A Water-filling Algorithm Maximizing the Volume of Submatrices Above the Rank

In this paper, we propose an algorithm to extract, from a given rectangular matrix, a submatrix with maximum volume, whose number of extracted columns is greater than the initial matrix rank. This problem arises in compression and summarization of databases, recommender systems, learning, numerical analysis or applied linear algebra. We use a continuous relaxation of the maximum volume matrix extraction problem, which admits a simple and closed form solution: the nonzero singular values of the extracted matrix must be equal. The proposed algorithm extracts matrices with singular values, which are close to be equal. It is inspired by a water-filling technique, traditionally dedicated to equalization strategies in communication channels. Simulations show that the proposed algorithm performs better than sampling methods based on determinantal point processes (DPPs) and achieves similar performance as the best known algorithm, but with a lower complexity.