# Prime Convolutional Model: Breaking the Ground for Theoretical Explainability

arXiv: https://arxiv.org/abs/2503.02773 · HTML: https://mlsanity.com/paper/2503.02773/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 35 of its 68 references that are in the map.
- First posted on arXiv 4 Mar 2025.

## Builds on (most cited first, 30 of 35)

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- [Music Generation by Deep Learning - Challenges and Directions](https://mlsanity.com/paper/1712.04371/) (2017)
- [A survey on Variational Autoencoders from a GreenAI perspective](https://mlsanity.com/paper/2103.01071/) (2021)
- [Distance-Restricted Explanations: Theoretical Underpinnings & Efficient Implementation](https://mlsanity.com/paper/2405.08297/) (2024)
- [A Comprehensive Survey on Data-Efficient GANs in Image Generation](https://mlsanity.com/paper/2204.08329/) (2022)
- [A survey on text generation using generative adversarial networks](https://mlsanity.com/paper/2212.11119/) (2022)

## Abstract (arXiv)

In this paper, we propose a new theoretical approach to Explainable AI. Following the Scientific Method, this approach consists in formulating on the basis of empirical evidence, a mathematical model to explain and predict the behaviors of Neural Networks. We apply the method to a case study created in a controlled environment, which we call Prime Convolutional Model (p-Conv for short). p-Conv operates on a dataset consisting of the first one million natural numbers and is trained to identify the congruence classes modulo a given integer m. Its architecture uses a convolutional-type neural network that contextually processes a sequence of B consecutive numbers to each input. We take an empirical approach and exploit p-Conv to identify the congruence classes of numbers in a validation set using different values for m and B. The results show that the different behaviors of p-Conv (i.e., whether it can perform the task or not) can be modeled mathematically in terms of m and B. The inferred mathematical model reveals interesting patterns able to explain when and why p-Conv succeeds in performing task and, if not, which error pattern it follows.