Title: Exploiting Group and Geometric Structures for Massive Data Analysis
Abstract: In the first part of the talk, I will introduce a new formulation for the statistical problem of estimating the alignment angles from noisy pairwise measurements as a nonconvex optimization problem that enforces consistency among the pairwise comparisons in multiple frequency channels. Inspired by harmonic retrieval in signal processing, we develop a simple yet efficient two-stage algorithm that leverages the multi-frequency information. We demonstrate in theory and practice that the proposed algorithm significantly outperforms state-of-the-art phase synchronization algorithms, at a computational costs incurred by using the extra frequency channels. The algorithmic framework can be generalized to general synchronization problems over compact Lie groups.
For the second part of the talk, we extend the idea of using multi-frequency channels to manifold data. We utilize a representation theoretic mechanism that canonically associates multiple independent vector bundles over a common base manifold, which provides multiple views for the geometry of the underlying manifold. The consistency across these group transformations provides a common base for performing unsupervised manifold learning through the redundancy created artificially across irreducible representations of the transformation group. I will show the application in cryo-electron microscopy image processing.
