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 $(1-\delta)$-confidence $\ell_2$-ball, is nearly the same as for the class of $s$-sparse signals, namely $O\left(s\log(en) + \log(\delta^{-1})\right) \cdot \log^2(es) \cdot \log(en/s).$ 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.

翻译：在独立同分布的复高斯噪声中观测到满足未知s阶线性递推关系的离散时间信号$(x_{1}, ..., x_{n}) \in \mathbb{C}^n$，其估计难度如何？所有此类信号的集合是参数化的，但极其丰富：它包含总次数为s的所有复指数多项式，包括具有任意s个频率的谐波振荡。从几何角度看，该类对应于所有s维$\mathbb{C}^\mathbb{Z}$平移不变子空间到$\mathbb{C}^{n}$上的投影的并集。我们证明，以$(1-\delta)$置信$\ell_2$球的最小最大半径平方作为统计复杂度度量，该类的统计复杂度与s稀疏信号类几乎相同，即$O\left(s\log(en) + \log(\delta^{-1})\right) \cdot \log^2(es) \cdot \log(en/s)$。此外，相应的近最小最大估计器是可处理的，并且可用于在相关检测问题中构建具有近最小最大检测阈值的检验统计量。这些统计结果基于一个逼近论结果：我们证明有限维平移不变子空间存在紧支撑的再生核，其傅里叶谱在所有$p \in [1,+\infty]$上同时具有近乎最小的$\ell_p$范数。