  This paper presents a bias-variance tradeoff of graph Laplacian regularizer,
which is widely used in graph signal processing and semi-supervised learning
tasks. The scaling law of the optimal regularization parameter is specified in
terms of the spectral graph properties and a novel signal-to-noise ratio
parameter, which suggests selecting a mediocre regularization parameter is
often suboptimal. The analysis is applied to three applications, including
random, band-limited, and multiple-sampled graph signals. Experiments on
synthetic and real-world graphs demonstrate near-optimal performance of the
established analysis.
