Combinatorics of bicubic maps with hard particles

We present a purely combinatorial solution of the problem of enumerating planar bicubic maps with hard particles. This is done by use of a bijection with a particular class of blossom trees with particles, obtained by an appropriate cutting of the maps. Although these trees have no simple local characterization, we prove that their enumeration may be performed upon introducing a larger class of "admissible" trees with possibly doubly-occupied edges and summing them with appropriate signed weights. The proof relies on an extension of the cutting procedure allowing for the presence on the maps of special non-sectile edges. The admissible trees are characterized by simple local rules, allowing eventually for an exact enumeration of planar bicubic maps with hard particles. We also discuss generalizations for maps with particles subject to more general exclusion rules and show how to re-derive the enumeration of quartic maps with Ising spins in the present framework of admissible trees. We finally comment on a possible interpretation in terms of branching processes.Comment: 41 pages, 19 figures, tex, lanlmac, hyperbasics, epsf. Introduction and discussion/conclusion extended, minor corrections, references adde

Deep Markov Random Field for Image Modeling

Markov Random Fields (MRFs), a formulation widely used in generative image modeling, have long been plagued by the lack of expressive power. This issue is primarily due to the fact that conventional MRFs formulations tend to use simplistic factors to capture local patterns. In this paper, we move beyond such limitations, and propose a novel MRF model that uses fully-connected neurons to express the complex interactions among pixels. Through theoretical analysis, we reveal an inherent connection between this model and recurrent neural networks, and thereon derive an approximated feed-forward network that couples multiple RNNs along opposite directions. This formulation combines the expressive power of deep neural networks and the cyclic dependency structure of MRF in a unified model, bringing the modeling capability to a new level. The feed-forward approximation also allows it to be efficiently learned from data. Experimental results on a variety of low-level vision tasks show notable improvement over state-of-the-arts.Comment: Accepted at ECCV 201

Cuts in matchings of 3-connected cubic graphs

We discuss conjectures on Hamiltonicity in cubic graphs (Tait, Barnette, Tutte), on the dichromatic number of planar oriented graphs (Neumann-Lara), and on even graphs in digraphs whose contraction is strongly connected (Hochst\"attler). We show that all of them fit into the same framework related to cuts in matchings. This allows us to find a counterexample to the conjecture of Hochst\"attler and show that the conjecture of Neumann-Lara holds for all planar graphs on at most 26 vertices. Finally, we state a new conjecture on bipartite cubic oriented graphs, that naturally arises in this setting.Comment: 12 pages, 5 figures, 1 table. Improved expositio

Consistent thermodynamic derivative estimates for tabular equations of state

Numerical simulations of compressible fluid flows require an equation of state (EOS) to relate the thermodynamic variables of density, internal energy, temperature, and pressure. A valid EOS must satisfy the thermodynamic conditions of consistency (derivation from a free energy) and stability (positive sound speed squared). When phase transitions are significant, the EOS is complicated and can only be specified in a table. For tabular EOS's such as SESAME from Los Alamos National Laboratory, the consistency and stability conditions take the form of a differential equation relating the derivatives of pressure and energy as functions of temperature and density, along with positivity constraints. Typical software interfaces to such tables based on polynomial or rational interpolants compute derivatives of pressure and energy and may enforce the stability conditions, but do not enforce the consistency condition and its derivatives. We describe a new type of table interface based on a constrained local least squares regression technique. It is applied to several SESAME EOS's showing how the consistency condition can be satisfied to round-off while computing first and second derivatives with demonstrated second-order convergence. An improvement of 14 orders of magnitude over conventional derivatives is demonstrated, although the new method is apparently two orders of magnitude slower, due to the fact that every evaluation requires solving an 11-dimensional nonlinear system.Comment: 29 pages, 9 figures, 16 references, submitted to Phys Rev

Learning Sparse High Dimensional Filters: Image Filtering, Dense CRFs and Bilateral Neural Networks

Bilateral filters have wide spread use due to their edge-preserving properties. The common use case is to manually choose a parametric filter type, usually a Gaussian filter. In this paper, we will generalize the parametrization and in particular derive a gradient descent algorithm so the filter parameters can be learned from data. This derivation allows to learn high dimensional linear filters that operate in sparsely populated feature spaces. We build on the permutohedral lattice construction for efficient filtering. The ability to learn more general forms of high-dimensional filters can be used in several diverse applications. First, we demonstrate the use in applications where single filter applications are desired for runtime reasons. Further, we show how this algorithm can be used to learn the pairwise potentials in densely connected conditional random fields and apply these to different image segmentation tasks. Finally, we introduce layers of bilateral filters in CNNs and propose bilateral neural networks for the use of high-dimensional sparse data. This view provides new ways to encode model structure into network architectures. A diverse set of experiments empirically validates the usage of general forms of filters

Solutions of the Bethe-Salpeter equation in Minkowski space and applications to electromagnetic form factors

We present a new method for solving the two-body Bethe-Salpeter equation in Minkowski space. It is based on the Nakanishi integral representation of the Bethe-Salpeter amplitude and on subsequent projection of the equation on the light-front plane. The method is valid for any kernel given by the irreducible Feynman graphs and for systems of spinless particles or fermions. The Bethe-Salpeter amplitudes in Minkowski space are obtained. The electromagnetic form factors are computed and compared to the Euclidean results.Comment: 20 pages, 14 figures, contribution to proceedings of the workshop: "Relativistic Description of Two- and Three-Body Systems in Nuclear Physics", ECT*, October 19-23, 2009. To be published in Few-Body System