Abstract: Establishing point correspondences between shapes is extremely challenging as it involves both finding sets of semantically persistent feature points, as well as their combinatorial matching. We focus on the latter and consider the Quadratic Assignment Matching (QAM) model. We suggest a novel convex relaxation for this NP-hard problem that builds upon a rank-one reformulation of the problem in a higher dimension, followed by relaxation into a semidefinite program (SDP). Our method is shown to be a certain hybrid of the popular spectral and doubly-stochastic relaxations of QAM and in particular we prove that it is tighter than both.Experimental evaluation shows that the proposed relaxation is extremely tight: in the majority of our experiments it achieved the certified global optimum solution for the problem, while other relaxations tend to produce sub-optimal solutions. This, however, comes at the price of solving an SDP in a higher dimension. Our approach is further generalized to the problem of Consistent Collection Matching (CCM), where we solve the QAM on a collection of shapes while simultaneously incorporating a global consistency constraint. Lastly, we demonstrate an application to metric learning of collections of shapes.