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Abstract:It is now known that the expressive power of graph convolutional neural nets (GCN) does not grow infinitely with the number of layers. Instead, the GCN output approaches a subspace spanned by the first eigenvector of the normalized graph Laplacian matrix with the convergence rate characterized by the "eigen-gap": the difference between the Laplacian's first two distinct eigenvalues. To promote a deeper GCN architecture with sufficient expressiveness, in this paper, given an empirical covariance matrix $\bar{C}$ computed from observable data, we learn a sparse graph Laplacian matrix $L$ closest to $\bar{C}^{-1}$ while maintaining a desirable eigen-gap that slows down convergence. Specifically, we first define a sparse graph learning problem with constraints on the first eigenvector (the most common signal) and the eigen-gap. We solve the corresponding dual problem greedily, where a locally optimal eigen-pair is computed one at a time via a fast approximation of a semi-definite programming (SDP) formulation. The computed $L$ with the desired eigen-gap is normalized spectrally and used for supervised training of GCN for a targeted task. Experiments show that our proposal produced deeper GCNs and smaller errors compared to a competing scheme without explicit eigen-gap optimization.