  Graph Structure Learning (GSL) recently has attracted considerable attentions
in its capacity of optimizing graph structure as well as learning suitable
parameters of Graph Neural Networks (GNNs) simultaneously. Current GSL methods
mainly learn an optimal graph structure (final view) from single or multiple
information sources (basic views), however the theoretical guidance on what is
the optimal graph structure is still unexplored. In essence, an optimal graph
structure should only contain the information about tasks while compress
redundant noise as much as possible, which is defined as "minimal sufficient
structure", so as to maintain the accurancy and robustness. How to obtain such
structure in a principled way? In this paper, we theoretically prove that if we
optimize basic views and final view based on mutual information, and keep their
performance on labels simultaneously, the final view will be a minimal
sufficient structure. With this guidance, we propose a Compact GSL architecture
by MI compression, named CoGSL. Specifically, two basic views are extracted
from original graph as two inputs of the model, which are refinedly reestimated
by a view estimator. Then, we propose an adaptive technique to fuse estimated
views into the final view. Furthermore, we maintain the performance of
estimated views and the final view and reduce the mutual information of every
two views. To comprehensively evaluate the performance of CoGSL, we conduct
extensive experiments on several datasets under clean and attacked conditions,
which demonstrate the effectiveness and robustness of CoGSL.
