Generalized Sampling on Graphs WithSubspace and Smoothness Priors

Generalized Sampling on Graphs With
Subspace and Smoothness Priors

Yuichi Tanaka and Yonina C. Eldar Y. Tanaka is with the Graduate School of BASE, Tokyo University of Agriculture and Technology, Koganei, Tokyo 184–8588, Japan. Y. Tanaka is also with PRESTO, Japan Science and Technology Agency, Kawaguchi, Saitama 332–0012, Japan (email: ytnk@cc.tuat.ac.jp).Y. C. Eldar is with Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, Rehovot 7610001, Israel (email: yonina.eldar@weizmann.ac.il).
Abstract

We consider a framework for generalized sampling of graph signals that parallels sampling in shift-invariant (SI) subspaces. This framework allows for arbitrary input signals, which are not constrained to be bandlimited. Furthermore, the sampling and reconstruction filters can be different. We present design methods of the correction filter that compensates for these differences and can be obtained in closed form in the graph frequency domain. This paper considers two priors on graph signals: The first is a subspace prior, where the signal is assumed to lie in a periodic graph spectrum (PGS) subspace. The PGS subspace is proposed as a counterpart of the SI subspace used in standard sampling theory. The second is a smoothness prior that imposes a smoothness requirement on the graph signal. We suggest recovery methods both for the case when the recovery filter can be optimized and in the setting in which a predefined filter must be used. Sampling is performed in the graph frequency domain, which is a counterpart of “sampling by modulation” in SI subspaces. We compare our approach with existing sampling methods in graph signal processing. The effectiveness of the proposed generalized sampling is validated numerically through several experiments.

I Introduction

Sampling theory for graph signals has been recently studied with the goal of building parallels of sampling results in standard signal processing [Anis2016, Chen2015, Marque2016, Chepur2018, Puy2018, Tsitsv2016, Valses2018, Pesens2008, Pesens2010, Sakiya2019a]. Since the pioneering Shannon–Nyquist sampling theorem [Shanno1949, Jerri1977], sampling theories that encompass more general signal spaces beyond that of bandlimited signals in shift-invariant (SI) spaces have been studied widely with many promising applications [Eldar2009, Eldar2003, Eldar2006, Unser2000, Unser1994, Eldar2004, Eldar2015]. More relaxed priors have also been considered such as smoothness priors. These theories allow for sampling and recovery of signals in arbitrary subspaces using almost arbitrary sampling and recovery kernels. These results are particularly useful in the SI setting in which sampling and recovery reduce to simple filtering operations.

Graph signal processing (GSP) [Shuman2013, Ortega2018] is a relatively new field of signal processing that studies discrete signals defined on a graph. Recent work on GSP ranges from theory to practical applications including wavelet/filter bank design [Hammon2011, Narang2013, Tanaka2014a, Sakiya2019], learning graphs from observed data [Dong2016, Kalofo2016, Egilme2017], restoration of graph signals [Onuki2016, Ono2015], image/point cloud processing [Cheung2018], and deep learning on graphs [Bronst2017].

One of the topics of interest in GSP is graph sampling theory [Anis2016, Chen2015, Marque2016, Chepur2018, Puy2018, Tsitsv2016, Valses2018, Pesens2008, Pesens2010, Sakiya2019a] which is aimed at recovering a graph signal from its sampled version. Most studies on sampling graph signals considered recovery of discrete graph signals from their sampled version [Anis2016, Chen2015, Marque2016, Chepur2018, Puy2018, Tsitsv2016, Valses2018, Sakiya2019a]. Current approaches generally rely on vertex subsampling. In contrast to time domain uniform sampling, graphs are naturally discrete and samples on the vertices are distributed nonuniformly. This fact implies that the maximum bandwidth, which is typically measured by the number of nonzero coefficients in the graph Fourier spectrum [Pesens2008, Anis2016], allowing for perfect recovery will differ depending on the sampling set. Therefore, sampling set selection for graph signals, often referred to as sensor position selection in machine learning and sensor network communities, is required in the context of GSP [Anis2016, Puy2018, Sakiya2019a, Krause2007, Krause2008]. While many deterministic and random sampling set selection methods have been studied, they still focus primarily on vertex subsampling.

Here, our goal is to build a generalized graph sampling framework that allows for (perfect) recovery of arbitrary graph signals beyond bandlimited signals, and parallels SI sampling for time domain signals. In SI sampling, the input subspace has a particular SI structure. Sampling is modeled by uniformly sampling the output of the signal convolved with an arbitrary sampling filter. Under a mild condition on the sampling filter, recovery is obtained by a correction filter having an explicit closed-form frequency response. Herein, we demonstrate how one can extend these ideas to graphs by defining an appropriate input space of graph signals and sampling in the graph frequency domain [Tanaka2018].

We consider two priors characterizing the graph signals:

  1. Subspace Prior: The signal lies in a known subspace characterized by a given generator; and

  2. Smoothness Prior: The signal is smooth on a given graph.

Both priors parallel those considered in SI sampling [Eldar2009, Eldar2015].

For the subspace prior, we define an input signal subspace which we define as a periodic graph spectrum (PGS) subspace, and is a counterpart of the SI subspace. This subspace maintains the repeated graph frequency spectra similar to that in SI signals. In particular, the spectral domain characteristics of such graph signals mimic that of SI time domain signals. In the smoothness prior, we assume that the quadratic form of the graph signal is small for a given smoothness function. In this setting, perfect recovery is no longer possible. Nonetheless, following the work in general Hilbert space sampling, we show how to design graph filters that allows us to best approximate the input signal under several different criteria [Eldar2005, Hiraba2007, Dvorki2009, Eldar2015].

