ENN: A Neural Network with DCT Adaptive Activation Functions

TL;DR

ENN tackles the limitation of fixed activation functions by modeling neuron non-linearities with a finite Discrete Cosine Transform (DCT) expansion $f(z) \\approx \sum_{q=1}^{Q/2} F_q \cos_q(z)$, and learning the coefficients jointly with linear weights via backpropagation. The approach preserves a small parameter budget due to DCT energy compaction and introduces explainability through the bump interpretation of activation responses. Empirical results on synthetic classification and regression tasks show ENN achieving up to 40% higher accuracy than fixed-activation baselines in several settings, highlighting strong task adaptability without widening the network. Overall, signal-processing-inspired, learnable non-linearities in ENN offer a principled path to greater expressiveness and gradient-based trainability with limited parameter growth.

Abstract

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.

Figures (12)

• Figure 1: A 2-layer perceptron with $M_1$ neurons in the hidden layer.
• Figure 2: Sigmoid function (dashed) and its DCT representation in the $[-1,1]$ range (solid).
• Figure 3: Bump generation. Mapping of a linear combination of two inputs over the response of a non-linear function.
• Figure 4: AAF in the hidden layer for (P7).
• Figure 5: Bumps corresponding to the AAF in Fig. \ref{['fig:lines_aaf']}.

Theorems & Definitions (1)

• Example 1: Linear discriminant