An Optimal Hybrid Nuclear Norm Regularization for Matrix Sensing With Subspace Prior Information

Matrix sensing refers to recovering a low-rank matrix from a few linear combinations of its entries. This problem naturally arises in many applications including recommendation systems, collaborative filtering, seismic data interpolation and wireless sensor networks. Recently, in these applications, it has been noted that exploiting additional subspace information might yield significant improvements in practical scenarios. This information is reflected by two subspaces forming angles with column and row spaces of the ground-truth matrix. Despite the importance of exploiting this information, there is limited theoretical guarantee for this feature. In this work, we aim to address this issue by proposing a novel hybrid nuclear norm regularization which besides low-rankness, encourages subspace prior information. Our proposed regularizer is a weighted combination of deformed nuclear norm functions. We derive a closed-form accurate expression for the mean squared error (MSE) of the proposed problem and obtain a closed-form relation for the optimal choice of weights. The analysis in this paper is based on the recent tools in convex geometry. Both theoretical analysis and simulations indicate the superiority of our method over the state-of-the-art methods.