# Construction of the similarity matrix for the spectral clustering method: numerical experiments

## TL;DR

This paper investigates how the construction of the similarity matrix, especially the choice of scale parameter and sparsity, affects the performance of spectral clustering through extensive numerical experiments.

## Contribution

It introduces methods for selecting the scale parameter and constructing the similarity matrix based on graph properties like the MST, improving spectral clustering accuracy.

## Key findings

- Optimal scale parameter selection enhances clustering performance.
- Graph-based similarity matrix construction outperforms traditional methods.
- Numerical experiments validate the proposed approaches on various datasets.

## Abstract

Spectral clustering is a powerful method for finding structure in a dataset through the eigenvectors of a similarity matrix. It often outperforms traditional clustering algorithms such as $k$-means when the structure of the individual clusters is highly non-convex. Its accuracy depends on how the similarity between pairs of data points is defined. Two important items contribute to the construction of the similarity matrix: the sparsity of the underlying weighted graph, which depends mainly on the distances among data points, and the similarity function. When a Gaussian similarity function is used, the choice of the scale parameter $\sigma$ can be critical. In this paper we examine both items, the sparsity and the selection of suitable $\sigma$'s, based either directly on the graph associated to the dataset or on the minimal spanning tree (MST) of the graph. An extensive numerical experimentation on artificial and real-world datasets has been carried out to compare the performances of the methods.