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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.
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 x i {\displaystyle x_{i}} is encoded by a linear combination of not more than T 0 {\displaystyle T_{0}} elements in a way identical to the MOD approach:
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.
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 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.
The trajectory matrix of multi-channel time series consists of linked trajectory matrices of separate times series. The rest of the algorithm is the same as in the univariate case. System of series can be forecasted analogously to SSA recurrent and vector algorithms (Golyandina and Stepanov, 2005). MSSA has many applications.
Latent semantic indexing (LSI) is an indexing and retrieval method that uses a mathematical technique called singular value decomposition (SVD) to identify patterns in the relationships between the terms and concepts contained in an unstructured collection of text. LSI is based on the principle that words that are used in the same contexts tend ...