Some families of increasing planar maps

Stack-triangulations appear as natural objects when one wants to define some increasing families of triangulations by successive additions of faces. We investigate the asymptotic behavior of rooted stack-triangulations with $2n$ faces under two different distributions. We show that the uniform distribution on this set of maps converges, for a topology of local convergence, to a distribution on the set of infinite maps. In the other hand, we show that rescaled by $n^{1/2}$, they converge for the Gromov-Hausdorff topology on metric spaces to the continuum random tree introduced by Aldous. Under a distribution induced by a natural random construction, the distance between random points rescaled by $(6/11)\log n$ converge to 1 in probability. We obtain similar asymptotic results for a family of increasing quadrangulations

A Sparse Stress Model

Force-directed layout methods constitute the most common approach to draw general graphs. Among them, stress minimization produces layouts of comparatively high quality but also imposes comparatively high computational demands. We propose a speed-up method based on the aggregation of terms in the objective function. It is akin to aggregate repulsion from far-away nodes during spring embedding but transfers the idea from the layout space into a preprocessing phase. An initial experimental study informs a method to select representatives, and subsequent more extensive experiments indicate that our method yields better approximations of minimum-stress layouts in less time than related methods.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

On Vertex- and Empty-Ply Proximity Drawings

We initiate the study of the vertex-ply of straight-line drawings, as a relaxation of the recently introduced ply number. Consider the disks centered at each vertex with radius equal to half the length of the longest edge incident to the vertex. The vertex-ply of a drawing is determined by the vertex covered by the maximum number of disks. The main motivation for considering this relaxation is to relate the concept of ply to proximity drawings. In fact, if we interpret the set of disks as proximity regions, a drawing with vertex-ply number 1 can be seen as a weak proximity drawing, which we call empty-ply drawing. We show non-trivial relationships between the ply number and the vertex-ply number. Then, we focus on empty-ply drawings, proving some properties and studying what classes of graphs admit such drawings. Finally, we prove a lower bound on the ply and the vertex-ply of planar drawings.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Spin network setting of topological quantum computation

The spin network simulator model represents a bridge between (generalised) circuit schemes for standard quantum computation and approaches based on notions from Topological Quantum Field Theories (TQFTs). The key tool is provided by the fiber space structure underlying the model which exhibits combinatorial properties closely related to SU(2) state sum models, widely employed in discretizing TQFTs and quantum gravity in low spacetime dimensions.Comment: Proc. "Foundations of Quantum Information", Camerino (Italy), 16-19 April 2004, to be published in Int. J. of Quantum Informatio

Hierarchies of Relaxations for Online Prediction Problems with Evolving Constraints

We study online prediction where regret of the algorithm is measured against a benchmark defined via evolving constraints. This framework captures online prediction on graphs, as well as other prediction problems with combinatorial structure. A key aspect here is that finding the optimal benchmark predictor (even in hindsight, given all the data) might be computationally hard due to the combinatorial nature of the constraints. Despite this, we provide polynomial-time \emph{prediction} algorithms that achieve low regret against combinatorial benchmark sets. We do so by building improper learning algorithms based on two ideas that work together. The first is to alleviate part of the computational burden through random playout, and the second is to employ Lasserre semidefinite hierarchies to approximate the resulting integer program. Interestingly, for our prediction algorithms, we only need to compute the values of the semidefinite programs and not the rounded solutions. However, the integrality gap for Lasserre hierarchy \emph{does} enter the generic regret bound in terms of Rademacher complexity of the benchmark set. This establishes a trade-off between the computation time and the regret bound of the algorithm

Who's Better? Who's Best? Pairwise Deep Ranking for Skill Determination

We present a method for assessing skill from video, applicable to a variety of tasks, ranging from surgery to drawing and rolling pizza dough. We formulate the problem as pairwise (who's better?) and overall (who's best?) ranking of video collections, using supervised deep ranking. We propose a novel loss function that learns discriminative features when a pair of videos exhibit variance in skill, and learns shared features when a pair of videos exhibit comparable skill levels. Results demonstrate our method is applicable across tasks, with the percentage of correctly ordered pairs of videos ranging from 70% to 83% for four datasets. We demonstrate the robustness of our approach via sensitivity analysis of its parameters. We see this work as effort toward the automated organization of how-to video collections and overall, generic skill determination in video.Comment: CVPR 201