Learning Fast Sparsifying Transforms

TL;DR

This paper introduces a method to learn fast, structured sparsifying transforms, both orthogonal and non-orthogonal, that are computationally efficient and suitable for resource-limited hardware, improving data representation.

Contribution

It proposes a novel approach to construct structured dictionaries as products of basic transformations, enabling fast manipulation and better data representation.

Findings

Orthogonal dictionaries constructed as products of generalized Givens rotations.

Non-orthogonal fast dictionaries as products of generalized transforms.

Proposed transforms balance data representation quality and computational efficiency.

Abstract

Given a dataset, the task of learning a transform that allows sparse representations of the data bears the name of dictionary learning. In many applications, these learned dictionaries represent the data much better than the static well-known transforms (Fourier, Hadamard etc.). The main downside of learned transforms is that they lack structure and therefore they are not computationally efficient, unlike their classical counterparts. These posse several difficulties especially when using power limited hardware such as mobile devices, therefore discouraging the application of sparsity techniques in such scenarios. In this paper we construct orthogonal and non-orthogonal dictionaries that are factorized as a product of a few basic transformations. In the orthogonal case, we solve exactly the dictionary update problem for one basic transformation, which can be viewed as a generalized Givens rotation, and then propose to construct orthogonal dictionaries that are a product of these transformations, guaranteeing their fast manipulation. We also propose a method to construct fast square but non-orthogonal dictionaries that are factorized as a product of few transforms that can be viewed as a further generalization of Givens rotations to the non-orthogonal setting. We show how the proposed transforms can balance very well data representation performance and computational complexity. We also compare with classical fast and learned general and orthogonal transforms.