidean distance (right). In both cases data points correspond to either word2vecmikolov2013efficient or FastTextbojanowski2016enriching embeddings.
  • Figure 2: The graph $P_D$ for $D=3$ representing the spatial location of the 8 octants of $\mathbb{R}^3$.
  • Figure 3: Construction of the graphs $G_0$ and $G_0^{d_0}$, the graph obtained by deleting the edges of $G_0$ such that $d_L(\phi_1,\phi_2)>d_0$. Elements of $V_0$ (that is, hyperoctants containing data points) are colored in red.
  • Figure 4: The same set of points under different coordinates.
  • Figure 5: Walk-through example illustrating the clustering process for a data set $A$ with no rotation performed.
  • ...and 5 more figures

Theorems & Definitions (3)

  • Proposition 1
  • Proposition 2
  • proof