On learning with shift-invariant structures
This work addresses signal processing challenges in domains like medical imaging, but appears incremental as it builds on prior results.
The paper tackles the problems of learning shift-invariant components from data and calculating shifts between signals, using circulant and convolutional matrices, and demonstrates effectiveness on synthetic, ECG, and image data.
We describe new results and algorithms for two different, but related, problems which deal with circulant matrices: learning shift-invariant components from training data and calculating the shift (or alignment) between two given signals. In the first instance, we deal with the shift-invariant dictionary learning problem while the latter bears the name of (compressive) shift retrieval. We formulate these problems 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, ECG signals and images.