Duality in Graphical Models

Graphical models have proven to be powerful tools for representing high-dimensional systems of random variables. One example of such a model is the undirected graph, in which lack of an edge represents conditional independence between two random variables given the rest. Another example is the bidirected graph, in which absence of edges encodes pairwise marginal independence. Both of these classes of graphical models have been extensively studied, and while they are considered to be dual to one another, except in a few instances this duality has not been thoroughly investigated. In this paper, we demonstrate how duality between undirected and bidirected models can be used to transport results for one class of graphical models to the dual model in a transparent manner. We proceed to apply this technique to extend previously existing results as well as to prove new ones, in three important domains. First, we discuss the pairwise and global Markov properties for undirected and bidirected models, using the pseudographoid and reverse-pseudographoid rules which are weaker conditions than the typically used intersection and composition rules. Second, we investigate these pseudographoid and reverse pseudographoid rules in the context of probability distributions, using the concept of duality in the process. Duality allows us to quickly relate them to the more familiar intersection and composition properties. Third and finally, we apply the dualization method to understand the implications of faithfulness, which in turn leads to a more general form of an existing result

On the Design of Cryptographic Primitives

The main objective of this work is twofold. On the one hand, it gives a brief overview of the area of two-party cryptographic protocols. On the other hand, it proposes new schemes and guidelines for improving the practice of robust protocol design. In order to achieve such a double goal, a tour through the descriptions of the two main cryptographic primitives is carried out. Within this survey, some of the most representative algorithms based on the Theory of Finite Fields are provided and new general schemes and specific algorithms based on Graph Theory are proposed

Face pairing graphs and 3-manifold enumeration

The face pairing graph of a 3-manifold triangulation is a 4-valent graph denoting which tetrahedron faces are identified with which others. We present a series of properties that must be satisfied by the face pairing graph of a closed minimal P^2-irreducible triangulation. In addition we present constraints upon the combinatorial structure of such a triangulation that can be deduced from its face pairing graph. These results are then applied to the enumeration of closed minimal P^2-irreducible 3-manifold triangulations, leading to a significant improvement in the performance of the enumeration algorithm. Results are offered for both orientable and non-orientable triangulations.Comment: 30 pages, 57 figures; v2: clarified some passages and generalised the final theorem to the non-orientable case; v3: fixed a flaw in the proof of the conical face lemm

Deep Expander Networks: Efficient Deep Networks from Graph Theory

Efficient CNN designs like ResNets and DenseNet were proposed to improve accuracy vs efficiency trade-offs. They essentially increased the connectivity, allowing efficient information flow across layers. Inspired by these techniques, we propose to model connections between filters of a CNN using graphs which are simultaneously sparse and well connected. Sparsity results in efficiency while well connectedness can preserve the expressive power of the CNNs. We use a well-studied class of graphs from theoretical computer science that satisfies these properties known as Expander graphs. Expander graphs are used to model connections between filters in CNNs to design networks called X-Nets. We present two guarantees on the connectivity of X-Nets: Each node influences every node in a layer in logarithmic steps, and the number of paths between two sets of nodes is proportional to the product of their sizes. We also propose efficient training and inference algorithms, making it possible to train deeper and wider X-Nets effectively. Expander based models give a 4% improvement in accuracy on MobileNet over grouped convolutions, a popular technique, which has the same sparsity but worse connectivity. X-Nets give better performance trade-offs than the original ResNet and DenseNet-BC architectures. We achieve model sizes comparable to state-of-the-art pruning techniques using our simple architecture design, without any pruning. We hope that this work motivates other approaches to utilize results from graph theory to develop efficient network architectures.Comment: ECCV'1

Bad News for Chordal Partitions

Reed and Seymour [1998] asked whether every graph has a partition into induced connected non-empty bipartite subgraphs such that the quotient graph is chordal. If true, this would have significant ramifications for Hadwiger's Conjecture. We prove that the answer is `no'. In fact, we show that the answer is still `no' for several relaxations of the question