The Geometry of Statistical Feature Learning in Mean-Field Langevin Dynamics

What the math gives to ML

The paper gives a constructive view of feature learning as a two-level object: a learned marginal distribution over feature parameters forms the base, while coefficient functions on that marginal form the fiber. The transferable asset is that mean-field Langevin dynamics can move the feature marginal toward statistically informative directions, while the induced kernel makes the resulting representation explicit and measurable. A practical adaptation is to train a finite particle approximation with entropy-controlled Langevin noise, then fit or jointly train output coefficients; temperature annealing should allow exploration early and concentration near useful features late. The single-index parity result also suggests eliminating antipodal redundancy through paired or quotient-aware neurons.