In this exercise you will apply PCA to a large library of facial images to extract dominant patterns across images. The dataset is called *Labeled Faces in the Wild*, or LFW, (source) and is popular in computer vision and facial recognition. It is made up of over a thousand 62 x 47 pixel face images from the internet, the first few of which are displayed below.

InÂ [1]:

```
import matplotlib.pyplot as plt
%matplotlib inline
import numpy as np
from sklearn.datasets import fetch_lfw_people # this will download images if
faces = fetch_lfw_people(min_faces_per_person=60) # you don't already have them
fig, ax = plt.subplots(4, 7, figsize=(12, 10))
for i, axi in enumerate(ax.flat):
axi.imshow(faces.images[i], cmap='pink')
axi.set(xticks=[], yticks=[], xlabel=faces.target_names[faces.target[i]])
```

**Task 1:** We refer to the principal components of face image datasets as *eigenfaces*.
Display the first 28 eigenfaces of this dataset. (They will have little resemblance to the first 28 images displayed above.)

**Task 2:** Let $N$ be the least number of dimensions to which can you reduce the dataset without exceeding 5% relative error in the Frobenius norm. Find $N$. (This requires you to combine what you learnt in the SVD lecture on the Frobenius norm of the error in best low-rank approximation with what you just learnt in the PCA lecture.)

**Task 3:** Repeat PCA, restricting to $N$ eigenfaces (with $N$ as in Task 2), holding back the last seven images in the dataset. Compute the representations of these last seven images in terms of the $N$ eigenfaces. How do they compare visually with the original seven images?

**Task 4:** Restricting to only images of Ariel Sharon and Hugo Chavez, represent (and plot) them as points on a three-dimensional space whose axes represent the principal axes 4, 5, and 6. Do you see the points somewhat clustered in two groups? (The principal directions 0, 1, 2, 3 are excluded in this task since they seem to reflect lighting, shadows, and generic facial features, so will likely not be useful in delineating individuals.)