SVD and Truncated SVD Explained: Theory, Python Examples, and Applications in Machine Learning & Deep Learning
Singular Value Decomposition (SVD) is a matrix factorization method widely used in mathematics, engineering, and economics. Since it's a crucial concept applied in accelerating matrix computations and data compression, it's worth studying at least once. SVD (Singular Value Decomposition) Theory Singular Value Decomposition (SVD) is a matrix factorization technique applicable to any real or complex matrix. Any matrix A (m×n) can be decomposed as follows: $A = U * \Sigma * V^T$ U : Orthogonal matrix composed of left singular vectors $(m \times m)$ Σ : Diagonal matrix $(m \times n)$ with singular values on the diagonal $V^T$ : Transposed matrix of right singular vectors $(n \times n)$ The singular values represent the energy or information content of matrix A, enabling tasks like dimensionality reduction or noise filtering. Truncated SVD Truncated SVD approximates the original matrix using only the top k singular values and corresponding singular vectors: $A \approx...