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An important difference between lasso regression and Tikhonov regularization is that lasso regression forces more entries of to actually equal 0 than would otherwise. In contrast, while Tikhonov regularization forces entries of w {\displaystyle w} to be small, it does not force more of them to be 0 than would be otherwise.
A regularization term (or regularizer) () is added to a loss function: = ((),) + where is an underlying loss function that describes the cost of predicting () when the label is , such as the square loss or hinge loss; and is a parameter which controls the importance of the regularization term.
Also known as Tikhonov regularization, named for Andrey Tikhonov, it is a method of regularization of ill-posed problems. [a] It is particularly useful to mitigate the problem of multicollinearity in linear regression, which commonly occurs in models with large numbers of parameters. [3]
In statistics and machine learning, lasso (least absolute shrinkage and selection operator; also Lasso, LASSO or L1 regularization) [1] is a regression analysis method that performs both variable selection and regularization in order to enhance the prediction accuracy and interpretability of the resulting statistical model. The lasso method ...
In statistics and, in particular, in the fitting of linear or logistic regression models, the elastic net is a regularized regression method that linearly combines the L 1 and L 2 penalties of the lasso and ridge methods. Nevertheless, elastic net regularization is typically more accurate than both methods with regard to reconstruction. [1]
Spectral Regularization is also used to enforce a reduced rank coefficient matrix in multivariate regression. [4] In this setting, a reduced rank coefficient matrix can be found by keeping just the top n {\displaystyle n} singular values, but this can be extended to keep any reduced set of singular values and vectors.
Manifold regularization is a type of regularization, a family of techniques that reduces overfitting and ensures that a problem is well-posed by penalizing complex solutions. In particular, manifold regularization extends the technique of Tikhonov regularization as applied to Reproducing kernel Hilbert spaces (RKHSs).
The data sets in the Anscombe's quartet are designed to have approximately the same linear regression line (as well as nearly identical means, standard deviations, and correlations) but are graphically very different. This illustrates the pitfalls of relying solely on a fitted model to understand the relationship between variables.