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Abstract:
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Fibre bundle is a natural geometric model for synchronization --- the statistical and computational problem of consistently registering or aligning a collection of objects. Phase synchronization, or the problem of recovering angles from noisy pairwise relative phase measurements, is among the most prototypical instantiations of synchronization problems, and can be modeled geometrically as inverse problems on complex line bundles. In this talk we investigate a "super-resolution" phenomenon for phase synchronization when multiple irreducible representations of the unitary group U(1) are jointly employed in the reconstruction algorithm; geometrically this amounts to considering an analogy of Fourier transform for functions on a principal bundle through a family of associated vector bundles. We will demonstrate the "unreasonable effectiveness" of the proposed algorithm, provide theoretical guarantees, and generalize the framework to synchronization problems over other compact Lie groups.
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