Learning a suitable graph is an important precursor to many graph signal processing (GSP) pipelines, such as graph signal compression and denoising. Previous graph learning algorithms either i) make assumptions on graph connectivity (e.g., graph sparsity), or ii) make edge weight assumptions such as positive edges only. In this paper, given an empirical covariance matrixC computed from data as input, we consider an eigen-structural assumption on the graph Laplacian matrix L: the first K eigenvectors of L are pre-selected,

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Transform coding to sparsify signal representations remains crucial in an image compression pipeline. While the Karhunen-Loève transform (KLT) computed from an empirical covariance matrix C is theoretically optimal for a stationary process, in practice, collecting sufficient statistics from a non-stationary image to reliably estimate C can be difficult. In this paper, to encode an intra-prediction residual block, we pursue a hybrid model-based / data-driven approach: the first K eigenvectors of a transform matrix are derived from a statistical model, e.g., the asymmetric discrete sine transform (ADST), for stability, while the remaining N − K are computed from C for data adaptivity. The transform computation is posed as a graph learning problem, where we seek a graph Laplacian matrix minimizing a graphical lasso objective inside a convex cone sharing the first K eigenvectors in a Hilbert space of real symmetric matrices. We efficiently solve the problem via augmented Lagrangian relaxation and proximal gradient (PG). Using open-source WebP as a baseline image codec, experimental results show that our hybrid graph transform achieved better coding performance than discrete cosine transform (DCT), ADST and KLT, and better stability than KLT.

In agronomics, predicting crop yield at a per field / county granularity is important for farmers to minimize uncertainty and plan seeding for the next crop cycle. While state-ofthe-art prediction techniques employ graph convolutional nets (GCN) to predict future crop yields given relevant features and crop yields of previous years, a dense underlying graph kernel requires long training and execution time. In this paper, we propose a graph sparsification method based on the Fiedler number to remove edges from a complete graph kernel, in order to lower the complexity of GCN training / execution. Specifically, we first show that greedily removing an edge at a time that induces the minimal change in the second eigenvalue leads to a sparse graph with good GCN performance. We then propose a fast method to choose an edge for removal per iteration based on an eigenvalue perturbation theorem. Experiments show that our Fiedler-based method produces a sparse graph with good GCN performance compared to other graph sparsification schemes in crop yield prediction.

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