Singular Value Decomposition

Singular Value Decomposition#

References#

[1]:

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

Dataset Preparation#

[2]:

ratings = pd.read_csv('./datasets/ratings.csv')
ratings.head()

[2]:
userId movieId rating timestamp
0 1 1 4.0 964982703
1 1 3 4.0 964981247
2 1 6 4.0 964982224
3 1 47 5.0 964983815
4 1 50 5.0 964982931 
[3]:

movies = pd.read_csv('./datasets/movies.csv')
movies.head(5)

[3]:
movieId title genres
0 1 Toy Story (1995) Adventure|Animation|Children|Comedy|Fantasy
1 2 Jumanji (1995) Adventure|Children|Fantasy
2 3 Grumpier Old Men (1995) Comedy|Romance
3 4 Waiting to Exhale (1995) Comedy|Drama|Romance
4 5 Father of the Bride Part II (1995) Comedy

Entity to Index Mappers#

[27]:

idx_to_userid_mapper = dict(enumerate(ratings.userId.unique()))
userid_to_idx_mapper = dict(zip(idx_to_userid_mapper.values(), idx_to_userid_mapper.keys()))

idx_to_movieid_mapper = dict(enumerate(ratings.movieId.unique()))
movieid_to_idx_mapper = dict(zip(idx_to_movieid_mapper.values(), idx_to_movieid_mapper.keys()))

[28]:

ratings['user_idx'] = ratings['userId'].map(userid_to_idx_mapper).apply(np.int32)
ratings['movie_idx'] = ratings['movieId'].map(movieid_to_idx_mapper).apply(np.int32)
ratings.head(5)

[28]:
userId movieId rating timestamp user_idx movie_idx
0 1 1 4.0 964982703 0 0
1 1 3 4.0 964981247 0 1
2 1 6 4.0 964982224 0 2
3 1 47 5.0 964983815 0 3
4 1 50 5.0 964982931 0 4 
[29]:

movies['movie_idx'] = movies['movieId'].map(movieid_to_idx_mapper).dropna()
movies.head(5)

[29]:
movieId title genres movie_idx
0 1 Toy Story (1995) Adventure|Animation|Children|Comedy|Fantasy 0.0
1 2 Jumanji (1995) Adventure|Children|Fantasy 481.0
2 3 Grumpier Old Men (1995) Comedy|Romance 1.0
3 4 Waiting to Exhale (1995) Comedy|Drama|Romance 482.0
4 5 Father of the Bride Part II (1995) Comedy 483.0

non mapped movies. i.e. non rated movie by any user.

[31]:

movies[movies.movie_idx.isna()].shape

[31]:

(18, 4)

Matrices#

User-Item#

[7]:

user_movie_matrix = ratings.pivot_table(values=['rating'] ,index=['user_idx'], columns=['movie_idx'])
user_movie_matrix.head(5)

[7]:
rating
movie_idx 0 1 2 3 4 5 6 7 8 9 ... 9714 9715 9716 9717 9718 9719 9720 9721 9722 9723
user_idx
0 4.0 4.0 4.0 5.0 5.0 3.0 5.0 4.0 5.0 5.0 ... NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN
1 NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN ... NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN
2 NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN ... NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN
3 NaN NaN NaN 2.0 NaN NaN NaN NaN NaN NaN ... NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN
4 4.0 NaN NaN NaN 4.0 NaN NaN 4.0 NaN NaN ... NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN

5 rows × 9724 columns

[8]:

X = user_movie_matrix.fillna(0).values

[ ]:

X.shape

(610, 9724)

Null values (Unwatched movies)#

[35]:

K = np.isnan(user_movie_matrix.values)
K

[35]:

array([[False, False, False, ...,  True,  True,  True],
       [ True,  True,  True, ...,  True,  True,  True],
       [ True,  True,  True, ...,  True,  True,  True],
       ...,
       [False, False,  True, ...,  True,  True,  True],
       [False,  True,  True, ...,  True,  True,  True],
       [False,  True, False, ..., False, False, False]])

Watched movies#

[36]:

M = ~K
M

[36]:

array([[ True,  True,  True, ..., False, False, False],
       [False, False, False, ..., False, False, False],
       [False, False, False, ..., False, False, False],
       ...,
       [ True,  True, False, ..., False, False, False],
       [ True, False, False, ..., False, False, False],
       [ True, False,  True, ...,  True,  True,  True]])

SVD Calculation#

[38]:

U, S, VT = np.linalg.svd(X, full_matrices=False)
U.shape, S.shape, VT.shape

[38]:

((610, 610), (610,), (610, 9724))

Eigen values plot#

[39]:

deciles = (np.linspace(0,1,11)*100).astype('int')
dims = np.linspace(1, len(S), 11).astype('int')
deciles, dims

[39]:

(array([  0,  10,  20,  30,  40,  50,  60,  70,  80,  90, 100]),
 array([  1,  61, 122, 183, 244, 305, 366, 427, 488, 549, 610]))

