  Sampling of signals belonging to a low-dimensional subspace has
well-documented merits for dimensionality reduction, limited memory storage,
and online processing of streaming network data. When the subspace is known,
these signals can be modeled as bandlimited graph signals. Most existing
sampling methods are designed to minimize the error incurred when
reconstructing the original signal from its samples. Oftentimes these
parsimonious signals serve as inputs to computationally-intensive linear
operators. Hence, interest shifts from reconstructing the signal itself towards
approximating the output of the prescribed linear operator efficiently. In this
context, we propose a novel sampling scheme that leverages graph signal
processing, exploiting the low-dimensional (bandlimited) structure of the input
as well as the transformation whose output we wish to approximate. We formulate
problems to jointly optimize sample selection and a sketch of the target linear
transformation, so when the latter is applied to the sampled input signal the
result is close to the desired output. Similar sketching as sampling ideas are
also shown effective in the context of linear inverse problems. Because these
designs are carried out off line, the resulting sampling plus
reduced-complexity processing pipeline is particularly useful for data that are
acquired or processed in a sequential fashion, where the linear operator has to
be applied fast and repeatedly to successive inputs or response signals.
Numerical tests showing the effectiveness of the proposed algorithms include
classification of handwritten digits from as few as 20 out of 784 pixels in the
input images and selection of sensors from a network deployed to carry out a
distributed parameter estimation task.
