A transversality theorem for semi-algebraic sets with application to signal recovery from the second moment and cryo-EM

TL;DR

This paper establishes a transversality theorem for semi-algebraic sets under group actions, enabling unique signal recovery from second moment measurements, with applications to cryo-EM and other inverse problems.

Contribution

It proves a general transversality theorem for semi-algebraic sets in group representations and applies it to signal recovery problems like cryo-EM, providing explicit bounds and theoretical insights.

Findings

Derived explicit bounds for molecular structure recovery from second moments.

Established theoretical limits for cryo-EM, factoring Gram matrices, and phase retrieval.

Provided bounds for permutation-invariant machine learning separators.

Abstract

Semi-algebraic priors are ubiquitous in signal processing and machine learning. Prevalent examples include a) linear models where the signal lies in a low-dimensional subspace; b) sparse models where the signal can be represented by only a few coefficients under a suitable basis; and c) a large family of neural network generative models. In this paper, we prove a transversality theorem for semi-algebraic sets in orthogonal or unitary representations of groups: with a suitable dimension bound, a generic translate of any semi-algebraic set is transverse to the orbits of the group action. This, in turn, implies that if a signal lies in a low-dimensional semi-algebraic set, then it can be recovered uniquely from measurements that separate orbits. As an application, we consider the implications of the transversality theorem to the problem of recovering signals that are translated by random group actions from their second moment. As a special case, we discuss cryo-EM: a leading technology to constitute the spatial structure of biological molecules, which serves as our prime motivation. In particular, we derive explicit bounds for recovering a molecular structure from the second moment under a semi-algebraic prior and deduce information-theoretic implications. We also obtain information-theoretic bounds for three additional applications: factoring Gram matrices, multi-reference alignment, and phase retrieval. Finally, we deduce bounds for designing permutation invariant separators in machine learning.