I was recently reading this arxiv paper, in the context of Machine Learning, and I was wondering what the notation $\odot$ and $\square^\ast_L$ are supposed to mean under section 3. I had read somewhere else that this notation is sometimes used for element-wise vector multiplication but I’m not entirely sure. I’m also curious if the polynomial $g_\theta(\lambda)$ is supposed to be referencing a characteristic polynomial. I’m somewhat new to Graph Fourier Transforms in general, so I’m also wondering how the filter functions numerically. If you have any insights please let me know! The referenced section is below.
Excerpt from the paper:
Given a graph $G$ with the Laplacian Matrix $\textbf{L} = \textbf{U}\Lambda \textbf{U}^\top$, $\Lambda = \operatorname{diag}(\lambda)$, a filter $t$, and a graph signal $f$ assigned on $G$, the spectral graph convolution of $f$ with $t$ on $G$ leveraging convolution theorem is defined as
\begin{align} f' &= t^\ast_Lf \\ &= U((U^\top t) \odot (U^\top f)) \\ &\approx U(g_\theta(\lambda) \odot(U^\top f)) \\ &= U(g_\theta(\Lambda)(U^\top f)) \\ &= g_\theta(L)f \end{align}
where $\hat{f} = U^\top f$ is the Graph Fourier Transform, and $f = U^\top \hat{f}$ is its inverse transform over the graph domain, $g_\theta(\lambda)$ is the polynomial over $\lambda$ and is used to approximate the transformed filter $\hat{t} = U^\top t$.