[Paper Review] Improved Cheeger's Inequality: Analysis of Spectral Partitioning Algorithms through Higher Order Spectral Gap

Tsz Chiu Kwok, Lap Chi Lau+3 more|arXiv (Cornell University)|Jan 23, 2013

Sparse and Compressive Sensing Techniques8 references16 citations

TL;DR

This paper improves Cheeger's inequality by showing that the spectral partitioning algorithm achieves an $ O(k) \frac{\lambda_2}{\sqrt{\lambda_k}} $ approximation ratio for the sparsest cut problem, where $ \lambda_2 $ and $ \lambda_k $ are eigenvalues of the normalized Laplacian. The result demonstrates that spectral partitioning is a constant-factor approximation when $ \lambda_k $ is constant for some $ k \geq 2 $, providing theoretical justification for its empirical success in clustering and image segmentation.

ABSTRACT

Let ϕ(G) be the minimum conductance of an undirected graph G, and let 0=λ_1 <= λ_2 <=... <= λ_n <= 2 be the eigenvalues of the normalized Laplacian matrix of G. We prove that for any graph G and any k >= 2, ϕ(G) = O(k) λ_2 / \sqrt{λ_k}, and this performance guarantee is achieved by the spectral partitioning algorithm. This improves Cheeger's inequality, and the bound is optimal up to a constant factor for any k. Our result shows that the spectral partitioning algorithm is a constant factor approximation algorithm for finding a sparse cut if λ_k$ is a constant for some constant k. This provides some theoretical justification to its empirical performance in image segmentation and clustering problems. We extend the analysis to other graph partitioning problems, including multi-way partition, balanced separator, and maximum cut.

Motivation & Objective

• To provide a tighter theoretical analysis of the spectral partitioning algorithm for finding sparse cuts in graphs.
• To generalize Cheeger's inequality by incorporating higher-order spectral gaps $ \lambda_k $, improving the approximation guarantee.
• To explain the empirical success of spectral partitioning in clustering and image segmentation through a refined theoretical framework.
• To extend the analysis to multi-way partitioning, balanced separator, and maximum cut problems using the same spectral gap framework.
• To establish connections between numerical stability of eigenspaces and algorithmic performance, particularly under perturbations.

Proposed method

• Propose a new upper bound on the conductance of the output set of the spectral partitioning algorithm using the second eigenvalue $ \lambda_2 $ and the $ k $-th eigenvalue $ \lambda_k $.
• Use a variational characterization of eigenvalues and energy lower bounds to derive a relationship between the Rayleigh quotient and conductance.
• Introduce a thresholding technique on the second eigenvector $ g $, normalized as $ f = g / \sqrt{d} $, to generate candidate cuts.
• Apply a novel analysis technique based on Dirichlet boundary conditions and energy decomposition to bound the conductance of threshold sets.
• Leverage the Davis-Kahan theorem to link spectral gap stability with robustness of the algorithm under edge weight perturbations.
• Extend the framework to multi-way partitioning and maximum cut by relating the bipartiteness ratio and cut expansion to higher eigenvalues.

Experimental results

Research questions

• RQ1 Can Cheeger's inequality be improved by incorporating higher eigenvalues $ \lambda_k $, rather than relying solely on $ \lambda_2 $?
• RQ2 Under what spectral conditions does the spectral partitioning algorithm achieve a constant-factor approximation for the sparsest cut problem?
• RQ3 How does the stability of the top $ k $-dimensional eigenspace relate to the performance of spectral partitioning?
• RQ4 Can the analysis of spectral partitioning be extended to multi-way partitioning and balanced separator problems using higher-order spectral gaps?
• RQ5 What is the relationship between the relaxed notion of $ (c,\epsilon) $-stability and the spectral gap $ \lambda_k $?

Key findings

• The spectral partitioning algorithm achieves an $ O(k) \frac{\lambda_2}{\sqrt{\lambda_k}} $ approximation ratio for the sparsest cut problem, improving upon the classical $ O(\sqrt{\lambda_2}) $ bound.
• The bound is optimal up to a constant factor, as shown by a cycle graph example where $ \phi(G) = \Omega(k\lambda_2 / \sqrt{\lambda_k}) $.
• When $ \lambda_k $ is a constant for some constant $ k \geq 2 $, the spectral partitioning algorithm provides a constant-factor approximation for the sparsest cut.
• The analysis extends to multi-way partitioning, showing that the algorithm's performance depends on the gap between $ \lambda_k $ and $ \lambda_{k+1} $.
• For the maximum cut problem, the paper derives improved bounds on the bipartiteness ratio and a spectral algorithm with performance $ O(k\lambda_2 / \sqrt{\lambda_k}) $.
• The paper establishes that instances with a large spectral gap $ \lambda_k $ relative to $ \lambda_{k+1} $ are $ (c, \Theta(c\lambda_2 / \lambda_3^{3/2})) $-stable, linking spectral stability to algorithmic robustness.