[66]:

fig, ax = plt.subplots(1, 2, figsize=(10, 5))

eig_value_coverage = S.cumsum()/S.sum()
coverage = np.round(eig_value_coverage[dims-1]*100, 0)
table = np.int32(np.c_[coverage, dims])

ax[0].plot(eig_value_coverage, "--", color='k')
ax[0].hlines(deciles/100, xmin=0, xmax=len(S), alpha=0.4, color='k', linestyle='--')
ax[0].vlines(dims, ymin=0, ymax=1, alpha=0.4, color='r', linestyle='--')

ax[0].set_xticks(dims)
ax[0].set_yticks(deciles/100)
ax[0].grid()

ax[1].table(cellText=table, colLabels=['coverage', 'dimensions'], loc='center')
ax[1].axis('off')

plt.show()
../_images/notebooks_svd_25_0.png

Low Rank Matrix#

[73]:

def low_rank_matrix(U, S, VT, rank):
    X_tilde = U[:,:rank] @ np.diag(S[:rank]) @ VT[:rank, :]
    return X_tilde

Loss : RMSE#

[74]:

def loss(X, U, S, VT, rank, M):
    X_tilde = low_rank_matrix(U, S, VT, rank)
    return np.sqrt(np.square(X - X_tilde, where=M).sum())

[75]:

l_losses = []

ranks = [2, 10, 20, 30, 50, 100, 150, 200, 300, 600]
for i in ranks:
    l_losses.append(loss(X, U, S, VT, i, M))

fig, ax = plt.subplots(1, 1, figsize=(8,5))

ax.plot(ranks, l_losses, 'ko-')

ax.grid()
plt.show()
../_images/notebooks_svd_30_0.png

single prediction#

[145]:

def get_prediction(user_idx, movie_idx, U, S, VT, rank=None):
    rank = rank or len(S)
    return np.round(U[user_idx,:rank] @ np.diag(S[:rank]) @ VT[: rank, movie_idx], 2)

[148]:

user_idx = 0
new_ratings = get_prediction(user_idx, ..., U, S, VT, 100)

[149]:

new_ratings

[149]:

array([ 3.48,  3.14,  3.8 , ..., -0.01, -0.01, -0.01])

user idx based top non watched/rated movie recommendations#

[192]:

user_idx = 10
rank = 100
top_n_movies = 10

[193]:

user_vector = user_movie_matrix.iloc[user_idx].values
non_rated_movies_idx = np.isnan(user_vector)
new_movies_ratings = get_prediction(user_idx, ..., U, S, VT, rank)

[194]:

top_n_idxs = np.c_[np.argsort(new_movies_ratings)][non_rated_movies_idx][::-1][:top_n_movies, 0]
top_n_idxs

[194]:

array([  7,  20,  25, 463,  34, 472, 232, 121,  42,  26])

[195]:

(
    ratings[ratings["user_idx"] == user_idx]
    .sort_values(by="rating", ascending=False).head(10)
    .join(other=movies[["movie_idx", "title", "genres"]], on=["movie_idx"], how="inner", rsuffix="x")
)

[195]:
userId movieId rating timestamp user_idx movie_idx movie_idxx title genres
1259 11 6 5.0 902154266 10 2 1.0 Grumpier Old Men (1995) Comedy|Romance
1272 11 349 5.0 902154342 10 19 3604.0 Money Train (1995) Action|Comedy|Crime|Drama|Thriller
1314 11 1784 5.0 902155043 10 729 3543.0 East of Eden (1955) Drama
1313 11 1721 5.0 902154989 10 987 359.0 This Is Spinal Tap (1984) Comedy
1311 11 1693 5.0 902155102 10 986 2145.0 Ben-Hur (1959) Action|Adventure|Drama
1304 11 1584 5.0 902155136 10 726 3732.0 To Be or Not to Be (1942) Comedy|Drama|War
1299 11 1408 5.0 902154383 10 94 296.0 Nobody Loves Me (Keiner liebt mich) (1994) Comedy|Drama
1293 11 1101 5.0 901200233 10 724 6541.0 Lost Horizon (1937) Drama
1287 11 593 5.0 902155102 10 34 1859.0 It Takes Two (1995) Children|Comedy
1285 11 529 5.0 902155043 10 974 86.0 Highlander (1986) Action|Adventure|Fantasy 
[196]:

movies[movies['movie_idx'].isin(top_n_idxs)]

[196]:
movieId title genres movie_idx
97 110 Braveheart (1995) Action|Drama|War 7.0
123 150 Apollo 13 (1995) Adventure|Drama|IMAX 463.0
277 318 Shawshank Redemption, The (1994) Crime|Drama 232.0
314 356 Forrest Gump (1994) Comedy|Drama|Romance|War 20.0
337 380 True Lies (1994) Action|Adventure|Comedy|Romance|Thriller 472.0
398 457 Fugitive, The (1993) Thriller 25.0
418 480 Jurassic Park (1993) Action|Adventure|Sci-Fi|Thriller 26.0
510 593 Silence of the Lambs, The (1991) Crime|Horror|Thriller 34.0
615 780 Independence Day (a.k.a. ID4) (1996) Action|Adventure|Sci-Fi|Thriller 42.0
1503 2028 Saving Private Ryan (1998) Action|Drama|War 121.0
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