Impact of regularization on Spectral Clustering (1312.1733v2)

Abstract: The performance of spectral clustering can be considerably improved via regularization, as demonstrated empirically in Amini et. al (2012). Here, we provide an attempt at quantifying this improvement through theoretical analysis. Under the stochastic block model (SBM), and its extensions, previous results on spectral clustering relied on the minimum degree of the graph being sufficiently large for its good performance. By examining the scenario where the regularization parameter $\tau$ is large we show that the minimum degree assumption can potentially be removed. As a special case, for an SBM with two blocks, the results require the maximum degree to be large (grow faster than $\log n$) as opposed to the minimum degree. More importantly, we show the usefulness of regularization in situations where not all nodes belong to well-defined clusters. Our results rely on a `bias-variance'-like trade-off that arises from understanding the concentration of the sample Laplacian and the eigen gap as a function of the regularization parameter. As a byproduct of our bounds, we propose a data-driven technique \textit{DKest} (standing for estimated Davis-Kahan bounds) for choosing the regularization parameter. This technique is shown to work well through simulations and on a real data set.

• The paper shows that regularization enhances spectral clustering performance by relaxing minimum degree assumptions and improving handling of weakly defined clusters.
• Regularization is demonstrated to mitigate the disruptive influence of weakly clustered nodes by appropriately modifying the sample eigenvectors and potentially increasing the eigen gap.
• A data-driven technique called DKest is introduced for practical selection of the optimal regularization parameter by minimizing estimated Davis-Kahan bounds.

Impact of Regularization on Spectral Clustering

The paper "Impact of Regularization on Spectral Clustering" by Antony Joseph and Bin Yu examines the theoretical underpinnings of regularized spectral clustering, particularly under the stochastic block model (SBM). Regularization is shown to enhance spectral clustering's performance, especially when applied to scenarios where graph nodes do not clearly belong to distinct clusters. This research introduces the RSC-$\tau$ algorithm and evaluates its efficacy and theoretical basis compared to traditional spectral clustering methods.

Key Contributions

1. Relaxation of Minimum Degree Assumptions: Traditional spectral clustering relies on the minimum degree being sufficiently large to ensure good cluster recovery under SBM. This paper demonstrates that, by utilizing regularization, the dependence on the minimum degree can potentially be removed. For two-block SBM, the maximum degree needs to grow faster than $\log n$, instead of the minimum degree condition traditional methods require.
2. Handling Weakly Defined Clusters: The research highlights regularization's capacity to handle nodes that do not belong to well-defined clusters. The standard spectral clustering algorithm can be disrupted by such nodes as their influence might mask the true cluster structure in the leading eigenvectors of the Laplacian. The paper shows that a suitably large regularization parameter can mitigate this.
3. Proposed Data-Driven Regularization Selection: A technique named DKest is proposed for selecting the regularization parameter. It estimates the optimal parameter by minimizing the estimated Davis-Kahan bounds across a range of potential values. This approach is shown to work effectively in simulations and on real data, providing a practical framework for applying regularization within spectral clustering setups.

Theoretical Insights

The paper outlines how regularization influences the bias-variance trade-off in spectral clustering. The regularized Laplacian's concentration and eigen gap's behavior regarding the regularization parameter are central to understanding the improvements offered by regularization:

• Concentration: Regularization contributes to better concentration of the sample Laplacian around its population counterpart, crucially affecting performance with low degree nodes.
• Eigen Gap: The introduction of a large regularization parameter modifies the sample eigenvectors, often increasing the gap, thereby enhancing cluster recovery.

The authors provide a comprehensive mathematical framework and derivation to support their claims, employing probability concentration results to establish high probability bounds.

Practical and Theoretical Implications

The theoretical improvements in spectral clustering due to regularization have significant implications for practical applications in community detection, text mining, image segmentation, and more. Regularization makes spectral clustering robust against non-ideal data conditions, such as low degrees or non-distinct cluster assignments. Practically, the DKest method allows practitioners to automate regularization parameter selection, potentially enhancing clustering performance in real-world datasets.

Future Directions

While the theoretical advantages of a large regularization parameter have been rigorously elucidated, future research could explore intermediate regularization values that might yield even better empirical results. Moreover, extending the paper to more complex models such as the degree-corrected stochastic block model (D-SBM) could provide deeper insights into handling heterogeneity in node degrees.

In conclusion, this paper offers a substantive contribution to the spectral clustering literature, emphasizing the role and benefit of regularization through both theoretical analysis and practical algorithm development. It presents an avenue for robust cluster detection under uncertainty and weak structural definitions within networked data.