Learning problems augmented with latent symmetries have attracted considerable interest in recent years. A significant class of such problems arises in experiments where a system is constrained to evolve in accordance with the rigid laws of nature, such as the celebrated technique of cryo electron microscopy (Cryo-EM). The constraint of such latent symmetries, given by group invariances or equivariances, precludes the possibility of having many repeated measurements of the exact same object, and poses a fundamental challenge for learning a signal in the presence of ambient noise. We will start with a gentle introduction to the problem of learning under latent symmetries, and explore its intriguing connections with a range of disparate topics — invariant theory, harmonic analysis, compressive sensing and Gaussian calculus. We will subsequently specialise to the Multi Reference Alignment (MRA) model, and explore the fundamental aspects of the recovery problem (such as sample complexity) in the presence of structural constraints on the signal (such as sparsity). In particular, we unveil a novel quartic dependence on noise level for the sample complexity of sparse MRA, leveraging a range of mathematical tools from uncertainty principles of Fourier analysis to techniques from combinatorial optimisation.

Based in part on the following works :

[1] Sparse Multi-Reference Alignment: Phase Retrieval, Uniform Uncertainty Principles and the Beltway Problem, S. Ghosh and P. Rigollet, Foundations of Computational Mathematics, 23(5), pp.1851-1898 (2023)

[2] Minimax-optimal estimation for sparse multi-reference alignment with collision-free signals, S. Ghosh, S.S. Mukherjee, J.B. Pan, arXiv preprint <arXiv:2312.07839>