Estimation of linear operators from scattered impulse responses

Jérémie Bigot (1) , Paul Escande (2, 3) , Pierre Weiss (4, 3)

1 IMB - Institut de Mathématiques de Bordeaux
2 DISC - Département d'Ingénierie des Systèmes Complexes
3 ITAV - Institut des Technologies Avancées en sciences du Vivant
4 IMT - Institut de Mathématiques de Toulouse UMR5219

We provide a new estimator of integral operators with smooth kernels, obtained from a set of scattered and noisy impulse responses. The proposed approach relies on the formalism of smoothing in reproducing kernel Hilbert spaces and on the choice of an appropriate regularization term that takes the smoothness of the operator into account. It is numerically tractable in very large dimensions. We study the estimator's robustness to noise and analyze its approximation properties with respect to the size and the geometry of the dataset. In addition, we show minimax optimality of the proposed estimator.

Mots clés: 94A20; 62H12; 65T60; 68W25; 41A25; 41A15; minimax; AMS classifications: 47A58; radial basis functions; estimator; Reproducing Kernel Hilbert Spaces; scattered approximation; Integral operator; convergence rate; complexity; numerical

Domaines: Théorie de l'information et codage [math.IT]; Analyse numérique [math.NA]; Statistiques [math.ST]; Traitement du signal et de l'image [eess.SP]

Jérémie Bigot, Paul Escande, Pierre Weiss. Estimation of linear operators from scattered impulse responses. Applied and Computational Harmonic Analysis, 2019, 47 (3), pp.730-758. ⟨10.1016/j.acha.2017.12.002⟩. ⟨hal-01380584v2⟩