43,472 research outputs found
Divide-and-Conquer Method for L1 Norm Matrix Factorization in the Presence of Outliers and Missing Data
The low-rank matrix factorization as a L1 norm minimization problem has
recently attracted much attention due to its intrinsic robustness to the
presence of outliers and missing data. In this paper, we propose a new method,
called the divide-and-conquer method, for solving this problem. The main idea
is to break the original problem into a series of smallest possible
sub-problems, each involving only unique scalar parameter. Each of these
subproblems is proved to be convex and has closed-form solution. By recursively
optimizing these small problems in an analytical way, efficient algorithm,
entirely avoiding the time-consuming numerical optimization as an inner loop,
for solving the original problem can naturally be constructed. The
computational complexity of the proposed algorithm is approximately linear in
both data size and dimensionality, making it possible to handle large-scale L1
norm matrix factorization problems. The algorithm is also theoretically proved
to be convergent. Based on a series of experiment results, it is substantiated
that our method always achieves better results than the current
state-of-the-art methods on matrix factorization calculation in both
computational time and accuracy, especially on large-scale applications such as
face recognition and structure from motion.Comment: 19 pages, 2 figures, 2 table
Global Existence of Solution for a Nonlinear Size-structured Population Model with Distributed Delay in the Recruitment
In this paper we study a nonlinear size-structured population model with
distributed delay in the recruitment. The delayed problem is reduced into an
abstract initial value problem of an ordinary differential equation in the
Banach space by using the delay semigroup techniques. The local existence and
uniqueness of solution as well as the continuous dependence on initial
conditions are obtained by using the general theory of quasi-linear evolution
equations in nonreflexive Banach spaces, while the global existence of solution
is obtained by the estimates of the solution and the extension theorem.Comment: 19 page
The estimation performance of nonlinear least squares for phase retrieval
Suppose that where is the target signal and is a noise
vector. The aim of phase retrieval is to estimate from
. A popular model for estimating is the nonlinear
least square . One already develops many efficient
algorithms for solving the model, such as the seminal error reduction
algorithm. In this paper, we present the estimation performance of the model
with proving that under the assumption of being a Gaussian random
matrix. We also prove the reconstruction error is
sharp. For the case where is sparse, we study the estimation
performance of both the nonlinear Lasso of phase retrieval and its
unconstrained version. Our results are non-asymptotic, and we do not assume any
distribution on the noise . To the best of our knowledge, our results
represent the first theoretical guarantee for the nonlinear least square and
for the nonlinear Lasso of phase retrieval.Comment: 22 page
Scalar perturbation of the viscosity dark fluid cosmological model
A general equation of state is used to model unified dark matter and dark
energy (dark fluid), and it has been proved that this model is equivalent to a
single fluid with time-dependent bulk viscosity. In this paper, we investigate
scalar perturbation of this viscosity dark fluid model. For particular
parameter selection, we find that perturbation quantity can be obtained exactly
in the future universe. We numerically solve the perturbation evolution
equations, and compare the results with those of CDM model.
Gravitational potential and the density perturbation of the model studied here
have the similar behavior with the standard model, though there exists
significant value differences in the late universe.Comment: 11 pages, 7 figure
Rational Solitons in the Parity-Time-Symmetric Nonlocal Nonlinear Schr\"{o}dinger Model
In this paper, via the generalized Darboux transformation, rational soliton
solutions are derived for the parity-time-symmetric nonlocal nonlinear
Schr\"{o}dinger (NLS) model with the defocusing-type nonlinearity. We find that
the first-order solution can exhibit the elastic interactions of rational
antidark-antidark, dark-antidark, and antidark-dark soliton pairs on a
continuous wave background, but there is no phase shift for the interacting
solitons. Also, we discuss the degenerate case in which only one rational dark
or antidark soliton survives. Moreover, we reveal that the second-order
rational solution displays the interactions between two solitons with
combined-peak-valley structures in the near-field regions, but each interacting
soliton vanishes or evolves into a rational dark or antidark soliton as |z|\ra
\infty. In addition, we numerically examine the stability of the first- and
second-order rational soliton solutions.Comment: 18 pages, 9 figures, 1 tabl
A recursive divide-and-conquer approach for sparse principal component analysis
In this paper, a new method is proposed for sparse PCA based on the recursive
divide-and-conquer methodology. The main idea is to separate the original
