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Prime Convolutional Model: Breaking the Ground for Theoretical Explainability

Francesco Panelli, Doaa Almhaithawi, Tania Cerquitelli, Alessandro Bellini

TL;DR

This work addresses the challenge of explainable AI by deriving a mathematical model of neural network behavior from empirical data, demonstrated on a controlled case study called Prime Convolutional Model (p-Conv). The method combines a prime-grid input representation with a locality-aware convolutional architecture to identify congruence classes modulo $m$ over the first $10^6$ natural numbers, revealing precise, testable relationships between hyperparameters and solvability. Key contributions include a structured experimental analysis across prime, prime-power, and splitting moduli, and a theoretical explanation that reduces the model's behavior to concrete arithmetic criteria (involving $B$, primes, and exponents) that predict when the task is solvable and what error patterns arise. The work advances a principled, a priori explainability framework with potential to generalize beyond this case study toward a broader theory of explainability in neural networks.

Abstract

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.

Prime Convolutional Model: Breaking the Ground for Theoretical Explainability

TL;DR

This work addresses the challenge of explainable AI by deriving a mathematical model of neural network behavior from empirical data, demonstrated on a controlled case study called Prime Convolutional Model (p-Conv). The method combines a prime-grid input representation with a locality-aware convolutional architecture to identify congruence classes modulo over the first natural numbers, revealing precise, testable relationships between hyperparameters and solvability. Key contributions include a structured experimental analysis across prime, prime-power, and splitting moduli, and a theoretical explanation that reduces the model's behavior to concrete arithmetic criteria (involving , primes, and exponents) that predict when the task is solvable and what error patterns arise. The work advances a principled, a priori explainability framework with potential to generalize beyond this case study toward a broader theory of explainability in neural networks.

Abstract

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 . Its architecture uses a convolutional-type neural network that contextually processes a sequence of 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 and . 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 and . 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.

Paper Structure

This paper contains 18 sections, 2 theorems, 13 equations, 15 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Methodology
  4. Input Data Representation
  5. Architecture
  6. The Task
  7. Evaluation Measures
  8. Impact of Hyperparameters on p-$\mathcal{C}$ $\mathscr{o}$ $\mathscr{n}$ $\mathscr{v}$ development
  9. Theoretical Observations and Experimental Results
  10. Prime Moduli
  11. Prime Power Moduli
  12. Splitting Moduli
  13. Towards Theoretical Explanation
  14. Conclusions and Future Work
  15. Hyperparameters Tuning
  16. ...and 3 more sections

Key Result

Theorem 1

Assume the validity of Experimental Observation EO_prime_power. Let $B\geq 2$ and let $p$ be a prime. Let $i_0\geq 0$ be the unique integer such that $p^{i_0}< B+2\leq p^{i_0+1}$. Then:

Figures (15)

  • Figure 1: Standard CNN architecture.
  • Figure 2: p-$\mathcal{C}$$\mathscr{o}$$\mathscr{n}$$\mathscr{v}$ final architecture.
  • Figure 3: Graphs of p-$\mathcal{C}$$\mathscr{o}$$\mathscr{n}$$\mathscr{v}$'s training losses when $m=7$, $B=8$ and (a) $k=3$, (b) $k=5$, (c) $k=7$.
  • Figure :
  • Figure :
  • ...and 10 more figures

Theorems & Definitions (7)

  • Example 1
  • Example 2
  • Example 3
  • Theorem 1
  • proof
  • Corollary 1
  • proof