New Fisher Widths Framework Explores Local Learning Geometry

This paper delves into the geometric properties of statistical manifolds, specifically through the lens of "Fisher widths." It defines two key functionals: the primal Fisher width, which quantifies local parameter variations within the Fisher information geometry, and the inverse-Fisher width, which characterizes the impact of anisotropic Gaussian measurements. These widths offer distinct perspectives on the learning and recovery processes in statistical models. The study demonstrates that for Fisher-regular losses, the primal Fisher width scales predictably within small Fisher balls. For sparse recovery, the inverse-Fisher width highlights how the geometry is influenced by both sparsity and the spectral position of active coordinates. A significant finding is a sharp inequality linking the primal and inverse-Fisher widths, suggesting that while Fisher anisotropy can shift complexity between these two geometric views, it cannot simultaneously reduce both relative to a Euclidean scale.