The goal of this paper is to propose novel strategies for adaptive learning of signals defined over graphs, which are observed over a (randomly time-varying) subset of vertices. We recast two classical adaptive algorithms in the graph signal processing framework, namely, the least mean squares (LMS) and the recursive least squares (RLS) adaptive estimation strategies. For both methods, a detailed mean-square analysis illustrates the effect of random sampling on the adaptive reconstruction capability and the steady-state performance. Then, several probabilistic sampling strategies are proposed to design the sampling probability at each node in the graph, with the aim of optimizing the tradeoff between steady-state performance, graph sampling rate, and convergence rate of the adaptive algorithms. Finally, a distributed RLS strategy is derived and is shown to be convergent to its centralized counterpart. Numerical simulations carried out over both synthetic and real data illustrate the good performance of the proposed sampling and reconstruction strategies for (possibly distributed) adaptive learning of signals defined over graphs. Adaptive Graph Signal Processing: Algorithms and Optimal Sampling Strategies

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