  This paper focuses on the statistical analysis of shapes of data objects
called shape graphs, a set of nodes connected by articulated curves with
arbitrary shapes. A critical need here is a constrained registration of points
(nodes to nodes, edges to edges) across objects. This, in turn, requires
optimization over the permutation group, made challenging by differences in
nodes (in terms of numbers, locations) and edges (in terms of shapes,
placements, and sizes) across objects. This paper tackles this registration
problem using a novel neural-network architecture and involves an unsupervised
loss function developed using the elastic shape metric for curves. This
architecture results in (1) state-of-the-art matching performance and (2) an
order of magnitude reduction in the computational cost relative to baseline
approaches. We demonstrate the effectiveness of the proposed approach using
both simulated data and real-world 2D and 3D shape graphs. Code and data will
be made publicly available after review to foster research.
