We study the problem of learning a partially observed matrix under the low rank assumption in the presence of fully observed side information that depends linearly on the true underlying matrix. This problem consists of an important generalization of the Matrix Completion problem, a central problem in Statistics, Operations Research and Machine Learning, that arises in applications such as recommendation systems, signal processing, system identification and image denoising. We formalize this problem as an optimization problem with an objective that balances the strength of the fit of the reconstruction to the observed entries with the ability of the reconstruction to be predictive of the side information. We derive a mixed-projection reformulation of the resulting optimization problem and present a strong semidefinite cone relaxation. We design an efficient, scalable alternating direction method of multipliers algorithm that produces high quality feasible solutions to the problem of interest. Our numerical results demonstrate that in the small rank regime ($k \leq 15$), our algorithm outputs solutions that achieve on average $79\%$ lower objective value and $90.1\%$ lower $\ell_2$ reconstruction error than the solutions returned by the experiment-wise best performing benchmark method. The runtime of our algorithm is competitive with and often superior to that of the benchmark methods. Our algorithm is able to solve problems with $n = 10000$ rows and $m = 10000$ columns in less than a minute.

学习具有部分观察到的矩阵在低秩假设下，与完全观察到的依赖于真实基础矩阵的线性侧向信息的问题。通过设计一种高效的交替方向乘子算法解决了兴趣问题。

部分观测下的预测低秩矩阵学习：混合投影ADMM