With-replacement leverage-score row sampling and rescaling

Input an orthonormal basis matrix $Q\in {\mathbb{K}}^{n\times d}$ with $d\ge 1$, a sample count $m\ge 1$, a parameter $0<\beta \le 1$, and a probability vector $\pi =({\pi }_0,\dots ,{\pi }_{n-1})$ satisfying ${\pi }_i\ge \beta {\mathrm{\ell }}_i(U)\mathrm{/}d$, where $U=\mathrm{Ran}(Q)$ and ${\mathrm{\ell }}_i(U)$ is from Row leverage scores of a finite-dimensional subspace. If the input checks fail, return invalid-input. Otherwise draw indices $J_1,\dots ,J_m$ independently with replacement from $\pi$. Form $S\in {\mathbb{K}}^{m\times n}$ by setting row $j$ to $e_{J_j}^{\ast }\mathrm{/}\sqrt{m{\pi }_{J_j}}$, and return success, the selected indices, and $S$.

Because $\beta >0$, ${\pi }_i=0$ implies ${\mathrm{\ell }}_i(U)=0$; such a row is never sampled, so the scaling never divides by zero. One canonical choice is ${\pi }_i={\mathrm{\ell }}_i(U)\mathrm{/}d$, which sets every zero-leverage row's probability to zero. Other admissible distributions may assign positive probability to a zero-leverage row, but that choice wastes samples. Repeated indices are allowed and are distinct rows of $S$.