Title: Orbit recovery for band-limited functions Show Chinese title

Title: 带限函数的轨道恢复

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. Show Chinese abstract

Abstract: 我们研究了任意紧李群上函数的三阶矩。 我们利用表示论技术，将经典傅里叶理论中的带限函数概念推广到紧群上的函数 $SU(n), SO(n), Sp(n)$ 上。 然后我们证明，对于一般的带限函数，三阶矩（或者它的傅里叶等价物，双谱）可以唯一确定该函数，至多通过一个酉矩阵的平移。 此外，如果 $G=SU(n)$ 或 $G=SO(2n+1)$，我们证明三阶矩可以确定带限函数的 $G$-轨道。 作为推论，我们得到了这些群的大类有限维表示，其中三阶矩可以确定一般向量的轨道。 当 $G=SO(3)$ 时，这给出了一个与冷冻电镜相关的结果，这是我们最初研究这个问题的动机。