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Non-unique games over compact groups and orientation estimation in cryo-EM
Inverse Problems ( IF 2.1 ) Pub Date : 2020-05-15 , DOI: 10.1088/1361-6420/ab7d2c
Afonso S Bandeira 1 , Yutong Chen 2 , Roy R Lederman 3 , Amit Singer 4
Affiliation  

Let $\mathcal{G}$ be a compact group and let $f_{ij} \in L^2(\mathcal{G})$. We define the Non-Unique Games (NUG) problem as finding $g_1,\dots,g_n \in \mathcal{G}$ to minimize $\sum_{i,j=1}^n f_{ij} \left( g_i g_j^{-1}\right)$. We devise a relaxation of the NUG problem to a semidefinite program (SDP) by taking the Fourier transform of $f_{ij}$ over $\mathcal{G}$, which can then be solved efficiently. The NUG framework can be seen as a generalization of the little Grothendieck problem over the orthogonal group and the Unique Games problem and includes many practically relevant problems, such as the maximum likelihood estimator} to registering bandlimited functions over the unit sphere in $d$-dimensions and orientation estimation in cryo-Electron Microscopy.

中文翻译:

冷冻电镜中紧凑组上的非唯一博弈和方向估计

令 $\mathcal{G}$ 是一个紧群,令 $f_{ij} \in L^2(\mathcal{G})$。我们将非唯一游戏 (NU​​G) 问题定义为寻找 $g_1,\dots,g_n \in \mathcal{G}$ 以最小化 $\sum_{i,j=1}^n f_{ij} \left( g_i g_j^{-1}\right)$。我们通过对 $\mathcal{G}$ 进行 $f_{ij}$ 的傅立叶变换,将 NUG 问题简化为半定规划 (SDP),然后可以有效地解决该问题。NUG 框架可以被看作是正交群上的小格洛腾迪克问题和唯一博弈问题的推广,包括许多实际相关的问题,例如最大似然估计}在单位球面上注册带限函数在 $d$-冷冻电子显微镜中的尺寸和方向估计。
更新日期:2020-05-15
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