Uniform Convergence of the Potential Function Algorithm

The identification problem of concern here is the estimation of a real function f (x) by means of noisy observations \ (Xᵢ, f (Xᵢ) + ᵢ (Xᵢ) ) \ of its pairs, the Xᵢ ’s being chosen independently according to some fixed law P. The approach taken for estimation is the “potential function” method (its sources are referenced herein), to wit: Choose f₀ arbitrarily and define the sequence \ fₙ \ by the recursive relation f₍ + ₁ (x) = fₙ (x) + ₙ (f (X₍ + ₁) + (X₍ + ₁) - fₙ (X₍ + ₁) ) K (X₍ + ₁, x), K being a positive symmetric kernel. From earlier publications it is known that under certain mild restrictions E\| fₙ - f \|² 0 in the L₂ (p) -norm. Rates of convergence have been obtained in the restrictive case that K (x, y) = ₈ - ₁N ᵢ² ᵢ (x) ᵢ (y) and f (x) span\ ᵢ, 1 i N\. The contribution of this paper is to prove that while no uniform bounds exist in the L₂ (p) -norm (we prove this) if \ ᵢ \ is an infinite set, we do have E\| f - fₙ \|ₖ ² < Cₙ (\| f \|) for the norm \| g \|ₖ² = g (x) g (y) K (x, y) p (x) p (y) dxdy and \ Cₙ (r) \ a sequence converging to 0 for each positive r. A final result concerns the rate at which increasing finite-dimensional projections of fₙ - f converge to 0 in the L₂ (p) -norm. From our methods it is seen that if f V = span (\ ᵢ \), then fₙ converges in the mean to the projection of f on V.