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I have a very simple yet fundamental question.

Suppose I have a vector of data $x \in \mathbb{R}^N$. Without additional information, I guess the majority of people think this vector as defined over some regular Euclidean domain: a time-vector, a grid, etc. In this case, we can process this vector by means of some (non-)linear transformation $Ax$ which takes into account the domain of the vector $x$. For instance, in the case of time, the matrix $A$ might represent the DFT matrix, hence expressing the time-vector $x$ into its frequency representation $\hat{x}$.

But now assume that my vector $x$ reside on a non-Euclidean domain and in my processing I want to incorporate it. Perhaps I would like also to express this data vector in a basis related to the non-Euclidean domain of interest.

My question is:

  1. How do I find a suitable basis to express my vector, which depends on a non-Euclidean domain?
  2. Does it make sense to define a basis expansion for a non-Euclidean domain through a matrix? This is the case for instance in graph signal processing, where graph signals (vectors) are expressed by using the eigenvectors of the Laplacian matrix of the graph. But also here: who does it tell me that this is the "right" basis to express my signal with?
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  • $\begingroup$ Answers to both 1 and 2 depend on the actual domain you're looking at. Generally, you'll find that "suitable" is hard and depends on what you want to do, and that answers often are ambiguous. Don't think your question is simple, at all! $\endgroup$
    – Marcus Müller
    Dec 15, 2021 at 14:05
  • $\begingroup$ 2. "Does it make sense to define …" well, that depends on the reason you're defining this, and how you define it. Your example, though, is, if I'm not totally mistaken, in an Euclidean domain, as Laplacian matrices form an integral domain that I think you could define an Euclidean function on, right? Honestly, this is more Algebra than I'm confident in, and not really signal processing.... $\endgroup$
    – Marcus Müller
    Dec 15, 2021 at 14:08
  • $\begingroup$ Thank you for your answer, first. A graph is a non-Euclidean domain, but you can represent it through adjacency and/or Laplacian, which are matrices and (in) general eigendecomposable. People in general use the eigenvector matrix as basis for the signal representation. $\endgroup$
    – VanBaffo
    Dec 15, 2021 at 14:19

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