Randomized coupled decompositions

Abstract: Coupled decompositions are a widely used tool for data fusion. As the volume of data increases, so does the dimensionality of matrices and tensors, highlighting the need for more efficient coupled decomposition algorithms. This paper studies the problem of coupled matrix factorization (CMF), where two matrices represented in low-rank form share a common factor. Additionally, it explores coupled matrix and tensor factorization (CMTF), where a matrix and a tensor are represented in low-rank form, also sharing a common factor matrix. We show that these problems can be solved using a direct approach with singular value decomposition (SVD), rather than relying on an iterative method. Knowing that matrices coming from real-world applications are often very large, the computational cost can be substantial. To address this issue and improve the efficiency, we propose new techniques for randomizing these algorithms. This includes a novel strategy for selecting a projection subspace that takes into account the contribution from both matrices involved in the decomposition equally. We present extensive results of numerical tests that confirm the efficiency of our algorithms. Furthermore, as a novel approach and with a high success rate, we apply our randomized algorithms to the face recognition problem.