Finite-sample error-rate control and power guarantees under the linear subspace model

Let $n,d,r\in\mathbb N$ with $r\le d$. Let $\mathcal S\subset\mathbb R^d$ be a known $r$-dimensional linear subspace, and fix an orthonormal basis matrix $U\in\mathbb R^{d\times r}$ so $\mathcal S=\{Ua:a\in\mathbb R^r\}$. Observe a 1D signal $Y\in\mathbb R^n$ generated by

where $K\ge 0$ is unknown, $\tau_j\in\{1,\dots,n-d+1\}$ are unknown object locations, $x_j\in\mathcal S\setminus\{0\}$ are unknown object shapes, $P_{\tau}$ inserts a length-$d$ vector into coordinates $\tau,\dots,\tau+d-1$ (zero elsewhere), and $\varepsilon\sim N(0,\sigma^2 I_n)$. Assume a non-overlap/separation condition such as $|\tau_j-\tau_\ell|\ge d$ for $j\ne \ell$, and a minimum signal strength condition $\|x_j\|_2\ge \mu_{\min}$ for all $j$.

For each candidate location $t\in\{1,\dots,n-d+1\}$, define the local hypothesis

Let $\mathcal T^\star=\{\tau_1,\dots,\tau_K\}$ be the true object set and $\widehat{\mathcal D}(Y)\subset\{1,\dots,n-d+1\}$ be the detection set returned by the specific procedure of Eldar-Mor Waknin-Davenport-Bendory-Schwartzman-Shkolnisky (with either its FWER-control or mFDR-control mode). Define $R=|\widehat{\mathcal D}(Y)|$, $V=|\widehat{\mathcal D}(Y)\setminus\mathcal T^\star|$, $\mathrm{FWER}=\Pr(V\ge 1)$, and