The Restricted Isometry Property (Low-Rank Matrices)
Let $\mathcal{A}:\mathbb{R}^{n\times n}\rightarrow \mathbb{R}^m$ be a linear operator. The constant $\delta_r:=\delta_r(\mathcal{A})$ is defined as follows:
\begin{equation} \delta_r:=\inf\{\delta>0:(1-\delta)\lVert\mathbf{X}\rVert_F\leq\lVert\mathcal{A}(\mathbf{X})\rVert_F\leq(1+\delta)\lVert\mathbf{X}\rVert_F \text{ for all } \mathbf{X}\in B_r\} \end{equation}
where $B_r=\{\mathbf{X}\neq\mathbf{0} \text{ , } \text{rank}(\mathbf{X})\leq r\}$ and $r\leq n$.
- An operator $\mathcal{A}$ is said to have the Restricted Isometry Property for a given $r\leq n$ if $\delta_r<1$.
My Question
Is there a deterministic (Not Random) linear operator $\mathcal{A}:\mathbb{R}^{n\times n}\rightarrow \mathbb{R}^m$ such that for a some $r\leq n$ and $m<n^2$ we have that $\delta_r <1$?