Recommender¶

In this exercise, you will take a look at Matrix Factorization. It can be used to build a straightforward recommender system.

Let’s consider 5 movies and 5 persons:

movies = ["Titanic", "Barbie", "Shrek", "Dune", "Star Trek"]
persons = ["Ada", "Bashir", "Choi", "Deryl", "Elena"]

Each person has rated some of the movies on a scale of 1 (worst) to 5 (best). You can store these conveniently in a pandas DataFrame:

R = np.array([
    [None, 3, 1, 4, 1],
    [2, 4, 1, 5, None],
    [3, 4, None, 5, None],
    [5, None, 4, 2, 5],
    [4, 1, 5, None, None],
])
R = pd.DataFrame(ratings, index=movies, columns=persons)

Matrix Factorization¶

Now we will build two other matrices P and Q. They add an extra dimension to the data. For better understanding, we will call this the “genre” of the movies. We will use two genres, “drama” and “sci-fi”.

The first matrix P indicates how much each movie belongs to each genre. Add the remaining values:

P = np.array([
    [5, 1],   # [drama, scifi] for Titanic
    ...       #
])
P.shape  # should be (5, 2)

The second matrix Q indicates how much each user likes each genre. Note that the dimensions are reversed here. Add the remaining values as well:

Q = np.array([
    [0, ...],   # drama
    [4, ...],   # sci-fi
])
Q.shape  # should be (2, 5)

Rebuild the rating matrix¶

Now comes the linear algebra trick: We use Matrix Multiplication to reconstruct the ratings matrix. Lets call this matrix S:

S = np.dot(P, Q)

It may help to normalize the values to a maximum of five:

S = 5 * S / S.max()
S.round(1)

Frobenius Norm¶

If we chose the values in P and Q well, the matrices should be similar. Next, we will use the Frobenius Norm to calculate their difference. It is practically the same as the L2 Norm.

Calculating the difference can be tricky because R contains missing values, but the pandas library takes that burden from us:

diff = R - S
np.sqrt((diff ** 2).sum().sum())

Optimize¶

Change the values in P and Q so that the Frobenius Norm becomes smaller.

Hint

There are many methods to factorize matrices automatically, like NMF or SVD. I kept the manual approach mainly for illustrative purposes.

Predict¶

Based on the predicted ratings in S, what movie would you recommend to Elena?