written by @marc_lelarge Probability recap We start with real random variables (r.v.). 1- Why is variance positive? Recall that $\text{Var}(X) = \mathbb{E}[X^2] - \mathbb{E}[X]^2$ so $\text{Var}(X)\geq 0$ means that $\mathbb{E}[X^2] \geq \mathbb{E}[X]^2$. answer: Start with
4/19/2023written by @marc_lelarge (part of the deep learning course) Jacobian Let ${\bf f}:\mathbb{R}^n\to \mathbb{R}^m$, we define its Jacobian as: \begin{align*} \newcommand{\bbx}{{\bf x}} \newcommand{\bbv}{{\bf v}} \newcommand{\bbw}{{\bf w}} \newcommand{\bbu}{{\bf u}} \newcommand{\bbf}{{\bf f}}
3/21/2023written by @marc_lelarge
3/5/2023written by @marc_lelarge For a graph $G$ with $n$ vertices, with symmetric adjacency matrix $A\in \mathbb{R}^{n\times n}$. For $i\in [n]$, we denote by $\lambda_i$ its eigenvalues and by $\psi_i$ its associated eigenvectors. The empirical measure of $G$ is defined as: \begin{align*} \mu_G = \frac{1}{n}\sum_{i=1}^n \delta_{\lambda_i} \end{align*} The rooted spectral measure of $G$ at vertex $v\in [n]$ is defined by:
2/26/2023or
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