We formulate the problem of matrix completion with and without side information as a non-convex optimization problem. We design fastImpute based on non-convex gradient descent and show it converges to a global minimum that is guaranteed to recover closely the underlying matrix while it scales to matrices of sizes beyond $10^5 \times 10^5$. We report experiments on both synthetic and real-world datasets that show fastImpute is competitive in both the accuracy of the matrix recovered and the time needed across all cases. Furthermore, when a high number of entries are missing, fastImpute is over $75\%$ lower in MAPE and $15$ times faster than current state-of-the-art matrix completion methods in both the case with side information and without.

翻译：我们以非曲线梯度下降为基础,设计快速投影,并显示它会达到一个全球最低值,保证能够紧紧地恢复基本矩阵,同时将基质缩放到超过10美元5分5秒的大小矩阵上。我们报告关于合成和真实世界数据集的实验,显示快速投影在所回收的矩阵的准确性和所有案例所需的时间方面都具有竞争力。此外,如果缺少大量条目,快速投影在MAPE中要低75美元,比目前最新的矩阵完成方法都快15美元,在有附带信息的情况下和没有附带信息的情况下都要快。