On learning with shift-invariant structures

Abstract

In this paper, we describe new results and algorithms, based on circulant matrices, for the task of learning shift-invariant components from training data. We deal with the shift-invariant dictionary learning problem which we formulate using circulant and convolutional matrices (including unions of such matrices), define optimization problems that describe our goals and propose efficient ways to solve them. Based on these findings, we also show how to learn a wavelet-like dictionary from training data. We connect our work with various previous results from the literature and we show the effectiveness of our proposed algorithms using synthetic as well as real ECG signals and images.