# Group Invariant Dictionary Learning

Symmetries and invariances arise naturally in a wide range of data analytical applications. Based on this observation, it is often desirable to learn data representations that respect such invariances. In this talk, we will describe a framework for the dictionary learning task -- this concerns learning sparse representations for data -- subject to the condition that the learned representation respects some pre-specified invariance. Our framework specializes to the widely studied convolutional dictionary learning problem when we consider integer shifts. Our procedure for learning such dictionaries relies on representing the symmetry as the action of a matrix group acting on the data and subsequently introducing a convex penalty function to induce sparsity with respect to the collection of matrix group elements. Our results suggest that incorporating such invariances as priors are useful when data is limited and when the full spectrum of symmetries are inadequately represented in the data.

*Yong Sheng Soh is an Assistant Professor in the Department of Mathematics at the National University of Singapore. He holds a concurrent appointment as a Research Scientist at the Institute of High Performance Computing, Singapore. He received his Ph.D. in Applied and Computational Mathematics from Caltech (2018) and his B.A. in Mathematics from the University of Cambridge (2011). He was awarded the W.P. Carey & Co. Prize for the best doctoral thesis in Applied Mathematics, the Ben P.C. Chou Prize for the best doctoral thesis in Information Science and Technology, and the 2018 INFORMS Optimization Society Student Paper Prize. His research interests are in mathematical optimization with a focus on applications in the data sciences.*