Hub seminar series
In the latest of the hub’s seminar series, Raphaël Tinarrage of the Institute of Science and Technology Austria visited Imperial College London on 11 November to give a talk on Linear orbits of compact Lie groups and machine learning.
When a problem involves continuous symmetries, such as rotations, one naturally expects a Lie group action. In some cases, this action is linear, that is, made of rigid Euclidean motions. As a matter of fact, linear actions arise in several corners of data analysis: in image processing, where standard embeddings commute with Euclidean isometries; in equivariant neural networks, where one structurally forces linear actions or favors them via optimization; or in physical systems, where representations are found sometimes through Noether’s theorem, and sometimes more unexpectedly.
However, most of the time, the representation is not observed directly, but only through its orbits. Recovering the underlying representation from a single orbit would not only allow one to verify the Lie linear orbit hypothesis, but also to improve existing data analysis techniques.
In his talk, Raphaël presented such an orbit-regression algorithm, developed with Henrique Ennes, PhD student at the Inria Centre at the Université Côte d’Azur. Building on previous work by Cahill, Mixon and Parshall, they tackle the problem at the level of Lie algebras, where it can be reformulated as a discrete-continuous optimization over the orthogonal group. In addition to presenting the algorithm and its theoretical guarantees, Raphaël’s talk also delved into the applications mentioned above.