Related papers: A Cheeger Inequality for Size-Specific Conductance

In the {\em nonuniform sparsest cut} problem, given two undirected graphs $G$ and $H$ over the same set of vertices $V$, we want to find a cut $(S,V-S)$ that minimizes the ratio between the fraction of $G$-edges that are cut and the…

Graph cuts are among the most prominent tools for clustering and classification analysis. While intensively studied from geometric and algorithmic perspectives, graph cut-based statistical inference still remains elusive to a certain…

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…

The O(d) Synchronization problem consists of estimating a set of unknown orthogonal transformations O_i from noisy measurements of a subset of the pairwise ratios O_iO_j^{-1}. We formulate and prove a Cheeger-type inequality that relates a…

We investigate a general question about the size and regularity of the data and the solutions in implicit function problems with loss of regularity. First, we give a heuristic explanation of the fact that the optimal data size found by…

In this article it is shown that the equilateral triangle maximizes the Cheeger constant and minimizes the torsional rigidity among shapes having a fixed minimal width. The proof techniques use direct comparisons with simpler shapes,…

The first nontrivial lower bound of the worst-case approximation ratio for the maxcut problem was achieved via the dual Cheeger problem, whose optimal value is referred to the dual Cheeger constant $h^+$, and later improved through its…

Let $\Gamma$ be a Cayley graph, or a Cayley sum graph, or a twisted Cayley graph, or a twisted Cayley sum graph, or a vertex-transitive graph. Denote the degree of $\Gamma$ by $d$, its edge Cheeger constant by $\mathfrak{h}_\Gamma$, and its…

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 aim to learn a sparse and connected graph from sparse data, where the number of observations K can be substantially smaller than the signal dimension N for signals x in R^N, and the underlying distribution is unknown. In this severely…

We investigate the homogenization of inclusions of infinite conductivity, randomly stationary distributed inside a homogeneous conducting medium. A now classical result by Zhikov shows that, under a logarithmic moment bound on the…

Testable weighted graph parameters and equivalent notions of testability are investigated based on papers of Laszlo Lovasz and coauthors. We prove that certain balanced minimum multiway cut densities are testable. Using this fact, quadratic…

We conclude our analysis of the linear response of charge transport in lattice systems of free fermions subjected to a random potential by deriving general mathematical properties of its conductivity at the macroscopic scale. The present…

This paper addresses the behavior of the Lov\'asz number for dense random circulant graphs. The Lov\'asz number is a well-known semidefinite programming upper bound on the independence number. Circulant graphs, an example of a Cayley graph,…

We propose a discrete approach for solving an inverse problem for Schr\"odinger's equation in two dimensions, where the unknown potential is to be determined from boundary measurements of the Dirichlet to Neumann map. For absorptive…

We define a new Cheeger-like constant for graphs and we use it for proving Cheeger-like inequalities that bound the largest eigenvalue of the normalized Laplace operator.

For a sequence $\{(M_i, g_i, x_i)\}$ of pointed Riemannian manifolds with boundary, the sequence $\{(M_i,\tilde g_i,x_i)\}$ is its conformal satellite if the metric $\tilde g_i$ is conformal to $g_i$, that is, $\tilde…

Let $\theta$ and $\mu$ denote the location and the size of the mode of a probability density. We study the joint convergence rates of semirecursive kernel estimators of $\theta$ and $\mu$. We show how the estimation of the size of the mode…

We consider supervised learning problems in which set predictions provide explicit uncertainty estimates. Using Choquet integrals (a.k.a. Lov{\'a}sz extensions), we propose a convex loss function for nondecreasing subset-valued functions…

In one-dimensional disordered wires electronic states are localized at any energy. Correlations of the states at close positive energies and the AC conductivity $\sigma(\omega)$ in the limit of small frequency are described by the…