61,495 research outputs found
Worst-case Optimal Submodular Extensions for Marginal Estimation
Submodular extensions of an energy function can be used to efficiently
compute approximate marginals via variational inference. The accuracy of the
marginals depends crucially on the quality of the submodular extension. To
identify the best possible extension, we show an equivalence between the
submodular extensions of the energy and the objective functions of linear
programming (LP) relaxations for the corresponding MAP estimation problem. This
allows us to (i) establish the worst-case optimality of the submodular
extension for Potts model used in the literature; (ii) identify the worst-case
optimal submodular extension for the more general class of metric labeling; and
(iii) efficiently compute the marginals for the widely used dense CRF model
with the help of a recently proposed Gaussian filtering method. Using synthetic
and real data, we show that our approach provides comparable upper bounds on
the log-partition function to those obtained using tree-reweighted message
passing (TRW) in cases where the latter is computationally feasible.
Importantly, unlike TRW, our approach provides the first practical algorithm to
compute an upper bound on the dense CRF model.Comment: Accepted to AISTATS 201
Dynamical Localization in Quasi-Periodic Driven Systems
We investigate how the time dependence of the Hamiltonian determines the
occurrence of Dynamical Localization (DL) in driven quantum systems with two
incommensurate frequencies. If both frequencies are associated to impulsive
terms, DL is permanently destroyed. In this case, we show that the evolution is
similar to a decoherent case. On the other hand, if both frequencies are
associated to smooth driving functions, DL persists although on a time scale
longer than in the periodic case. When the driving function consists of a
series of pulses of duration , we show that the localization time
increases as as the impulsive limit, , is
approached. In the intermediate case, in which only one of the frequencies is
associated to an impulsive term in the Hamiltonian, a transition from a
localized to a delocalized dynamics takes place at a certain critical value of
the strength parameter. We provide an estimate for this critical value, based
on analytical considerations. We show how, in all cases, the frequency spectrum
of the dynamical response can be used to understand the global features of the
motion. All results are numerically checked.Comment: 7 pages, 5 figures included. In this version is that Subsection III.B
and Appendix A on the quasiperiodic Fermi Accelerator has been replaced by a
reference to published wor
Localized Manifold Harmonics for Spectral Shape Analysis
The use of Laplacian eigenfunctions is ubiquitous in a wide range of computer graphics and geometry processing applications. In particular, Laplacian eigenbases allow generalizing the classical Fourier analysis to manifolds. A key drawback of such bases is their inherently global nature, as the Laplacian eigenfunctions carry geometric and topological structure of the entire manifold. In this paper, we introduce a new framework for local spectral shape analysis. We show how to efficiently construct localized orthogonal bases by solving an optimization problem that in turn can be posed as the eigendecomposition of a new operator obtained by a modification of the standard Laplacian. We study the theoretical and computational aspects of the proposed framework and showcase our new construction on the classical problems of shape approximation and correspondence. We obtain significant improvement compared to classical Laplacian eigenbases as well as other alternatives for constructing localized bases
Functional Maps Representation on Product Manifolds
We consider the tasks of representing, analyzing and manipulating maps
between shapes. We model maps as densities over the product manifold of the
input shapes; these densities can be treated as scalar functions and therefore
are manipulable using the language of signal processing on manifolds. Being a
manifold itself, the product space endows the set of maps with a geometry of
its own, which we exploit to define map operations in the spectral domain; we
also derive relationships with other existing representations (soft maps and
functional maps). To apply these ideas in practice, we discretize product
manifolds and their Laplace--Beltrami operators, and we introduce localized
spectral analysis of the product manifold as a novel tool for map processing.
Our framework applies to maps defined between and across 2D and 3D shapes
without requiring special adjustment, and it can be implemented efficiently
with simple operations on sparse matrices.Comment: Accepted to Computer Graphics Foru
Multi-Modal Mean-Fields via Cardinality-Based Clamping
Mean Field inference is central to statistical physics. It has attracted much
interest in the Computer Vision community to efficiently solve problems
expressible in terms of large Conditional Random Fields. However, since it
models the posterior probability distribution as a product of marginal
probabilities, it may fail to properly account for important dependencies
between variables. We therefore replace the fully factorized distribution of
Mean Field by a weighted mixture of such distributions, that similarly
minimizes the KL-Divergence to the true posterior. By introducing two new
ideas, namely, conditioning on groups of variables instead of single ones and
using a parameter of the conditional random field potentials, that we identify
to the temperature in the sense of statistical physics to select such groups,
we can perform this minimization efficiently. Our extension of the clamping
method proposed in previous works allows us to both produce a more descriptive
approximation of the true posterior and, inspired by the diverse MAP paradigms,
fit a mixture of Mean Field approximations. We demonstrate that this positively
impacts real-world algorithms that initially relied on mean fields.Comment: Submitted for review to CVPR 201
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