In the field of statistical learning theory, matrix regularization generalizes notions of vector regularization to cases where the object to be learned is a matrix. The purpose of regularization is to enforce conditions, for example sparsity or smoothness, that can produce stable predictive functions. For example, in the more common vector framework, Tikhonov regularization optimizes over
to find a vector that is a stable solution to the regression problem. When the system is described by a matrix rather than a vector, this problem can be written as
where the vector norm enforcing a regularization penalty on has been extended to a matrix norm on .
Matrix regularization has applications in matrix completion, multivariate regression, and multi-task learning. Ideas of feature and group selection can also be extended to matrices, and these can be generalized to the nonparametric case of multiple kernel learning.
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matrixregularization generalizes notions of vector regularization to cases where the object to be learned is a matrix. The purpose of regularization...
engineering. Also known as Tikhonov regularization, named for Andrey Tikhonov, it is a method of regularization of ill-posed problems. It is particularly...
point of view, the matrix completion problem is an application of matrixregularization which is a generalization of vector regularization. For example, in...
Look up regularization, regularisation, or regularizations in Wiktionary, the free dictionary. Regularization may refer to: Regularization (linguistics)...
and generalization of the extension method of covariance matrix inversion by regularization". Imaging Spectrometry IX. Proceedings of SPIE. 5159: 299...
Regularized least squares (RLS) is a family of methods for solving the least-squares problem while using regularization to further constrain the resulting...
Spectral regularization is any of a class of regularization techniques used in machine learning to control the impact of noise and prevent overfitting...
for regularization". BIT. 27 (4): 534–553. doi:10.1007/BF01937276. S2CID 37591557. Horn, Roger A.; Johnson, Charles R. (1985). "Section 7.3". Matrix Analysis...
estimator can be derived both from a regularization and a Bayesian perspective. The main assumption in the regularization perspective is that the set of functions...
theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix or discrete Laplacian, is a matrix representation of a...
Manifold regularization adds a second regularization term, the intrinsic regularizer, to the ambient regularizer used in standard Tikhonov regularization. Under...
and other metrics. Regularization perspectives on support-vector machines interpret SVM as a special case of Tikhonov regularization, specifically Tikhonov...
codes. The regularization and kernel theory literature for vector-valued functions followed in the 2000s. While the Bayesian and regularization perspectives...
case where no regularization has been integrated, by the singular values of matrix F {\displaystyle F} . Of course, the use of regularization (or other kinds...
generalized least squares, when all the off-diagonal entries of the covariance matrix of the errors, are null. The fit of a model to a data point is measured...
projection matrix P of the fan-beam geometry, which is constrained by the data fidelity term. This may contain noise and artifacts as no regularization is performed...
noisy inputs. L1 with L2 regularization can be combined; this is called elastic net regularization. Another form of regularization is to enforce an absolute...
.} The matrix X T X {\displaystyle \mathbf {X} ^{\operatorname {T} }\mathbf {X} } is known as the normal matrix or Gram matrix and the matrix X T y {\displaystyle...
likewise independent of v {\displaystyle v} . For the second term (the regularization term), since v {\displaystyle v} is orthogonal to ∑ i = 1 n α i φ (...
functions. In some contexts, a regularized version of the least squares solution may be preferable. Tikhonov regularization (or ridge regression) adds a...
\lVert f\rVert _{\mathcal {H}}<k} . This is equivalent to imposing a regularization penalty R ( f ) = λ k ‖ f ‖ H {\displaystyle {\mathcal {R}}(f)=\lambda...
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