The singular value decomposition (SVD) is a crucial concept for machine learning that builds the foundation of many algorithms you will use throughout your machine learning journey. It’s not only used in ML or data science but in other fields too. At its core, SVD breaks down complex datasets into simpler, more interpretable components, opening the door to a world of insights. It also allows you to find solutions to various problems such as image compression, text mining, and multidimensional data analysis.
That’s why it’s so important to understand. For machine learning, you don’t need to learn how to implement SVD in Python because so many libraries allow you to perform SVD. Another reason why you don’t need to know its implementation is because it can be a very complex project which will deviate you from the main goal of actually learning how to use it. However, to use SVD you need to understand the basics of SVD so that you can make the best of it and understand more advanced concepts such as PCA which I will explain in another post. So, let’s get started and learn SVD.
Table of Contents
Prerequisites
- Understanding Of Linear Algebra Until Eigendecomposition & Spectral Decomposition
- Python & Numpy
What You Will Learn:
- Basics Of SVD – Definition, Proofs & Examples
- Basics of Low Rank Approximation – Definition, Proof & Examples
- Example of SVD Application
- How to perform SVD & Low-Rank Approximation Using Numpy & Python
What is SVD?
You must have heard about the eigendecomposition and we use eigendecomposition to understand various underlying features. The problem with eigendecomposition is the matrix should be square but in the real world, our data points are not in a square matrix form all the time.
So, we need a way to decompose such matrices so that we can know more about the underlying features of such matrices. So, SVD or singular value decomposition is a way to dissect any matrices further down so that we can work with such real-world matrices.
A formal definition can be: The singular value decomposition of a matrix is the factorization of A into the product of three matrices A = UΣVT where the columns of U and VT (V transpose) are orthonormal and the matrix Σ is diagonal with positive real entries. σ1,…,σr being the singular values of A satisfying σ1 ≥ σ2 ≥ ··· ≥ σr > 0.
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