Matrix Factorization for Recommender Systems

Matrix Factorization

In the vast world of digital experiences, recommender systems have become our everyday guides—whether we’re watching movies on Netflix, shopping on Amazon, or discovering music on Spotify. But what fuels these recommendations? Matrix factorisation (MF) is a potent method at the core of many of these systems.

Matrix factorization helps us extract hidden patterns—called latent features—from large, sparse datasets of user-item interactions. Especially within collaborative filtering, MF stands out by efficiently learning from user behavior and preferences without requiring explicit content details.

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🔍 Understanding Matrix Factorization

Let’s dissect this process in detail.

📊 1. Representation of Data

A user-item interaction matrix R is at the heart of the procedure. In this matrix:

• Rows represent users.
• Columns represent items (e.g., movies, books, or products).
• Cell values show user interactions like ratings or views.

But here’s the catch: most users interact with only a small portion of available items. So, the matrix is typically sparse—full of missing or zero values.

🧩 2. Breaking Down the Matrix

Splitting the matrix R into two smaller matrices is the goal of matrix factorisation:

• U (User Matrix): contains user vectors (latent features).
• V (Item Matrix): contains item vectors (latent features).

These vectors capture hidden traits such as a user’s genre preference or an item’s popularity.

🧠 3. Discovering Hidden Elements

The goal of the decomposition is to reconstruct the original matrix as accurately as possible by learning the latent features. This is done using iterative optimization techniques (like stochastic gradient descent) to minimize the difference between the predicted and actual ratings.

🧮 4. Making Predictions

Once the latent factors are learned, we multiply user and item vectors to estimate the missing values in the original matrix. This allows us to predict how a user might rate or interact with an unseen item.

🎯 5. Generating Recommendations

Finally, we recommend the items with the highest predicted ratings for each user. These predictions are personalized, as they are derived from the learned preferences and behaviors of individual users.

💻 Let’s Implement It Using TensorFlow & Keras

Here’s a practical demonstration of matrix factorization using embeddings in Keras.

▶️ Step 1: Import Required Libraries

import numpy as np  
import pandas as pd  
from matplotlib import pyplot as plt  
import tensorflow as tf  
from tensorflow import keras  
import os  
import random  
  
tf.set_random_seed(1)  
np.random.seed(1)  
random.seed(1)

📁 Step 2: Load and Prepare the Dataset

input_dir = '../input/movielens-preprocessing'  
ratings_path = os.path.join(input_dir, 'rating.csv')  
ratings_df = pd.read_csv(ratings_path, usecols=['userId', 'movieId', 'rating', 'y'])  
df = ratings_df  

movies_df = pd.read_csv(os.path.join(input_dir, 'movie.csv'), usecols=['movieId', 'title'])

🧱 Step 3: Build the Matrix Factorization Model

movie_embedding_size = user_embedding_size = 8

user_id_input = keras.Input(shape=(1,), name='user_id')  
movie_id_input = keras.Input(shape=(1,), name='movie_id')  

user_embedded = keras.layers.Embedding(df.userId.max()+1, user_embedding_size, input_length=1)(user_id_input)  
movie_embedded = keras.layers.Embedding(df.movieId.max()+1, movie_embedding_size, input_length=1)(movie_id_input)  

dotted = keras.layers.Dot(2)([user_embedded, movie_embedded])  
out = keras.layers.Flatten()(dotted)  

model = keras.Model(inputs=[user_id_input, movie_id_input], outputs=out)  
model.compile(tf.train.AdamOptimizer(0.001), loss='MSE', metrics=['MAE'])  
model.summary(line_length=88)

🏋️ Step 4: Train the Model

history = model.fit(  
    [df.userId, df.movieId],  
    df.y,  
    batch_size=5000,  
    epochs=20,  
    verbose=0,  
    validation_split=0.05,  
)

📊 Step 5: Visualize the Results

history_dir = '../input/embedding-layers'  
path = os.path.join(history_dir, 'history-1.csv')  
hdf = pd.read_csv(path)

fig, ax = plt.subplots(figsize=(15, 8))  
c1 = 'blue'  
ax.plot(history.epoch, history.history['val_mean_absolute_error'], '--', label='Validation MAE', color=c1)  
ax.plot(history.epoch, history.history['mean_absolute_error'], label='Training MAE', color=c1)

c2 = 'orange'  
ax.plot(hdf.epoch, hdf.val_mae, '--', label='Validation MAE (DNN)', color=c2)  
ax.plot(hdf.epoch, hdf.train_mae, label='Training MAE (DNN)', color=c2)

ax.set_xlabel('Epoch')  
ax.set_ylabel('Mean Absolute Error')  
ax.set_xlim(left=0)  
baseline_mae = 0.73  
ax.axhline(baseline_mae, ls='-.', label='Baseline', color='#002255', alpha=.5)  
ax.grid()  
fig.legend()

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📌 Conclusion: Why Matrix Factorization Matters

Matrix factorisation is still a fundamental method for creating accurate and scalable recommender systems, despite its simplicity. As our experiment shows, even with compact embeddings, the model performs competitively and learns meaningful patterns. While overfitting can occur, regularization and hybrid approaches can help mitigate it.

Matrix factorization allows us to bridge the gap between users and the unknown—delivering recommendations that feel less like code and more like intuition.

✨ Stay tuned to Updategadh for more hands-on tutorials and deep dives into AI, ML, and data science trends!

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