"The analysis of datasets taking the form of simple, undirected graphs continues to gain in importance across a variety of disciplines. Two choices of null model, the logistic-linear model and the implicit log-linear model, have come into common use for analyzing such network data, in part because each accounts for the heterogeneity of network node degrees typically observed in practice. Here we show how these both may be viewed as instances of a broader class of null models, with the property that all members of this class give rise to essentially the same likelihood-based estimates of link probabilities in sparse graph regimes. This facilitates likelihood-based computation and inference, and enables practitioners to choose the most appropriate null model from this family based on application context. Comparative model fits for a variety of network datasets demonstrate the practical implications of our results."

"We address the statistical issue of determining the maximal spaces (maxisets) where model selection procedures attain a given rate of convergence. By considering first general dictionaries, then orthonormal bases, we characterize these maxisets in terms of approximation spaces. These results are illustrated by classical choices of wavelet model collections. For each of them, the maxisets are described in terms of functional spaces. We take a special care of the issue of calculability and measure the induced loss of performance in terms of maxisets."

"It is proposed that we use the term “approximation” for inexact description of a target system and “idealization” for another system whose properties also provide an inexact description of the target system. Since systems generated by a limiting process can often have quite unexpected—even inconsistent—properties, familiar limit processes used in statistical physics can fail to provide idealizations but merely provide approximations."

In this paper we address sampling and approximation of functions on combinatorial graphs. We develop filtering on graphs by using Schr"odinger's group of operators generated by combinatorial Laplace operator. Then we construct a sampling theory by proving Poincare and Plancherel-Polya-type inequalities for functions on graphs. These results lead to a theory of sparse approximations on graphs and have potential applications to filtering, denoising, data dimension reduction, image processing, image compression, computer graphics, visualization and learning theory.

"This article studies exponential families $mathcal{E}$ on finite sets such that the information divergence $D(P|mathcal{E})$ of an arbitrary probability distribution from $mathcal{E}$ is bounded by some constant $D>0$. A particular class of low-dimensional exponential families that have low values of $D$ can be obtained from partitions of the state space. The main results concern optimality properties of these partition exponential families. Exponential families where $D=log(2)$ are studied in detail. This case is special, because if $D<log(2)$, then $mathcal{E}$ contains all probability measures with full support."

"Chains of infinite order are generalizations of Markov chains that constitute a flexible class of models in the general theory of stochastic processes. These processes can be naturally studied using approximating Markov chains. Here we derive new upper bounds for the $bar{d}$-distance and the K"ullback-Leibler divergence between chains of infinite order and their canonical $k$-step Markov approximations. In contrast to the bounds available in the literature our results apply to chains of infinite order compatible with general classes of probability kernels. In particular, we allow kernels with discontinuities and null transition probabilities.""  (Pedantry: Pretty sure Kullback did not spell his name with an umlaut!)

"I construct a new universal differential equation (B), in the sense of Rubel. That is, its solutions approximate to arbitrary accuracy any continuous function on any interval of the real line."