2015/03/30 by Clément Marteau, Theofanis Sapatinas, Marteau, Clément +1 · 3 citations
Engineering · Mathematics · #FOS: Mathematics #Numerical methods in inverse problems #Statistical and numerical algorithms #Statistics Theory (math.ST) #Ultrasonics and Acoustic Wave Propagation #math.ST #stat.TH
paper · pdf · doi:10.48550/arxiv.1503.08562
openalex publication_date 2015/03/30 · arxiv created 2015/06/19 · arxiv updated 2015/06/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a Gaussian sequence model that contains ill-posed inverse problems as special cases. We assume that the associated operator is partially unknown in the sense that its singular functions are known and the corresponding singular values are unknown but observed with Gaussian noise. For the considered model, we study the minimax goodness-of-fit testing problem. Working with certain ellipsoids in the space of squared-summable sequences of real numbers, with a ball of positive radius removed, we obtain lower and upper bounds for the minimax separation radius in the non-asymptotic framework, i.e., for fixed values of the involved noise levels. Examples of mildly and severely ill-posed inverse problems with ellipsoids of ordinary-smooth and super-smooth sequences are examined in detail and minimax rates of goodness-of-fit testing are obtained for illustrative purposes.