```
In [1]:
```# %load ../../preconfig.py
%matplotlib inline
import matplotlib.pyplot as plt
import seaborn as sns
sns.set(color_codes=True)
plt.rcParams['axes.grid'] = False
#import numpy as np
#import pandas as pd
#import sklearn
#import itertools
import logging
logger = logging.getLogger()
def show_image(filename, figsize=None, res_dir=True):
if figsize:
plt.figure(figsize=figsize)
if res_dir:
filename = './res/{}'.format(filename)
plt.imshow(plt.imread(filename))

$M e = \lambda e$, where $\lambda$ is an *eigenvalue* and $e$ is the corresponding *eigenvector*.

to make the eigenvector unique, we require that:

every eigenvector is a

*unit vector*.the first nonzero component of an eigenvector is positive.

To find *principal* eigenvector, we use **power iteration** method:

start with any unit vector $v$, and compute $M^i v$ iteratively until it converges. When $M$ is a stochastic matrix, the limiting vector is the *principal* eigenvector, and its corresponding eigenvalue is 1.

Another method $O(n^3)$: algebra solution.

idea:

Start by computing the pricipal eigenvector, then modify the matrix to, in effect, remove the principal eigenvector.

To find the pricipal eigenpair

power iteration:

Start with any nonzero vector $x_0$ and then iterate: $$x_{k+1} := \frac{M x_k}{\| M x_k\|}$$$x$ is the pricipal eigenvector when it reaches convergence, and $\lambda_1 = x^T M x$.

Attention: Since power iteration will introduce small errors, inaccuracies accumulate when we try to compute all eigenpairs.

To find the second eigenpair

we create $M^* = M - \lambda_1 x x^T$, then use power iteration again.

proof: the second eigenpair of $M^*$ is also that of $M$.

略

```
In [3]:
```plt.imshow(plt.imread('./res/fig11_2.png'))

```
Out[3]:
```

Since $M^T M e = \lambda e = e \lambda$,

Let $E$ be the matrix whose columns are the eigenvectors, ordered as largest eigenvalue first. Define the matrix $L$ to have the eigenvalues of $M^T M$ along the diagonal, largest first, and 0's in all other entries.

$M^T M E = E L$

Let $E_k$ be the first $k$ columns of $E$. Then $M E_k$ is a $k$-dimensional representation of $M$.

Let $M$ be an $m \times n$ matrix, and let the rank of $M$ be $r$.

$$M = U \Sigma V^T$$$U$ is an $m \times r$

*column-orthnormal matrix*(each of its columns is a**unit vector**and the dot product of any two columns is 0).$V$ is an $n \times r$ column-orthnormal maxtrix.

$\Sigma$ is a diagonal matrix. The elements of $\Sigma$ are called the

*singular values*of $M$.

```
In [4]:
```show_image('fig11_5.png')

```
```

```
In [6]:
```show_image('fig11_7.png', figsize=(8, 10))

```
```

viewing the $r$ columns of $U$, $\Sigma$, and $V$ as representing *concepts* that are hidden in the original matrix $M$.

In Fig 11.7:

*concepts*: "science fiction" and "romance".$U$ connects peopel to concepts.

$V$ connects movies to concepts.

$\Sigma$ give the strength of each of the concepts.

```
In [7]:
```show_image('fig11_9.png', figsize=(8, 10))

```
```

```
In [8]:
```show_image('fig11_10.png', figsize=(8, 10))

```
```

Let $M = P Q R$, $m_{i j} = \sum_k \sum_l p_{ik} q_{kl} r_{lj}$.

Then \begin{align} \| M \|^2 &= \sum_i \sum_j (m_{ij})^2 \\ &= \sum_i \sum_j (\sum_k \sum_l p_{ik} q_{kl} r_{lj})^2 \\ &= \sum_i \sum_j \left ( \sum_k \sum_l \sum_n \sum_m p_{ik} q_{kl} r_{lj} p_{in} q_{nm} r_{mj} \right) \\ &\text{as $Q$ is diagonal matrix, $q_{kl}$ and $q_{nm}$ will be 0 unless $k = l$ and $n = m$.} \\ &= \sum_i \sum_j \sum_k \sum_n p_{ik} q_{kk} r_{kj} p_{in} q_{nn} r_{nj} \\ &= \sum_j \sum_k \sum_n \color{blue}{\sum_i p_{ik} p_{in}} q_{kk} r_{kj} q_{nn} r_{nj} \\ &\text{as } P = U, \sum_i p_{ik} p_{in} = 1 \text{ if } k = n \text{ and 0 otherwise} \\ &= \sum_j \sum_k q_{kk} r_{kj} q_{kk} r_{kj} \\ &= \sum_k (q_{kk})^2 \end{align}

How many singular values should we retain?

A useful rule of thumb is to retain enough singular values to make up 90\% of the *energy* in $\Sigma$.

The SVD of a matrix $M$ is strongly connected to the eigenvalues of the symmetric matrices $M^T M$and $M M^T$.

$M^T = (U \Sigma V^T)^T = V \Sigma^T U^T = V \Sigma U^T$

\begin{align} M^T M &= V \Sigma U^T U \Sigma V^T \\ &= V \Sigma^2 V^T \\ M^T M V &= V \Sigma^2 V^T V \\ & = V \Sigma^2 \end{align}similar, $M M^T U = U \Sigma^2$.

SVD: even if $M$ is sparse, $U$ and $V$ will be dense.

CUR: if $M$ is sparse, $C$ and $R$ will be sparse.

Let $f = \sum_{i,j} m_{ij}^2$.

find $C$

- pick $r$ rows, and each row is picked with $p_i = \frac{\sum_j m_{ij}^2}{f}$.
- normailize: each row is divided by $\sqrt{r p_i}$.

find $R$ selected in the analogous way.

Counstructing $U$.

- find $M$, that is the intersection of the chosen columns of $C$ and $R$.
- compute the SVD of $W$: $W = X \Sigma Y^T$.
- compute $\Sigma^+$, the
*Moore-Penrose pseudoinverse*of the diagonal matrix $\Sigma$: replace $\sigma$ by $1/\sigma$ if $\sigma \neq 0$. - $U = Y (\Sigma^+)^2 X^T$.

```
In [4]:
```show_image('fig11_13.png')

```
```

```
In [3]:
```show_image('ex11_17.png')

```
```

```
In [5]:
``````
#Exercise
```

```
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```