Abstract
A symmetry group can be used to eliminate a source of variability in the data, which does not provide information for a regression or classification problem. The lecture introduced the notion of Lie group. It considers the case of the multiplicative and translational group, as well as the group of diffeomorphisms that distort the temporal or spatial supports of signals. Knowledge of a symmetry group can be used to define a data representation that is invariant to the action of the group. The notion of canonical invariant is defined from the orbits of the group action.
When you don't know the symmetry group, but know that it belongs to a certain high-dimensional group, you can construct invariants with linear projectors if you have linearized the action of the largest group. This is the strategy used to learn invariants by deformation and therefore relative to the action of a subgroup of the diffeomorphism group. This linearization can be carried out to1st order with a limited development on regular functions, but this is not sufficient when the functions are irregular.
Dimension reduction can also be achieved by constructing parsimonious representations. Such representations are obtained by decomposing the data with a linear operator, called a dictionary, and selecting the coefficients of greatest amplitude. These dictionaries, which can be learned, are interpreted as sets of discriminating patterns in a classification problem.
