分野別セミナー

Abstract: Motivated by cutting-edge applications like cryo-electron microscopy (cryo-EM), learning problems in the presence of latent symmetries has gained salience in recent years. Such latent symmetries preclude the possibility of having many repeated measurements of the exact same object, and pose a fundamental challenge for statistical learning. We will start with a gentle introduction to the problem of learning under latent symmetries, focussing on the setting of Multi Reference Alignment (MRA). Despite significant interest, a clear picture for understanding rates of estimation in this model has emerged only recently, especially in the practically important regime of high ambient noise (sigma >> 1), where the best known results exhibit an asymptotic sample complexity of order sigma^6, whereas in the absence of latent symmetries it is known to be of order sigma^2. In recent work, we investigate this problem for sparse signals, where we unveil a remarkable intermediate sample complexity of order sigma^4. Our results explore and exploit connections to classical topics, such as crystallographic phase retrieval, the beltway problem from combinatorial optimization, and uniform uncertainty principles from harmonic analysis. Based on joint work with P. Rigollet.

Ref: [1] Sparse Multi-Reference Alignment: Phase Retrieval, Uniform Uncertainty Principles and the Beltway Problem,
S. Ghosh and P. Rigollet,
Foundations of Computational Math. (2022).
https://doi.org/10.1007/s10208-022-09584-6