Higher-order Projected Power Iterations for Scalable Multi-Matching

The matching of multiple objects (e.g. shapes or images) is a fundamental problem in vision and graphics. In order to robustly handle ambiguities, noise and repetitive patterns in challenging real-world settings, it is essential to take geometric consistency between points into account. Computationally, the multi-matching problem is difficult. It can be phrased as simultaneously solving multiple (NP-hard) quadratic assignment problems (QAPs) that are coupled via cycle-consistency constraints. The main limitations of existing multi-matching methods are that they either ignore geometric consistency and thus have limited robustness, or they are restricted to small-scale problems due to their (relatively) high computational cost. We address these shortcomings by introducing a Higher-order Projected Power Iteration method, which is (i) efficient and scales to tens of thousands of points, (ii) straightforward to implement, (iii) able to incorporate geometric consistency, (iv) guarantees cycle-consistent multi-matchings, and (iv) comes with theoretical convergence guarantees. Experimentally we show that our approach is superior to existing methods

Adaptively Transforming Graph Matching

Recently, many graph matching methods that incorporate pairwise constraint and that can be formulated as a quadratic assignment problem (QAP) have been proposed. Although these methods demonstrate promising results for the graph matching problem, they have high complexity in space or time. In this paper, we introduce an adaptively transforming graph matching (ATGM) method from the perspective of functional representation. More precisely, under a transformation formulation, we aim to match two graphs by minimizing the discrepancy between the original graph and the transformed graph. With a linear representation map of the transformation, the pairwise edge attributes of graphs are explicitly represented by unary node attributes, which enables us to reduce the space and time complexity significantly. Due to an efficient Frank-Wolfe method-based optimization strategy, we can handle graphs with hundreds and thousands of nodes within an acceptable amount of time. Meanwhile, because transformation map can preserve graph structures, a domain adaptation-based strategy is proposed to remove the outliers. The experimental results demonstrate that our proposed method outperforms the state-of-the-art graph matching algorithms

A Functional Representation for Graph Matching

Graph matching is an important and persistent problem in computer vision and pattern recognition for finding node-to-node correspondence between graph-structured data. However, as widely used, graph matching that incorporates pairwise constraints can be formulated as a quadratic assignment problem (QAP), which is NP-complete and results in intrinsic computational difficulties. In this paper, we present a functional representation for graph matching (FRGM) that aims to provide more geometric insights on the problem and reduce the space and time complexities of corresponding algorithms. To achieve these goals, we represent a graph endowed with edge attributes by a linear function space equipped with a functional such as inner product or metric, that has an explicit geometric meaning. Consequently, the correspondence between graphs can be represented as a linear representation map of that functional. Specifically, we reformulate the linear functional representation map as a new parameterization for Euclidean graph matching, which is associative with geometric parameters for graphs under rigid or nonrigid deformations. This allows us to estimate the correspondence and geometric deformations simultaneously. The use of the representation of edge attributes rather than the affinity matrix enables us to reduce the space complexity by two orders of magnitudes. Furthermore, we propose an efficient optimization strategy with low time complexity to optimize the objective function. The experimental results on both synthetic and real-world datasets demonstrate that the proposed FRGM can achieve state-of-the-art performance