The Nonnegative Matrix Factorization and atomic deconvolution

The Nonnegative Matrix Factorization is an unsupervised maching learning technique that finds a representation of measured data in terms of two low-rank factors. It has recently gained popularity in various applications as a feature selection and dimension reduction tool, e.g. text mining, signal processing, and image processing. Thus, the nonnegative matrix factorization is an increasingly important tool in big data analysis as data continues to grow not only in size but also in complexity. In this paper, we advance the NMF analysis in the case of the convolution. That is, the two factors have the clear roles of convolution kernel and signal. Specifically, for the case of the point-spread function, atoms are the weights that describe the kernel. Using proper atoms, we develop a method for the blind deconvolution based on an NMF representation so that we obtain an estimate of the signal and the kernel. In addition, with Magnetic Resonance Imaging (MRI). Specifically, we extend the idea of the two factors to the Fourier transform and develop a coordinate-descent method in order to determine phases.