What is Dirichlet norm?

Question

While I was reading this paper, there was a word Dirichlet norm. And it represents $$||X||^2_{G_r} = \operatorname{trace}(X^T\Delta_r X)$$ where $X$ is the columns of matrix, $\Delta_r$ is the Laplacian of row graphs. The point is, what is Dirichlet norm? Can someone explain me what actually it is ? Thank you

Answer 1

That is the definition of the Dirichlet (semi)norm, taken as an analogue of the corresponding concept from complex/harmonic analysis. Essentially, if you recall the graph Laplacian is the analogue of the Laplacian in $\mathbb{R}^n$, then it is reasonable to take as analogue of $$\require{cancel} \lVert f\rVert_{\mathcal{D}}^2= \int \lvert f'\rvert^2\,\mathrm{d}A=\int f\Delta f\,\mathrm{d}A+\cancel{\text{boundary term}} $$ as a possible measure of how much $f$ deviates from constant, so we have the quantity $\operatorname{trace}(X^T\Delta X)$.