svd-decomposition

An implementation of Singular Value Decomposition (SVD) in Python, along with some of its applications.

Overview

This repository demonstrates the computation of Singular Value Decomposition (SVD) in Python, leveraging numpy.linalg.eigh for eigenvalue decomposition of symmetric matrices and numpy.linalg.norm for vector norms. The primary goal is to provide a clear and educational implementation of SVD using established numerical routines for core linear algebra operations, rather than a fully from-scratch implementation of all underlying algorithms. This approach balances clarity and efficiency, making the code easier to understand while still providing reasonable performance.

Implementation

The SVD is computed using the following steps:

Compute the eigenvalues and eigenvectors of A^TA using numpy.linalg.eigh.
Construct Σ (the diagonal matrix of singular values) and V^T (the transpose of the right singular vectors) from the eigenvalues and eigenvectors.
Compute U (the left singular vectors) using the relationship U = (A * V) / Σ.

The implementation is encapsulated within an SVD class for better organization and maintainability.

Key features:

Uses numpy.linalg.eigh and numpy.linalg.norm: Leverages NumPy's efficient routines for core linear algebra operations.
SVD Class: Encapsulates the SVD computation logic.
Handles Rank Deficiency: Includes a function to extend the set of left singular vectors to a full orthonormal basis when the matrix is rank-deficient.
Clear Documentation: The code is thoroughly documented with docstrings to explain the purpose of each function and method.

Applications

This repository also includes examples demonstrating the use of SVD in various applications:

Image Compression: Demonstrates how SVD can be used to compress images by keeping only the most significant singular values.
Basic Usage: Shows how to compute the SVD of a simple matrix and reconstruct the original matrix.

More applications will be added in the future.

Getting Started

Clone the repository:

git clone https://github.com/John-6670/svd-decomposition

Navigate to the project directory:
```
cd svd-decomposition
```

(Optional) Create a virtual environment (recommended):

python3 -m venv .venv
source .venv/bin/activate  # On Linux/macOS
.venv\Scripts\activate     # On Windows

Install requirements:
```
pip install -r requirements.txt
```