Rethinking Sketching as Sampling: A Graph Signal Processing Approach

Sampling of bandlimited graph signals has welldocumented merits for dimensionality reduction, affordable storage, and online processing of streaming network data. 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 operator (e.g., graph filters and transforms). Hence, interest shifts from reconstructing the signal itself towards instead approximating the output of the prescribed linear operator efficiently. In this context, we propose a novel sampling scheme that leverages the bandlimitedness 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 affordably applied to the sampled input signal the result is close to the desired output. These designs are carried out off line, and several heuristic (sub)optimal solvers are proposed to accommodate high-dimensional problems, especially when computational resources are at a premium. Similar sketching as sampling ideas are also shown effective in the context of linear inverse problems. The developed sampling plus reduced-complexity processing pipeline is particularly useful for streaming data, where the linear transform has to be applied fast and repeatedly to successive inputs or response signals. Numerical tests show the effectiveness of the proposed algorithms in classifying handwritten digits from as few as 20 out of 784 pixels in the input images, as well as in accurately estimating the frequency components of bandlimited graph signals sampled at few nodes.