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In mathematics, low-rank approximation refers to the process of approximating a given matrix by a matrix of lower rank. More precisely, it is a minimization problem, in which the cost function measures the fit between a given matrix (the data) and an approximating matrix (the optimization variable), subject to a constraint that the approximating matrix has reduced rank.
Mathematical applications of the SVD include computing the pseudoinverse, matrix approximation, and determining the rank, range, and null space of a matrix. The SVD is also extremely useful in all areas of science, engineering, and statistics, such as signal processing, least squares fitting of data, and process control.
Low-rank matrix approximations are essential tools in the application of kernel methods to large-scale learning problems. [ 1 ] Kernel methods (for instance, support vector machines or Gaussian processes [ 2 ] ) project data points into a high-dimensional or infinite-dimensional feature space and find the optimal splitting hyperplane.
A common case is finding the inverse of a low-rank update A + UCV of A (where U only has a few columns and V only a few rows), or finding an approximation of the inverse of the matrix A + B where the matrix B can be approximated by a low-rank matrix UCV, for example using the singular value decomposition.
Non-negative matrix factorization (NMF or NNMF), also non-negative matrix approximation [1] [2] is a group of algorithms in multivariate analysis and linear algebra where a matrix V is factorized into (usually) two matrices W and H, with the property that all three matrices have no negative elements. This non-negativity makes the resulting ...
In practice, we can construct one specific rank factorization as follows: we can compute , the reduced row echelon form of .Then is obtained by removing from all non-pivot columns (which can be determined by looking for columns in which do not contain a pivot), and is obtained by eliminating any all-zero rows of .
Given an input matrix and a desired low rank , the randomized LU returns permutation matrices , and lower/upper trapezoidal matrices , of size and respectively, such that with high probability ‖ ‖ +, where is a constant that depends on the parameters of the algorithm and + is the (+)-th singular value of the input matrix .
Low-rank approximation — find best approximation, constraint is that rank of some matrix is smaller than a given number; Meta-optimization — optimization of the parameters in an optimization method; Multidisciplinary design optimization; Optimal computing budget allocation — maximize the overall simulation efficiency for finding an ...