Assume $\bf X$ is an unknown $30 \times 4$ high-dimension matrix. Assume ${\bf B}_n$ is a known $8 \times 30$ matrix for $n=1,2,...,10$.
If we can observe the $8 \times 4$ low-dimension matrix ${\bf Y}_n = {\bf B}_n {\bf X}$, we can perfectly recover $\bf X$ with least square \begin{equation} \min_{\bf X} \sum_{n=1}^{10}||{\bf B}_n {\bf X}-{\bf Y}_n||_{\rm F}^2 \end{equation}
If we can only observe the $8 \times 4$ left singular vector ${\bf U}_{n}$ of ${\bf Y}_n$, where ${\bf Y}_n = {\bf U}_n {\bf \Lambda}_n {\bf V}_n^H$ is singular value decomposition, we can still recover $\bf X$ with least square \begin{align} &\min_{{\bf X},{\bf \Lambda}_n,{\bf V}_n} \sum_{n=1}^{10}||{\bf B}_n {\bf X}-{\bf U}_n {\bf \Lambda}_n {\bf V}_n^H||_{\rm F}^2 \\ &{\text {s.t. }} {\bf \Lambda}_n {\text { is diagonal}}, {\bf V}_n {\text { is unitary}}, {{\rm tr}({\bf X}^H{\bf X})=1} \end{align} The last constraint is to avoid the ordinary solution ${\bf X} = {\bf 0}$ and ${\bf \Lambda}_n = {\bf 0}$. The above problem can be solved by alternatively solve one of ${\bf X}$, ${\bf \Lambda}_n$, and ${\bf V}_n$ while keeping the other two fixed. Note that, there will be a amplitude ambiguity and right unitary rotation ambiguity of recovered $\bf X$, i.e., ${\bf X}_{\text{est}} = a{\bf X}{\bf V}$, where $\bf V$ is a $4 \times 4$ unitary matrix. If we only focus on the left singular vectors of $\bf X$, then the impact of $a$ and $\bf V$ can be ignored.
If we can only observe the $8 \times 1$ 1st left singular vector ${\bf u}_{n,1}$ of ${\bf Y}_n$, where ${\bf u}_{n,1}$ is the 1st column of ${\bf U}_{n}$, can we recover $\bf X$ to some certain extent?