CURSOR: Scalable Mixed-Order Hypergraph Matching with CUR Decomposition

TL;DR

CURSOR addresses the scalability bottleneck in high-order hypergraph matching by leveraging CUR decomposition to form a sparse, low-rank approximation of the compatibility tensor. It cascades a CUR-based second-order matching to obtain a rough alignment and then builds a third-order tensor via fiber-CUR guided by that result, followed by a PRL-based refinement for fast convergence. The method substantially reduces memory and computation while improving or preserving matching accuracy across large-scale synthetic and real datasets, and it can be integrated with existing hypergraph matching algorithms. This work enables practical large-scale hypergraph matching in big-data contexts and offers a flexible framework for combining low-rank tensor approximations with probabilistic refinement.

Abstract

To achieve greater accuracy, hypergraph matching algorithms require exponential increases in computational resources. Recent kd-tree-based approximate nearest neighbor (ANN) methods, despite the sparsity of their compatibility tensor, still require exhaustive calculations for large-scale graph matching. This work utilizes CUR tensor decomposition and introduces a novel cascaded second and third-order hypergraph matching framework (CURSOR) for efficient hypergraph matching. A CUR-based second-order graph matching algorithm is used to provide a rough match, and then the core of CURSOR, a fiber-CUR-based tensor generation method, directly calculates entries of the compatibility tensor by leveraging the initial second-order match result. This significantly decreases the time complexity and tensor density. A probability relaxation labeling (PRL)-based matching algorithm, especially suitable for sparse tensors, is developed. Experiment results on large-scale synthetic datasets and widely-adopted benchmark sets demonstrate the superiority of CURSOR over existing methods. The tensor generation method in CURSOR can be integrated seamlessly into existing hypergraph matching methods to improve their performance and lower their computational costs.

Figures (11)

• Figure 1: The comparison between the traditional ANN-based framework and CURSOR. Instead of calculating the whole tensor block (the light orange area in \ref{['subfig:traditional']}) and extracting the highest compatibilities in each block (the blue cubes), CURSOR only calculates a small number of block fibers (the light orange area in \ref{['subfig:curann']}) and retains fewer elements in these fibers, effectively reducing computational costs for large-scale hypergraph matching. The method chooses the fibers based on the second-order matching result. CURSOR calculates fibers in all three tensor modes, and only one is shown in \ref{['subfig:curann']} for clarity.
• Figure 2: Matching result with deformation \ref{['rotation_acc']} rotation with angle $[-30^\circ, 30^\circ]$, \ref{['scale_acc']} scale on $x$-coordinate with scale factor $1.1^\beta$ where $\beta\in[-5,5]$, \ref{['noise_acc']} adding noise with $\sigma/\sigma_0=[0,0.1]$, and \ref{['outlier_acc']} adding $n_l$ outliers where the outlier ratio$=n_l/n_1$.
• Figure 3: Comparison results on the House and Hotel dataset with various matching algorithms. The dashed curves represent the matching results on the compatibility tensors using ANN. The solid curves with the same color denote the matching accuracy on tensors generated by CURSOR with the same hypergraph matching algorithms.
• Figure 4: The Cars and Motorbikes datasets with CURSOR and state-of-the-art hypergraph matching algorithms. \ref{['subfig:nnz']} The number of non-zero compatibilities with ANN-based methods and CURSOR. The matching accuracy on the \ref{['subfig:motor']} Motorbikes and \ref{['subfig:car']} Cars datasets.
• Figure 5: Car and Motorbike matching examples. Top row Car dataset, bottom row Motorbike dataset. Each example shows the matched results with the highest accuracy among trials. The green and red lines denote matches and mismatches, respectively.