Tamir Bendory

Noam Janco, Tamir Bendory

The group synchronization problem involves estimating a collection of group elements from noisy measurements of their pairwise ratios. This task is a key component in many computational problems, including the molecular reconstruction problem in single-particle cryo-electron microscopy (cryo-EM). The standard methods to estimate the group elements are based on iteratively applying linear and non-linear operators, and are not necessarily optimal. Motivated by the structural similarity to deep neural networks, we adopt the concept of algorithm unrolling, where training data is used to optimize the algorithm. We design unrolled algorithms for several group synchronization instances, including synchronization over the group of 3-D rotations: the synchronization problem in cryo-EM. We also apply a similar approach to the multi-reference alignment problem. We show by numerical experiments that the unrolling strategy outperforms existing synchronization algorithms in a wide variety of scenarios.

Michael Fraiman, Paulina Hoyos, Tamir Bendory et al.

Principal component analysis (PCA) is a fundamental technique for dimensionality reduction and denoising; however, its application to three-dimensional data with arbitrary orientations -- common in structural biology -- presents significant challenges. A naive approach requires augmenting the dataset with many rotated copies of each sample, incurring prohibitive computational costs. In this paper, we extend PCA to 3D volumetric datasets with unknown orientations by developing an efficient and principled framework for SO(3)-invariant PCA that implicitly accounts for all rotations without explicit data augmentation. By exploiting underlying algebraic structure, we demonstrate that the computation involves only the square root of the total number of covariance entries, resulting in a substantial reduction in complexity. We validate the method on real-world molecular datasets, demonstrating its effectiveness and opening up new possibilities for large-scale, high-dimensional reconstruction problems.