New AI Math Breakthrough Could Make Tech Smarter and Cheaper

A new paper studies kernel ridge regression when the input data is anisotropic, meaning different directions carry different amounts of information, with a power-law decay structure. The authors derive sharp asymptotic formulas for the kernel spectrum and generalization error, showing how anisotropy reshapes learning curves. For weak anisotropy, some features of the isotropic case remain—variance peaks at integer sample complexities, but these peaks are damped—while bias transitions can decouple from those peaks. For strong anisotropy, the effective dimension becomes constant, variance stops depending on sample size under ridgeless interpolation or vanishes at an explicit rate with fixed ridge penalty, and bias transitions depend on the target's decay rate. The results clarify how the input geometry shapes kernel features and impacts generalization properties.