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A note on the almost one half holomorphic pinching
Motivated by a previous work of Zheng and the second named author, we study
pinching constants of compact K\"ahler manifolds with positive holomorphic
sectional curvature. In particular we prove a gap theorem following the work of
Petersen and Tao on Riemannian manifolds with almost quarter-pinched sectional
curvature.Comment: 6 pages. This is the version which the authors submitted to a journal
for consideration for publication in June 2017. The reference has not been
updated since the
Poisson Matrix Completion
We extend the theory of matrix completion to the case where we make Poisson
observations for a subset of entries of a low-rank matrix. We consider the
(now) usual matrix recovery formulation through maximum likelihood with proper
constraints on the matrix , and establish theoretical upper and lower bounds
on the recovery error. Our bounds are nearly optimal up to a factor on the
order of . These bounds are obtained by adapting
the arguments used for one-bit matrix completion \cite{davenport20121}
(although these two problems are different in nature) and the adaptation
requires new techniques exploiting properties of the Poisson likelihood
function and tackling the difficulties posed by the locally sub-Gaussian
characteristic of the Poisson distribution. Our results highlight a few
important distinctions of Poisson matrix completion compared to the prior work
in matrix completion including having to impose a minimum signal-to-noise
requirement on each observed entry. We also develop an efficient iterative
algorithm and demonstrate its good performance in recovering solar flare
images.Comment: Submitted to IEEE for publicatio
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