Orbit recovery for band-limited functions
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866910547268599808 |
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| author | Edidin, Dan Satriano, Matthew |
| author_facet | Edidin, Dan Satriano, Matthew |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_00155 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Orbit recovery for band-limited functions Edidin, Dan Satriano, Matthew Information Theory Representation Theory 94A12, 22D10 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. |
| title | Orbit recovery for band-limited functions |
| topic | Information Theory Representation Theory 94A12, 22D10 |
| url | https://arxiv.org/abs/2306.00155 |