Rethinking Sketching as Sampling: A Graph Signal Processing Approach

Abstract

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.