Related papers: Cheeger $N$-clusters

We study partition problems based on two ostensibly different kinds of energy functionals defined on $k$-partitions of metric graphs: Cheeger-type functionals whose minimisers are the $k$-Cheeger cuts of the graph, and the corresponding…

For any subgraph of a graph, the Laplacian with Neumann boundary condition was introduced by Chung and Yau [CY94]. In this paper, motivated by the Riemannian case, we introduce the Cheeger constants for Neumann problems and prove…

This paper establishes the consistency of a family of graph-cut-based algorithms for clustering of data clouds. We consider point clouds obtained as samples of a ground-truth measure. We investigate approaches to clustering based on…

Spectral clustering is one of the most popular clustering algorithms that has stood the test of time. It is simple to describe, can be implemented using standard linear algebra, and often finds better clusters than traditional clustering…

In this paper we prove a new extremal property of the Reuleaux triangle: it maximizes the Cheeger constant among all bodies of (same) constant width. The proof relies on a fine analysis of the optimality conditions satisfied by an optimal…

In this paper, we introduce Cheeger type constants via isocapacitary constants introduced by Maz'ya to estimate first Dirichlet, Neumann and Steklov eigenvalues on a finite subgraph of a graph. Moreover, we estimate the bottom of the…

We prove that the planar hexagonal honeycomb is asymptotically optimal for a large class of optimal partition problems, in which the cells are assumed to be convex, and the criterion is to minimize either the sum or the maximum among the…

The coboundary expansion generalizes the classical graph expansion to the case of the general simplicial complexes, and allows the definition of the higher-dimensional Cheeger constants $h_k(X)$ for an arbitrary simplicial complex $X$, and…

We show a sufficient criterion to determine if a planar set $\Omega$ is a minimizer of the prescribed curvature functional among all of its subsets. As a special case, we derive a sufficient criterion to determine if $\Omega$ is a…

We study planar $N$-clusters that minimize, under an area constraint, a weighted perimeter $P_\varepsilon$ depending on a small parameter $\varepsilon>0$. Specifically we weight $2-\varepsilon$ the boundary between the interior chambers and…

We construct two minimal Cheeger sets in the Euclidean plane, i.e. unique minimizers of the ratio "perimeter over area" among their own measurable subsets. The first one gives a counterexample to the so-called weak regularity property of…

We study a rather broad class of optimal partition problems with respect to monotone and coercive functional costs that involve the Dirichlet eigenvalues of the partitions. We show a sharp regularity result for the entire set of minimizers…

We prove a lower bound for the $k$-th Steklov eigenvalues in terms of an isoperimetric constant called the $k$-th Cheeger-Steklov constant in three different situations: finite spaces, measurable spaces, and Riemannian manifolds. These…

Cheeger-type inequalities in which the decomposability of a graph and the spectral gap of its Laplacian mutually control each other play an important role in graph theory and network analysis, in particular in the context of expander…

In this paper, we show how sparse or isoperimetric cuts of a probability density function relate to Cheeger cuts of its principal eigenfunction, for appropriate definitions of `sparse cut' and `principal eigenfunction'. We construct these…

We prove a uniform version of the Tits alternative. As a consequence, we obtain uniform lower bounds for the Cheeger constant of Cayley grahs of finitely generated non virtually solvable linear groups in arbitrary characteristic. Also we…

We initiate the study of the following general clustering problem. We seek to partition a given set $P$ of data points into $k$ clusters by finding a set $X$ of $k$ centers and assigning each data point to one of the centers. The cost of a…

This paper investigates links between the eigenvalues and eigenfunctions of the Laplace-Beltrami operator, and the higher Cheeger constants of smooth Riemannian manifolds, possibly weighted and/or with boundary. The higher Cheeger constants…

In this paper, we study the Steklov eigenvalue of a Riemannian manifold (M, g) with smooth boundary. For compact M , we establish a Cheeger-type inequality for the first Steklov eigenvalue by the isocapacitary constant. For non-compact M ,…

We consider non-negative $\sigma$-finite measure spaces coupled with a proper functional $P$ that plays the role of a perimeter. We introduce the Cheeger problem in this framework and extend many classical results on the Cheeger constant…