Time and Frequency Domain Optimization with Shift, Convolution and Smoothness in Factor Analysis Type Decompositions

We propose the Time Frequency Gradient Method (TFGM) which forms a framework for optimization of models that are constrained in the time domain while having efficient representations in the frequency domain. Since the constraints in the time domain in general are not transparent in a frequency representation we demonstrate how the class of objective functions that are separable in either time or frequency instances allow the gradient in the time or frequency domain to be converted to the opposing domain. We further demonstrate the usefulness of this framework for three different models; Shifted Non-negative Matrix Factorization, Convolutive Sparse Coding as well as Smooth and Sparse Matrix Factorization. Matlab implementation of the proposed algorithms are available for download at www.erpwavelab.org.