Transformed Schatten-1 Iterative Thresholding Algorithms for Low Rank Matrix Completion

We study a non-convex low-rank promoting penalty function, the transformed Schatten-1 (TS1), and its applications in matrix completion. The TS1 penalty, as a matrix quasi-norm defined on its singular values, interpolates the rank and the nuclear norm through a nonnegative parameter a. We consider the unconstrained TS1 regularized low-rank matrix recovery problem and develop a fixed point representation for its global minimizer. The TS1 thresholding functions are in closed analytical form for all parameter values. The TS1 threshold values differ in subcritical (supercritical) parameter regime where the TS1 threshold functions are continuous (discontinuous). We propose TS1 iterative thresholding algorithms and compare them with some state-of-the-art algorithms on matrix completion test problems. For problems with known rank, a fully adaptive TS1 iterative thresholding algorithm consistently performs the best under different conditions with ground truth matrix being multivariate Gaussian at varying covariance. For problems with unknown rank, TS1 algorithms with an additional rank estimation procedure approach the level of IRucL-q which is an iterative reweighted algorithm, non-convex in nature and best in performance

Confronting brane inflation with Planck and pre-Planck data

In this paper, we compare brane inflation models with the Planck data and the pre-Planck data (which combines WMAP, ACT, SPT, BAO and H0 data). The Planck data prefer a spectral index less than unity at more than 5\sigma confidence level, and a running of the spectral index at around 2\sigma confidence level. We find that the KKLMMT model can survive at the level of 2\sigma only if the parameter $\beta$ (the conformal coupling between the Hubble parameter and the inflaton) is less than $\mathcal{O}(10^{-3})$, which indicates a certain level of fine-tuning. The IR DBI model can provide a slightly larger negative running of spectral index and red tilt, but in order to be consistent with the non-Gaussianity constraints from Planck, its parameter also needs fine-tuning at some level.Comment: 10 pages, 8 figure

ℓ1\ell_1-minimization method for link flow correction

A computational method, based on $\ell_1$-minimization, is proposed for the problem of link flow correction, when the available traffic flow data on many links in a road network are inconsistent with respect to the flow conservation law. Without extra information, the problem is generally ill-posed when a large portion of the link sensors are unhealthy. It is possible, however, to correct the corrupted link flows \textit{accurately} with the proposed method under a recoverability condition if there are only a few bad sensors which are located at certain links. We analytically identify the links that are robust to miscounts and relate them to the geometric structure of the traffic network by introducing the recoverability concept and an algorithm for computing it. The recoverability condition for corrupted links is simply the associated recoverability being greater than 1. In a more realistic setting, besides the unhealthy link sensors, small measurement noises may be present at the other sensors. Under the same recoverability condition, our method guarantees to give an estimated traffic flow fairly close to the ground-truth data and leads to a bound for the correction error. Both synthetic and real-world examples are provided to demonstrate the effectiveness of the proposed method