Coherence-band gradient similarity conjecture for one-bit millimetre-wave channel estimation

Let $h$ be the objective function, let $x_i$ and $x_j$ denote its $i$-th and $j$-th channel coefficients, and let $abla h(x_i)$ and $abla h(x_j)$ denote the corresponding \partial gradients. For sensing vectors $\mathbf{a}_i$ and $\mathbf{a}_j$, define their coherence by

$$\mu(i,j)=\frac{|\mathbf{a}_i^{\mathrm{H}}\mathbf{a}_j|}{\\|\mathbf{a}_i\\|\\|\mathbf{a}_j\\|}.$$

For $\eta\in(0,1)$, the $\eta$-coherence band of $i$ is

$$B_{\eta}(i)=\\{j\mid \mu(i,j)\geq\eta\\}.$$

Coherence-band gradient similarity conjecture. For sufficiently large $\eta$, if $j\in B_{\eta}(i)$ and $x_i=x_j$, then

$$\nabla h(x_i)\approx\nabla h(x_j).$$

The claim is intended to explain support-identification failure when the sensing matrix has highly coherent columns: coefficients with equal values and highly coherent associated sensing vectors are expected to have approximately equal gradients. The source presents this as a conjecture, without supplying a resolution.