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Guarantees of Riemannian Optimization for Low Rank Matrix Completion
We study the Riemannian optimization methods on the embedded manifold of low
rank matrices for the problem of matrix completion, which is about recovering a
low rank matrix from its partial entries. Assume entries of an
rank matrix are sampled independently and uniformly with replacement. We
first prove that with high probability the Riemannian gradient descent and
conjugate gradient descent algorithms initialized by one step hard thresholding
are guaranteed to converge linearly to the measured matrix provided
\begin{align*} m\geq C_\kappa n^{1.5}r\log^{1.5}(n), \end{align*} where
is a numerical constant depending on the condition number of the
underlying matrix. The sampling complexity has been further improved to
\begin{align*} m\geq C_\kappa nr^2\log^{2}(n) \end{align*} via the resampled
Riemannian gradient descent initialization. The analysis of the new
initialization procedure relies on an asymmetric restricted isometry property
of the sampling operator and the curvature of the low rank matrix manifold.
Numerical simulation shows that the algorithms are able to recover a low rank
matrix from nearly the minimum number of measurements
Micromachined Millimetre-Wave Passive Components at 38 and 77 GHz
A precision micro-fabrication technique has been developed for millimetre-wave components of air-filled three-dimensional structures, such as rectangular coaxial lines or waveguides. The devices are formed by bonding several layers of micromachining defined slices with a thickness of a few hundred micrometres. The slices are thickphotoresist SU8 defined by photolithography, or silicon with a pattern defined by deep reactive ion etching; both are coated with gold by evaporation. The process is simple, and low-cost, as compared with conventional precision metal machining, but yields mm-wave components with good performance. The components are light weight and truly airfilled with no dielectric support. This paper reviews several of these micromachined mm-wave components at 38 and 77 GHz for communications and radar applications
Multidimensional optical fractionation with holographic verification
The trajectories of colloidal particles driven through a periodic potential
energy landscape can become kinetically locked in to directions dictated by the
landscape's symmetries. When the landscape is realized with forces exerted by a
structured light field, the path a given particle follows has been predicted to
depend exquisitely sensitively on such properties as the particle's size and
refractive index These predictions, however, have not been tested
experimentally. Here, we describe measurements of colloidal silica spheres'
transport through arrays of holographic optical traps that use holographic
video microscopy to track individual spheres' motions in three dimensions and
simultaneously to measure each sphere's radius and refractive index with
part-per-thousand resolution. These measurements confirm previously untested
predictions for the threshold of kinetically locked-in transport, and
demonstrate the ability of optical fractionation to sort colloidal spheres with
part-per-thousand resolution on multiple characteristics simultaneously.Comment: 4 pages, 2 figures. Accepted for publication in Physical Review
Letter
Spin-transfer torques in anti-ferromagnetic metals from first principles
In spite of the absence of a macroscopic magnetic moment, an anti-ferromagnet
is spin-polarized on an atomic scale. The electric current passing through a
conducting anti-ferromagnet is polarized as well, leading to spin-transfer
torques when the order parameter is textured, such as in anti-ferromagnetic
non-collinear spin valves and domain walls. We report a first principles study
on the electronic transport properties of anti-ferromagnetic systems. The
current-induced spin torques acting on the magnetic moments are comparable with
those in conventional ferromagnetic materials, leading to measurable angular
resistances and current-induced magnetization dynamics. In contrast to
ferromagnets, spin torques in anti-ferromagnets are very nonlocal. The torques
acting far away from the center of an anti-ferromagnetic domain wall should
facilitate current-induced domain wall motion.Comment: The paper has substantially been rewritten, 4 pages, 5 figure
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