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In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix into a rotation, followed by a rescaling followed by another rotation. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any m × n {\displaystyle m\times n} matrix.
The singular value decomposition (SVD) of is given as = † where and are unitary matrices and is a diagonal matrix of size , that holds the singular values from the largest (top left) in descending order.
The term higher order singular value decomposition (HOSVD) was coined be DeLathauwer, but the algorithm referred to commonly in the literature as the HOSVD and attributed to either Tucker or DeLathauwer was developed by Vasilescu and Terzopoulos. [6] [7] [8] Robust and L1-norm-based variants of HOSVD have also been proposed. [9] [10] [11] [12]
In linear algebra, the generalized singular value decomposition (GSVD) is the name of two different techniques based on the singular value decomposition (SVD).The two versions differ because one version decomposes two matrices (somewhat like the higher-order or tensor SVD) and the other version uses a set of constraints imposed on the left and right singular vectors of a single-matrix SVD.
K-SVD is an algorithm that performs SVD at its core to update the atoms of the dictionary one by one and basically is a generalization of K-means. It enforces that each element of the input data is encoded by a linear combination of not more than elements in a way identical to the MOD approach:
In applied mathematics, k-SVD is a dictionary learning algorithm for creating a dictionary for sparse representations, via a singular value decomposition approach. k-SVD is a generalization of the k-means clustering method, and it works by iteratively alternating between sparse coding the input data based on the current dictionary, and updating the atoms in the dictionary to better fit the data.
The first idea behind the Proper Orthogonal Decomposition (POD), as it was originally formulated in the domain of fluid dynamics to analyze turbulences, is to decompose a random vector field u(x, t) into a set of deterministic spatial functions Φ k (x) modulated by random time coefficients a k (t) so that:
A simple polynomial-time algorithm exists for certifying that a tensor is of rank 1, namely the higher-order singular value decomposition. The rank of the tensor of zeros is zero by convention. The rank of a tensor a 1 ⊗ ⋯ ⊗ a M {\displaystyle \mathbf {a} _{1}\otimes \cdots \otimes \mathbf {a} _{M}} is one, provided that a m ∈ F I m ∖ ...