# A Quadratic Penalty Method for Hypergraph Matching

## TL;DR

This paper introduces a quadratic penalty method for hypergraph matching, reformulating it as a sparse tensor optimization problem and demonstrating effective support set recovery with theoretical guarantees and efficient algorithms.

## Contribution

It presents a novel quadratic penalty approach for hypergraph matching, including theoretical analysis and practical algorithms for support set identification.

## Key findings

- Supports exact support set recovery under certain conditions
- The proposed method is efficient in accuracy and speed
- Numerical results validate the effectiveness of the approach

## Abstract

Hypergraph matching is a fundamental problem in computer vision. Mathematically speaking, it maximizes a polynomial objective function, subject to assignment constraints. In this paper, we reformulate the hypergraph matching problem as a sparse constrained tensor optimization problem. The optimality conditions are characterized based on the sparse constrained optimization theory. By dropping the sparsity constraint, we show that the resulting relaxation problem can recover the global minimizer of the original problem. The critical step in solving the original problem is to identify the location of nonzero entries (referred as support set) in a global minimizer. Inspired by such observations, we penalize the equality constraints and apply the quadratic penalty method to solve the relaxation problem. Under reasonable assumptions, we show that the support set of the global minimizer in hypergraph matching problem can be correctly identified when the number of iterations is sufficiently large. A projected gradient method is applied as a subsolver to solve the quadratic penalty subproblem. Numerical results demonstrate that the exact recovery of support set indeed happens, and the proposed algorithms are efficient in terms of both accuracy and speed.