Impression

Impression is an open-source library for recommender systems with collaborative, content-based filtering, matrix factorization, and hybrid models.

Collaborative filtering

• User-based memory model
• Item-based memory model
• Matrix factorization

Collaborative filtering

• Takes an arbitraty m x n rating matrix with discrete or continuous observed ratings.
• Computes similarity score matrix in the offline mode.
• Estimates user (item) similarity score as Pearson correlation, (adj) cosine similarity, z-score of mutually observed ratings.
• Predicts the rating of an item for a user as a weighted dot product of positive similarity scores and observed peer ratings.
• Tune the model by fitting similarity sensitivity with alpha > 1 and min significant similarity score threshold.

User-based memory model

The UserMemoryModel class provides an abstraction of a neighborhood-based collaborative filtering method for predicting the target item rating for a user from observed ratings of similar users.

Parameters:

• sim_method: (cosine, pearson, z-score): a method for fitting similarity scores and predicting target ratings.
• min_similarity: (≥0.1) minimum significant similarity score of a peer rating for prediction.
• alpha: (≥1.0) an exponent amplifies the weight of high sim scores.

>>> from impression.collaborative_filtering import UserMemoryModel

>>> rating_matrix
array([[nan,  2.,  0., nan],
       [-2., nan, nan,  0.],
       [ 1., -1., nan, nan],
       [ 1.,  0., -1., nan]])

>>> umm = UserMemoryModel(sim_method='pearson',
                          alpha=1.0,
                          min_similarity=0.1)

>>> umm.fit(rating_matrix)

>>> umm.predict(user_id=0, item_id=0)
2.0

>>> umm.top_k_items(user_id=0, k=3)
[0, 1, 2]

Predict missing ratings for all users. Note the model fails to predict ratings for some user-item pairs due to the sparsity of observed ratings and lack of similar peers.

>>> umm.complete_rating_matrix()
array([[ 2.,  2.,  0., nan],
       [-2., nan, nan,  0.],
       [ 1., -1., -1., nan],
       [ 1.,  0., -1., nan]])

Estimated m x m similarity score matrix of users.

>>> umm.sim_scores.round(1)
array([[ 1. ,  0. , -1. ,  0.7],
       [ nan,  1. , -1. , -1. ],
       [ nan,  nan,  1. ,  0.7],
       [ nan,  nan,  nan,  1. ]])

---

Item-based memory model

The ItemMemoryModel class provides an abstraction of a neighborhood-based collaborative filtering method for predicting the target item rating for a user from observed ratings of similar users. Unlike the user-based model, the item-based model estimates similarity scores between items (columns).

>>> from impression.collaborative_filtering import ItemMemoryModel

>>> rating_matrix
array([[nan,  2.,  0., nan,  1., -1.],
       [-2., nan, nan,  0., nan,  1.],
       [ 1., -1., nan, nan,  0., nan],
       [ 1.,  0., -1., nan,  2., -2.]])

>>> imm = ItemMemoryModel(alpha=1.0, min_similarity=0.1)

>>> imm.fit(rating_matrix)

>>> imm.predict(user_id=3, item_id=3)
-2.0

>>> imm.top_k_items(user_id=3, k=3)
[4, 0, 1]

Predict missing ratings for all users. The item-based model requires more observed items to solve a matrix completion problem than the user-based model. Since the item-based model predicts the rating of a target item from the other rated items by the same user, it is argued [ref.] to provide more personalized prediction at the cost of a required number of rated items by the target user.

>>> imm.complete_rating_matrix()
array([[ 1.,  2.,  0., -1.,  1., -1.],
       [-2., nan,  1.,  0., -2.,  1.],
       [ 1., -1., nan, nan,  0., nan],
       [ 1.,  0., -1., -2.,  2., -2.]])

Estimated n x n similarity score matrix of items.

>>> imm.sim_scores.round(1)
array([[ 1. , -0.7, -1. , -1. ,  0.7, -0.9],
       [ nan,  1. , -0.4,  0. ,  0.2, -0.6],
       [ nan,  nan,  1. ,  0. , -1. ,  1. ],
       [ nan,  nan,  nan,  1. ,  0. ,  1. ],
       [ nan,  nan,  nan,  nan,  1. , -0.9],
       [ nan,  nan,  nan,  nan,  nan,  1. ]])

---

Matrix factorization

The neighborhood-based model supports a dimensionality reduction of the observed rating matrix with SVD and PCA methods for speeding up the fitting similarity matrix. Both compression methods provide more robust predictions with mean-centering the rating matrix across users' ratings (rows) and items' ratings (cols).

Parameters:

• factorization_method: (none, svd, pca) a dimensionality reduction method for observed ratings.
• approximate_factorization: (True, False) approximate matrix factorization for truncated rating representation m x d, where d = min(n_users, n_items).
• compression_rate: (0 ≤ r < 1) controls the rating matrix dimensionality. Set to 0.0 for lossless compression.

>>> from impression.collaborative_filtering import UserMemoryModel

>>> rating_matrix
array([[nan,  2.,  0., nan,  1., -1.],
       [-2., nan, nan,  0., nan,  1.],
       [ 1., -1., nan, nan,  0., nan],
       [ 1.,  0., -1., nan,  2., -2.]])

>>> umm = UserMemoryModel(factorization_method='pca',
                          compression_rate=0.5)

>>> umm.fit(rating_matrix)

>>> umm.complete_rating_matrix().round(2)
array([[-1.17,  2.  ,  0.  ,  0.83,  1.  , -1.  ],
       [-2.  ,  1.17, -0.83,  0.  ,  0.17,  1.  ],
       [ 1.  , -1.  , -1.  ,   nan,  0.  , -2.  ],
       [ 1.  ,  0.  , -1.  ,   nan,  2.  , -2.  ]])

>>> umm.sim_scores.round(2)
array([[ 1.  ,  0.88, -0.94, -0.45],
       [  nan,  1.  , -0.66, -0.82],
       [  nan,   nan,  1.  ,  0.11],
       [  nan,   nan,   nan,  1.  ]])

Both dimensionality reduction methods support approximate matrix factorization that limits dimensionality of the rating matrix representation to m x d, where d = min(num_users, num_items).

>>> umm = UserMemoryModel(factorization_method='svd',
                          approximate_factorization=True)

>>> umm.fit(rating_matrix)

>>> umm.sim_scores.round(2)
array([[ 1.  , -0.48, -0.53,  0.14],
       [  nan,  1.  , -0.17, -0.9 ],
       [  nan,   nan,  1.  ,  0.18],
       [  nan,   nan,   nan,  1.  ]])

Dependencies

Install environment and dependencies with pipenv sync --dev

• pipenv >= 2022.5.2
• pyenv >= 2.2.5

References

Aggarwal, C. C. (2016), Recommender Systems - The Textbook, Springer.