Shift-invariant spaces, bandlimited spaces and reproducing kernel spaces with shift-invariant kernels on undirected finite graphs

TL;DR

This paper demonstrates that every GSIS is a reproducing kernel Hilbert space with a shift-invariant kernel, and every GSIS is a principal GSIS, and proposes a novel sampling and reconstruction algorithm with finite steps.

Abstract

In this paper, we introduce the concept of graph shift-invariant space (GSIS) on an undirected finite graph, which is the linear space of graph signals being invariant under graph shifts, and we study its bandlimiting, kernel reproducing and sampling properties. Graph bandlimited spaces have been widely applied where large datasets on networks need to be handled efficiently. In this paper, we show that every GSIS is a bandlimited space, and every bandlimited space is a principal GSIS. Functions in a reproducing kernel Hilbert space with shift-invariant kernel could be learnt with significantly low computational cost. In this paper, we demonstrate that every GSIS is a reproducing kernel Hilbert space with a shift-invariant kernel. Based on the nested Krylov structure of GSISs in the spatial domain, we propose a novel sampling and reconstruction algorithm with finite steps, with its performance tested for well-localized signals on circulant graphs and flight delay dataset of the 50 busiest airports in the USA.

Figures (2)

• Figure 1: Plotted on the top are the damped cosine wave signal ${\bf x}_0$ (in blue) and the reconstruction error ${\bf x}_{n, W}-{\bf x}_0$ (in red), where $A=1, n=6, N=100, P=\lfloor N/6\rfloor=16, \sigma=0.1$, and $(\lambda, \omega)=(1/4, 2\pi/5)$ (top left) and $(1/8, 2\pi/10)$ (top right). The relative maximal reconstruction error ${\rm RE}(6, 16)$ and the relative maximal sampling error ${\rm SE}(6, 16)$ are $(0.0680, 0.0680)$ (top left) and $(0.2865, 0.0679)$ (top right) respectively. Shown in the middle and bottom rows are the average of the relative maximal signal reconstruction error ${\rm RE}(n, P)$ (left) and the relative maximal sampling error ${\rm SE}(n, P)$ (right) over $M=100$ trials, where $1\le n\le 18, 1\le P\le 45$, ${\bf x}_0$ is the damped cosine wave signal with $A=1, \lambda=1/4$ and $\omega=2\pi/5$, and the noise level $\sigma= 0$ (the middle row) and $\sigma=0.1$ (the bottom row).
• Figure 2: Plotted on the top are the flight delay data ${\bf x}_d$ (left) and absolute error data $|{\bf x}_{K; d, n}-{\bf x}_d|$ (right) of the top 50 US airports on 29 August 2024, except Honolulu International airport (HNL) and Luis Munoz Marin International airport (SJU), where $d=60$, ${\bf x}_{K; d, n}$ is reconstructed from Algorithm \ref{['kylovsampling.alg']} in the noiseless environment with adaptive GSIS model and Krylov level $n=2$, and maximal approximation error is $\|{\bf x}_{K; 60, 2}-{\bf x}_{60}\|_\infty=7.9382$. Shown on the bottom are the average of maximal approximation error $F_{K, n}$ by Krylov subspaces of adaptive GSIS (in green) and of non-adaptive GSIS (in blue) and the maximal approximation error $F_{B, n}$ by bandlimited spaces (in red) over 184 days, where the subsampling set contains the whole 50 airports for the left plot and the top 30 airports with most delay on average over 184 days for the right plot.

Theorems & Definitions (14)

• Proposition 2.2
• Corollary 2.3
• Theorem 3.1
• Theorem 3.2
• Proposition 3.3
• Proposition 3.4
• Proposition 3.5
• Remark 3.6
• Theorem 4.1
• Theorem 4.2