The Singular Value Decomposition (SVD) is a factorization of a matrix A into three matrices:
- U: an orthogonal matrix (i.e., its transpose is its inverse)
- Σ (Sigma): a diagonal matrix containing the singular values of A
- V (Vee): an orthogonal matrix (i.e., its transpose is its inverse)
The SVD can be written as:
A = U Σ V*
The singular values in Σ are non-negative and represent the importance of each column of A. The columns of U and V are orthonormal, meaning they have length 1 and are perpendicular to each other.
The Cholesky Decomposition is a factorization of a Hermitian positive-definite matrix A into a lower triangular matrix L and its conjugate transpose L*:
A = L L*
Here, L is a lower triangular matrix with real and positive diagonal entries. The Cholesky decomposition is useful for:
- Generating Gaussian random variables with a given covariance matrix
- Solving systems of linear equations with positive-definite matrices
- Computing the inverse of a positive-definite matrix
The Cholesky decomposition is specific to Hermitian positive-definite matrices, whereas the SVD can be applied to any matrix (although it may not be unique for non-square matrices).
Singular Value Decomposition and Cholesky Decomposition
Purpose: SVD is primarily used for dimensionality reduction, feature extraction, and data compression, while Cholesky decomposition is used for solving systems of linear equations, inverting matrices, and computing covariance matrices.
Matrix Properties: SVD can be applied to any matrix, while Cholesky decomposition is specifically designed for symmetric positive-definite (SPD) matrices. SVD can handle non-SPD matrices, but Cholesky decomposition fails when the input matrix is singular or not SPD.
Decomposition Structure: SVD decomposes a matrix into three factors: U (orthogonal), Σ (diagonal), and V (orthogonal). Cholesky decomposition, on the other hand, factorizes a SPD matrix into L (lower triangular) and L^T (transposed lower triangular).
Computational Cost: SVD is generally more computationally expensive than Cholesky decomposition, especially for large matrices. However, SVD can be parallelized and optimized for modern computing architectures, making it a viable option for large-scale data analysis.
Reusability: Cholesky decomposition produces reusable matrices (L and L^T), which can be leveraged for solving multiple linear systems with the same coefficient matrix. SVD also produces reusable matrices (U and V), but they are less straightforward to reuse for solving linear systems.
Applications: SVD has numerous applications in data science, machine learning, and signal processing, such as principal component analysis (PCA), independent component analysis (ICA), and matrix factorization. Cholesky decomposition is commonly used in statistics, linear algebra, and numerical analysis for tasks like covariance matrix estimation, linear regression, and Gaussian processes.