sparse PCA problem into a series of much simpler sub-problems, each having a
closed-form solution. By recursively solving these sub-problems in an
analytical way, an efficient algorithm is constructed to solve the sparse PCA
problem. The algorithm only involves simple computations and is thus easy to
implement. The proposed method can also be very easily extended to other sparse
PCA problems with certain constraints, such as the nonnegative sparse PCA
problem. Furthermore, we have shown that the proposed algorithm converges to a
stationary point of the problem, and its computational complexity is
approximately linear in both data size and dimensionality. The effectiveness of
the proposed method is substantiated by extensive experiments implemented on a
series of synthetic and real data in both reconstruction-error-minimization and
data-variance-maximization viewpoints.Comment: 35 pages, 4 figure
Improved bounds for the RIP of Subsampled Circulant matrices
In this paper, we study the restricted isometry property of partial random
circulant matrices. For a bounded subgaussian generator with independent
entries, we prove that the partial random circulant matrices satisfy -order
RIP with high probability if one chooses rows
randomly where is the vector length. This improves the previously known
bound .Comment: 9 page
On the Performance of Sparse Recovery via L_p-minimization (0<=p <=1)
It is known that a high-dimensional sparse vector x* in R^n can be recovered
from low-dimensional measurements y= A^{m*n} x* (m<n) . In this paper, we
investigate the recovering ability of l_p-minimization (0<=p<=1) as p varies,
where l_p-minimization returns a vector with the least l_p ``norm'' among all
the vectors x satisfying Ax=y. Besides analyzing the performance of strong
recovery where l_p-minimization needs to recover all the sparse vectors up to
certain sparsity, we also for the first time analyze the performance of
``weak'' recovery of l_p-minimization (0<=p<1) where the aim is to recover all
the sparse vectors on one support with fixed sign pattern. When m/n goes to 1,
we provide sharp thresholds of the sparsity ratio that differentiates the
success and failure via l_p-minimization. For strong recovery, the threshold
strictly decreases from 0.5 to 0.239 as p increases from 0 to 1. Surprisingly,
for weak recovery, the threshold is 2/3 for all p in [0,1), while the threshold
is 1 for l_1-minimization. We also explicitly demonstrate that l_p-minimization
(p<1) can return a denser solution than l_1-minimization. For any m/n<1, we
provide bounds of sparsity ratio for strong recovery and weak recovery
respectively below which l_p-minimization succeeds with overwhelming
probability. Our bound of strong recovery improves on the existing bounds when
m/n is large. Regarding the recovery threshold, l_p-minimization has a higher
threshold with smaller p for strong recovery; the threshold is the same for all
p for sectional recovery; and l_1-minimization can outperform l_p-minimization
for weak recovery. These are in contrast to traditional wisdom that
l_p-minimization has better sparse recovery ability than l_1-minimization since
it is closer to l_0-minimization. We provide an intuitive explanation to our
findings and use numerical examples to illustrate the theoretical predictions
The Limits of Error Correction with lp Decoding
An unknown vector f in R^n can be recovered from corrupted measurements y =
Af + e where A^(m*n)(m>n) is the coding matrix if the unknown error vector e is
sparse. We investigate the relationship of the fraction of errors and the
recovering ability of lp-minimization (0 < p <= 1) which returns a vector x
minimizing the "lp-norm" of y - Ax. We give sharp thresholds of the fraction of
errors that determine the successful recovery of f. If e is an arbitrary
unknown vector, the threshold strictly decreases from 0.5 to 0.239 as p
increases from 0 to 1. If e has fixed support and fixed signs on the support,
the threshold is 2/3 for all p in (0, 1), while the threshold is 1 for
l1-minimization.Comment: 5 pages, 1 figure. ISIT 201
Hadronic weak decays of the charmed baryon
Two-body hadronic weak decays of the charmed baryon , including
Cabibbo-favored (CF), singly Cabibbo-suppressed (SCS) and doubly
Cabibbo-suppressed (DCS) modes, are studied systematically in this work. To
estimate nonfactorizable contributions, we work in the pole model for the
-wave amplitudes and current algebra for the -wave amplitudes. Among all
the channels decaying into a baryon octet and a pseudoscalar meson,
is the only allowed CF mode. The predicted
branching fraction of order and large and positive decay asymmetry of
order indicate that a measurement of this mode in the near future is
promising. Proceeding through purely nonfactorizable contributions, the SCS
mode and DCS mode are predicted to have branching fractions as large as and
, respectively. The two DCS modes and
are suggested to serve as new physics searching
channels for their vanishing SM background.Comment: V1: 20 pages, 1 figure, 5 tables. arXiv admin note: text overlap with
arXiv:2001.04553, arXiv:1910.13626; V2: version accepted by PRD, 24 pages,
references added, footnotes adde
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