Quantum Semi Non-negative Matrix Factorization

Kaoutar Benlamine, Younès Bennani, Basarab Matei, Nistor Grozavu

Abstract: Semi Non-negative Matrix Factorization (SNMF) is a machine learning algorithm that is used to decompose large data matrices where the data matrix is unconstrained (i.e., it may have mixed signs). We develop the quantum version of SNMF using quantum gradient descent, and we show that the quantum version of SNMF provides an exponential speedup compared to its classical version. Our approach is well adapted for high-dimensional problems since in the context of using a quantum approach only a few number of iterations is enough to obtain a close approximation of the factorization. From the algorithmic analysis point of view, the proposed algorithm for Quantum Semi Non-negative Matrix Factorization (QSNMF) yields a performance \(T\mathcal {O}( polylog K (M+N)),\) while the classical version of Semi Non-negative Matrix Factorization (SNMF) yields a performance \(T\mathcal {O}(poly (K M N))\) where M and N are the dimensions of the matrix, K is the number of clusters and T is the maximum number of iterations to prepare the quantum states.