We present a multiresolution construction for efficiently computing, compressing and applying large powers of operators that have high powers with low numerical rank. This allows the fast computation of functions of the operator, notably the associated Green’s function, in compressed form, and their fast application. Classes of operators satisfying these conditions include diffusion-like operators, in any dimension, on manifolds, graphs, and in non-homogeneous media. In this case our construction can be viewed as a far-reaching generalization of Fast Multipole Methods, achieved through a different point of view, and of the non-standard wavelet representation of Calderón-Zygmund and pseudodifferential operators, achieved through a different multiresolution analysis adapted to the operator. We show how the dyadic powers of an operator can be used to induce a multiresolution analysis, as in classical Littlewood-Paley and wavelet theory, and we show how to construct, with fast and stable algorithms, scaling function and wavelet bases associated to this multiresolution analysis, and the corresponding downsampling operators, and use them to compress the corresponding powers of the operator. This allows to extend multiscale signal processing to general spaces (such as manifolds and graphs) in a very natural way, with corresponding fast algorithms.

Harmonic analysis and diffusion on discrete data has been shown to lead to state-of-the-art algorithms for machine learning tasks, especially in the context of semi-supervised and transductive learning. The success of these algorithms rests on the assumption that the function(s) to be studied (learned, interpolated, etc.) are smooth with respect to the geometry of the data. In this paper we present a method for modifying the given geometry so the function(s) to be studied are smoother with respect to the modified geometry, and thus more amenable to treatment using harmonic analysis methods. Among the many possible applications, we consider the problems of image denoising and transductive classification. In both settings, our approach improves on standard diffusion based methods.

We construct a multiscale tight frame based on an arbitrary orthonormal basis for the L 2 space of an arbitrary sigma finite measure space. The approximation properties of the resulting multiscale are studied in the context of Besov approximation spaces, which are characterized both in terms of suitable K–functionals and the frame transforms. The only major condition required is the uniform boundedness of a summabilility operator. We give sufficient conditions for this to hold in the context of a very general class of metric measure spaces. The theory is illustrated using the approximation of characteristic functions of caps on a dumbell manifold, and applied to the problem of recognition of hand–written digits. Our methods outperforms comparable methods for semi–supervised learning.

For many mathematicians and physicists, the Internet has become a popular realworld domain for the application and/or development of new theories related to the organization and behavior of large-scale, complex, and dynamic systems. In some cases, the Internet has served both as inspiration and justification for the popularization of new models and mathematics within the scientific enterprise. For example, scale-free network models of the preferential attachment type [8] have been claimed to describe the Internet’s connectivity structure, resulting in surprisingly general and strong claims about the network’s resilience to random failures of its components and its vulnerability to targeted attacks against its infrastructure [2]. These models have, as their trademark, power-law type node degree distributions that drastically distinguish them from the classical Erdős-Rényi type random graph models [13]. These “scale-free ” network models have attracted significant attention within the scientific community and have been partly responsible for launching and fueling the new field of network science [42, 4]. To date, the main role that mathematics has played in network science has been to put the physicists’ largely empirical findings on solid grounds Walter Willinger is at AT&T Labs-Research in Florham Park, NJ. His email address is walter@research.att. com.

In this paper we introduce a novel graph classification algorithm and demonstrate its efficacy in drug design. In our method, we use graphs to model chemical structures and apply a wavelet analysis of graphs to create features capturing graph local topology. We design a novel graph kernel function to utilize the created feature to build predictive models for chemicals. We call the new graph kernel a graph wavelet-alignment kernel. We have evaluated the efficacy of the wavelet-alignment kernel using a set of chemical structure-activity prediction benchmarks. Our results indicate that the use of the kernel function yields performance profiles comparable to, and sometimes exceeding that of the existing state-of-the-art chemical classification approaches. In addition, our results also show that the use of wavelet functions significantly decreases the computational costs for graph kernel computation with more than 10 fold speed up.

Online Social Networks (OSNs) provide a unique opportunity for researchers to study how a combination of technological, economical, and social forces have been conspiring to provide a service that has attracted the largest user population in the history of the Internet. With more than half a billion of users and counting, OSNs have the potential to impact almost every aspect of networking, including measurements and performance modeling/analysis, network architecture and system design, and privacy and user behavior, to name just a few. However, much of the existing OSN research literature seems to have lost sight of this unique opportunity and has avoided dealing with the new challenges posed by OSNs. We argue in this position paper that it is high time for OSN researcher to exploit and face these opportunities and challenges to provide a basic understanding of the OSN eco-system as a whole. Such an understanding has to reflect the key role users play in this system and must focus on the system’s dynamics, purpose and functionality when trying to illuminate the main technological, economic, and social forces at work in the current OSN revolution. 1.