Near-Optimal and Tractable Estimation under Shift-Invariance
Dmitrii M. Ostrovskii
TL;DR
This paper addresses the statistical complexity of estimating discrete-time signals that obey an unknown linear recurrence of order $s$, observed in complex Gaussian noise. It introduces adaptive estimators based on reproducing filters for shift-invariant subspaces (SIS), showing near-minimax performance with a rate that matches sparse-signal benchmarks up to polylogarithmic factors. The authors construct compactly supported reproducing kernels whose Fourier spectra have nearly minimal $\ell_p$ norms for all $p$, enabling tractable optimization-based estimators (solvable as second-order cone programs) and a multiscale full-domain recovery scheme. They also develop a near-minimax detection threshold for the associated hypothesis testing problem. Collectively, the results provide a principled, computationally feasible framework for estimating and detecting broad classes of signals—including harmonic oscillations with arbitrary frequencies—under shift-invariance, bridging super-resolution, sparse recovery, and spectral estimation in a unified theory.
Abstract
How hard is it to estimate a discrete-time signal $(x_{1}, ..., x_{n}) \in \mathbb{C}^n$ satisfying an unknown linear recurrence relation of order $s$ and observed in i.i.d. complex Gaussian noise? The class of all such signals is parametric but extremely rich: it contains all exponential polynomials over $\mathbb{C}$ with total degree $s$, including harmonic oscillations with $s$ arbitrary frequencies. Geometrically, this class corresponds to the projection onto $\mathbb{C}^{n}$ of the union of all shift-invariant subspaces of $\mathbb{C}^\mathbb{Z}$ of dimension $s$. We show that the statistical complexity of this class, as measured by the squared minimax radius of the $(1-δ)$-confidence $\ell_2$-ball, is nearly the same as for the class of $s$-sparse signals, namely $O\left(s\log(en) + \log(δ^{-1})\right) \cdot \log^2(es) \cdot \log(en/s).$ 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 $\ell_p$-norms, for all $p \in [1,+\infty]$ at once.