Greedy MAXCUT Algorithms and their Information Content

MAXCUT defines a classical NP-hard problem for graph partitioning and it serves as a typical case of the symmetric non-monotone Unconstrained Submodular Maximization (USM) problem. Applications of MAXCUT are abundant in machine learning, computer vision and statistical physics. Greedy algorithms to approximately solve MAXCUT rely on greedy vertex labelling or on an edge contraction strategy. These algorithms have been studied by measuring their approximation ratios in the worst case setting but very little is known to characterize their robustness to noise contaminations of the input data in the average case. Adapting the framework of Approximation Set Coding, we present a method to exactly measure the cardinality of the algorithmic approximation sets of five greedy MAXCUT algorithms. Their information contents are explored for graph instances generated by two different noise models: the edge reversal model and Gaussian edge weights model. The results provide insights into the robustness of different greedy heuristics and techniques for MAXCUT, which can be used for algorithm design of general USM problems.Comment: This is a longer version of the paper published in 2015 IEEE Information Theory Workshop (ITW

Data-driven modelling of biological multi-scale processes

Biological processes involve a variety of spatial and temporal scales. A holistic understanding of many biological processes therefore requires multi-scale models which capture the relevant properties on all these scales. In this manuscript we review mathematical modelling approaches used to describe the individual spatial scales and how they are integrated into holistic models. We discuss the relation between spatial and temporal scales and the implication of that on multi-scale modelling. Based upon this overview over state-of-the-art modelling approaches, we formulate key challenges in mathematical and computational modelling of biological multi-scale and multi-physics processes. In particular, we considered the availability of analysis tools for multi-scale models and model-based multi-scale data integration. We provide a compact review of methods for model-based data integration and model-based hypothesis testing. Furthermore, novel approaches and recent trends are discussed, including computation time reduction using reduced order and surrogate models, which contribute to the solution of inference problems. We conclude the manuscript by providing a few ideas for the development of tailored multi-scale inference methods.Comment: This manuscript will appear in the Journal of Coupled Systems and Multiscale Dynamics (American Scientific Publishers

Learning representations from dendrograms

We propose unsupervised representation learning and feature extraction from dendrograms. The commonly used Minimax distance measures correspond to building a dendrogram with single linkage criterion, with defining specific forms of a level function and a distance function over that. Therefore, we extend this method to arbitrary dendrograms. We develop a generalized framework wherein different distance measures and representations can be inferred from different types of dendrograms, level functions and distance functions. Via an appropriate embedding, we compute a vector-based representation of the inferred distances, in order to enable many numerical machine learning algorithms to employ such distances. Then, to address the model selection problem, we study the aggregation of different dendrogram-based distances respectively in solution space and in representation space in the spirit of deep representations. In the first approach, for example for the clustering problem, we build a graph with positive and negative edge weights according to the consistency of the clustering labels of different objects among different solutions, in the context of ensemble methods. Then, we use an efficient variant of correlation clustering to produce the final clusters. In the second approach, we investigate the combination of different distances and features sequentially in the spirit of multi-layered architectures to obtain the final features. Finally, we demonstrate the effectiveness of our approach via several numerical studies

Proceedings of the Fifth Workshop on Information Theoretic Methods in Science and Engineering

These are the online proceedings of the Fifth Workshop on Information Theoretic Methods in Science and Engineering (WITMSE), which was held in the Trippenhuis, Amsterdam, in August 2012