Networks,molecular pathways in particular, are increasingly looked at as a better organized and richer version of gene signatures. However, high variability can be injected by the different methods that are typically used in system biology to define a cellular wiring diagramat diverse levels of organization, from transcriptomics to signalling,of the functional design.

Despite its common use even in biological contexts, the problem of quantitatively comparing networks (\textit{e.g.}, using a metric instead of evaluating network properties) is a still an open issue in many scientific disciplines. The central problem is of course which network metric should be used to evaluate stability, whether focusing on local changes or global structural changes.  As widely known, the classic distances in the edit family focus only on the portions of the network interested by the differences in the presence/absence of matching links. Spectral distances - based on the list of eigenvalues of the Laplacian matrix of the underlying graph - are instead particularly effective for studying global structures. In particular, the Ipsen-Mikhailov distance was found robust in a wide range of situations. However, global distances can be tricked by isomorph or close to isomorph graphs. Here both approaches are improved by proposing a glocal measure which combines a spectral distance with a typical Hamming local editing component.

Current projects:

• Definition and analysis of spectral and edit distances for graph comparison
• Analysis/implementation/extensions of algorithms for molecular pathway profiling
• Mathematical methods for the analysis of network properties (e.g. modularity)
• Generation of networks with given characteristics

Supervised discrimination and feature ranking are essential steps throughout any data analysis task. The high-throughput (and beyond) data for functional genomics we are dealing with and the need for a methodology warranting honest estimate of the performances require algorithms able to cope with the p>>n issue, adapting to very unbalanced problems and robustly detecting the relevant features: all this has also to be realized within reasonable time. To such aim, we are constantly considering state-of-art classifiers and ranking methods, finding possible improvements, implementing them in our library mply and benchmarking them against more standard choices.

Current projects:

• Analysis/implementation/extensions of classifiers in the family of Discriminant Analysis
• Analysis/implementation/extensions of feature ranking evolutionary algorithms: Genetic Algorithms, Particle Swarm Optimization, Ant Colony Optimization
• Use of Discrete Wavelet Transform in feature ranking

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We are investigating the mathematical aspects of a few problems in Statistical Machine Learning, linked to some common issues emerging while analyzing datasets coming from the biological side. Dealing with datasets of very small sample size, or with unbalanced classes is very common; many hints along these directions have been proposed in literature, for instance the use of virtual samples. Furthermore, if the samples are described by many features, there is a non negligible probability that for a feature to result discriminant just by chance. Finally, we are interested in studying the concept of (leave-one-out) stability for ranking algorithm, in parallel to the similar concept for classifiers.

Current projects:

• Use of virtual samples in small-size datasets analysis
• Mathematical aspects of the p>>n problems: the problem of the fake discriminant features
• (Leave-one-out) stability for feature ranking algorithms

Application of algebraic and combinatorial techniques to statistical problem is becoming a fascinating new workfield. A relevant role within this topic is played by the ranking problem, which comes naturally also from computational biology when trying to detect a panel of important features. We proposed a solution based on symmetric group theory, which is now being extended to partial lists; moreover, the mathematical aspects of the proposal are currenly under investigation.

A completely different topic concerns the study of longitudinal data, in particular, microarray time series. We devised a methodology to detect and reconstruct major trends in order to clarify the underlying networks of gene relations: a few novel techniques based on algebraic structures have recently appeared, and we are planning to integrate them in the existing framework.

Current projects:

• Algebraic methods for ranked lists comparison, with extensions to partial lists
• Mathematical aspects of the lists stability methods
• Combinatorial and algebraic solutions for the study of microarray time series

Another important class of modular graded Lie algebras exists, namely the thin Lie algebras, which represents a possible generalization of the algebras of maximal class. In an algebra of maximal class, the first homogeneous component is two-dimensional, while all other components have dimension one: in a thin algebra, also components other than the first one have dimension two. These two-dimensional components are called diamonds. The quotient coming from factorizing a thin algebra by its homogeneous components following the second diamond is an algebra of maximal class. In a joint work with Avitabile we derived some restrictions on the possible positions of the second diamond, while in another joint work with Caranti we studied the structure of such quotient in the odd characteristic case: I have also dealt with the same problem in the case of characteristic two, where interesting results are emerging from a collaboration with Avitabile and Mattarei. Further present research lines concern the construction of more thin algebras aiming to a classification of all these structures. During all the previous studies, the computational side always played a fundamental role: some algebraic manipulation packages as GAP, Magma and p-Quotient have been invaluable in providing suggestions on the directions to follow in developing the above theories, although none of the results rely on such computations.

Current projects:

• (Finite) presentation of Bi-Zassenhaus algebras
• Thin Lie algebras with non-metabelian quotients of maximal class
• Nottingham algebras with diamonds of finite and infinite type.

Details on the computational aspects are described  here.