Minimal Dirichlet energy partitions for graphs

Braxton Osting (1) , Chris D. White (2) , Edouard Oudet (3)

1 UCLA - University of California [Los Angeles]
2 University of Texas at Austin [Austin]
3 EDP - Equations aux Dérivées Partielles

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.

Mots clés: clustering; graph partition; graph Laplacian; Dirichlet eigenvalues; rearrangement algorithm

Domaines: Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]

Braxton Osting, Chris D. White, Edouard Oudet. Minimal Dirichlet energy partitions for graphs. SIAM Journal on Scientific Computing, 2014, 36 (4), pp.A1635-A1651. ⟨10.1137/130934568⟩. ⟨hal-00978056⟩