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## Solution to Mathematics for Machine Learning Exercise 4.12

Solution: By SVD (Singular Value Decomposition), we have $A=U\Sigma V^T$. Let $y=V^T x$, then $\|y\|_2^2=y^Ty=(V^Tx)^TV^Tx=x^TVV^Tx=x^Tx=\|x\|_2^2.$Then we have\begin{align*}\|A x\|_2^2=&\ (Ax)^T(Ax)=x^TA^TAx\\ = &\ x^T V \begin{bmatrix}\sigma_1^2 & 0 & 0\\ 0 &…

## Solution to Mathematics for Machine Learning Exercise 7.4

Consider whether the following statements are true or false: (a) The sum of any two convex functions is convex. Solution: True. Let $f_1(\mathbf x)$ and $f_2(\mathbf x)$ be two convex…

## Solution to Mathematics for Machine Learning Exercise 7.3

Consider whether the following statements are true or false: (a) The intersection of any two convex sets is convex. Solution: True. Let $\mathcal C_1$ and $\mathcal C_2$ be two convex…

## Solution to Mathematics for Machine Learning Exercise 7.1

Consider the univariate function $$f(x)=x^3+6x^2-3x-5.$$Find its stationary points and indicate whether they are maximum, minimum, or saddle points. I will assume you are familiar with Calculus, see the book Calculus…

## Solution to Mathematics for Machine Learning Exercise 5.4

Compute the Taylor polynomials $T_n$, $n=0,\dots,5$ of $f(x)=\sin x+\cos x$ at $x_0=0$. Solution: By Definition 5.3, we have to compute $f(x_0)$, $f’(x_0)$, $f^{(2)}(x_0)$, $f^{(3)}(x_0)$, $f^{(4)}(x_0)$, $f^{(5)}(x_0)$. It is not hard…

## Solution to Mathematics for Machine Learning Exercise 5.3

Compute the derivative $f’(x)$ of the function $$f(x)=\exp\left(-\frac{1}{2\sigma^2}(x-\mu)^2\right).$$ Solution: Clearly, we have $$\left(-\frac{1}{2\sigma^2}(x-\mu)^2\right)’=-\frac{1}{2\sigma^2}2(x-\mu)=\frac{-(x-\mu)}{\sigma^2}.$$Therefore, by Chain rule (5.32), we have \begin{align*}f’(x)=&\ \exp\left(-\frac{1}{2\sigma^2}(x-\mu)^2\right)\cdot \left(-\frac{1}{2\sigma^2}(x-\mu)^2\right)’\\=&\ \frac{-(x-\mu)}{\sigma^2}\cdot\exp\left(-\frac{1}{2\sigma^2}(x-\mu)^2\right)\end{align*}