Anatoli JUDITSKY (Université de Grenoble) – “Adaptive estimation from indirect observations “
February 3, 2:00 pm - 3:15 pm
The Statistical Seminar: Every Monday at 2:00 pm.
Time: 2:00 pm – 3:15 pm
Date: 3rd of February 2020
Place: Room 3001.
Anatoli JUDITSKI (Université de Grenoble) – “Adaptive estimation from indirect observations “
Abstract: We consider the problem of estimating of unknown “signal” known to belong to the union of finitely many convex compact sets from indirect noisy observations of the signal. Our main assumption is that the observation scheme in question is “good” in the sense of Goldenshluger, J., Nemirovski, 2015, the simplest example being the Gaussian o.s., where the observation is the sum of linear image of the signal and the standard Gaussian noise. We show that proposed estimates obey some optimality guarantees (e.g., the “aggregation cost” is bounded by the minimax risk of estimation on the union), leading to (quasi-)minimax (up to a moderate factor) estimates on the union of convex sets in the situation where (quasi-)minimax optimal estimates are available for each set of the family (our working example being quasi-minimax estimation on unions of centered at the origin ellipsoids/ellitopes/spectratopes). The estimates (and in the case of ellitopes/spectratopes, upper bounds on their worst-case risks) stem from solutions to explicit convex optimization problems, making the estimates “computation-friendly.” We also discuss application of the proposed approach to adaptive estimation.
Joint work with A. Goldenshluger and A. Nemirovski.
Cristina BUTUCEA, Alexandre TSYBAKOV, Julie JOSSE, Eric MOULINES, Mathieu ROSENBAUM