Modified HALS Algorithm for Image Completion and Recommendation System

The paper is concerned with the task of reconstructing missing values in an observed incomplete matrix, assuming its low-rank approximation. The problem has important applications, especially in image processing and social sciences. In our approach, we focus on the problem of recovering missing pixels in images perturbed with impulse noise in a transmission channel as well as estimating unknown ratings in a recommendation system. For solving these problems, we used the modified version of the Hierarchical Least Squares Algorithm (HALS), including the smoothed version, and compared them with other algorithm, such as the SPC-QV. The numerical experiments are carried out for various cases of incomplete data. For image processing, the incomplete images are obtained by removing random pixels and regular grid lines from test images. For recommendation systems, we used real rating matrices from the MovieLens database that contains five-star movie recommendation ratings. The best performance is obtained if nonnegativity and smoothing constraints are imposed onto the estimated low-rank factors.