rankN

RankN is a term used in mathematics to denote rank-related properties associated with an N-th level in linear objects such as matrices and tensors. In practice, RankN often refers to either a matrix or tensor having rank N or to a rank-N approximation or decomposition that expresses the object as a sum of N simpler components.

Matrix rank: For an m-by-n matrix A, rank(A) is the dimension of the vector space spanned by its rows or columns. A has rank N if there are N linearly independent rows (or columns), and no greater. Rank-N matrices can be factored as A = U V where U is m×N and V is N×n. This factorization reflects the idea that the column space and row space have dimension N.

Rank-N approximation: Among all matrices of rank N, the one closest to A in the Frobenius norm is given by truncating the singular value decomposition to its top N singular values. This A_N minimizes ||A − A_N||_F among all rank-N matrices and provides a practical way to compress or denoise data.

Tensor rank: For a d-way tensor T, the rank is the smallest number r such that T can be written as a sum of r rank-1 tensors. Rank-N here means r = N. Computing tensor rank is generally hard and is related to decompositions such as the CP decomposition, which seeks a compact, low-rank representation.

Applications: Rank-N concepts underpin data compression, noise reduction, system identification, and collaborative filtering. They provide a framework for low-rank approximations in machine learning and signal processing.