Adaptive Least Mean Squares Estimation of Graph Signals

Abstract

The aim of this paper is to propose a least mean squares (LMS) strategy for adaptive estimation of signals defined over graphs. Assuming the graph signal to be band-limited, over a known bandwidth, the method enables reconstruction, with guaranteed performance in terms of mean-square error, and tracking from a limited number of observations over a subset of vertices. A detailed mean square analysis provides the performance of the proposed method, and leads to several insights for designing useful sampling strategies for graph signals. Numerical results validate our theoretical findings, and illustrate the performance of the proposed method. Furthermore, to cope with the case where the bandwidth is not known beforehand, we propose a method that performs a sparse online estimation of the signal support in the (graph) frequency domain, which enables online adaptation of the graph sampling strategy. Finally, we apply the proposed method to build the power spatial density cartography of a given operational region in a cognitive network environment

Inference of spatiotemporal processes over graphs via kernel kriged Kalman filtering

“…The tests compare the following reconstruction algorithms: (i) The least mean-squares (LMS) algorithm in [9] with step size µ LMS ; (ii) the distributed least-squares reconstruction (DLSR) algorithm [8] with step sizes µ DLSR and β DLSR (both LMS and DLSR can track slowly time-varying B-bandlimited graph signals); (iii) The B-bandlimited instantaneous estimator (BL-IE) which uses the estimator in [3], [4] per slot t; and (iv) Algorithm 1 with the following configuration: a diffusion kernel (cf. Table I) with parameter σ; a state noise covariance Σ η [t] = s η Σ η with parameter s η > 0 and Σ η := N N a positive definite matrix with N ∈ R N ×N a random matrix with standardized Gaussian entries; and a transition matrix…”

“…Distributed reconstruction methods are reported in [8] and [9]. However, they rely on the bandlimited model, whose effectiveness in capturing the dynamics of real-world graph functions may not hold.…”

Optimal sampling strategies for adaptive learning of graph signals

“…Then, several reconstruction methods have been proposed, either iterative as in [13], [14], or batch, as in [8], [10], [15]. Recently, adaptive strategies for online reconstruction and learning of graph signals were also proposed in [16]- [18], and paved the way to the development of novel adaptive GSP tools. In particular, reference [16] proposed an LMS estimation strategy for adaptive reconstruction of graph signals from a subset of samples smartly collected over the graph.…”

“…Recently, adaptive strategies for online reconstruction and learning of graph signals were also proposed in [16]- [18], and paved the way to the development of novel adaptive GSP tools. In particular, reference [16] proposed an LMS estimation strategy for adaptive reconstruction of graph signals from a subset of samples smartly collected over the graph. The method was then extended to the distributed setting in [17].…”

“…In this paper, we first extend the LMS algorithm of [16] to incorporate a probabilistic sampling mechanism, where each node in the graph has an assigned probability to be sampled at each time instant. We also derive a mean-square analysis of the proposed method that illustrates the role played by the sampling strategy on the performance of the LMS algorithm.…”