Shift-Coupling of Random Rooted Graphs and Networks

In this paper, we present a result similar to the shift-coupling result of Thorisson (1996) in the context of random graphs and networks. The result is that a given random rooted network can be obtained by changing the root of another given one if and only if the distributions of the two agree on the invariant sigma-field. Several applications of the result are presented for the case of unimodular networks. In particular, it is shown that the distribution of a unimodular network is uniquely determined by its restriction to the invariant sigma-filed. Also, the theorem is applied to the existence of an invariant transport kernel that balances between two given (discrete) measures on the vertices. An application is the existence of a so called extra head scheme for the Bernoulli process on an infinite unimodular graph. Moreover, a construction is presented for balancing transport kernels that is a generalization of the Gale-Shapley stable matching algorithm in bipartite graphs. Another application is on a general method that covers the situations where some vertices and edges are added to a unimodular network and then, to make it unimodular, the probability measure is biased and then a new root is selected. It is proved that this method provides all possible unimodularizations in these situations. Finally, analogous existing results for stationary point processes and unimodular networks are discussed in detail.Comment: 34 page

A Unified Framework for Generalizing the Gromov-Hausdorff Metric

In this paper, a general approach is presented for generalizing the Gromov-Hausdorff metric to consider metric spaces equipped with some additional structure. A special case is the Gromov-Hausdorff-Prokhorov metric which considers measured metric spaces. This abstract framework also unifies several existing generalizations which consider metric spaces equipped with a measure, a point, a closed subset, a curve or a tuple of such structures. It can also be useful for studying new examples of additional structures. The framework is provided both for compact metric spaces and for boundedly-compact pointed metric spaces. In addition, completeness and separability of the metric is proved under some conditions. This enables one to study random metric spaces equipped with additional structures, which is the main motivation of this work.Comment: 40 pages. The previous version of the paper is now split into two papers: The current one and `Metrization of the Gromov-Hausdorff (-Prokhorov) Topology for Boundedly-Compact Metric Spaces

Outer Bounds on the Admissible Source Region for Broadcast Channels with Correlated Sources

Two outer bounds on the admissible source region for broadcast channels with correlated sources are presented: the first one is strictly tighter than the existing outer bound by Gohari and Anantharam while the second one provides a complete characterization of the admissible source region in the case where the two sources are conditionally independent given the common part. These outer bounds are deduced from the general necessary conditions established for the lossy source broadcast problem via suitable comparisons between the virtual broadcast channel (induced by the source and the reconstructions) and the physical broadcast channel

Stable transports between stationary random measures

We give an algorithm to construct a translation-invariant transport kernel between ergodic stationary random measures $\Phi$ and $\Psi$ on $\mathbb R^d$, given that they have equal intensities. As a result, this yields a construction of a shift-coupling of an ergodic stationary random measure and its Palm version. This algorithm constructs the transport kernel in a deterministic manner given realizations $\varphi$ and $\psi$ of the measures. The (non-constructive) existence of such a transport kernel was proved in [8]. Our algorithm is a generalization of the work of [3], in which a construction is provided for the Lebesgue measure and an ergodic simple point process. In the general case, we limit ourselves to what we call constrained densities and transport kernels. We give a definition of stability of constrained densities and introduce our construction algorithm inspired by the Gale-Shapley stable marriage algorithm. For stable constrained densities, we study existence, uniqueness, monotonicity w.r.t. the measures and boundedness.Comment: In the second version, we change the way of presentation of the main results in Section 4. The main results and their proofs are not changed significantly. We add Section 3 and Subsection 4.6. 25 pages and 2 figure

Safe Linear Stochastic Bandits

We introduce the safe linear stochastic bandit framework---a generalization of linear stochastic bandits---where, in each stage, the learner is required to select an arm with an expected reward that is no less than a predetermined (safe) threshold with high probability. We assume that the learner initially has knowledge of an arm that is known to be safe, but not necessarily optimal. Leveraging on this assumption, we introduce a learning algorithm that systematically combines known safe arms with exploratory arms to safely expand the set of safe arms over time, while facilitating safe greedy exploitation in subsequent stages. In addition to ensuring the satisfaction of the safety constraint at every stage of play, the proposed algorithm is shown to exhibit an expected regret that is no more than $O(\sqrt{T}\log (T))$ after $T$ stages of play

Fuzzy Lattice Reasoning for Pattern Classification Using a New Positive Valuation Function

This paper describes an enhancement of fuzzy lattice reasoning (FLR) classifier for pattern classification based on a positive valuation function. Fuzzy lattice reasoning (FLR) was described lately as a lattice data domain extension of fuzzy ARTMAP neural classifier based on a lattice inclusion measure function. In this work, we improve the performance of FLR classifier by defining a new nonlinear positive valuation function. As a consequence, the modified algorithm achieves better classification results. The effectiveness of the modified FLR is demonstrated by examples on several well-known pattern recognition benchmarks