Let Xi, i ∈ V form a Markov random field (MRF) represented by an undirected graph G = (V, E), and V’ be a subset of V. We determine the smallest graph that can always represent the subfield Xi, i ∈ V’ as an MRF. Based on this result, we obtain a necessary and suffcient condition for a subfield of a Markov tree to be also a Markov tree. When G is a path so that Xi, i ∈ V form a Markov chain, it is known that the I-Measure is always nonnegative (Kawabata and Yeung, 1992). We prove that Markov chain is essentially the only MRF such that the I-Measure is always nonnegative. By applying our characterization of the smallest graph representation of a subfield of an MRF, we develop a recursive approach for constructing information diagrams for MRFs. Our work is built on the set-theoretic characterization of an MRF (Yeung et al., 2002).

We study the change of multivariate mutual information among a set of random variables when some common randomness is added to or removed from a subset of the random variables. This is formulated more precisely as two new multiterminal secret key agreement problems that, respectively, ask how one can increase the secrecy capacity efficiently by adding common randomness to a small subset of users, and how one can simplify the source model by removing redundant common randomness that does not contribute to the secrecy capacity. Characterizations and strongly polynomial-time computations are derived for the rates of change, maximum usable increment, and redundancy. These results can be applied to study the communication complexity for secret key agreement.

Algebraic topology studies topological spaces with the help of tools from abstract algebra. The main focus of this paper is to show that many concepts from algebraic topology can be conveniently expressed in terms of (normal) factor graphs. As an application, we give an alternative proof of a classical duality result of Kramers and Wannier, which expresses the partition function of the 2-D Ising model at a low temperature in terms of the partition function of the 2-D Ising model at a high temperature. Moreover, we discuss analogous results for the 3-D Ising model and the Potts model.

We consider the problem of learning a neural network classifier. Under the information bottleneck (IB) principle, we associate with this classification problem a representation learning problem, which we call “IB learning”. We show that IB learning is, in fact, equivalent to a special class of the quantization problem. The classical results in rate-distortion theory then suggest that IB learning can benefit from a “vector quantization” approach, namely, simultaneously learning the representations of multiple input objects. Such an approach assisted with some variational techniques, result in a novel learning framework, “Aggregated Learning”, for classification with neural network models. In this framework, several objects are jointly classified by a single neural network. The effectiveness of this framework is verified through extensive experiments on standard image recognition and text classification tasks.

In the light of the recent interest in approximating the partition function of the Ising model using the dual normal factor graph, we revisit the classical duality result of Kramers and Wannier, where the dual normal factor graph may be viewed as an intermediate step in establishing such a result.

We show that, under the info-clustering framework, correlated random variables can be clustered in an agglomerative manner. While the existing divisive approach successively segregates the random variables into subsets with increasing multivariate mutual information, our agglomerative approach successively merges subsets of random variables sharing a large amount of normalized total correlation. We show that both approaches result in the same hierarchy of clusters, but the agglomerative approach is an order of magnitude faster than the divisive one. The uniqueness of the hierarchy produced by the two approaches is due to a fundamental connection that we uncover between the well-known total correlation and the recently proposed measure of multivariate mutual information. We implement the new algorithm and provide a data structure for efficient storage and retrieval of the hierarchical clustering solution.