  Graph signal processing (GSP) is a prominent framework for analyzing signals
on non-Euclidean domains. The graph Fourier transform (GFT) uses the
combinatorial graph Laplacian matrix to reveal the spectral decomposition of
signals in the graph frequency domain. However, a common challenge in applying
GSP methods is that in many scenarios the underlying graph of a system is
unknown. A solution in such cases is to construct the unobserved graph from
available data, which is commonly referred to as graph or network inference.
Although different graph inference methods exist, these are restricted to
learning from either smooth graph signals or simple additive Gaussian noise.
Other types of noisy data, such as discrete counts or binary digits, are rather
common in real-world applications, yet are underexplored in graph inference. In
this paper, we propose a versatile graph inference framework for learning from
graph signals corrupted by exponential family noise. Our framework generalizes
previous methods from continuous smooth graph signals to various data types. We
propose an alternating algorithm that jointly estimates the graph Laplacian and
the unobserved smooth representation from the noisy signals. We also extend our
approach to a variational form to account for the inherent stochasticity of the
latent smooth representation. Finally, since real-world graph signals are
frequently non-independent and temporally correlated, we further adapt our
original setting to a time-vertex formulation. We demonstrate on synthetic and
real-world data that our new algorithms outperform competing Laplacian
estimation methods that suffer from noise model mismatch.
