lowerrank

Lowerrank is a term rarely used in formal mathematics, where the standard term is low-rank. It refers to matrices whose rank is small relative to their dimensions. A matrix A in R^{m×n} is low-rank if its rank r is much less than min(m, n). Equivalently, A has only a few nonzero singular values; the remaining values are often negligible in the presence of noise.

In practice, low-rank structure is exploited through low-rank approximation. For any matrix, the best approximation of a given rank k in the Frobenius or spectral norm is provided by its truncated singular value decomposition (SVD), according to the Eckart-Young theorem. This underpins many dimensionality reduction and data compression techniques.

Common methods for obtaining and working with low-rank models include singular value decomposition, truncated SVD, and nuclear norm minimization, the latter being a convex relaxation of rank minimization used in optimization problems. Low-rank models appear across fields, including machine learning, statistics, and signal processing.

Applications span collaborative filtering (recommender systems), image and video compression and denoising, background subtraction in video analytics, system identification, and analysis of biological or genetic data. Related topics include Robust Principal Component Analysis (RPCA), which combines low-rank plus sparse components, matrix completion, rank estimation, numerical rank, and perturbation theory.

Limitations include the possibility that data do not exhibit true low-rank structure, the challenge of selecting an appropriate rank, and sensitivity to noise and outliers that can obscure the intrinsic rank.