written by @marc_lelarge :::info Cost function: for a partition $C_1, \dots, C_k$ of $n$ observations in $k$ clusters, the quality of the clustering is given by the following cost function: \begin{eqnarray*} \sum_{j=1}^k \sum_{i\in C_j} |x_i-\mu_j|^2 \text{ where, } \mu_j =\frac{1}{|C_j|}\sum_{i\in C_j} x_i. \end{eqnarray*} ::: :::success
4/25/2023written 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
3/5/2023or
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