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In signal processing (SP) often we need to compute sample-wise distance between two signals, e.g. computing MSE. SP on graphs concerns the case when the domain of the signal has some intrinsic structure represented as weighted undirected graphs. The weighs could represent, for example, the correlation between the signal values in each dimension. I was wondering if there is a generalize notion of sample-wise euclidean distance for SP on graphs which takes into account the structure of the graph?

p.s. “The Emerging Field of Signal Processing on Graphs” discusses the distances as open issues.

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  • $\begingroup$ Can you please clarify what you mean by "...takes into account the structure of the graph"? The distance metric is " shaping" the graph itself. Are you referring to alternative to euclidean distance metrics? (Of which there are many). $\endgroup$ – A_A Jul 13 '16 at 22:52
  • $\begingroup$ @A_A I can put it in this way, assuming we have two different graphs on which we define two signals. how I would compare the energy of two signals? Note the energy is related to euclidean distance. My intuition is that the calculated energy (distance) should be normalized with respect to the graph structure (edges, weights) but I don't know how. $\endgroup$ – Vahid Jul 14 '16 at 11:59
  • $\begingroup$ Can you please edit your question to provide some more information? (Such as what was just mentioned in this comment). It would be good to have a broader view of what you are trying to achieve if possible. In the meantime, please note that energy as the sum of all squared samples of a signal would be related to the graph's distance matrix (Distance "defines" the structure of the graph, so, it sort of cancels itself to try to define a distance metric that takes into account the structure of the graph that results from the metric itself.). $\endgroup$ – A_A Jul 14 '16 at 13:45

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