Let $x_1, x_2, x_3, x_4, x_5$ be a system of orthonormal vectors in $\mathbb{R}^{10}$. Consider the matrix $A=x_1 x_1^T+\ldots . .+x_5 x_5^T$. Which of the following statements is/are correct?

Consider designing a linear binary classifier $f(x)=\operatorname{sign} g\left(w^T x+b\right), x \in \mathbb{R}^2$ on the following training data:

Class -1: $\left\{\binom{2}{0},\binom{0}{2},\binom{2}{2}\right\}$, Class - 2: $\left\{\binom{0}{0}\right\}$

Hard-margin support vector machine (SVM) formulation is solved to obtain $w$ and $b$. Which of the following options is/are correct?

Consider a two-class problem in $R^d$ with class labels red and green. Let $\mu_{\text {red }}$ and $\mu_{\text {green }}$ be the means of the two classes.

Given test sample $x \in R^d$, a classifier calculates

the squared Euclidean distance (denoted by $\|\cdot\|^2$ ) between $x$ and the means of the two classes and assigns the class label that the sample x is closest to. That is, the classifier computes

$$ f(x)=\left\|\mu_{\text {red }}-x\right\|^2-\left\|\mu_{\text {green }}-x\right\|^2 $$

and assigns the label red to $x$ if $f(x)<0$, and green otherwise. Which of the following statements is/are correct?

Let $D=\left\{x^{(1)}, \ldots ., x^{(n)}\right\}$ be a dataset of $n$ observations where each $x^i \in \mathbb{R}^{100}$. It is given that $\sum_{i=1}^n x^{(\mathrm{i})}=0$ The covariance matrix computed from $D$ has eigenvalues $\lambda_i=100^{2-i}, 1 \leq i \leq 100$. Let $u \in \mathbb{R}^{100}$ be the direction of maximum variance with $u^T u=1$.

The value of $\frac{1}{n} \sum_{i=1}^n\left(u^T x^{(i)}\right)^2= $_________