In this paper, we bring forward a completely perturbed nonconvex Schatten $p$-minimization to address a model of completely perturbed low-rank matrix recovery. The paper that based on the restricted isometry property generalizes the investigation to a complete perturbation model thinking over not only noise but also perturbation, gives the restricted isometry property condition that guarantees the recovery of low-rank matrix and the corresponding reconstruction error bound. In particular, the analysis of the result reveals that in the case that $p$ decreases $0$ and $a>1$ for the complete perturbation and low-rank matrix, the condition is the optimal sufficient condition $δ_{2r}<1$ \cite{Recht et al 2010}. The numerical experiments are conducted to show better performance, and provides outperformance of the nonconvex Schatten $p$-minimization method comparing with the convex nuclear norm minimization approach in the completely perturbed scenario.
Over the past decades, many individual nonconvex methods have been proposed to achieve better sparse recovery performance in various scenarios. However, how to construct a valid nonconvex regularization function remains open in practice. In this paper, we fill in this gap by presenting a unified framework for constructing the nonconvex regularization based on the probability density function. Meanwhile, a new nonconvex sparse recovery method constructed via the Weibull distribution is studied.
Fast Signal Recovery from Saturated Measurements by Linear Loss and Nonconvex Penalties
Institute of Image Processing and Pattern Recognition, Shanghai Jiao Tong University（上海交通大学图像处理与模式识别研究所）；University of Electronic Science and Technology of China（电子科技大学）；Michigan State University（密歇根州立大学）
In this paper we theoretically study exact recovery of sparse vectors from compressed measurements by minimizing a general nonconvex function that can be decomposed into the sum of single variable functions belonging to a class of smooth nonconvex sparsity promoting functions. Null space property (NSP) and restricted isometry property (RIP) are used as key theoretical tools. The notion of \emph{scale function} associated to a sparsity promoting function is introduced to generalize the state-of-the-art analysis technique of the $l_p$ minimization problem. The analysis is used to derive an upper…
An Optimal Condition of Robust Low-rank Matrices Recovery
本文研究了低秩矩阵核范数最小化的重构条件。我们得到了充分条件$δ_{tr}<t/(4-t)$与$0<t<4/3$，以保证鲁棒重建$(z\neq0)$或精确重建$(z=0)$的所有秩$r$矩阵$X\in\mathbb{R}^{m\times n}$从$b=\mathcal{A}(X)+z$通过核范数最小化。此外，我们不仅证明了当$t=1$时，$δ_r<1/3$的上界与蔡和张\cite{Cai and Zhang}的结果相同，而且证明了所得到的关于恢复误差的上界更好.此外，我们证明了限制等距性条件是尖锐的。此外，数值实验表明，核范数最小化方法对于恢复低秩矩阵是稳定和鲁棒的。
We prove new results about the robustness of well-known convex noise-blind optimization formulations for the reconstruction of low-rank matrices from underdetermined linear measurements. Our results are applicable for symmetric rank-one measurements as used in a formulation of the phase retrieval problem. We obtain these results by establishing that with high probability rank-one measurement operators defined by i.i.d. Gaussian vectors exhibit the so-called Schatten-1 quotient property, which corresponds to a lower bound for the inradius of their image of the nuclear norm (Schatten-1) unit bal…
We study a noisy tensor completion problem of broad practical interest, namely, the reconstruction of a low-rank tensor from highly incomplete and randomly corrupted observations of its entries. While a variety of prior work has been dedicated to this problem, prior algorithms either are computationally too expensive for large-scale applications, or come with sub-optimal statistical guarantees. Focusing on "incoherent" and well-conditioned tensors of a constant CP rank, we propose a two-stage nonconvex algorithm -- (vanilla) gradient descent following a rough initialization -- that achieves th…
Carnegie Mellon University（卡内基梅隆大学）；University of Chicago（芝加哥大学）；Air Force Research Laboratory（美国空军研究实验室）
Tensors, which provide a powerful and flexible model for representing multi-attribute data and multi-way interactions, play an indispensable role in modern data science across various fields in science and engineering. A fundamental task is to faithfully recover the tensor from highly incomplete measurements in a statistically and computationally efficient manner. Harnessing the low-rank structure of tensors in the Tucker decomposition, this paper develops a scaled gradient descent (ScaledGD) algorithm to directly recover the tensor factors with tailored spectral initializations, and shows tha…
Finding the Sparsest Vectors in a Subspace: Theory, Algorithms, and Applications
New York University（纽约大学）；University of Denver（丹佛大学）；Chinese University of Hong Kong（香港中文大学）；ShanghaiTech University（上海科技大学）；Columbia University（哥伦比亚大学）；Johns Hopkins University（约翰斯·霍普金斯大学）
This paper delivers improved theoretical guarantees for the convex programming approach in low-rank matrix estimation, in the presence of (1) random noise, (2) gross sparse outliers, and (3) missing data. This problem, often dubbed as robust principal component analysis (robust PCA), finds applications in various domains. Despite the wide applicability of convex relaxation, the available statistical support (particularly the stability analysis vis-à-vis random noise) remains highly suboptimal, which we strengthen in this paper. When the unknown matrix is well-conditioned, incoherent, and of co…
