ACDC: A Structured Efficient Linear Layer

TL;DR

This paper introduces ACDC, a structured linear layer that reduces parameters and computational costs using diagonal matrices and the discrete cosine transform, enabling efficient deep learning models.

Contribution

The paper presents a novel ACDC module with $O(N)$ parameters and $O(N \, log N)$ operations, and demonstrates its effectiveness in deep neural networks for image recognition.

Findings

ACDC approximates linear layers effectively.

Interleaving ACDC with ReLU modules maintains performance.

Training factors like initialization and depth are critical.

Abstract

The linear layer is one of the most pervasive modules in deep learning representations. However, it requires $O(N^2)$ parameters and $O(N^2)$ operations. These costs can be prohibitive in mobile applications or prevent scaling in many domains. Here, we introduce a deep, differentiable, fully-connected neural network module composed of diagonal matrices of parameters, $\mathbf{A}$ and $\mathbf{D}$, and the discrete cosine transform $\mathbf{C}$. The core module, structured as $\mathbf{ACDC^{-1}}$, has $O(N)$ parameters and incurs $O(N log N )$ operations. We present theoretical results showing how deep cascades of ACDC layers approximate linear layers. ACDC is, however, a stand-alone module and can be used in combination with any other types of module. In our experiments, we show that it can indeed be successfully interleaved with ReLU modules in convolutional neural networks for image recognition. Our experiments also study critical factors in the training of these structured modules, including initialization and depth. Finally, this paper also provides a connection between structured linear transforms used in deep learning and the field of Fourier optics, illustrating how ACDC could in principle be implemented with lenses and diffractive elements.