Abstract

Co-authors: Paul Escande (Université de Toulouse), Jérémie Bigot (Université de Toulouse)

In this talk, I will propose a variational method to reconstruct operators with smooth kernels from 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 and yields a representation that can be used for achieving fast matrix-vector products. We study the estimator's robustness to noise and analyze its approximation properties with respect to the size and the geometry of the dataset. It turns out to be minimax optimal.

We finally show applications of the proposed algorithms to reconstruction of spatially varying blur operators in microscopy imaging.

Related Links

• weiss%2FPublis%2FJournals%2F2016%2FEstimation_Linear_Operators_Bigot_Escande_Weiss_2016.pdf">https://www.math.univ-toulouse.fr/weiss/Publis/Journals/2016/Estimation_Linear_Operators_Bigot_Escande_Weiss_2016.pdf – Main ideas
• weiss%2FPublis%2FJournals%2F2016%2FApproximation_Integral_Operators_Convolution-Product_Expansion_Escande_Weiss_2016.pdf">https://www.math.univ-toulouse.fr/weiss/Publis/Journals/2016/Approximation_Integral_Operators_Convolution-Product_Expansion_Escande_Weiss_2016.pdf – Preliminaries