Collaborative Filtering using SVD, CUR, and PQ Matrix Decomposition

This project demonstrates collaborative filtering for recommendation systems using three matrix decomposition techniques: Singular Value Decomposition (SVD), CUR decomposition, and PQ decomposition (Matrix Factorization). These techniques help in dimensionality reduction and latent feature extraction, improving the scalability and accuracy of recommendation systems.

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Methods

• SVD Decomposition: Computes singular values and explores the loss of data with varying latent factors.
• CUR Decomposition: Approximates the original matrix using selected columns and rows, with tunable latent dimensions.
• PQ Matrix Factorization: Learns user and item latent vectors using gradient descent to minimize prediction error.

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Prerequisites

Before running the code, ensure you have the following dependencies installed:

• numpy
• pandas
• matplotlib
• seaborn
• scikit-learn

To install them, you can run:

pip install numpy pandas matplotlib seaborn scikit-learn

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Dataset

The project uses the MovieLens dataset for movie ratings:

• Path: Place the dataset in the data/ratings.csv.
• Structure: The dataset should include userId, movieId, and rating columns.

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How to Run

1. Clone this repository:

   git clone https://github.com/yourusername/collaborative-filtering.git
   cd collaborative-filtering

2. Place the dataset file (ratings.csv) in the data/ml-latest-small/ directory.

3. Run the script:

   python collaborative_filtering.py

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Outputs

1. SVD Decomposition

• Displays the top 20 singular values.
• Plots the loss of data against the number of latent factors (k).
• Time taken for SVD decomposition is logged.

2. CUR Decomposition

• Computes CUR approximation of the matrix.
• Plots the reconstruction loss for varying k (latent factors).
• Time taken for CUR decomposition is logged.

3. PQ Decomposition

• Performs matrix factorization using gradient descent.
• Logs training and test mean squared errors (MSE).
• Time taken for PQ decomposition is logged.

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Results

Top 20 singular values are [9032.38102201 4265.13020478 2962.83432586 2856.37494764 2441.34461236 2269.55931732 2169.8992637 1848.47223494 1701.69413469 1528.15832014 1476.74413397 1449.77168211 1432.00119537 1413.20720491 1319.28764566 1281.82058619 1213.72797731 1203.99401803 1198.55552737 1135.24246017]

Visualization

Two key plots are generated:

1. Loss vs. Latent Factors (SVD):

   • Visualizes data reconstruction loss as latent factors (k) increase.

2. Loss vs. Latent Factors (CUR):

   • Visualizes CUR reconstruction loss for varying k.

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