  Here we consider the problem of denoising features associated to complex
data, modeled as signals on a graph, via a smoothness prior. This is motivated
in part by settings such as single-cell RNA where the data is very
high-dimensional, but its structure can be captured via an affinity graph. This
allows us to utilize ideas from graph signal processing. In particular, we
present algorithms for the cases where the signal is perturbed by Gaussian
noise, dropout, and uniformly distributed noise. The signals are assumed to
follow a prior distribution defined in the frequency domain which favors
signals which are smooth across the edges of the graph. By pairing this prior
distribution with our three models of noise generation, we propose Maximum A
Posteriori (M.A.P.) estimates of the true signal in the presence of noisy data
and provide algorithms for computing the M.A.P. Finally, we demonstrate the
algorithms' ability to effectively restore signals from white noise on image
data and from severe dropout in single-cell RNA sequence data.
