# Solving Partial Assignment Problems using Random Clique Complexes

## TL;DR

This paper introduces a novel approach to partial assignment problems by leveraging random clique complexes, which capture higher-order structures, and demonstrates superior accuracy and robustness over existing methods through theoretical analysis and experiments.

## Contribution

It proposes a new formulation of partial assignment problems using random clique complexes and provides theoretical and empirical validation of its effectiveness.

## Key findings

- Outperforms existing matching algorithms significantly.
- Effective on datasets with severe occlusions and distortions.
- Provides theoretical analysis of runtime and asymptotic behavior.

## Abstract

We present an alternate formulation of the partial assignment problem as matching random clique complexes, that are higher-order analogues of random graphs, designed to provide a set of invariants that better detect higher-order structure. The proposed method creates random clique adjacency matrices for each k-skeleton of the random clique complexes and matches them, taking into account each point as the affine combination of its geometric neighbourhood. We justify our solution theoretically, by analyzing the runtime and storage complexity of our algorithm along with the asymptotic behaviour of the quadratic assignment problem (QAP) that is associated with the underlying random clique adjacency matrices. Experiments on both synthetic and real-world datasets, containing severe occlusions and distortions, provide insight into the accuracy, efficiency, and robustness of our approach. We outperform diverse matching algorithms by a significant margin.