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- W3201418504 abstract "We study the phase synchronization problem with measurements <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>${Y}= {z}^{ast} {z}^{ast{mathrm {H}}}+sigma {W}in mathbb {C}^{n}times {n}$ </tex-math></inline-formula> , where <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>${z}^{ast}$ </tex-math></inline-formula> is an <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>${n}$ </tex-math></inline-formula> -dimensional complex unit-modulus vector and <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>${W}$ </tex-math></inline-formula> is a complex-valued Gaussian random matrix. It is assumed that each entry <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>${Y}_{jk}$ </tex-math></inline-formula> is observed with probability <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>${p}$ </tex-math></inline-formula> . We prove that the minimax lower bound of estimating <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>${z}^{ast}$ </tex-math></inline-formula> under the squared <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$ell _{2}$ </tex-math></inline-formula> loss is <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$(1- {o}(1))frac {sigma ^{2}}{2p}$ </tex-math></inline-formula> . We also show that both generalized power method and maximum likelihood estimator achieve the error bound <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$(1+ {o}(1))frac {sigma ^{2}}{2p}$ </tex-math></inline-formula> . Thus, <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$frac {sigma ^{2}}{2p}$ </tex-math></inline-formula> is the exact asymptotic minimax error of the problem. Our upper bound analysis involves a precise characterization of the statistical property of the power iteration. The lower bound is derived through an application of van Trees’ inequality." @default.
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- W3201418504 date "2021-12-01" @default.
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- W3201418504 title "Exact Minimax Estimation for Phase Synchronization" @default.
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- W3201418504 doi "https://doi.org/10.1109/tit.2021.3112712" @default.
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