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Tensor rank decomposition
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<p>In <a href="/facts/Multilinear_algebra/LrwpUFsM">multilinear algebra</a>, the tensor rank decomposition or rank-<i>R</i> decomposition is the decomposition of a tensor as a sum of <i>R</i> rank-1 tensors, where <i>R</i> is minimal. Computing this decomposition is an open problem. </p><p>Canonical polyadic decomposition (CPD) is a variant of the tensor rank decomposition, in which the tensor is approximated as a sum of <i>K</i> rank-1 tensors for a user-specified <i>K</i>. The CP decomposition has found some applications in <a href="/facts/Linguistics/oQuamvf2">linguistics</a> and <a href="/facts/Chemometrics/xEgJ6fYV">chemometrics</a>. It was introduced by <a href="/facts/Frank_Lauren_Hitchcock/0jf4asvo">Frank Lauren Hitchcock</a> in 1927 and later rediscovered several times, notably in psychometrics. The CP decomposition is referred to as CANDECOMP, PARAFAC, or CANDECOMP/PARAFAC (CP). Note that the PARAFAC2 rank decomposition is a variation of the CP decomposition. </p><p>Another popular generalization of the matrix SVD known as the <a href="/facts/Higher-order_singular_value_decomposition/JTtTDy39">higher-order singular value decomposition</a> computes orthonormal mode matrices and has found applications in <a href="/facts/Econometrics/7U52gzHt">econometrics</a>, <a href="/facts/Signal_processing/Npaja6zb">signal processing</a>, <a href="/facts/Computer_vision/Tl2Yyk66">computer vision</a>, <a href="/facts/Computer_graphics/lapBE8wv">computer graphics</a>, and <a href="/facts/Psychometrics/56X26MDI">psychometrics</a>. </p>