Generalized sampling for standard and graph sampling paradigms allows for the use of arbitrary sampling and reconstruction filters that are not necessarily ideal low-pass filters. It also allows for the use of fixed recovery filters that may have implementation advantages. In all settings, and under all recovery criteria considered, we show that reconstruction is given by spectral graph filters, whose response has a closed form solution that depends on the generator function, smoothness, and sampling/reconstruction filters.

Our framework relies on graph sampling performed in the graph frequency domain [Tanaka2018] as a counterpart of “sampling by modulation” in the SI setting [Eldar2015, Vetter2014]. This sampling method maintains the shape of the graph spectrum. Whereas in SI sampling, sampling in the time and frequency domains coincide, in the graph setting, vertex domain sampling and frequency domain sampling are in general different. Sampling by modulation enables a generalized graph sampling framework that is analogous to SI sampling—exhibiting a symmetric structure where the sampling and reconstruction steps contain similar building blocks as those in SI sampling. Our approach reduces to the standard SI results in the case of a graph representing the conventional time axis whose graph Fourier basis is the discrete Fourier transform (DFT).

In the context of subspace sampling with a PGS prior, our results allow for perfect recovery of graph signals beyond bandlimited ones for almost all signal and sampling spaces. In particular, we require these subspaces to satisfy a direct-sum (DS) condition, as in generalized sampling. When the DS condition does not hold, we design a correction filter that best approximates the input under both least-squares (LS) and minimax (MX) criteria. We then introduce LS and MX strategies for recovery under a smoothness prior. In all cases, the graph filters have explicit graph frequency responses that parallel those in the SI setting.

Due to the generality of our results, they allow in particular for recovery of non-bandlimited graph signals. This is in contrast to most studies on graph sampling theory [Anis2016, Chen2015, Marque2016, Puy2018] which focus on recovery of bandlimited signals or recovery of noisy bandlimited signals [Anis2016, Marque2016, Puy2018, Sakiya2019]. We validate the reconstruction error of our generalized sampling framework for non-bandlimited graph signals through numerical experiments. In the special case in which the underlying graph is bipartite, we show that perfect recovery of a non-bandlimited graph signal is possible with vertex domain sampling and reconstruction.

An earlier work focused on generalized sampling of graph signals has been reported in [Chepur2018]. This approach is based on the framework of generalized Hilbert space sampling [Eldar2009, Eldar2015] and demonstrates the possibility of perfect recovery of graph signals that are not necessarily bandlimited. However, sampling operator inversion is in general required for the reconstruction process. Similar matrix inversions can be found in many graph sampling studies [Anis2016, Chen2015, Marque2016, Chepur2018, Puy2018, Tsitsv2016, Valses2018, Sakiya2019a]. Such inversion can be computationally demanding especially for large graphs. In addition, most previous work considered vertex domain subsampling, which also leads to different building blocks in the sampling and reconstruction steps. Our framework, in contrast, leads to simple closed form recovery methods based on graph filters in both the sampling and recovery steps. We expand on the similarities and differences between our work and previous approaches in Section LABEL:sec:relationship.

Our preliminary work [Tanaka2019b] studies generalized graph sampling with a subspace prior with a DS condition. This paper significantly expands its results by introducing an integrated framework, studying different criteria, and considering the smoothness prior.

The remainder of this paper is organized as follows. Section II reviews generalized sampling in Hilbert spaces and in the SI setting. The notations and basics of GSP are introduced in Section LABEL:sec:relatedwork. A framework for generalized graph sampling is presented in Section LABEL:sec:framework. Section LABEL:sec:subspace proposes signal recovery methods assuming a PGS subspace prior. We relax this prior in Section LABEL:sec:smoothness to a smoothness prior. Section LABEL:sec:relationship describes the relationships between our work and existing methods. Numerical experiments are presented in Section LABEL:sec:exp. Finally, Section LABEL:sec:conclusion concludes the paper.

Ii Generalized Sampling in Hilbert Space

This section introduces prior results on generalized sampling in Hilbert spaces [Eldar2003, Eldar2006, Eldar2015] and corresponding results in the SI setting, which are fundamental for our generalized graph sampling approach. Detailed derivations of the results can be found in [Eldar2015] and the references therein. Table II summarizes the main results of this section in the SI setting.

\adl@mkpreamc——\@addtopreamble\@arstrut\@preamble \adl@mkpreamc\@addtopreamble\@arstrut\@preamble

\adl@mkpreamc—\@addtopreamble\@arstrut\@preamble \adl@mkpreamc——\@addtopreamble\@arstrut\@preamble \adl@mkpreamc—\@addtopreamble\@arstrut\@preamble \adl@mkpreamc\@addtopreamble\@arstrut\@preamble
Filter CF RF \adl@mkpreamc——\@addtopreamble\@arstrut\@preamble CF RF \adl@mkpreamc\@addtopreamble\@arstrut\@preamble
Subspace DS, MX DS, MX
Prior LS LS
Smoothness LS LS
Prior MX MX
TABLE I: Correction and Reconstruction Filters for Shift-Invariant and Graph Spectral Filters where CF and RF are abbreviations of correction filter and reconstruction filter, respectively. DS, LS, and MX refer to direct-sum, least squares, and minimax solutions, respectively. Spectra and are defined in (LABEL:eqn:sampled_cc) and (LABEL:eqn:graph_cc), respectively.
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