Matrix completion under interval uncertainty

•Matrix completion under interval uncertainty can be encoded using box constraints.•E.g. in collaborative filtering, a rating of 2 can be replaced by an interval 1–3.•Present-best results in terms of statistical performance are provided for a benchmark.•Convergence of randomized coordinate-descent methods to stationary points is proven. Matrix completion under interval uncertainty can be cast as a matrix completion problem with element-wise box constraints. We present an efficient alternating-direction parallel coordinate-descent method for the problem. We show that the method outperforms any other known method on a benchmark in image in-painting in terms of signal-to-noise ratio, and that it provides high-quality solutions for an instance of collaborative filtering with 100,198,805 recommendations within 5 minutes on a single personal computer.