Gradient Descent as Implicit EM in Distance-Based Neural Models

This paper provides a direct mathematical derivation showing that gradient descent on objectives with log-sum-exp structure over distances or energies implicitly performs Expectation-Maximization (EM). This unifies various learning regimes, including unsupervised mixture modeling, attention mechanisms, and cross-entropy classification, under a single mechanism. The key contribution is the algebraic identity that the gradient with respect to each distance is the negative posterior responsibility. This offers a new perspective on understanding the Bayesian behavior observed in neural networks, suggesting it's a consequence of the objective function's geometry rather than an emergent property.