Guaranteeing generalisation in neural networks

Neural networks need to be able to guarantee their intrinsic generalisation abilities if they are to be used reliably. Mitchell's concept and version spaces technique is able to guarantee generalisation in the symbolic concept-learning environment in which it is implemented. Generalisation, according to Mitchell, is guaranteed when there is no alternative concept that is consistent with all the examples presented so far, except the current concept, given the bias of the user. A form of bidirectional convergence is used by Mitchell to recognise when the no-alternative situation has been reached. Mitchell's technique has problems of search and storage feasibility in its symbolic environment. This thesis aims to show that by evolving the technique further in a neural environment, these problems can be overcome. Firstly, the biasing factors which affect the kind of concept that can be learned are explored in a neural network context. Secondly, approaches for abstracting the underlying features of the symbolic technique that enable recognition of the no-alternative situation are discussed. The discussion generates neural techniques for guaranteeing generalisation and culminates in a neural technique which is able to recognise when the best fit neural weight state has been found for a given set of data and topology

Genuinely nonabelian partial difference sets

Strongly regular graphs (SRGs) provide a fertile area of exploration in algebraic combinatorics, integrating techniques in graph theory, linear algebra, group theory, finite fields, finite geometry, and number theory. Of particular interest are those SRGs with a large automorphism group. If an automorphism group acts regularly (sharply transitively) on the vertices of the graph, then we may identify the graph with a subset of the group, a partial difference set (PDS), which allows us to apply techniques from group theory to examine the graph. Much of the work over the past four decades has concentrated on abelian PDSs using the powerful techniques of character theory. However, little work has been done on nonabelian PDSs. In this paper we point out the existence of \textit{genuinely nonabelian} PDSs, i.e., PDSs for parameter sets where a nonabelian group is the only possible regular automorphism group. We include methods for demonstrating that abelian PDSs are not possible for a particular set of parameters or for a particular SRG. Four infinite families of genuinely nonabelian PDSs are described, two of which -- one arising from triangular graphs and one arising from Krein covers of complete graphs constructed by Godsil \cite{Godsil_1992} -- are new. We also include a new nonabelian PDS found by computer search and present some possible future directions of research.Comment: 24 page

Nonabelian partial difference sets constructed using abelian techniques

A $(v,k,\lambda, \mu)$-partial difference set (PDS) is a subset $D$ of a group $G$ such that $|G| = v$, $|D| = k$, and every nonidentity element $x$ of $G$ can be written in either $\lambda$ or $\mu$ different ways as a product $gh^{-1}$, depending on whether or not $x$ is in $D$. Assuming the identity is not in $D$ and $D$ is inverse-closed, the corresponding Cayley graph ${\rm Cay}(G,D)$ will be strongly regular. Partial difference sets have been the subject of significant study, especially in abelian groups, but relatively little is known about PDSs in nonabelian groups. While many techniques useful for abelian groups fail to translate to a nonabelian setting, the purpose of this paper is to show that examples and constructions using abelian groups can be modified to generate several examples in nonabelian groups. In particular, in this paper we use such techniques to construct the first known examples of PDSs in nonabelian groups of order $q^{2m}$, where $q$ is a power of an odd prime $p$ and $m \ge 2$. The groups constructed can have exponent as small as $p$ or as large as $p^r$ in a group of order $p^{2r}$. Furthermore, we construct what we believe are the first known Paley-type PDSs in nonabelian groups and what we believe are the first examples of Paley-Hadamard difference sets in nonabelian groups, and, using analogues of product theorems for abelian groups, we obtain several examples of each. We conclude the paper with several possible future research directions.Comment: 26 page

e-Social Science and Evidence-Based Policy Assessment : Challenges and Solutions

Peer reviewedPreprin

The use of phylogeny to interpret cross-cultural patterns in plant use and guide medicinal plant discovery: an example from Pterocarpus (Leguminosae)

The study of traditional knowledge of medicinal plants has led to discoveries that have helped combat diseases and improve healthcare. However, the development of quantitative measures that can assist our quest for new medicinal plants has not greatly advanced in recent years. Phylogenetic tools have entered many scientific fields in the last two decades to provide explanatory power, but have been overlooked in ethnomedicinal studies. Several studies show that medicinal properties are not randomly distributed in plant phylogenies, suggesting that phylogeny shapes ethnobotanical use. Nevertheless, empirical studies that explicitly combine ethnobotanical and phylogenetic information are scarce.In this study, we borrowed tools from community ecology phylogenetics to quantify significance of phylogenetic signal in medicinal properties in plants and identify nodes on phylogenies with high bioscreening potential. To do this, we produced an ethnomedicinal review from extensive literature research and a multi-locus phylogenetic hypothesis for the pantropical genus Pterocarpus (Leguminosae: Papilionoideae). We demonstrate that species used to treat a certain conditions, such as malaria, are significantly phylogenetically clumped and we highlight nodes in the phylogeny that are significantly overabundant in species used to treat certain conditions. These cross-cultural patterns in ethnomedicinal usage in Pterocarpus are interpreted in the light of phylogenetic relationships.This study provides techniques that enable the application of phylogenies in bioscreening, but also sheds light on the processes that shape cross-cultural ethnomedicinal patterns. This community phylogenetic approach demonstrates that similar ethnobotanical uses can arise in parallel in different areas where related plants are available. With a vast amount of ethnomedicinal and phylogenetic information available, we predict that this field, after further refinement of the techniques, will expand into similar research areas, such as pest management or the search for bioactive plant-based compounds