Stability of matrix factorization for collaborative filtering

We study the stability vis a vis adversarial noise of matrix factorization algorithm for matrix completion. In particular, our results include: (I) we bound the gap between the solution matrix of the factorization method and the ground truth in terms of root mean square error; (II) we treat the matrix factorization as a subspace fitting problem and analyze the difference between the solution subspace and the ground truth; (III) we analyze the prediction error of individual users based on the subspace stability. We apply these results to the problem of collaborative filtering under manipulator attack, which leads to useful insights and guidelines for collaborative filtering system design.Comment: ICML201

Maximal Quantum Fisher Information in a Mach-Zehnder Interferometer without initial parity

Mach-Zehnder interferometer is a common device in quantum phase estimation and the photon losses in it are an important issue for achieving a high phase accuracy. Here we thoroughly discuss the precision limit of the phase in the Mach-Zehnder interferometer with a coherent state and a superposition of coherent states as input states. By providing a general analytical expression of quantum Fisher information, the phase-matching condition and optimal initial parity are given. Especially, in the photon loss scenario, the sensitivity behaviors are analyzed and specific strategies are provided to restore the phase accuracies for symmetric and asymmetric losses.Comment: 10 pages, 3 figure

Chandra Observation of a Weak Shock in the Galaxy Cluster A2556

Based on a 21.5 ks \chandra\ observation of A2556, we identify an edge on the surface brightness profile (SBP) at about 160$h_{71}^{-1}$ kpc northeast of the cluster center, and it corresponds to a shock front whose Mach number $\mathcal{M}$ is calculated to be $1.25_{-0.03}^{+0.02}$. No prominent substructure, such as sub-cluster, is found in either optical or X-ray band that can be associated with the edge, suggesting that the conventional super-sonic motion mechanism may not work in this case. As an alternative solution, we propose that the nonlinear steepening of acoustic wave, which is induced by the turbulence of the ICM at the core of the cluster, can be used to explain the origin of the shock front. Although nonlinear steepening weak shock is expected to occur frequently in clusters, why it is rarely observed still remains a question that requires further investigation, including both deeper X-ray observation and extensive theoretical studies.Comment: 15 pages, 4 figures, accepted by Ap