The converse measurement-rate conjecture for two-sensor distributed compressive sensing

Let $J=2$. Fix the sparsity rate of the common component $S_C$ and the innovation sparsity rates $S_1=S_2=S_I$. Define the measurement function $c'(S):=S\,c(S)$, where $c(S)$ is the oversampling factor used for single-source $\ell_1$-norm minimization. Let $R_1$ and $R_2$ be the measurement rates of the two sensors.

Converse measurement-rate conjecture. The following conditions on the measurement rates are necessary for recovery with probability one:

$$R_j\geq c'\left(S_I+S_CS_I-S_CS_I^2\right),\qquad j=1,2,$$ $$R_1+R_2\geq c'\left(S_I+S_CS_I-S_CS_I^2\right)+c'\left(S_C+S_I-S_CS_I\right).$$

This conjecture gives individual and sum-rate converse bounds for the two-sensor case. The source explains that the proposed derivation relies on Gaussian measurement matrices, whereas the distributed sensing matrix has block structure; the bounds therefore remain to be proved rigorously.