A Graph Trend Filtering Interpretations

Here we give some insight for our definition of the family of graph difference operators (5) and (6), based on the idea of piecewise polynomials over graphs. In the univariate case, sparsity of β under the difference operator D implies a specific kth order piecewise polynomial structure for the components of β [6, 8]. Since the components of β correspond to (real-valued) input locations x = (x1, . . . xn), the interpretation of a piecewise polynomial here is unambiguous. But for a graph, does sparsity of ∆β mean that the components of β are piecewise polynomial? And what does the latter even mean, as the components of β are defined over the nodes? To address these questions, we intuitively define a piecewise polynomial over a graph, and show that it implies sparsity under our constructed graph difference operators.