Low-rank approximation

<p>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 <a href="/facts/Mathematical_optimization/oRn8Iv5I">minimization</a> problem, in which the <a href="/facts/Loss_function/xv5ozuhl">cost function</a> 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 <a href="/facts/Rank_(linear_algebra)/Yp9b5LK3">rank</a>. The problem is used for <a href="/facts/Mathematical_model/CGhyuxgz">mathematical modeling</a> and <a href="/facts/Data_compression/gMFuY4FT">data compression</a>. The rank constraint is related to a constraint on the complexity of a model that fits the data. In applications, often there are other constraints on the approximating matrix apart from the rank constraint, e.g., <a href="/facts/Nonnegative_matrix_factorization/DIBLH8B0">non-negativity</a> and <a href="/facts/Hankel_matrix/96ePFS7m">Hankel structure</a>.
</p><p>Low-rank approximation is closely related to numerous other techniques, including <a href="/facts/Principal_component_analysis/pCbVTqdR">principal component analysis</a>, <a href="/facts/Factor_analysis/LT6B9I7D">factor analysis</a>, <a href="/facts/Total_least_squares/1jRjpM8f">total least squares</a>, <a href="/facts/Latent_semantic_analysis/IugrMyMl">latent semantic analysis</a>, <a href="/facts/Orthogonal_regression/j4bFDpub">orthogonal regression</a>, and <a href="/facts/Dynamic_mode_decomposition/IH8Dp0xj">dynamic mode decomposition</a>.
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Low-rank approximation open-in-new

Low-rank approximation