# ENN: A Neural Network with DCT Adaptive Activation Functions

arXiv: https://arxiv.org/abs/2307.00673 · HTML: https://mlsanity.com/paper/2307.00673/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 38 references that are in the map.
- First posted on arXiv 2 Jul 2023.

## Builds on (most cited first, 7 of 7)

- [The Kolmogorov-Arnold representation theorem revisited](https://mlsanity.com/paper/2007.15884/) (2020)
- [Why Deep Neural Networks for Function Approximation?](https://mlsanity.com/paper/1610.04161/) (2016)
- [Learning Activation Functions to Improve Deep Neural Networks](https://mlsanity.com/paper/1412.6830/) (2014)
- [Deep Learning with S-shaped Rectified Linear Activation Units](https://mlsanity.com/paper/1512.07030/) (2015)
- [Learning Combinations of Activation Functions](https://mlsanity.com/paper/1801.09403/) (2018)
- [Neural Networks with Smooth Adaptive Activation Functions for Regression](https://mlsanity.com/paper/1608.06557/) (2016)
- [DCT-based Air Interface Design for Function Computation](https://mlsanity.com/paper/2306.16321/) (2023)

## Built on by (most cited first, 4 of 4)

- [Zorro: A Flexible and Differentiable Parametric Family of Activation Functions That Extends ReLU and GELU](https://mlsanity.com/paper/2409.19239/) (2024)
- [Polynomial, trigonometric, and tropical activations](https://mlsanity.com/paper/2502.01247/) (2025)
- [Developing Training Procedures for Piecewise-linear Spline Activation Functions in Neural Networks](https://mlsanity.com/paper/2509.18161/) (2025)
- [Efficient Neural Networks with Discrete Cosine Transform Activations](https://mlsanity.com/paper/2511.03531/) (2025)

## Abstract (arXiv)

The expressiveness of neural networks highly depends on the nature of the activation function, although these are usually assumed predefined and fixed during the training stage. Under a signal processing perspective, in this paper we present Expressive Neural Network (ENN), a novel model in which the non-linear activation functions are modeled using the Discrete Cosine Transform (DCT) and adapted using backpropagation during training. This parametrization keeps the number of trainable parameters low, is appropriate for gradient-based schemes, and adapts to different learning tasks. This is the first non-linear model for activation functions that relies on a signal processing perspective, providing high flexibility and expressiveness to the network. We contribute with insights in the explainability of the network at convergence by recovering the concept of bump, this is, the response of each activation function in the output space. Finally, through exhaustive experiments we show that the model can adapt to classification and regression tasks. The performance of ENN outperforms state of the art benchmarks, providing above a 40% gap in accuracy in some scenarios.