From graph cuts to isoperimetric inequalities: Convergence rates of Cheeger cuts on data clouds

增强机器人定位网络鲁棒性的谱图理论方法

Spectral Graph Theoretic Methods for Enhancing Network Robustness in Robot Localization

本文通过最大化代数连通性增强机器人定位网络鲁棒性，将问题表述为NP难的MISDP，并利用谱图理论引入Cheeger割结合MILP外逼近算法与贪婪k-opt启发式高效求解，显著提升了计算效率。

Higher order Cheeger inequalities for Steklov eigenvalues

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 lower bounds can be considered as higher order Cheeger type inequalities for the Steklov eigenvalues. In particular it extends the Cheeger type inequality for the first nonzero Steklov eigenvalue previously studied by Escobar in 1997 and by Jammes in 2015 to higher order Steklov eigenvalues. The technique we develop to get this lower bound is based on considerin…

Isoperimetric Inequalities in Simplicial Complexes

In graph theory there are intimate connections between the expansion properties of a graph and the spectrum of its Laplacian. In this paper we define a notion of combinatorial expansion for simplicial complexes of general dimension, and prove that similar connections exist between the combinatorial expansion of a complex, and the spectrum of the high dimensional Laplacian defined by Eckmann. In particular, we present a Cheeger-type inequality, and a high-dimensional Expander Mixing Lemma. As a corollary, using the work of Pach, we obtain a connection between spectral properties of complexes an…

$p$-Norm Flow Diffusion for Local Graph Clustering

Local graph clustering and the closely related seed set expansion problem are primitives on graphs that are central to a wide range of analytic and learning tasks such as local clustering, community detection, nodes ranking and feature inference. Prior work on local graph clustering mostly falls into two categories with numerical and combinatorial roots respectively. In this work, we draw inspiration from both fields and propose a family of convex optimization formulations based on the idea of diffusion with p-norm network flow for $p\in (1,\infty)$. In the context of local clustering, we char…

Hamiltonian surgery: Cheeger-type gap inequalities for nonpositive (stoquastic), real, and Hermitian matrices

Cheeger inequalities bound the spectral gap $γ$ of a space by isoperimetric properties of that space and vice versa. In this paper, I derive Cheeger-type inequalities for nonpositive matrices (aka stoquastic Hamiltonians), real matrices, and Hermitian matrices. For matrices written $H = L+W$, where $L$ is either a combinatorial or normalized graph Laplacian, I show that: (1) when $W$ is diagonal and $L$ has maximum degree $d_{\max}$, $2h \geq γ\geq \sqrt{h^2 + d_{\max}^2}-d_\max$; (2) when $W$ is real, we can often route negative-weighted edges along positive-weighted edges such that the Cheeg…

Spectral Estimates for Infinite Quantum Graphs

We investigate the bottom of the spectra of infinite quantum graphs, i.e., Laplace operators on metric graphs having infinitely many edges and vertices. We introduce a new definition of the isoperimetric constant for quantum graphs and then prove the Cheeger-type estimate. Our definition of the isoperimetric constant is purely combinatorial and thus it establishes connections with the combinatorial isoperimetric constant, one of the central objects in spectral graph theory and in the theory of simple random walks on graphs. The latter enables us to prove a number of criteria for quantum graphs…

一个多对数阶的高阶 Cheeger 不等式

A sharp higher-order Cheeger inequality

本文证明了归一化拉普拉斯算子的第 k 个特征值与 k 个不相交集合的最大电导之间存在多对数阶的 Cheeger 不等式，并给出了构造性证明，为谱聚类提供了理论保证。

离散到连续的扩展：Lovász扩展、优化与特征值问题

Discrete-to-Continuous Extensions: Lovász extension, optimizations and eigenvalue problems

本文利用Lovász扩展系统地将离散数学问题转化为连续优化问题，涵盖最大k割、Cheeger常数、独立数、色数等，并首次给出色数与挫折指数的连续表示，建立非线性特征值联系。