The use of group-aware algorithms is nothing new to the machine-learning community. Different methods, ranging from invariant kernels to G-CNN, exist in which information about the underlying symmetry of data is used to equally improve modeling accuracy and explainability. Unfortunately, all these techniques rely on the previous manual identification of the groups acting on data, meaning that they can be implemented only when the user already knows what type of symmetry to expect. And, on the other hand, although in several problems, such as in image classification, these symmetries are clear, they may also be hidden within the data’s geometric structure. We then propose an alternative framework, in which the algebraic information, here restricted to linear representations of compact Lie groups, is detected directly from data, and which may be subsequently used by other steps of symmetric learning. Although there are other works, some of which we make use of, that aim at inquiring Lie objects from point clouds, as far as we can tell, ours is the first algorithm in literature to determine how close these points lie to an orbit of a representation of a compact Lie group, giving for free a basis of the most likely derived representation in a canonical set of coordinates. The methods developed are fully expressed and implemented for detecting orbits of $\SO(2)$, $\SO(3)$, $\SU(2)$, and their respective products, although they can be applied to other compact Lie groups by simply providing their lists of real irreducible representations. As a secondary goal of this thesis, we aim to provide a review of Lie theory to the machine learning community. Consequently, we present some of the already existing applications of this theory to the field, with a special interest in computer vision, generalized harmonic analysis, physics-related data, and group-oriented machine learning. We subsequently indicate how our algorithm can be incorporated into problems when the symmetries are not too clear, additionally suggesting alternative supervised techniques when data is scarce, or learning is restricted. Finally, we present ideas of further applications of our methods and directions for generalization to non-linear actions, both of which shall be considered in future works.
Link do zoom: https://fgv-br.zoom.us/j/91054693899
Meeting ID: 910 5469 3899
3 de abril de 2023, às 10h