Orbit recovery for band-limited functions

05/31/2023

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by   Dan Edidin, et al.

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We study the third moment for functions on arbitrary compact Lie groups. We use techniques of representation theory to generalize the notion of band-limited functions in classical Fourier theory to functions on the compact groups SU(n), SO(n), Sp(n). We then prove that for generic band-limited functions the third moment or, its Fourier equivalent, the bispectrum determines the function up to translation by a single unitary matrix. Moreover, if G=SU(n) or G=SO(2n+1) we prove that the third moment determines the G-orbit of a band-limited function. As a corollary we obtain a large class of finite-dimensional representations of these groups for which the third moment determines the orbit of a generic vector. When G=SO(3) this gives a result relevant to cryo-EM which was our original motivation for studying this problem.