  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.
