# Compact and Interpretable Neural Networks Using Lehmer Activation Units

## TL;DR

This paper introduces a new type of neural network activation unit that combines feature weighting and nonlinearity, leading to compact and interpretable models.

## Contribution

The novel contribution is the development of Lehmer Activation Units (LAUs) that unify feature aggregation and nonlinearity in a single differentiable operator.

## Key findings

- LAUs enable compact neural network architectures with strong predictive performance.
- The complex-valued formulation of LAUs enhances expressive capacity through phase-sensitive interactions.
- LAU-based networks are proven to be universally approximable with formal guarantees.

## Abstract

We introduce Lehmer Activation Units (LAUs), a class of aggregation-based neural activations derived from the Lehmer transform that unify feature weighting and nonlinearity within a single differentiable operator. Unlike conventional pointwise activations, LAUs operate on collections of features and adapt their aggregation behavior through learnable parameters, yielding intrinsically interpretable representations. We develop both real-valued and complex-valued formulations, with the complex extension enabling phase-sensitive interactions and enhanced expressive capacity. We establish a universal approximation theorem for LAU-based networks, providing formal guarantees of expressive completeness. Empirically, we show that LAUs enable highly compact architectures to achieve strong predictive performance under tightly controlled experimental settings, demonstrating that expressive power can be concentrated within individual neurons rather than architectural depth. These results position LAUs as a principled, interpretable, and efficient alternative to conventional activation functions.