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 $L1$ 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 $\mathbf{y}=\lvert A\mathbf{x_0}\rvert+\eta$ where $\mathbf{x_0} \in \mathbb{R}^d$ is the target signal and $\eta\in \mathbb{R}^m$ is a noise vector. The aim of phase retrieval is to estimate $\mathbf{x_0}$ from $\mathbf{y}$. A popular model for estimating $\mathbf{x_0}$ is the nonlinear least square $\widehat{\mathbf{x}}:={\rm argmin}_{\mathbf{x}} \| \lvert A \mathbf{x}\rvert-\mathbf{y}\|_2$. 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 $\|\widehat{\mathbf{x}}-\mathbf{x_0} \|\lesssim {\|\eta\|_2}/{\sqrt{m}}$ under the assumption of $A$ being a Gaussian random matrix. We also prove the reconstruction error ${\|\eta\|_2}/{\sqrt{m}}$ is sharp. For the case where $\mathbf{x_0}$ 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 $\eta$. 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 $\Lambda$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 $s$-order RIP with high probability if one chooses $m\gtrsim s \log^2(s)\log (n)$ rows randomly where $n$ is the vector length. This improves the previously known bound $m \gtrsim s \log^2 s\log^2 n$.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 Ωc\Omega_c

Two-body hadronic weak decays of the charmed baryon $\Omega_c$, 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 $P$-wave amplitudes and current algebra for the $S$-wave amplitudes. Among all the channels decaying into a baryon octet and a pseudoscalar meson, $\Omega_c\to \Xi^0\overline{K}^0$ is the only allowed CF mode. The predicted branching fraction of order $3.8\%$ and large and positive decay asymmetry of order $0.50$ indicate that a measurement of this mode in the near future is promising. Proceeding through purely nonfactorizable contributions, the SCS mode $\Omega_c\to\Lambda^0\overline{K}^0$ and DCS mode $\Omega_c\to\Lambda^0 \eta$ are predicted to have branching fractions as large as $0.8\%$ and $0.4\%$, respectively. The two DCS modes $\Omega_c\to\Sigma^0\eta$ and $\Omega_c\to\Lambda^0\pi^0$ 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