# Computing the $p$-Spectral Radii of Uniform Hypergraphs with Applications

## TL;DR

This paper introduces a novel conjugate gradient algorithm, CSRH, for efficiently computing the p-spectral radius of uniform hypergraphs, with applications in large-scale data analysis and hypergraph optimization.

## Contribution

The paper develops a globally convergent algorithm, CSRH, for calculating the p-spectral radius of hypergraphs, outperforming existing methods especially on large-scale problems.

## Key findings

- CSRH efficiently computes p-spectral radii for hypergraphs with millions of vertices.
- CSRH reliably finds the global maximizer due to the semialgebraic structure.
- The method effectively ranks real-world datasets based on hypergraph spectral properties.

## Abstract

The $p$-spectral radius of a uniform hypergraph covers many important concepts, such as Lagrangian and spectral radius of the hypergraph, and is crucial for solving spectral extremal problems of hypergraphs. In this paper, we establish a spherically constrained maximization model and propose a first-order conjugate gradient algorithm to compute the $p$-spectral radius of a uniform hypergraph (CSRH). By the semialgebraic nature of the adjacency tensor of a uniform hypergraph, CSRH is globally convergent and obtains the global maximizer with a high probability. When computing the spectral radius of the adjacency tensor of a uniform hypergraph, CSRH stands out among existing approaches. Furthermore, CSRH is competent to calculate the $p$-spectral radius of a hypergraph with millions of vertices and to approximate the Lagrangian of a hypergraph. Finally, we show that the CSRH method is capable of ranking real-world data set based on solutions generated by the $p$-spectral radius model.