Convergence analysis of Tikhonov regularization for non-linear statistical inverse problems

Abhishake Rastogi (1) , Gilles Blanchard (2, 3) , Peter Mathé (4)

1 Institut für Mathematik [Potsdam]
2 LMO - Laboratoire de Mathématiques d'Orsay
3 DATASHAPE - Understanding the Shape of Data
4 WIAS - Weierstraß-Institut für Angewandte Analysis und Stochastik = Weierstrass Institute for Applied Analysis and Stochastics [Berlin]

We study a non-linear statistical inverse problem, where we observe the noisy image of a quantity through a non-linear operator at some random design points. We consider the widely used Tikhonov regularization (or method of regularization) approach to estimate the quantity for the non-linear ill-posed inverse problem. The estimator is defined as the minimizer of a Tikhonov functional, which is the sum of a data misfit term and a quadratic penalty term. We develop a theoretical analysis for the minimizer of the Tikhonov regularization scheme using the concept of reproducing kernel Hilbert spaces. We discuss optimal rates of convergence for the proposed scheme, uniformly over classes of admissible solutions, defined through appropriate source conditions.

Mots clés: general source condition; Tikhonov regularization; reproducing kernel Hilbert space; Statistical inverse problem; minimax convergence rates

Domaines: Statistiques [math.ST]; Apprentissage [cs.LG]; Intelligence artificielle [cs.AI]

Abhishake Rastogi, Gilles Blanchard, Peter Mathé. Convergence analysis of Tikhonov regularization for non-linear statistical inverse problems. Electronic Journal of Statistics , 2020, 14 (2), pp.2798-2841. ⟨10.1214/20-EJS1735⟩. ⟨hal-03082290v2⟩