# On Consistency of Compressive Spectral Clustering

## TL;DR

This paper analyzes the consistency of spectral clustering via graph filtering on the stochastic block model, focusing on how sparsity, dimensionality, and approximation errors affect community detection accuracy.

## Contribution

It provides a theoretical analysis of spectral clustering with graph filtering, highlighting conditions for consistent community recovery in large graphs.

## Key findings

- Sparsity impacts the accuracy of spectral clustering.
- Dimensionality reduction influences clustering consistency.
- Approximation errors in graph filtering affect community detection performance.

## Abstract

Spectral clustering is one of the most popular methods for community detection in graphs. A key step in spectral clustering algorithms is the eigen decomposition of the $n{\times}n$ graph Laplacian matrix to extract its $k$ leading eigenvectors, where $k$ is the desired number of clusters among $n$ objects. This is prohibitively complex to implement for very large datasets. However, it has recently been shown that it is possible to bypass the eigen decomposition by computing an approximate spectral embedding through graph filtering of random signals. In this paper, we analyze the working of spectral clustering performed via graph filtering on the stochastic block model. Specifically, we characterize the effects of sparsity, dimensionality and filter approximation error on the consistency of the algorithm in recovering planted clusters.