# Learning Multiplication-free Linear Transformations

## TL;DR

This paper introduces novel dictionary learning algorithms that produce structured, multiplication-free or multiplication-efficient linear transformations, significantly reducing computational complexity for sparse representations.

## Contribution

It presents new algorithms for learning structured dictionaries based on factorizations into simple, numerically efficient transformations with closed-form solutions.

## Key findings

- Effective in image data compression
- Outperforms traditional transforms in computational efficiency
- Achieves comparable or better sparse representation quality

## Abstract

In this paper, we propose several dictionary learning algorithms for sparse representations that also impose specific structures on the learned dictionaries such that they are numerically efficient to use: reduced number of addition/multiplications and even avoiding multiplications altogether. We base our work on factorizations of the dictionary in highly structured basic building blocks (binary orthonormal, scaling and shear transformations) for which we can write closed-form solutions to the optimization problems that we consider. We show the effectiveness of our methods on image data where we can compare against well-known numerically efficient transforms such as the fast Fourier and the fast discrete cosine transforms.