Rethinking sketching as sampling: A graph signal processing approach

Abstract

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 operators (e.g., graph filters and transforms). 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 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 solvers are proposed to accommodate highdimensional 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 data that reside on a low-dimensional subspace and is acquired in a streaming fashion, where the linear transform 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.

Controllability of Bandlimited Graph Processes Over Random Time Varying Graphs

“…The GFTx K ofx writes asx K = V H K x. This representation connects the signal bandwidth with the sampling and reconstruction strategies as shown in [12], [14], [15], [29], [30]. We will also exploit bandlimitedness in Section IV to control the network from a few driving nodes.…”

“…Lead by the promising results of bandlimited graph signal reconstruction from samples on a few nodes [12], [14], [15], [29], [30], we aim to control x t through a fixed, time-invariant, set of nodes S of cardinality |S| = M ≤ N . Let then B = C T denote a binary matrix that selects these nodes.…”

“…Proposition 2. Consider the linear system (12) describing a process over a sequence of RES(p) graphs G t with inband mean evolution (14). A necessary condition to control the mean system in finite time T towards a target frequency contentx * K over the K frequencies of interest is to select M ≥ K/T driving nodes.…”

“…the graph signal). Recent evidence from graph signal processing (GSP) [9]- [11] has shown that this coupling can bring substantial benefits in graph signal sampling [12], interpolation [13], [14], adaptive reconstruction [15], and observability of diffusion processes [16]. A common point that unifies [10]- [16] is the so-called graph Fourier transform (GFT).…”

“…But the bandlimitedness prior poses a powerful and parsimonious model to develop practical tools. This prior is exploited in GSP for sampling [10], [11], [18], where signals arising in economic [19], handwritten digits [12], [14] or brain functional imaging [20], are approximately bandlimited. In the Zachary's Karate club example, a bandlimited network state corresponds to opinions polarized into a few clusters of like-minded members [12].…”