Alan Oursland

Alan Oursland

Neural networks trained with standard objectives exhibit behaviors characteristic of probabilistic inference: soft clustering, prototype specialization, and Bayesian uncertainty tracking. These phenomena appear across architectures -- in attention mechanisms, classification heads, and energy-based models -- yet existing explanations rely on loose analogies to mixture models or post-hoc architectural interpretation. We provide a direct derivation. For any objective with log-sum-exp structure over distances or energies, the gradient with respect to each distance is exactly the negative posterior responsibility of the corresponding component: $\partial L / \partial d_j = -r_j$. This is an algebraic identity, not an approximation. The immediate consequence is that gradient descent on such objectives performs expectation-maximization implicitly -- responsibilities are not auxiliary variables to be computed but gradients to be applied. No explicit inference algorithm is required because inference is embedded in optimization. This result unifies three regimes of learning under a single mechanism: unsupervised mixture modeling, where responsibilities are fully latent; attention, where responsibilities are conditioned on queries; and cross-entropy classification, where supervision clamps responsibilities to targets. The Bayesian structure recently observed in trained transformers is not an emergent property but a necessary consequence of the objective geometry. Optimization and inference are the same process.

Alan Oursland

This paper introduces a theoretical framework that connects neural network linear layers with the Mahalanobis distance, offering a new perspective on neural network interpretability. While previous studies have explored activation functions primarily for performance optimization, our work interprets these functions through statistical distance measures, a less explored area in neural network research. By establishing this connection, we provide a foundation for developing more interpretable neural network models, which is crucial for applications requiring transparency. Although this work is theoretical and does not include empirical data, the proposed distance-based interpretation has the potential to enhance model robustness, improve generalization, and provide more intuitive explanations of neural network decisions.

Alan Oursland

Neural networks may naturally favor distance-based representations, where smaller activations indicate closer proximity to learned prototypes. This contrasts with intensity-based approaches, which rely on activation magnitudes. To test this hypothesis, we conducted experiments with six MNIST architectural variants constrained to learn either distance or intensity representations. Our results reveal that the underlying representation affects model performance. We develop a novel geometric framework that explains these findings and introduce OffsetL2, a new architecture based on Mahalanobis distance equations, to further validate this framework. This work highlights the importance of considering distance-based learning in neural network design.

Alan Oursland

We present empirical evidence that neural networks with ReLU and Absolute Value activations learn distance-based representations. We independently manipulate both distance and intensity properties of internal activations in trained models, finding that both architectures are highly sensitive to small distance-based perturbations while maintaining robust performance under large intensity-based perturbations. These findings challenge the prevailing intensity-based interpretation of neural network activations and offer new insights into their learning and decision-making processes.