[Paper Review] Nearest Descent, In-Tree, and Clustering

Teng Qiu, Kai-Fu Yang+2 more|arXiv (Cornell University)|Dec 7, 2014

Advanced Clustering Algorithms Research3 citations

TL;DR

This paper proposes a novel graph-theoretic clustering method based on a physically inspired rule called Nearest Descent (ND), which constructs an In-Tree by iteratively selecting parent nodes in the direction of steepest potential descent. The method reduces clustering to removing a few distinguishable edges from the In-Tree, enabling automatic or interactive cluster detection with high accuracy across diverse data shapes and high-dimensional datasets.

ABSTRACT

In this paper, we propose a physically inspired graph-theoretical clustering method, which first makes the data points organized into an attractive graph, called In-Tree, via a physically inspired rule, called Nearest Descent (ND). In particular, the rule of ND works to select the nearest node in the descending direction of potential as the parent node of each node, which is in essence different from the classical Gradient Descent or Steepest Descent. The constructed In-Tree proves a very good candidate for clustering due to its particular features and properties. In the In-Tree, the original clustering problem is reduced to a problem of removing a very few of undesired edges from this graph. Pleasingly, the undesired edges in In-Tree are so distinguishable that they can be easily determined in either automatic or interactive way, which is in stark contrast to the cases in the widely used Minimal Spanning Tree and k-nearest-neighbor graph. The cluster number in the proposed method can be easily determined based on some intermediate plots, and the cluster assignment for each node is easily made by quickly searching its root node in each sub-graph (also an In-Tree). The proposed method is extensively evaluated on both synthetic and real-world datasets. Overall, the proposed clustering method is a density-based one, but shows significant differences and advantages in comparison to the traditional ones. The proposed method is simple yet efficient and reliable, and is applicable to various datasets with diverse shapes, attributes and any high dimensionality

Motivation & Objective

• To develop a clustering method that is robust to arbitrary data shapes and high dimensionality.
• To address the limitations of traditional clustering methods such as k-means and DBSCAN in handling complex, non-spherical clusters.
• To provide a method where cluster number can be easily determined through intermediate visualizations.
• To enable efficient and reliable cluster assignment via root-finding in sub-trees.
• To reduce the complexity of cluster boundary detection by transforming it into a sparse edge removal problem on a structured graph.

Proposed method

• Applies the Nearest Descent (ND) rule to construct an In-Tree, where each node selects its parent as the nearest node in the direction of steepest potential descent.
• Uses a potential function derived from data point density to guide the descent process, ensuring hierarchical organization.
• Constructs the In-Tree as a directed acyclic graph with a single root per connected component, preserving topological structure.
• Identifies and removes a small number of 'undesired' edges that connect distinct clusters, based on their structural and topological distinguishability.
• Performs cluster assignment by tracing each node to its root in the In-Tree, with each root representing a distinct cluster.
• Employs intermediate plots (e.g., edge weight distributions or tree depth profiles) to guide automatic or interactive determination of optimal cluster count.

Experimental results

Research questions

• RQ1 Can a physically inspired graph construction rule like Nearest Descent produce a hierarchical structure suitable for clustering?
• RQ2 Can the undesired edges separating clusters be reliably and distinctly identified in the In-Tree compared to traditional graphs like MST or k-NN?
• RQ3 Does the method enable automatic and accurate determination of cluster count without prior knowledge?
• RQ4 How does the method perform on datasets with complex, non-spherical, or high-dimensional cluster structures?
• RQ5 Can the cluster assignment process be efficiently computed via simple root-finding in the In-Tree structure?

Key findings

• The In-Tree structure effectively captures the intrinsic clustering hierarchy of data, enabling accurate and efficient cluster assignment.
• The undesired edges separating clusters are highly distinguishable in the In-Tree, allowing for reliable removal with minimal human or algorithmic intervention.
• The method automatically determines the number of clusters by analyzing intermediate structural plots such as edge weight or depth distributions.
• The approach achieves high clustering accuracy across diverse datasets, including those with irregular shapes and high dimensionality.
• The method outperforms traditional density-based clustering techniques in terms of robustness and consistency on complex data distributions.
• The computational cost is low due to the sparsity of the In-Tree and the efficiency of root-finding operations in the tree structure.