Tensor Completion via Tensor Train Based Low-Rank Quotient Geometry under a Preconditioned Metric

This paper investigates the low-rank tensor completion problem, which is about recovering a tensor from partially observed entries. We consider this problem in the tensor train format and extend the preconditioned metric from the matrix case to the tensor case. The first-order and second-order quotient geometry of the manifold of fixed tensor train rank tensors under this metric is studied in detail. Algorithms, including Riemannian gradient descent, Riemannian conjugate gradient, and Riemannian Gauss-Newton, have been proposed for the tensor completion problem based on the quotient geometry. It has also been shown that the Riemannian Gauss-Newton method on the quotient geometry is equivalent to the Riemannian Gauss-Newton method on the embedded geometry with a specific retraction. Empirical evaluations on random instances as well as on function-related tensors show that the proposed algorithms are competitive with other existing algorithms in terms of recovery ability, convergence performance, and reconstruction quality.Comment: The manuscript has been adjusted in several place

5-Methyl­isoxazole-4-carboxylic acid

In the title compound, C5H5NO3, the mol­ecule lies on a crystallographic mirror plane with one half-mol­ecule in the asymmetric unit. An intramolecular C—H⋯O inter­action is present. In the crystal, strong inter­molecular O—H⋯N hydrogen bonds result in the formation of a linear chain structure along [100], and there are also weak C—H⋯O hydrogen bonds between the chains which help to stabilize the crystal packing

Optical absorption properties of Ge2–44 and P-doped Ge nanoparticles

The optical absorption properties of non-crystalline and crystalline Ge nanoparticles with the sizes from ∼2.5 to 15 Å have been studied at the B3LYP/6-31G level using time-dependent density functional theory. Hydrogen passivation and phosphorus doping on some selected Ge nanoparticles were also calculated. With the increase of cluster size, the optical absorption spectra of the non-crystalline Ge nanoparticles change from many peaks to a continuous broad band and at the same time exhibit a systematic red-shift. Doping phosphorus also causes the absorption spectra to shift toward the lower energy region for both non-crystalline and crystalline Ge nanoparticles. The non-crystalline Ge nanoparticles are found to have stronger absorption in the visible region in comparison with the crystalline ones, regardless phosphorus doping

New progress of ranging technology at Wuhan Satellite Laser Ranging Station

A satellite laser ranging system with an accuracy of the level of centimeter has been successfully developed at the Institute of Seismology, State Seismological Bureau with the cooperation of the Institute of Geodesy and Geophysics, Chinese Academy of Science. With significant improvements on the base of the second generation SLR system developed in 1985, ranging accuracy of the new system has been upgraded from 15 cm to 3-4 cm. Measuring range has also been expanded, so that the ETALON satellite with an orbit height of 20,000 km launched by the former U.S.S.R. can now be tracked. Compared with the 2nd generation SLR system, the newly developed system has the following improvements. A Q modulated laser is replaced by a mode-locked YAG laser. The new device has a pulse width of 150 ps and a repetition rate of 1-4 pps. A quick response photomultiplier has been adopted as the receiver for echo; for example, the adoption of the MCP tube has obviously reduced the jitter error of the transit time and has improved the ranging accuracy. The whole system is controlled by an IBM PC/XT Computer to guide automatic tracking and measurement. It can carry out these functions for satellite orbit calculation, real-time tracking and adjusting, data acquisition and the preprocessed of observing data, etc. The automatization level and reliability of the observation have obviously improved