Lecture 12 (Mar 15)

Example on low rank matrix completion.

Why 1-norm minimization promotes sparsity?

Fourier Series

Consider
$$
\mathcal{B}=\left\{ \frac{1}{\sqrt{2}} , \cos x, \cos 2x, \cdots, \sin x, \sin 2x , \cdots \right\} \, .
$$

With respect to the inner product
$$
<f,g> = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) g(x) dx \, ,
$$
$\mathcal{B}$ is orthonormal.

Fourier series:
$$
f(x)=\frac{a_0}{2} + \sum_{n=1}^{\infty} ( a_n \cos nx + b_n \sin nx ) \, ,
$$
where
$$
a_0=\frac{1}{\pi} \int_{-\pi}^{\pi} f(x) dx \, , \, a_n=\frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos nx dx \, , \, b_n=\frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin nx dx \, .
$$

Example: $f(x)=x$ on $x\in(-\pi,\pi)$ with periodic extension and is normalized. The partial sum of the Fourier series is given by
$$
s_N(x)= 2\sum_{n=1}^N \frac{(-1)^{n+1}\sin nx}{n} \, .
$$

Reading materials

Lecture Notes: p.67-71