MSI Weekly Bulletin - Week starting Monday 20 August, 2007

The statistical machine learning model is a widely-used model for describing a pattern recognition problem but in many situations it is not known whether the model can be solved in a data-efficient way. We will consider one particular algorithm (the empirical minimisation algorithm) for attacking this model and we will study how the parameters of the model affect our ability to "learn" well. A brief description of the model follows: Given a family of functions F from X to R, let T be a function from X to R and $\mu$ be a probability measure on X. Without knowing T or $\mu$, we want to pick a function $f \in F$ that almost minimises some measure-dependent distance between f and T (for example, the expectation of $(f - T)^2$). Our only information on T is a sample of its values $(x_1, T(x_1)), \dots, (x_n, T(x_n))$ where the $x_i$ are distributed independently according to $\mu$.

In this talk some results about classes of groups which can be characterised by means of existence of some relevant subgroups are presented. Some local versions are also analysed.

A persistent feature of complex systems in engineering and science is the emergence of macroscopic, coarse grained, coherent behaviour from the interactions of microscopic agents (molecules, cells) and with their environment. In current modelling, ranging from ecology to materials science, the underlying microscopic mechanisms are known, but the closures to translate microscale knowledge to a system level macroscopic description are rarely available in closed form. Kevrekidis proposes new ‘equation free’ computational methodologies to circumvent this stumbling block in multiscale modeling. Nonlinear coordinate transforms underpin analytic techniques that support these computational methodologies. But to do so we must cross multiple space and time scales, in both deterministic and stochastic systems, and where the microstructure is either smooth or detailed. Using examples, I describe progress in using nonlinear coordinate transforms to illuminate such multiscale modelling issues.