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In order to give preference to a particular solution with desirable properties, a regularization term can be included in this minimization: ‖ ‖ + ‖ ‖ for some suitably chosen Tikhonov matrix. In many cases, this matrix is chosen as a scalar multiple of the identity matrix ( Γ = α I {\displaystyle \Gamma =\alpha I} ), giving preference ...
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.
In probability theory and directional statistics, the von Mises distribution (also known as the circular normal distribution or the Tikhonov distribution) is a continuous probability distribution on the circle. It is a close approximation to the wrapped normal distribution, which is the circular analogue of the normal distribution.
The learning problem with the least squares loss function and Tikhonov regularization can be solved analytically. Written in matrix form, the optimal w {\displaystyle w} is the one for which the gradient of the loss function with respect to w {\displaystyle w} is 0.
For instance, Tikhonov regularization corresponds to a normally distributed prior on that is centered at 0. To see this, first note that the OLS objective is proportional to the log-likelihood function when each sampled y i {\displaystyle y^{i}} is normally distributed around w T ⋅ x i {\displaystyle w^{\mathsf {T}}\cdot x^{i}} .
The connection between the regularized least squares (RLS) estimation problem (Tikhonov regularization setting) and the theory of ill-posed inverse problems is an example of how spectral regularization algorithms are related to the theory of ill-posed inverse problems.
Regularization perspectives on support-vector machines interpret SVM as a special case of Tikhonov regularization, specifically Tikhonov regularization with the hinge loss for a loss function. This provides a theoretical framework with which to analyze SVM algorithms and compare them to other algorithms with the same goals: to generalize ...
Manifold regularization can extend a variety of algorithms that can be expressed using Tikhonov regularization, by choosing an appropriate loss function and hypothesis space . Two commonly used examples are the families of support vector machines and regularized least squares algorithms.