Noisy population recovery in polynomial time

In the noisy population recovery problem of Dvir et al., the goal is to learn an unknown distribution $f$ on binary strings of length $n$ from noisy samples. For some parameter $\mu \in [0,1]$, a noisy sample is generated by flipping each coordinate of a sample from $f$ independently with probability $(1-\mu)/2$. We assume an upper bound $k$ on the size of the support of the distribution, and the goal is to estimate the probability of any string to within some given error $\varepsilon$. It is known that the algorithmic complexity and sample complexity of this problem are polynomially related to each other. We show that for $\mu > 0$, the sample complexity (and hence the algorithmic complexity) is bounded by a polynomial in $k$, $n$ and $1/\varepsilon$ improving upon the previous best result of $\mathsf{poly}(k^{\log\log k},n,1/\varepsilon)$ due to Lovett and Zhang. Our proof combines ideas from Lovett and Zhang with a \emph{noise attenuated} version of M\"{o}bius inversion. In turn, the latter crucially uses the construction of \emph{robust local inverse} due to Moitra and Saks

Yibing Zhang, Piano

Piano Sonata Hob. XVI/50 / Joseph Haydn; Islamey op. 18 / Mily Balakirev; Piano Sonata No. 11 in A major, K. 331 / W. A. Mozart; Rigoletto de Verdi - Paraphrase de Concert, S434 / Franz Lisz

On Real Solutions of the Equation Φ\u3csup\u3e\u3cem\u3et\u3c/em\u3e\u3c/sup\u3e (\u3cem\u3eA\u3c/em\u3e) = 1/\u3cem\u3en\u3c/em\u3e J\u3csub\u3e\u3cem\u3en\u3c/em\u3e\u3c/sub\u3e

For a class of n × n-matrices, we get related real solutions to the matrix equation Φt (A) = 1/n Jn by generalizing the approach of and applying the results of Zhang, Yang, and Cao [SIAM J. Matrix Anal. Appl., 21 (1999), pp. 642–645]. These solutions contain not only those obtained by Zhang, Yang, and Cao but also some which are neither diagonally nor permutation equivalent to those obtained by Zhang, Yang, and Cao. Therefore, the open problem proposed by Zhang, Yang, and Cao in the cited paper is solved

Hadronic effects in leptonic systems: muonium hyperfine structure and anomalous magnetic moment of muon

Contributions of hadronic effects to the muonium physics and anomalous magnetic moment of muon are considered. Special attention is paid to higher-order effects and the uncertainty related to the hadronic contribution to the hyperfine structure interval in the ground state of muonium.Comment: Presented at PSAS 2002 (St. Petersburg