Information Geometry

Information geometry treats probability distributions as points on a statistical manifold, with the Fisher information metric measuring distinguishability and KL or Bregman divergences describing separation. Its core tools include natural gradients, exponential-family duality, curvature analysis, and pullback metrics that account for how models map parameters or representations to distributions. These methods are used in settings such as deep-network optimization and steering, generative-model sampling, Bayesian inference, and robustness under distribution shift. A central challenge is that Fisher geometry can be expensive or noisy to estimate, and the value of its invariant, task-sensitive metrics depends on which approximation and model setting are used.