3 research outputs found
Quasi-universality of Reeb graph distances
We establish bi-Lipschitz bounds certifying quasi-universality (universality
up to a constant factor) for various distances between Reeb graphs: the
interleaving distance, the functional distortion distance, and the functional
contortion distance. The definition of the latter distance is a novel
contribution, and for the special case of contour trees we also prove strict
universality of this distance. Furthermore, we prove that for the special case
of merge trees the functional contortion distance coincides with the
interleaving distance, yielding universality of all four distances in this
case.Comment: 17 pages + 6 pages appendix, 5 figures; this version includes the
appendix to the conference paper for SoCG 2022 with the same content
otherwis
Maximum entropy methods for texture synthesis: theory and practice
Recent years have seen the rise of convolutional neural network techniques in
exemplar-based image synthesis. These methods often rely on the minimization of
some variational formulation on the image space for which the minimizers are
assumed to be the solutions of the synthesis problem. In this paper we
investigate, both theoretically and experimentally, another framework to deal
with this problem using an alternate sampling/minimization scheme. First, we
use results from information geometry to assess that our method yields a
probability measure which has maximum entropy under some constraints in
expectation. Then, we turn to the analysis of our method and we show, using
recent results from the Markov chain literature, that its error can be
explicitly bounded with constants which depend polynomially in the dimension
even in the non-convex setting. This includes the case where the constraints
are defined via a differentiable neural network. Finally, we present an
extensive experimental study of the model, including a comparison with
state-of-the-art methods and an extension to style transfer