HW 3 for CS 4220

You may (and probably should) talk about problems with the each other, with the TAs, and with me, providing attribution for any good ideas you might get. Your final write-up should be your own.

1: Artificial inference

In the code block below, we create an artificial data set for a linear function on the unit square in ${\mathbb{R}}^2$:

$$f(x,y)=c_1+c_{2}x+c_{3}y$$

Based on noisy measurements $b_i=f(x_i,y_i)+{ϵ}_i$, we want to use least squares to find an estimator $\hat{c}$ for the underlying $c$. Fill in the TODO line to do this least squares computation.

2: Retargeting Reflections

Given a nonzero vector $x\in {\mathbb{R}}^n$ and a target unit vector $u$, find a unit vector $v$ such that

$$(I-2v{v}^T)x=\sigma u$$

3: Continuous least squares

For a general inner product, the normal equations for finding the closest point $p\in \mathcal{V}$ to a target $f$ is

$$⟨q,p-f⟩=0,\text{ for all }q\in \mathcal{V}.$$

Use this approach to find the cubic polynomial $p(x)=\alpha x+\beta x^3$ that minimizes

$$\parallel p-\sin (\cdot ){\parallel }_2^2={\int }_{-1}^1(p(x)-\sin (x){)}^{2}dx$$

You may use without further note the following integrals:

$$\begin{array}{rl}{\int }_{-1}^{1}x\sin (x)dx& =2\sin (1)-2\cos (1)\\ {\int }_{-1}^1{x}^3\sin (x)dx& =10\cos (1)-6\sin (1)\end{array}$$

We also provide a sanity check – this should be down around ${10}^{-16}$ if everything is done correctly.