  Optimizing a network of maps among a collection of objects/domains (or map
synchronization) is a central problem across computer vision and many other
relevant fields. Compared to optimizing pairwise maps in isolation, the benefit
of map synchronization is that there are natural constraints among a map
network that can improve the quality of individual maps. While such
self-supervision constraints are well-understood for undirected map networks
(e.g., the cycle-consistency constraint), they are under-explored for directed
map networks, which naturally arise when maps are given by parametric maps
(e.g., a feed-forward neural network). In this paper, we study a natural
self-supervision constraint for directed map networks called path-invariance,
which enforces that composite maps along different paths between a fixed pair
of source and target domains are identical. We introduce path-invariance bases
for efficient encoding of the path-invariance constraint and present an
algorithm that outputs a path-variance basis with polynomial time and space
complexities. We demonstrate the effectiveness of our approach on optimizing
object correspondences, estimating dense image maps via neural networks, and
semantic segmentation of 3D scenes via map networks of diverse 3D
representations. In particular, for 3D semantic segmentation, our approach only
requires 8% labeled data from ScanNet to achieve the same performance as
training a single 3D segmentation network with 30% to 100% labeled data.
