Optimal data approximation with group invariances

Time: 12.11.2019, 11am-12pm (Time has changed!)
Room: 03.323 – Besprechungsraum Chair in Applied Analysis (Alexander von Humboldt-Professorschip)

Optimal data approximation with group invariances
Prof. Davide Barbieri (Universidad Autónoma de Madrid, Spain)

Suppose we are given a finite, typically large, dataset of L^2 functions on Rn, or on a discrete space, such as a dataset of digital images. In order to efficiently deal with the dataset, it may be useful to possess a small dictionary of elementary functions, together with a set of simple composition rules, that allow us to express with a good approximation each data as a composition of the elements of the dictionary. This problem can be expressed as an optimization problem for the L^2 approximation of the data, constrained by a predetermined set of composition rules. When these rules are chosen as basic linear combinations, the problem is equivalent to the well-known Principal Component Analysis. However, when the dataset possesses some symmetries, the set of rules can be effectively enlarged to include group actions on the dictionary, and the constraints can be written in terms of a projection of the data onto an invariant subspace. This is the case for the example of natural images, for which we can consider invariance under translations and rotations. A valuable set of tools to address this problem is provided by the so-called abstract wavelet analysis, which focuses on the geometric representation of data in Hilbert spaces as functions on a given group, extending the traditional and successful approaches of wavelet and time-frequency analysis. In this talk, we will provide a gentle introduction to this topic, and outline the optimization problem in general dataset. Numerical results will finally be presented on datasets commonly used for testing Artificial Intelligence systems such as the image-net.