SVDBasis

SVDBasis refers to a basis for a data space formed from the singular vectors obtained via the singular value decomposition of a matrix. Given a real or complex matrix A of size m-by-n, the SVD expresses A = U Σ V^*, where U is m-by-r with left singular vectors, Σ is an m-by-n diagonal-like matrix of nonzero singular values, and V is n-by-r with right singular vectors, with r equal to the rank of A. The columns of U and V form orthonormal bases for the column space and row space of A, respectively. The singular values indicate the importance of each corresponding basis vector in representing A.

In practice, SVDBasis is often truncated: selecting the top k singular vectors yields a reduced basis that approximates A by a rank-k matrix: A_k ≈ U_k Σ_k V_k^*. This truncation underpins many dimensionality reduction tasks, such as low-rank approximation, feature extraction, and data compression. In data analysis, SVDBasis can serve as the basis for projecting data into a subspace that captures the most variance when data are centered, paralleling principal component analysis. The basis vectors remain orthonormal, which simplifies inner products and reconstructions.

Limitations include sensitivity to scaling, the computational cost of full or partial SVD, and linearity constraints. SVDBasis is most effective when the data lie close to a linear subspace; nonlinear structure may require kernel methods or manifold learning. It is widely used in statistics, signal processing, and machine learning for noise reduction, compact representation, and interpretable feature spaces.