Compactification of Infinite Graphs and Sampling

Opuscula Mathematica, 2011

We prove two sampling theorems for infinite (countable discrete) weighted graphs G; one example being "large grids of resistors" i.e., networks and systems of resistors. We show that there is natural ambient continuum X containing G, and there are Hilbert spaces of functions on X that allow interpolation by sampling values of the functions restricted only on the vertices in G. We sample functions on X from their discrete values picked in the vertex-subset G. We prove two theorems that allow for such realistic ambient spaces X for a fixed graph G, and for interpolation kernels in function Hilbert spaces on X, sampling only from points in the subset of vertices in G. A continuum is often not apparent at the outset from the given graph G. We will solve this problem with the use of ideas from stochastic integration.

Proceedings of the American Mathematical Society, 2005

The notion of band limited functions is introduced on a quantum graph. The main results of the paper are a uniqueness theorem and a reconstruction algorithm of such functions from discrete sets of values. It turns out that some of our band limited functions can have compact supports and their frequencies can be localized on the “time" side. It opens an opportunity to consider signals of a variable band width and to develop a sampling theory with variable rate of sampling.

2015

We propose a sampling theory for signals that are supported on either directed or undirected graphs. The theory follows the same paradigm as classical sampling theory. We show that the perfect recovery is possible for graph signals bandlimited under the graph Fourier transform, and the sampled signal coefficients form a new graph signal, whose corresponding graph structure is constructed from the original graph structure, preserving frequency contents. By imposing a specific structure on the graph, graph signals reduce to finite discrete-time signals and the proposed sampling theory works reduces to classical signal processing. We further establish the connection to frames with maximal robustness to erasures as well as compressed sensing, and show how to choose the optimal sampling operator, how random sampling works on circulant graphs and Erdős-Rényi graphs, and how to handle full-band graph signals by using graph filter

IEEE Transactions on Signal Processing, 2019

Graph signal processing (GSP) has become an important tool in many areas such as image processing, networking learning and analysis of social network data. In this paper, we propose a broader framework that not only encompasses traditional GSP as a special case, but also includes a hybrid framework of graph and classical signal processing over a continuous domain. Our framework relies extensively on concepts and tools from functional analysis to generalize traditional GSP to graph signals in a separable Hilbert space with infinite dimensions. We develop a concept analogous to Fourier transform for generalized GSP and the theory of filtering and sampling such signals.

2020

A continuous-time graph signal can be viewed as a continuous time-series of graph signals. It generalizes both the classical continuous-time signal and ordinary graph signal. In this paper, we consider the sampling theory of bandlimited continuous-time graph signals. We describe an explicit procedure to determine a discrete sampling set for perfect signal recovery. Moreover, in analogous to the Nyquist-Shannon sampling theorem, we give an explicit formula for the minimal sample rate.

2020

We present recent advances in harmonic analysis on infinite graphs. Our approach combines combinatorial tools with new results from the theory of unbounded Hermitian operators in Hilbert space, geometry, boundary constructions, and spectral invariants. We focus on particular classes of infinite graphs, including such weighted graphs which arise in electrical network models, as well as new diagrammatic graph representations. We further stress some direct parallels between our present analysis on infinite graphs, on the one hand, and, on the other, specific areas of potential theory, probability, harmonic functions, and boundary theory. The limit constructions, finite to infinite, and local to global, can be used in various applications.

2018 IEEE Global Conference on Signal and Information Processing (GlobalSIP)

In this paper, we extend the sampling theory on graphs by constructing a framework that exploits the structure in product graphs for efficient sampling and recovery of bandlimited graph signals that lie on them. Product graphs are graphs that are composed from smaller graph atoms; we motivate how this model is a flexible and useful way to model richer classes of data that can be multi-modal in nature. Previous works have established a sampling theory on graphs for bandlimited signals. Importantly, the framework achieves significant savings in both sample complexity and computational complexity.

Applied and Computational Harmonic Analysis, 2006

A notion of band-limited functions is introduced in terms of a Hamiltonian on a quantum graph Γ. It is shown that a bandlimited function is uniquely determined and can be reconstructed in a stable way from a countable set of "measurements" {Φ i (f)}, i ∈ N, where {Φ i } is a sequence of compactly supported measures whose supports are "small" and "densely" distributed over the graph. In particular, {Φ i }, i ∈ N, can be a sequence of Dirac measures δ x i , x i ∈ Γ. A reconstruction method in terms of frames is given which is a generalization of the classical result of Duffin-Schaeffer about exponential frames on intervals. The second reconstruction algorithm is based on an appropriate generalization of average variational splines to the case of quantum graphs. To obtain all these results we establish some analogs of Poincaré and Plancherel-Polya inequalities on quantum graphs.

2015 23rd European Signal Processing Conference (EUSIPCO), 2015

Continuous-time signals are well known for not being perfectly localized in both time and frequency domains. Conversely, a signal defined over the vertices of a graph can be perfectly localized in both vertex and frequency domains. We derive the conditions ensuring the validity of this property and then, building on this theory, we provide the conditions for perfect reconstruction of a graph signal from its samples. Next, we provide a finite step algorithm for the reconstruction of a band-limited signal from its samples and then we show the effect of sampling a non perfectly band-limited signal and show how to select the bandwidth that minimizes the mean square reconstruction error.

2010

Sampling and reconstruction of functions is a central tool in science. A key result is given by the sampling theorem for bandlimited functions attributed to Whittaker, Shannon, Nyquist, and Kotelnikov. We develop an analogous sampling theory for operators which we call bandlimited if their Kohn-Nirenberg symbols are bandlimited. We prove sampling theorems for such operators and show that they are extensions of the classical sampling theorem.