Near-optimal lower-bound conjecture for one-sided estimation under shift-invariance

Let $X(w_1,\ldots,w_s)$ denote the shift-invariant subspace generated by frequencies $w_1,\ldots,w_s\in\mathbb{T}$. For an estimator $\widehat{x}=\widehat{x}(y_{-3n},\ldots,y_0)$, let $\operatorname{Risk}_{n,\delta}(\widehat{x},X)$ denote the paper's risk at scale $n$ and confidence parameter $\delta$, and let $\sigma^2$ be the noise variance.

One-sided estimation lower-bound conjecture. There exist constants $c_0,c,r>0$ such that, for every $s,n\in\mathbb{N}$ satisfying

$$n\geq c_0s^2\log^r(n),$$

there is an SIS $X(w_1,\ldots,w_s)$ with $w_1,\ldots,w_s\in\mathbb{T}$ such that every estimator based on $y_{-3n},\ldots,y_0$ satisfies

$$\operatorname{Risk}_{n,\delta}(\widehat{x},X)\geq c\sigma^2\left(s^2\log^r(n)+\log(\delta^{-1})\right).$$

This is intended to show that the one-sided estimation guarantee cannot be improved by more than a logarithmic factor. The claim is presented as a conjectural lower bound motivated by the near-optimality of the causal reproducing-filter result; its resolution is not supplied here.