当前位置: X-MOL 学术Appl. Math. Comput. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
Robust recovery of low-rank matrices with non-orthogonal sparse decomposition from incomplete measurements
Applied Mathematics and Computation ( IF 4 ) Pub Date : 2021-03-01 , DOI: 10.1016/j.amc.2020.125702
Massimo Fornasier , Johannes Maly , Valeriya Naumova

We consider the problem of recovering an unknown effectively $(s_1,s_2)$-sparse low-rank-$R$ matrix $X$ with possibly non-orthogonal rank-$1$ decomposition from incomplete and inaccurate linear measurements of the form $y = \mathcal A (X) + \eta$, where $\eta$ is an ineliminable noise. We first derive an optimization formulation for matrix recovery under the considered model and propose a novel algorithm, called Alternating Tikhonov regularization and Lasso (A-T-LA$\text{S}_{2,1}$), to solve it. The algorithm is based on a multi-penalty regularization, which is able to leverage both structures (low-rankness and sparsity) simultaneously. The algorithm is a fast first order method, and straightforward to implement. We prove global convergence for any linear measurement model to stationary points and local convergence to global minimizers. By adapting the concept of restricted isometry property from compressed sensing to our novel model class, we prove error bounds between global minimizers and ground truth, up to noise level, from a number of subgaussian measurements scaling as $R(s_1+s_2)$, up to log-factors in the dimension, and relative-to-diameter distortion. Simulation results demonstrate both the accuracy and efficacy of the algorithm, as well as its superiority to the state-of-the-art algorithms in strong noise regimes and for matrices, whose singular vectors do not possess exact (joint-) sparse support.

中文翻译:

用非正交稀疏分解从不完整测量中稳健恢复低秩矩阵

我们考虑从 $(s_1,s_2)$-sparse low-rank-$R$ 矩阵 $X$ 和可能的非正交秩-$1$ 分解有效地恢复未知数的问题,该分解来自 $y 形式的不完整和不准确的线性测量= \mathcal A (X) + \eta$,其中 $\eta$ 是无法消除的噪音。我们首先在所考虑的模型下推导出矩阵恢复的优化公式,并提出一种新算法,称为交替 Tikhonov 正则化和套索 (AT-LA$\text{S}_{2,1}$),以解决它。该算法基于多重惩罚正则化,能够同时利用两种结构(低秩和稀疏性)。该算法是一种快速的一阶方法,并且易于实现。我们证明了任何线性测量模型的全局收敛到静止点和局部收敛到全局极小值。通过将受限等距属性的概念从压缩感知应用到我们的新模型类,我们证明了全局最小化器和地面实况之间的误差界限,直到噪声水平,从许多次高斯测量缩放为 $R(s_1+s_2)$,尺寸中的对数因子,以及相对于直径的失真。仿真结果证明了该算法的准确性和有效性,以及它在强噪声范围和矩阵中优于最先进算法的优越性,矩阵的奇异向量不具有精确的(联合)稀疏支持。尺寸中的对数因子,以及相对于直径的失真。仿真结果证明了该算法的准确性和有效性,以及它在强噪声范围和矩阵中优于最先进算法的优越性,矩阵的奇异向量不具有精确的(联合)稀疏支持。尺寸中的对数因子,以及相对于直径的失真。仿真结果证明了该算法的准确性和有效性,以及它在强噪声范围和矩阵中优于最先进算法的优越性,矩阵的奇异向量不具有精确的(联合)稀疏支持。
更新日期:2021-03-01
down
wechat
bug