Principal Component Analysis (PCA) is a well-established technique for dimensionality reduction in statistics and machine learning. However, its well-known limitations include sensitivity to outliers and an inability to handle incomplete data which is quite common in real-world datasets. To address these two limitations simultaneously, the problem of Robust Matrix Completion (RMC) has been proposed. In this study, we introduce a deep-learning-augmented approach to RMC, called Learned Robust Matrix Completion (LRMC), through a process known as Learning to Optimize or Deep Unfolding.

 The method of Deep Unfolding unrolls a conventional iterative algorithm as a deep neural network (DNN) where each iteration becomes a layer of the DNN. The hyperparameters of the original algorithm (i.e., stepsizes, thresholding values, etc.) then become the trainable parameters (i.e., weights) of the DNN. This approach allows us to optimize the performance of the proposed algorithm for a specific class of RMC problems through the techniques of DNN training. Through extensive empirical experiments on synthetic datasets and real-world applications, we demonstrate that LRMC outperforms state-of-the-art methods, suggesting its potential as an attractive choice for addressing PCA problems with both outliers and missing entries.

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