On the discrepancy principle and generalised maximum likelihood for regularisation

Source abstract

Let f n λ be the regularised solution of a general, linear operator equation, K f 0 = g , from discrete, noisy data y i = g ( x i ) + ε i , i = 1, …, n , where ε i are uncorrelated random errors with variance σ 2 . In this paper, we consider the two well–known methods – the discrepancy principle and generalised maximum likelihood (GML), for choosing the crucial regularisation parameter λ. We investigate the asymptotic properties as n → ∞ of the “expected” estimates λ D and λ M corresponding to these two methods respectively. It is shown that if f 0 is sufficiently smooth, then λ D is weakly asymptotically optimal (ao) with respect to the risk and an L 2 norm on the output error. However, λ D oversmooths for all sufficiently large n and also for all sufficiently small σ 2 . If f 0 is not too smooth relative to the regularisation space W , then λ D can also be weakly ao with respect to a whole class of loss functions involving stronger norms on the input error. For the GML method, we show that if f 0 is smooth relative to W (for example f 0 ∈ W θ, 2 , θ > m , if W = W m , 2 ), then λ M is asymptotically sub-optimal and undersmoothing with respect to all of the loss functions above.