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Graph semi-supervised learning (GSSL) is a technique that uses a combination of labeled and unlabeled nodes on a graph to determine a classifier for new, incoming data. This problem can be analyzed through the lens of graph signal processing. In particular, the penalty functions used in the optimization formulation of standard GSSL algorithms can be interpreted as appropriately-defined filters in the Graph Fourier domain. We propose a wavelet-regularized semi-supervised learning algorithm using suitably-defined spline-like graph wavelets. These wavelets are critically-sampled, perfect-reconstruction basis representations, in contrast to much of the existing work proposing overcomplete representations. Critical sampling is essential for controlling the complexity in applications dealing with large scale datasets. We are also interested in understanding when wavelet-regularized approaches perform better than traditional Fourier-based regularizers. We compare the performance of our proposed spline-like, wavelet-regularized learning algorithm (as well as other existing graph wavelet designs) to some standard graph semi-supervised learning techniques on synthetic and real-world datasets.