New Fisher Widths Framework Links Local Learning Geometry to Anisotropic Recovery

In a 38-page paper with three figures, Vu Khac Ky proposes a novel framework using Fisher widths to understand local learning geometry and anisotropic recovery. The primal Fisher width, w_G(T) = w(G^{1/2}T), measures the size of parameter fluctuations within the Fisher metric geometry. For Fisher-regular losses, the author proves that the scale w_G(H_r)/√n is attained on sufficiently small Fisher balls, providing a precise characterization of local learning complexity. This result offers a principled way to quantify how the Fisher information matrix shapes the effective parameter space in statistical models. The framework extends classic Gaussian-width analysis to curved statistical manifolds, capturing the intrinsic geometry of learning problems.

Complementing the primal width, the inverse-Fisher width w_{G^{-1}}(T) = w(G^{-1/2}T) captures the effect of anisotropic Gaussian measurements whose covariance is determined by the inverse Fisher information. In sparse recovery, the geometry depends not only on sparsity but also on the position of active coordinates within the Fisher spectrum. Khac Ky derives a two-sided estimate for the statistical dimension along with support-sensitive recovery guarantees. A sharp inequality connects the two widths: on any compact coordinate set T, w_G(T) w_{G^{-1}}(T) ≥ w(T)^2. This means Fisher anisotropy can transfer complexity between geometries but cannot reduce both widths simultaneously relative to the Euclidean scale. The work has implications for designing algorithms in high-dimensional statistics, particularly where local curvature and anisotropic noise interact.