Consistency of Dirichlet Partitions
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A Dirichlet k-partition of a domain U ⊆R^d is a collection of k pairwise disjoint open subsets such that the sum of their first Laplace-Dirichlet eigenvalues is minimal. A discrete version of Dirichlet partitions has been posed on graphs with applications in data analysis. Both versions admit variational formulations: solutions are characterized by minimizers of the Dirichlet energy of mappings from U into a singular space Σ_k ⊆R^k. In this paper, we extend results of N.García Trillos and D. Slepčev to show that there exist solutions of the continuum problem arising as limits to solutions of a sequence of discrete problems. Specifically, a sequence of points {x_i}_i ∈N from U is sampled i.i.d. with respect to a given probability measure ν on U and for all n ∈N, a geometric graph G_n is constructed from the first n points x_1, x_2, ..., x_n and the pairwise distances between the points. With probability one with respect to the choice of points {x_i}_i ∈N, we show that as n →∞ the discrete Dirichlet energies for functions G_n →Σ_k Γ-converge to (a scalar multiple of) the continuum Dirichlet energy for functions U →Σ_k with respect to a metric coming from the theory of optimal transport. This, along with a compactness property for the aforementioned energies that we prove, implies the convergence of minimizers. When ν is the uniform distribution, our results also imply the statistical consistency statement that Dirichlet partitions of geometric graphs converge to partitions of the sampled space in the Hausdorff sense.
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