Problem
Research questions and friction points this paper is trying to address.
Hypergraphs
Spectral Techniques
Cheeger's Inequality
Non-uniform Hypergraphs
Conductance
A Cheeger Inequality for Hypergraphs and Its Applications
🤖 AI Summary
本文通过构建超图的谱框架和确立Cheeger不等式，解决了非均匀超图中谱技术开发难题，并创建了一类新的最优超图扩展器。
📝 Abstract
Hypergraphs provide a natural framework for modeling higher-order relationships, but the development of spectral techniques with provable guarantees for general non-uniform hypergraphs remains challenging. Building on Banerjee's normalized adjacency matrix and Spiro's averaging-based diffusion framework, we develop a spectral framework for non-uniform hypergraphs and establish Cheeger's inequality for their conductance. A fundamental result in the spectral theory of hypergraphs asserts that, for every non-covering hypergraph, the second-smallest eigenvalue of its normalized Laplacian is at most one. This spectral characterization yields an improved Cheeger's inequality for non-covering hypergraphs, and we show that the resulting inequality is tight on both sides using cycle and cube hypergraphs. Our framework further yields higher-order Cheeger inequalities and provides theoretical guarantees for Fiedler's spectral partitioning algorithm, all in the setting of hypergraphs. Finally and most notably, we construct a new family of optimal hypergraph expanders that is tight for the Alon--Boppana bound.
Innovation
Methods, ideas, or system contributions that make the work stand out.
spectral framework
Cheeger's inequality
non-uniform hypergraphs
optimal hypergraph expanders
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