Singular Value Decomposition (SVD)

To actively decompose a given matrix, Singular Value Decomposition (SVD) utilizes three matrices. The SVD technique is widely used in machine learning for dimensionality reduction. By utilizing the decomposed matrices, we can actively approximate the original matrix with a lower-rank representation. The process involves passively decomposing the original matrix into three matrices using SVD. Then actively using them to obtain the lower-rank approximation.

Given an m x n matrix A, SVD factorizes A into three matrices:

A = U * Σ * V^T

where U is an m x r matrix whose columns are the left singular vectors of A, and Σ is an r x r diagonal matrix containing the singular values of A. V is an n x r matrix whose columns are the right singular vectors of A.

To obtain a lower-rank approximation of matrix A, we can use the singular values in Σ. It is arranged in decreasing order. By selecting the first k singular values and their corresponding left and right singular vectors. We can obtain a set of matrices: the first k columns of U, the first k rows of Σ, and the first k rows of V^T. These matrices are multiplying together using matrix multiplication to obtain a new matrix B with dimensions k x n.

B = Uk * Σk * Vk^T

In Singular Value Decomposition (SVD), we can use it to reduce dimensionality by selecting a subset of the largest singular values and their corresponding singular vectors. This will allow us to approximate the original matrix with a lower-rank representation. The process involves actively selecting the subset of singular values and singular vectors. Then using the obtain the lower-rank approximation of the original matrix. This can be useful for reducing the computational complexity of a dataset and identifying important features.