# On learning with shift-invariant structures

## TL;DR

This paper introduces new algorithms for learning shift-invariant structures and calculating shifts between signals using circulant matrices, with applications to dictionary learning and signal alignment.

## Contribution

It presents novel methods for shift-invariant dictionary learning and shift retrieval, including algorithms for convolutional and circulant matrices, and demonstrates their effectiveness on real data.

## Key findings

- Algorithms successfully recover shift-invariant components.
- Effective shift retrieval from synthetic and real signals.
- Learned wavelet-like dictionaries from data.

## Abstract

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.