Orbit recovery for band-limited functions (Q6594423)

The main result of the paper states that if \(G\) is one of the classical (compact) groups then a generic band-limited function can be recovered up to translation by a single unitary matrix from its third moment. Here, the orbit of any generic band-limited function in \(L^2(SU(n))\) or real-valued band-limited function in \(L^2(SO(2n+1))\) can be recovered from its third moment.\N\NFor unitary representation \(V\) of compact group \(G\), the \(d\)th moment function \(m_d:V\to \underbrace{V\otimes\cdots\otimes V}_{d-1\,\text{ times}}\otimes\ V^\ast\) is\N\[\Nm_c(f)=\int_G \underbrace{g\cdot f\otimes \cdots \otimes g\cdot f}_{d -1\text{ times}}\otimes\ \overline{g\cdot f}\, dg\, .\N\]\NWhen \(f:D\to\mathbb{C}\) is defined as a function on some domain \(D\subset G\) on which \(G\) acts ,then \(m_d\) is a function from \(D^d\to\mathbb{C}\) and \(g\cdot f(x)=f(g^{-1} x)\).\N\NAfter a discussion of moments in the context of irreducible representations and quantities that are determined by the first and second moments, the Fourier transform is reviewed in the context of finite groups. If \(D_V\) is the unitary transformation defined by \(v\to gv\) then the matrix \(F(f)(V)=\int_G f(g) D_V(g)^\ast\, dg\in \text{ End}(V)\) is the Fourier coefficient of \(f\) at \(V\). The inverse Fourier transform of \(T\in \text{End}(V)\) is \(\frac{1}{\dim}(V) \text{ Tr} (T D_V(g)^\ast)\). Higher order spectra for \(V=L^2(G^{d-1})\) are defined as follows: The \((d-1)\)st higher spectrum \(a_d(f)\) is defined as the Fourier transform of \(m_d(f)\in L^2(G^{d-1})\). The bispectrum (corresponding to \(d=3\)) is\N\[\Na_2(f)(V\otimes W)=[F(f)(V)\otimes F(f)(W)][F(f)(V\otimes W)^\ast]\, .\N\]\NThe bispectrum is further described in terms of decomposition of \(V\otimes W\) into irreducible representations and the following lemma is proved: if \(V,W\) are irreducibles and \(V\otimes W\simeq V_1\oplus\cdots\oplus V_r\) is a direct sum of irreducibles, if \(f\in L^2(G)\) has invertible Fourier coefficients \(F(f)(V)\) and \(F(f)(W)\) then for each summand, \(F(f)(V_i)\) is determined by \(F(f)(V)\), \(F(f)(W)\) and \(a_2(f)(V\otimes W)\).\N\NBand-limited functions for compact Lie groups are defined in terms of decompositions of irreducible representations in terms of weight spaces. This decomposition is reviewed (Sect.~3.0.1). When \(G\) is simply connected, the band of \(V_\lambda\) is defined as the sum of the weights \(a_i\) of the so-called highest weight vector \(\lambda=a_1\omega_1+\dots+a_n\omega_n\). The concept is reviewed concretely in the contexts of \(SU(n)\), \(Sp(n)\) and \(SO(n)\) and explicitly in \(SO(2n+1,\mathbb{R})\) and \(SO(2n,\mathbb{R})\).\N\NIn the case of a general compact Lie group whose irreducible representations can be banded, the lemma yields that if \(f\in L^2(G)\) and \(W\) is irreducible with \(b>1\), then \(F(f)(W)\) is determined from the bispectrum coefficients \(a_2(f)(W_1\otimes W_{b-1})\) along with \(F(f)(W_1)\) and \(F(f)(W_{b-1})\) for some \(W_1\) of band one and \(W_{b-1}\) of band \(b-1\), provided these coefficients are invertible (Prop.~3.5).\N\NThe irreducible representation corresponding to weight \(\omega_1\) above is called the \emph{defining representation}. For the cases of \(SU(n)\), \(Sp(n)\) and \(SO(n)\), a case analysis is done to show that the Fourier coefficients of any band-one representation \(V_\ell\) are determined by those of the defining representation, denoted \(V_1\), and the bispectra of \(V_1\otimes V_k\) where \(k<\ell\) above (Thm.~3.7). It is shown in the cases \(G=SU(n)\) or \(G=SO(2n+1)\), if \(f\in L^2(G)\) is band-limited with band \(b\geq 1\) and the Fourier coefficients of all irreducible representations whose bands are at most \(\lceil b/2\rceil\) are invertible, then the \(G\)-orbit of \(f\) is determined by its bispectrum (Thm.~4.1.).\N\NExamples (and counterexamples) for specific classical groups and specific \(n\), and applications are presented in Sect.~5 along with comparisons with earlier related results. The case of \(SO(3)\) is discussed in some detail, pointing out the relevance of results here in the context of cryo-EM.