10,007 research outputs found
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 (the conformal coupling between the Hubble parameter and the
inflaton) is less than , 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
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