Supervised learning with incomplete data via sparse representations

This paper addresses the problem of training a classifier on incomplete data and its application to a complete or incomplete test dataset. A supervised learning method is developed to train a general classifier, such as a logistic regression or a deep neural network, using only a limited number of observed entries, assuming sparse representations of data vectors on an unknown dictionary. The proposed method simultaneously learns the classifier, the dictionary and the corresponding sparse representations of each input data sample. A theoretical analysis is also provided comparing this method with the standard imputation approach, which consists on performing data completion followed by training the classifier based on their reconstructions. The limitations of this last "sequential" approach are identified, and a description of how the proposed new "simultaneous" method can overcome the problem of indiscernible observations is provided. Additionally, it is shown that, if it is possible to train a classifier on incomplete observations so that its reconstructions are well separated by a hyperplane, then the same classifier also correctly separates the original (unobserved) data samples. Extensive simulation results are presented on synthetic and well-known reference datasets that demonstrate the effectiveness of the proposed method compared to traditional data imputation methods.