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Orbit recovery for band-limited functions

Dan Edidin, Matthew Satriano

TL;DR

This work develops a representation-theoretic framework to recover orbits of band-limited vectors under compact classical groups from the third moment, via the bispectrum and higher-order spectra. By defining band-limited representations through highest-weight theory and exploiting tensor-product decompositions, the authors show that, for $SU(n)$ and $SO(2n+1)$ (and related groups), the third moment uniquely determines the $G$-orbit of a generic band-limited function under suitable invertibility conditions on low-band Fourier coefficients. They prove, in particular, that the third moment can recover all Fourier coefficients from the defining representation through a generalized frequency-marching procedure, yielding sharp results for certain groups and practical algorithmic insights. These results have direct implications for orbit recovery problems in MRA and cryo-EM and suggest scalable approaches for finite-dimensional approximations of $L^2(G)$ in applications that involve symmetry and group actions.

Abstract

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.

Orbit recovery for band-limited functions

TL;DR

This work develops a representation-theoretic framework to recover orbits of band-limited vectors under compact classical groups from the third moment, via the bispectrum and higher-order spectra. By defining band-limited representations through highest-weight theory and exploiting tensor-product decompositions, the authors show that, for and (and related groups), the third moment uniquely determines the -orbit of a generic band-limited function under suitable invertibility conditions on low-band Fourier coefficients. They prove, in particular, that the third moment can recover all Fourier coefficients from the defining representation through a generalized frequency-marching procedure, yielding sharp results for certain groups and practical algorithmic insights. These results have direct implications for orbit recovery problems in MRA and cryo-EM and suggest scalable approaches for finite-dimensional approximations of in applications that involve symmetry and group actions.

Abstract

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 . 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 or we prove that the third moment determines the -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 this gives a result relevant to cryo-EM which was our original motivation for studying this problem.
Paper Structure (35 sections, 15 theorems, 51 equations)

This paper contains 35 sections, 15 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Moments of representations of compact groups
  3. Moments and the decomposition into irreducibles
  4. The Fourier transform on compact groups
  5. The regular representation and higher-order spectra
  6. The bispectrum
  7. Banding functions for simple compact Lie groups
  8. Fundamental representations and banding for simply connected groups
  9. Type $A_{n-1}$ - the group $\mathop{\mathrm{SU}}\nolimits(n)$
  10. Type $C_n$- the group $\mathop{\mathrm{Sp}}\nolimits(n)$
  11. The non-simply connected groups $\mathop{\mathrm{SO}}\nolimits(n)$
  12. Type $B_n$ - the group $\mathop{\mathrm{SO}}\nolimits(2n+1,\mathbb{R})$
  13. Type $D_n$ - the group $\mathop{\mathrm{SO}}\nolimits(2n,\mathbb{R})$
  14. Reduction to representations of band one
  15. Reduction to the defining representation
  16. ...and 20 more sections

Key Result

Theorem 1.1

(i) If $G$ is one of the classical groups then the generic band-limited function can be recovered up to translation by a single unitary matrix from its third moment. (ii) The orbit of any generic band-limited function in $L^2(\mathop{\mathrm{SU}}\nolimits(n))$ or real-valued band-limited function in

Theorems & Definitions (33)

  • Theorem 1.1: Informal
  • Remark 1.2
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Example 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 23 more