Understanding Regularized Spectral Clustering via Graph Conductance (1806.01468v4)

Abstract: This paper uses the relationship between graph conductance and spectral clustering to study (i) the failures of spectral clustering and (ii) the benefits of regularization. The explanation is simple. Sparse and stochastic graphs create a lot of small trees that are connected to the core of the graph by only one edge. Graph conductance is sensitive to these noisy dangling sets'. Spectral clustering inherits this sensitivity. The second part of the paper starts from a previously proposed form of regularized spectral clustering and shows that it is related to the graph conductance on aregularized graph'. We call the conductance on the regularized graph CoreCut. Based upon previous arguments that relate graph conductance to spectral clustering (e.g. Cheeger inequality), minimizing CoreCut relaxes to regularized spectral clustering. Simple inspection of CoreCut reveals why it is less sensitive to small cuts in the graph. Together, these results show that unbalanced partitions from spectral clustering can be understood as overfitting to noise in the periphery of a sparse and stochastic graph. Regularization fixes this overfitting. In addition to this statistical benefit, these results also demonstrate how regularization can improve the computational speed of spectral clustering. We provide simulations and data examples to illustrate these results.

• The paper reveals that regularization mitigates spectral clustering’s sensitivity to noisy, dangling sets, leading to more balanced partitions.
• Regularization via the CoreCut concept relaxes the spectral clustering problem, boosting computational speed compared to the unregularized approach.
• Empirical results demonstrate that regularized spectral clustering outperforms traditional methods in social network and brain graph applications.

Understanding Regularized Spectral Clustering via Graph Conductance

This paper, authored by Yilin Zhang and Karl Rohe, focuses on the domain of spectral clustering, aiming to address its limitations and propose enhancements through the lens of graph conductance and regularization techniques. Spectral clustering, a widely-used method for partitioning graph nodes based on the eigenvectors of the graph Laplacian, often suffers from issues in applied research. Specifically, it tends to produce uninteresting partitions characterized by large clusters containing most nodes and several small clusters with minimal representation. This limitation is notably observed in applications involving brain graphs and social networks.

Key Contributions

1. Failures of Spectral Clustering: Spectral clustering is susceptible to unbalanced partitions due to its sensitivity to noisy, dangling sets found in sparse and stochastic graphs. These sets are connected to the graph's core by only one edge, and the graph conductance of such sets is notably small. This sensitivity leads to overfitting, where spectral clustering captures noise rather than essential graph structure.
2. Regularization Benefits: Regularization mitigates the failures of spectral clustering by addressing its sensitivity to noise. By enhancing computational speed and preventing overfitting, regularized spectral clustering provides more balanced partitions that yield meaningful clusters.
3. CoreCut Concept: The paper introduces CoreCut, a variation of conductance defined on a regularized graph, which is less sensitive to small cuts. The paper demonstrates that minimizing CoreCut relaxes to regularized spectral clustering, providing an insightful explanation of why regularization helps achieve better results.
4. Empirical Evaluations: Through simulations and real-data examples, the authors illustrate the effectiveness of regularized spectral clustering over the vanilla approach. Regularized spectral clustering avoids the overfitting observed in unregularized methods and consistently produces more balanced partitions. Moreover, the computational efficiency improvements are quantified, showcasing that Regularized-SC runs faster than Vanilla-SC in practical implementations.

Implications and Future Directions

The paper underscores the significance of regularization in improving spectral clustering's accuracy and computational efficiency. These findings have practical implications for clustering applications where sparse and stochastic noise pervades, such as in social network analysis and brain graph modeling. Additionally, the insights offered by this paper open up potential avenues for exploring regularization techniques in other graph-based machine learning tasks. One possibility is in the design of neural network architectures that account for graph structure, where understanding and mitigating peripheral noise could be crucial.

More broadly, this research highlights the importance of considering graph structures in developing machine learning methods, suggesting the integration of regularization concepts into broader applications. As AI progresses, leveraging these insights in developing new models that generalize beyond traditional convolutional networks will likely see further exploration. The proposed regularization strategies in spectral clustering may serve as foundational work in adapting neural networks to graph-based data.

In conclusion, the paper provides a detailed examination of spectral clustering failures and the benefits offered by regularization, offering a comprehensive approach to understanding and enhancing graph-based clustering techniques. Through rigorous analysis and empirical validation, it sets the stage for future research and application in both theoretical and practical domains.