Minimal Dirichlet energy partitions for graphs

Abstract: Motivated by a geometric problem, we introduce a new non-convex graph partitioning objective where the optimality criterion is given by the sum of the Dirichlet eigenvalues of the partition components. A relaxed formulation is identified and a novel rearrangement algorithm is proposed, which we show is strictly decreasing and converges in a finite number of iterations to a local minimum of the relaxed objective function. Our method is applied to several clustering problems on graphs constructed from synthetic data, MNIST handwritten digits, and manifold discretizations. The model has a semi-supervised extension and provides a natural representative for the clusters as well.

• The paper introduces a new graph partitioning method that minimizes the sum of Dirichlet eigenvalues to yield balanced, well-connected clusters.
• The method employs a relaxation and rearrangement algorithm rooted in spectral theory, achieving high cluster purity and effective semi-supervised performance.
• Empirical evaluations on synthetic, standard, and manifold datasets demonstrate robust performance and geometric interpretability of the proposed framework.

Minimal Dirichlet Energy Partitions for Graphs

Introduction and Background

The work introduces a novel non-convex graph partitioning objective centered on the sum of Dirichlet eigenvalues for the components of a graph partition. For a graph $G=(V,E)$ with prescribed non-negative edge weights, the objective is to partition the vertex set into $k$ subsets such that the sum of the first Dirichlet eigenvalues $\sum_{i=1}^k \lambda(V_i)$ is minimized, where $\lambda(V_i)$ is the first Dirichlet eigenvalue of the component $V_i$.

This approach diverges from traditional objectives (e.g., normalized cut, Cheeger cut) by avoiding perimeter-based measures in favor of an interior-sensitive, spectral quantity that inherently promotes balanced and well-connected clusters. The formulation is motivated by analogous partitioning problems in geometric PDE and spectral optimal design, yielding highly interpretable objectives with connections to continuum eigenvalue partitioning.

Mathematical Formulation

Given a subset $S \subset V$, the Dirichlet energy is defined as

$$\lambda(S) = \inf_{\substack{\|\psi\|_V=1 \ \psi|_{S^c}=0}} \|\nabla\psi\|^2_{w,E},$$

which is the first eigenvalue of the graph Laplacian (parametrized by $r\in[0,1]$) subject to Dirichlet boundary conditions on $S^c$. The global partitioning problem is then posed as

$$\min_{V= \amalg_{i=1}^k V_i} \sum_{i=1}^k \lambda(V_i).$$

The manuscript develops a relaxation of this combinatorial problem, adopting a functional perspective where vertex functions $\phi_i : V \to [0,1]$ (with $\sum_i \phi_i = 1$) serve as soft cluster indicators. The relaxed energy incorporates a penalization with parameter $\alpha$ that approximates the Dirichlet constraint,

$$\lambda^\alpha(\phi) = \inf_{\|\psi\| = 1}\|\nabla\psi\|^2_{w,E} + \alpha \|\psi\|^2_{1-\phi}.$$

For large $\alpha$, the formulation is mathematically shown to recover the original Dirichlet eigenvalue on hard partitions.

The key theoretical result is that every (local) minimizer of the relaxed energy over the assignment simplex is a collection of indicator functions, ensuring consistency between the relaxed and combinatorial formulations.

Algorithmic Contributions

The authors propose a rearrangement algorithm that iteratively updates partition indicator functions. At each step, for each candidate cluster, the ground state of the corresponding Schrödinger operator is computed; vertices are then reassigned to the cluster whose ground-state eigenvector is maximized at that vertex. This procedure provably decreases the objective in each step and converges to a local minimum in a finite number of iterations, as established by rigorous analysis.

This approach differs fundamentally from gradient-based or convex relaxation methods and shares conceptual kinship with rearrangement and thresholding techniques utilized in variational calculus and curvature-motion algorithms. Notably, the method provides natural confidence values for label assignments via the values of the ground state eigenvectors.

The framework is robustly extensible: a semi-supervised variant is supported by fixing label assignments for a subset of nodes, maintaining convergence guarantees.

