- Series
- Stochastics Seminar
- Time
- Thursday, January 16, 2025 - 3:30pm for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Dmitrii Ostrovskii – Georgia Tech – ostrov@gatech.edu – https://ostrodmit.github.io/
- Organizer
- Dmitrii Ostrovskii
Please Note: This talk is hosted jointly with the Analysis Seminar.
In the 1990s, Arkadi Nemirovski asked the question:
How hard is it to estimate a solution to an unknown homogeneous linear difference equation with constant coefficients of order S, observed in the Gaussian noise on [0,N]?
The class of all such solutions, or "signals," is parametric -- described by 2S complex parameters -- but extremely rich: it contains all exponential polynomials over C with total degree S, including harmonic oscillations with S arbitrary frequencies. Geometrically, this class corresponds to the projection onto C^n of the union of all shift-invariant subspaces of C^Z of dimension S. We show that the statistical complexity of this class, quantified by the squared minimax radius of the (1-P)-confidence Euclidean norm ball, is nearly the same as for the class of S-sparse signals, namely (S log(N) + log(1/P)) log^2(S) log(N/S) up to a constant factor. Moreover, the corresponding near-minimax estimator is tractable, and it can be used to build a test statistic with a near-minimax detection threshold in the associated detection problem. These statistical results rest upon an approximation-theoretic one: we show that finite-dimensional shift-invariant subspaces admit compactly supported reproducing kernels whose Fourier spectra have nearly the smallest possible p-norms, for all p ≥ 1 at once.
The talk is based on the recent preprint https://arxiv.org/pdf/2411.03383.