Under-Counted Matrix Completion Without Detection Features

Under-counted matrix completion (UC-MC) has many important applications, especially in epidemiology and ecology where the observed data are often smaller than the actual numbers. Existing works model the under-counting effects using entry-wise miss detection probabilities, which are usually formulated as functions of detection-related side information or features (e.g., weather conditions for observing a certain species). However, such features are not always available. This work proposes a model for UC-MC that circumvents using such side information. By assuming that the detection probabilities for a large proportion of entries are similar, the under-counting probabilities are approximated by a specially structured (i.e., rank-one plus sparse) matrix. This way, a structure-regularized UC-MC formulation is attained, and no detection-related side information is used. A re-parameterization-based implementation is proposed, allowing us to employ off-the-shelf gradient-based optimizers to tackle the loss function of interest. Simulations are used to demonstrate the effectiveness of the proposed approach.