Calibrated Elastic Regularization in Matrix Completion
Calibrated Elastic Regularization in Matrix Completion
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发表时间：
2012-11
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通讯作者：
Tingni Sun;Cun-Hui Zhang
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作者：
Tingni Sun;Cun-Hui Zhang
本文研究矩阵补全问题，即从指标的一个小子集的观测值估计一个矩阵。我们提出了一种校正谱弹性网方法，其中包含核惩罚和Frobenius惩罚，并开发了一种迭代算法来解决凸最小化问题。迭代算法交替进行两种算法，一种是通过当前猜测来估算不完整矩阵中的缺失项，另一种是通过对估算矩阵进行缩放软阈值奇异值分解来估计矩阵，直到得到的矩阵收敛。校正步骤是为了校正由Frobenius罚引起的偏置。在适当的相干条件和适当的惩罚水平下，我们证明了所提出的估计器达到了接近最优阶的误差界，并且与噪声水平成比例。这提供了对有噪声和无噪声矩阵补全问题的统一分析。最后给出了仿真结果，与前人的方案进行了比较。
This paper concerns the problem of matrix completion, which is to estimate a matrix from observations in a small subset of indices. We propose a calibrated spectrum elastic net method with a sum of the nuclear and Frobenius penalties and develop an iterative algorithm to solve the convex minimization problem. The iterative algorithm alternates between imputing the missing entries in the incomplete matrix by the current guess and estimating the matrix by a scaled soft-thresholding singular value decomposition of the imputed matrix until the resulting matrix converges. A calibration step follows to correct the bias caused by the Frobenius penalty. Under proper coherence conditions and for suitable penalties levels, we prove that the proposed estimator achieves an error bound of nearly optimal order and in proportion to the noise level. This provides a unified analysis of the noisy and noiseless matrix completion problems. Simulation results are presented to compare our proposal with previous ones.