Relation to Existing Methods

The proposed objective generalizes previous spectral objectives such as Cheeger cuts and is naturally related to normalized cuts, but with a crucial distinction: Dirichlet eigenvalues account for both boundary conductance and connectivity within clusters. The relationship between the Dirichlet energy and local Cheeger constants is quantitatively established; thus, the method subsumes and generalizes perimeter-focused objectives.

A key observation is the equivalence, for regular graphs and the normalized Laplacian case, between the relaxed Dirichlet partition objective and a certain nonnegative matrix factorization (NMF) problem. This connection positions the method within the context of spectral clustering and NMF-based clustering approaches, but with optimization carried out over piecewise Dirichlet eigenvectors rather than algebraic factorizations.

Empirical Evaluation

The rearrangement algorithm is benchmarked on diverse tasks:

• Clustering Synthetic Data: Unlike traditional spectral clustering, the method accurately detects nonconvex clusters (e.g., "moons" datasets), exhibiting strong impurity reductions over normalized cuts for the same affinity graphs.
• Standard Small Datasets: Across 12 datasets, the method achieves purity values competitive with—often superior to—those reported by state-of-the-art clustering algorithms, finding lower-energy partitions than ground truth labels, highlighting the strength of the objective in unsupervised contexts.
• MNIST Digits: On the MNIST dataset with semi-supervision (3% labeled points), the algorithm yields purity near 0.96, with the confidence structure readily facilitating representative selection for each cluster as the maximal eigenvector locations. The confusion matrix indicates high fidelity with isolated confusions (notably between digits 6 and 8).
• Manifold Discretizations: Partitions on discretized tori and spheres reveal that the method is capable of generating geometrically meaningful and symmetric segmentations, recovering structures like the Y-partition for the sphere and hexagonal tiling for the torus, concordant with continuum geometric conjectures.

For all cases, the algorithm converges in modest iterations and the primary computational cost is dominated by repeated ground-state eigenvalue computations—a cost that is manageable for graphs of moderate size and potentially addressable using fast eigensolvers or parallelization.

Parameter Selection and Practical Guidance

The parameter $\alpha$ must be tuned to enforce an appropriate trade-off between localization and diffusivity of the eigenvectors: too large $\alpha$ instantaneously localizes the eigenfunctions and impedes learning, while too small values underconstrain cluster supports. The heuristic adopted is to choose $\alpha \approx k\lambda_2$ (the product of the number of clusters and the Fiedler value of the Laplacian), with empirical support provided. The choice of Laplacian normalization ($r=0$ vs $r=1$) is also application-dependent, affecting volume or cardinality balancing.

Theoretical and Practical Implications

The introduction of the sum of Dirichlet eigenvalues as a partition energy enriches the toolkit for graph-based clustering with a notion sensitive to both cluster connectivity and interior structure. The strict minimization of this objective enforces clusters that are not merely weakly separated but also internally coherent, a desideratum not uniformly satisfied by cut-based objectives.

From a theoretical perspective, the results unify graph partitioning, spectral clustering, geometric PDE methods, and NMF clustering within a mathematically principled framework. The method's extension to graphs constructed from manifold discretizations paves the way to study convergence to continuum partitioning problems and the spectral geometry of optimal partitions.

On the practical side, the method produces not only high-purity label assignments but also natural prototypes/representatives via the eigenvector maxima. The semi-supervised extension is straightforward and practically useful for applications with partial labeling.

Future Directions

Open problems include establishing convergence of the discrete partitions (as the graph density increases) to their continuum analogues, devising improved eigensolvers to accelerate large-scale application, and exploring extensions to other boundary conditions (e.g., Neumann). The presented rearrangement algorithm could potentially be adapted to other NMF objectives or spectral functionals.

Conclusion

This research advances the field of unsupervised and semi-supervised clustering with a principled, variational approach founded on the spectral geometry of graphs. By focusing on the minimization of the sum of Dirichlet eigenvalues and developing an efficient, globally convergent rearrangement algorithm, it combines insights from spectral theory, variational analysis, and cluster analysis, offering both a theoretically coherent and empirically robust alternative to perimeter-based partitioning and classic spectral clustering methods. The algorithm's ability to produce partition representatives with confidence scores, its performance across a variety of datasets, and its geometric adaptability highlight its utility and invite further investigation into both its theoretical properties and practical capabilities.