Product graphs are a useful way to model richer forms of graph-structured data that can be multi-modal in nature. In this work, we study the reconstruction or estimation of smooth signals on product graphs from noisy measurements. We motivate and present representations and algorithms that exploit the inherent structure in product graphs for better and more computationally efficient recovery. These contributions stem from the key insight that smooth graph signals on product graphs can be structured as low-rank tensors. We develop and present algorithms primarily based on two approaches, the first of which is the Tucker decomposition for tensors, while the second is a flexible convex optimization formulation. We further present numerical experiments that exhibit the superior performance of these methods with respect to existing methods for smooth signal recovery on graphs.
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