On a class of non-Hermitian matrices with positive definite Schur complements

Given a positive definite nXn matrix A and a Hermitian mXm matrix D, we characterize under which conditions there exists a strictly contractive matrix K such that the non-Hermitian block-matrix with the enties A and -AK in the first row and K^*A and D in the second has a positive definite Schur complement with respect to its submatrix A. Additionally, we show that K can be chosen such that diagonalizability of the block-matrix is guaranteed and we compute its spectrum. Moreover, we show a connection to the recently developed frame theory for Krein spaces

KNIME: the Konstanz Information Miner

The Konstanz Information Miner is a modular environment which enables easy visual assembly and interactive execution of a data pipeline. It is designed as a teaching, research and collaboration platform, which enables easy integration of new algorithms, data manipulation or visualization methods as new modules or nodes. In this paper we describe some of the design aspects of the underlying architecture and briefly sketch how new nodes can be incorporated

Maximum-Score Diversity Selection

This thesis discusses the problem of Maximum-Score Diversity Selection (MSDS). Pure diversity selection, as it is often performed e.g. in early drug discovery, is the selection of a subset of available objects that is as diverse as possible. MSDS adds a second objective, which additionally tries to maximize the "score'' of the subset, which usually is the sum of scores of all elements in the subset. Thus, this problem is a classical multi-objective optimization problem since both objectives -- maximizing score and maximizing diversity -- tend to conflict with each other. In this thesis several methods are presented, developed, and evaluated to efficiently solve this special multi-objective optimization problem. After a more detailed discussion about the application of MSDS in drug discovery, the question of suitable definitions of diversity is considered. This is essential for later application domains, where users have only a vague feeling of diversity. Then the Maximum-Score Diversity Selection problem is formalized and shown to be an NP-hard optimization problem. Therefore no exact solution can be computed efficiently for all but the smallest cases. After putting MSDS into the context of multi-objective optimization, the usage of evolutionary algorithms -- specifically genetic algorithms -- for solving the problem is evaluated. This also includes the presentation of novel genetic operators for evolving subsets or combinations of objects. However, being a universal tool, genetic algorithms may not be the best technique for the actual problem. Hence, several problem-specific heuristics are discussed, two of them motivated by the transformation of MSDS into a graph-theoretic problem used in the NP-hardness proof, and a novel heuristics methods, known as Score Erosion. The comparison of all approaches on various synthetic and real-world datasets reveals that all heuristics find solutions of similar quality, given the right measures of diversity, with Score Erosion being the fastest of all presented algorithm as a result of its linear time complexity. Also the questions are investigated as to how the structure of the search space influences the results and whether the application of MSDS pays off in practice

Hybrid Fragment Mining with MoFa and FSG

Abstract – In the last few years a number of different subgraph mining algorithms have been proposed. They are often used for nding frequent fragments in molecular databases. All these algorithms behave quite well when used on small datasets of not more than a few thousand molecules. However, they all fail on larger amounts of data because they are either time consuming or have enormous memory requirements. In this paper we present a hybrid mining technique that overcomes the individual problems of the underlying algorithms and outperforms the individual methods impressively on large databases

Full Perfect Extension Pruning for Frequent Graph Mining

Mining graph databases for frequent subgraphs has recently developed into an area of intensive research. Its main goals are to reduce the execution time of the existing basic algorithms and to enhance their capability to find meaningful graph fragments. Here we present a method to achieve the former, namely an improvement of what we called perfect extension pruning in an earlier paper [2]. With it the number of generated fragments and visited search tree nodes can be reduced, thus accelerating the search

Subgraph Mining

Graphs are often used as models in very different application areas ranging from networks to molecules and proteins. Having graphs in a graph database it is an interesting problem to find small graph parts, so called subgraphs, that appear in a certain number of graphs within the database. Possible subgraphs of a set of graphs form a lattice that must be searched to find the subgraphs that appear most frequently. Two steps are necessary for this search: first new possible subgraphs must be generated, secondly it must be checked how often a newly generated subgraph appears in the database. Additionally intelligent pruning methods, inexact graph matching and background knowledge can be incorporated in the mining algorithms