General-rank intrinsic Cramér–Rao lower bound on Tucker manifolds

Let $m\ge 2$ and $d_1,\dots,d_m\in\mathbb N$. Set $d^*:=\prod_{j=1}^m d_j$ and let $\mathbb R^{d_1\times\cdots\times d_m}$ be equipped with the Frobenius inner product $\langle A,B\rangle=\sum_{i_1,\dots,i_m}A_{i_1,\dots,i_m}B_{i_1,\dots,i_m}$ and norm $\|A\|_F^2=\langle A,A\rangle$. For a tensor $T\in\mathbb R^{d_1\times\cdots\times d_m}$, denote by $\operatorname{rank}_j(T)$ its mode-$j$ matricization rank. For a fixed multilinear rank vector $r=(r_1,\dots,r_m)$ with $1\le r_j\le d_j$, define the Tucker manifold

At $T\in\mathcal M_r$, let $\mathsf T_T\mathcal M_r$ be the tangent space and let $\mathcal P_T:\mathbb R^{d_1\times\cdots\times d_m}\to \mathsf T_T\mathcal M_r$ be the orthogonal projector (Frobenius metric).

This setup follows Ma & Xia (2024).

Data are $n$ i.i.d. observations $(\omega_t,Y_t)$, $t=1,\dots,n$, generated by

with $\omega_t$ independent of $\xi_t$. The parameter is $T\in\mathcal M_r$. For a fixed query tensor $I\in\mathbb R^{d_1\times\cdots\times d_m}$, the target functional is $g_I(T):=\langle T,I\rangle$.

An estimator $\widehat g_I=\widehat g_I((\omega_t,Y_t)_{t=1}^n)$ is unbiased if $\mathbb E_T[\widehat g_I]=g_I(T)$ for all $T\in\mathcal M_r$, and locally unbiased at $T_0\in\mathcal M_r$ if for every smooth curve $T(s)\subset\mathcal M_r$ with $T(0)=T_0$ and $\dot T(0)\in\mathsf T_{T_0}\mathcal M_r$,

Source-proven part (under the paper's assumptions): Theorem 1 in the source establishes the intrinsic Cramér-Rao lower-bound form in the rank-one Tucker setting.