  A significant increase in the number of reconfigurable intelligent surface
(RIS) elements results in a spherical wavefront in the near field of extremely
large-scale RIS (XL-RIS). Although the channel matrix of the cascaded two-hop
link may become sparse in the polar-domain representation, their accurate
estimation of these polar-domain parameters cannot be readily guaranteed. To
tackle this challenge, we exploit the sparsity inherent in the cascaded
channel. To elaborate, we first estimate the significant path-angles and
distances corresponding to the common paths between the BS and the XL-RIS.
Then, the individual path parameters associated with different users are
recovered. This results in a two-stage channel estimation scheme, in which
distinct learning-based networks are used for channel training at each stage.
More explicitly, in stage I, a denoising convolutional neural network (DnCNN)
is employed for treating the grid mismatches as noise to determine the true
grid index of the angles and distances. By contrast, an iterative shrinkage
thresholding algorithm (ISTA) based network is proposed for adaptively
adjusting the column coherence of the dictionary matrix in stage II. Finally,
our simulation results demonstrate that the proposed two-stage learning-based
channel estimation outperforms the state-of-the-art benchmarks.
