  This paper proposes a learning-based framework for reconstructing 3D shapes
from functional operators, compactly encoded as small-sized matrices. To this
end we introduce a novel neural architecture, called OperatorNet, which takes
as input a set of linear operators representing a shape and produces its 3D
embedding. We demonstrate that this approach significantly outperforms previous
purely geometric methods for the same problem. Furthermore, we introduce a
novel functional operator, which encodes the extrinsic or pose-dependent shape
information, and thus complements purely intrinsic pose-oblivious operators,
such as the classical Laplacian. Coupled with this novel operator, our
reconstruction network achieves very high reconstruction accuracy, even in the
presence of incomplete information about a shape, given a soft or functional
map expressed in a reduced basis. Finally, we demonstrate that the
multiplicative functional algebra enjoyed by these operators can be used to
synthesize entirely new unseen shapes, in the context of shape interpolation
and shape analogy applications.
