  Cycle consistency has long been exploited as a powerful prior for jointly
optimizing maps within a collection of shapes. In this paper, we investigate
its utility in the approaches of Deep Functional Maps, which are considered
state-of-the-art in non-rigid shape matching. We first justify that under
certain conditions, the learned maps, when represented in the spectral domain,
are already cycle consistent. Furthermore, we identify the discrepancy that
spectrally consistent maps are not necessarily spatially, or point-wise,
consistent. In light of this, we present a novel design of unsupervised Deep
Functional Maps, which effectively enforces the harmony of learned maps under
the spectral and the point-wise representation. By taking advantage of cycle
consistency, our framework produces state-of-the-art results in mapping shapes
even under significant distortions. Beyond that, by independently estimating
maps in both spectral and spatial domains, our method naturally alleviates
over-fitting in network training, yielding superior generalization performance
and accuracy within an array of challenging tests for both near-isometric and
non-isometric datasets. Codes are available at
https://github.com/rqhuang88/Spatiallyand-Spectrally-Consistent-Deep-Functional-Maps.
