5,271 research outputs found
Curvature Matrix Models for Dynamical Triangulations and the Itzykson-DiFrancesco Formula
We study the large-N limit of a class of matrix models for dually weighted
triangulated random surfaces using character expansion techniques. We show that
for various choices of the weights of vertices of the dynamical triangulation
the model can be solved by resumming the Itzykson-Di Francesco formula over
congruence classes of Young tableau weights modulo three. From this we show
that the large-N limit implies a non-trivial correspondence with models of
random surfaces weighted with only even coordination number vertices. We
examine the critical behaviour and evaluation of observables and discuss their
interrelationships in all models. We obtain explicit solutions of the model for
simple choices of vertex weightings and use them to show how the matrix model
reproduces features of the random surface sum. We also discuss some general
properties of the large-N character expansion approach as well as potential
physical applications of our results.Comment: 37 pages LaTeX; Some clarifying comments added, last Section
rewritte
Surface Networks
We study data-driven representations for three-dimensional triangle meshes,
which are one of the prevalent objects used to represent 3D geometry. Recent
works have developed models that exploit the intrinsic geometry of manifolds
and graphs, namely the Graph Neural Networks (GNNs) and its spectral variants,
which learn from the local metric tensor via the Laplacian operator. Despite
offering excellent sample complexity and built-in invariances, intrinsic
geometry alone is invariant to isometric deformations, making it unsuitable for
many applications. To overcome this limitation, we propose several upgrades to
GNNs to leverage extrinsic differential geometry properties of
three-dimensional surfaces, increasing its modeling power.
In particular, we propose to exploit the Dirac operator, whose spectrum
detects principal curvature directions --- this is in stark contrast with the
classical Laplace operator, which directly measures mean curvature. We coin the
resulting models \emph{Surface Networks (SN)}. We prove that these models
define shape representations that are stable to deformation and to
discretization, and we demonstrate the efficiency and versatility of SNs on two
challenging tasks: temporal prediction of mesh deformations under non-linear
dynamics and generative models using a variational autoencoder framework with
encoders/decoders given by SNs
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