Tropical secant graphs of monomial curves

The first secant variety of a projective monomial curve is a threefold with an action by a one-dimensional torus. Its tropicalization is a three-dimensional fan with a one-dimensional lineality space, so the tropical threefold is represented by a balanced graph. Our main result is an explicit construction of that graph. As a consequence, we obtain algorithms to effectively compute the multidegree and Chow polytope of an arbitrary projective monomial curve. This generalizes an earlier degree formula due to Ranestad. The combinatorics underlying our construction is rather delicate, and it is based on a refinement of the theory of geometric tropicalization due to Hacking, Keel and Tevelev.Comment: 30 pages, 8 figures. Major revision of the exposition. In particular, old Sections 4 and 5 are merged into a single section. Also, added Figure 3 and discussed Chow polytopes of rational normal curves in Section

Automated construction of a hierarchy of self-organized neural network classifiers

This paper documents an effort to design and implement a neural network-based, automatic classification system which dynamically constructs and trains a decision tree. The system is a combination of neural network and decision tree technology. The decision tree is constructed to partition a large classification problem into smaller problems. The neural network modules then solve these smaller problems. We used a variant of the Fuzzy ARTMAP neural network which can be trained much more quickly than traditional neural networks. The research extends the concept of self-organization from within the neural network to the overall structure of the dynamically constructed decision hierarchy. The primary advantage is avoidance of manual tedium and subjective bias in constructing decision hierarchies. Additionally, removing the need for manual construction of the hierarchy opens up a large class of potential classification applications. When tested on data from real-world images, the automatically generated hierarchies performed slightly better than an intuitive (handbuilt) hierarchy. Because the neural networks at the nodes of the decision hierarchy are solving smaller problems, generalization performance can really be improved if the number of features used to solve these problems is reduced. Algorithms for automatically selecting which features to use for each individual classification module were also implemented. We were able to achieve the same level of performance as in previous manual efforts, but in an efficient, automatic manner. The technology developed has great potential in a number of commercial areas, including data mining, pattern recognition, and intelligent interfaces for personal computer applications. Sample applications include: fraud detection, bankruptcy prediction, data mining agent, scalable object recognition system, email agent, resource librarian agent, and a decision aid agent

A Large Scale Dataset for the Evaluation of Ontology Matching Systems

Recently, the number of ontology matching techniques and systems has increased significantly. This makes the issue of their evaluation and comparison more severe. One of the challenges of the ontology matching evaluation is in building large scale evaluation datasets. In fact, the number of possible correspondences between two ontologies grows quadratically with respect to the numbers of entities in these ontologies. This often makes the manual construction of the evaluation datasets demanding to the point of being infeasible for large scale matching tasks. In this paper we present an ontology matching evaluation dataset composed of thousands of matching tasks, called TaxME2. It was built semi-automatically out of the Google, Yahoo and Looksmart web directories. We evaluated TaxME2 by exploiting the results of almost two dozen of state of the art ontology matching systems. The experiments indicate that the dataset possesses the desired key properties, namely it is error-free, incremental, discriminative, monotonic, and hard for the state of the art ontology matching systems. The paper has been accepted for publication in "The Knowledge Engineering Review", Cambridge Universty Press (ISSN: 0269-8889, EISSN: 1469-8005)

Algorithmic Aspects of a General Modular Decomposition Theory

A new general decomposition theory inspired from modular graph decomposition is presented. This helps unifying modular decomposition on different structures, including (but not restricted to) graphs. Moreover, even in the case of graphs, the terminology ``module'' not only captures the classical graph modules but also allows to handle 2-connected components, star-cutsets, and other vertex subsets. The main result is that most of the nice algorithmic tools developed for modular decomposition of graphs still apply efficiently on our generalisation of modules. Besides, when an essential axiom is satisfied, almost all the important properties can be retrieved. For this case, an algorithm given by Ehrenfeucht, Gabow, McConnell and Sullivan 1994 is generalised and yields a very efficient solution to the associated decomposition problem