Body: <p><strong>Context:</strong> I was reading the following set of notes (page 83): <a href="https://www.cs.mcgill.ca/%7Ewlh/grl_book/files/GRL_Book.pdf" rel="nofollow noreferrer">here</a> and it says:</p>
  <blockquote>
  <p>Thus, the Fourier transform of signal (or function) <span class="math-container">$ \mathbf{f} \in R^{|V|} $</span> on a
  graph can be computed as <span class="math-container">$$ \mathbf{s} = \mathbf{U}^T \mathbf{f} $$</span></p>
  </blockquote>
  <p><strong>Question:</strong> What happens if each node has multiple 'signals'? Are the Fourier transforms on each signal independent of one another?</p>
  <p><strong>Attempt:</strong> I assume that each signal is denoted as a column vector, and thus multiple signals may be written as a matrix <span class="math-container">$\mathbf{F} = [\mathbf{f_1}, \mathbf{f_2}, ..., \mathbf{f_n}]$</span> (where <span class="math-container">$ \mathbf{f_i} \in R^{|V|} $</span>) for a graph with <span class="math-container">$n$</span> signals. Thus, the graph Fourier transform would be <span class="math-container">$$ \mathbf{S} = \mathbf{U}^T \mathbf{F} $$</span> and thus each of the Fourier transforms would be independent of one another. Is this the correct way to think about this.</p>
  <p>Many thanks in advance!</p>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2021-09-10T07:54:03.923Z
Comments.Text: Single node can't have multiple signals they have $\mathbb{R}^n$ dimensional feature vector, so in this context each node have $n$ number of channel. The signal is define over the entire graph. Take the first channel of each node, you'll get a graph signal, then take second channel you'll get another signal(Think of RGB image where we have 3 channel). Now when we perform GFT, we perform it over individual graph signal.
Comments.UserDisplayName: Swakshar Deb
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2021-09-11T03:29:51.277Z
Comments.Text: Thanks @SwaksharDeb! Okay cool, yes I was thinking exactly along the lines of RGB (but didn't know how to word it). Okay so each nodes can have $n$-dimensional feature vectors and when taking the GFT, we just consider one layer at a time (over the whole graph).? That makes more sense now
Comments.UserDisplayName: Rocky the Owl
Tags: <geometric-deep-learning><graph-neural-networks><spectral-analysis>
Title: How does graph Fourier transform work when multiple signals present on each node?
UserDisplayName: Rocky the Owl
ThreadText: # How does graph Fourier transform work when multiple signals present on each node?
  
  (Asked by: username_0 on 2021-09-10)
  
  **Context:** I was reading the following set of notes (page 83): [here](https://www.cs.mcgill.ca/%7Ewlh/grl_book/files/GRL_Book.pdf) and it says:
  
  > Thus, the Fourier transform of signal (or function) $ \mathbf{f} \in R^{|V|} $ on a
  > graph can be computed as $$ \mathbf{s} = \mathbf{U}^T \mathbf{f} $$
  
  **Question:** What happens if each node has multiple 'signals'? Are the Fourier transforms on each signal independent of one another?
  
  **Attempt:** I assume that each signal is denoted as a column vector, and thus multiple signals may be written as a matrix $\mathbf{F} = [\mathbf{f\_1}, \mathbf{f\_2}, ..., \mathbf{f\_n}]$ (where $ \mathbf{f\_i} \in R^{|V|} $) for a graph with $n$ signals. Thus, the graph Fourier transform would be $$ \mathbf{S} = \mathbf{U}^T \mathbf{F} $$ and thus each of the Fourier transforms would be independent of one another. Is this the correct way to think about this.
  
  Many thanks in advance!
  
  **Tags:** geometric-deep-learning, graph-neural-networks, spectral-analysis
  
  > **Comments:**
  > **username_1**: Single node can't have multiple signals they have $\mathbb{R}^n$ dimensional feature vector, so in this context each node have $n$ number of channel. The signal is define over the entire graph. Take the first channel of each node, you'll get a graph signal, then take second channel you'll get another signal(Think of RGB image where we have 3 channel). Now when we perform GFT, we perform it over individual graph signal.  
  > **username_0**: Thanks @username_1! Okay cool, yes I was thinking exactly along the lines of RGB (but didn't know how to word it). Okay so each nodes can have $n$-dimensional feature vectors and when taking the GFT, we just consider one layer at a time (over the whole graph).? That makes more sense now  
  
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  ## 0 answers
  
  