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Group Fairness via Group Consensus
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Abstract We study the statistical estimation problem of orthogonal group synchronization and rotation group synchronization. The model is $$Y_{ij} = Z_i^* Z_j^{*T} + \sigma W_{ij}\in{\mathbb{R}}^{d\times d}$$ where $$W_{ij}$$ is a Gaussian random matrix and $$Z_i^*$$ is either an orthogonal matrix or a rotation matrix, and each $$Y_{ij}$$ is observed independently with probability $$p$$. We analyze an iterative polar decomposition algorithm for the estimation of $Z^*$ and show it has an error of $$(1+o(1))\frac{\sigma ^2 d(d-1)}{2np}$$ when initialized by spectral methods. A matching minimax lower bound is further established that leads to the optimality of the proposed algorithm as it achieves the exact minimax risk.more » « less
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null (Ed.)We first develop some basic facts about hypergeometric sheaves on the multiplicative group [Formula: see text] in characteristic [Formula: see text]. Certain of their Kummer pullbacks extend to irreducible local systems on the affine line in characteristic [Formula: see text]. One of these, of rank [Formula: see text] in characteristic [Formula: see text], turns out to have the Conway group [Formula: see text], in its irreducible orthogonal representation of degree [Formula: see text], as its arithmetic and geometric monodromy groups.more » « less
