Questions tagged [tikhonov-regularization]

Tikhonov regularization, named for Andrey Tikhonov, is the most commonly used method of regularization of ill-posed problems, and is a generalization of ridge regression.

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How does regularization affect the offset term and weights for OLS?

I have a couple of questions regarding regularization that I am confused on, and searches on this forum doesn't seem to answer my specific questions. When we regularize with $\lambda ||w||^2$ or $\...
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Regularization and multicollinearity

Multicollinearity/collinearity, from what I understand, occurs when 2 or more predictor/independent variables (variates) are strongly linearly dependent. This leads to overfitting, and we could use ...
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Trace of the Hat Matrix in Ridge Regression

Generally, I know that the trace of the hat matrix ($H$) is equal to the rank of H since it is an orthogonal projection. If I wanted to show the trace of $H$ in ridge regression, would I be able to ...
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Why we only use L1 and L2 in penalized regression [duplicate]

I understand we do not want to go lower than 1 because the problem is no longer convex. But why not use L3 for example? Is there any reason why L2 is so popular but not higher norm?
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Why does $l_2$ norm regularization not have a square root?

Specifically talking about Ridge Regression's cost function since Ridge Regression is based off of the $l_2$ norm. We should expect the cost function to be: $$J(\theta)=MSE(\theta) + \alpha\sqrt{\...
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Derivation of ridge regression for multi-value-target vectors

At university, I learned with these slides about ridge regression and its derivation with the assumption that the target- and predicted values have the dimensions $1\times1$. However, now I need to ...
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non-linear tikhonov/ridge regularization?

For traditional ridge regression, the loss function is $loss\_function = ||A\mathbf{x}-\mathbf{b}||_2^2 + ||\Gamma\mathbf{x}||_2^2$ https://en.wikipedia.org/wiki/Tikhonov_regularization Is there a ...
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$L^2$ Regularization and Hessian Matrix [duplicate]

In the second paragraph it is mentioned that eigenvector of $H$ is rescaled by a factor of $\frac{\lambda_i} {\lambda_i +\alpha}$ What exactly meant by that ?
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The origin of the term “regularization”

When I introduce concepts to my students, I often find it fun to tell them where the terminology originates ("regression", for example, is a term with an interesting origin). I haven't been able to ...
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Sequential Least Squares for Tikhonov Regularization [duplicate]

Given a Weighted Linear Least Squares problem where the cost function is given by: $$ J = { \left( x - H \Theta \right) }^{T} {C}^{-1} { \left( x - H \Theta \right) } $$ There is a Sequential ...
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Applying L1, L2 and Tikhonov Regularization to Neural Nets: Possible Misconceptions

I'm interested in applying several different types of regularization to neural nets and want to make sure I haven't learned the material incorrectly. I have successfully coded Weight Decay and Dropout,...
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Difference Between Two Tikhonov Regularization Schemes

For the solution of $Ax = b$, where $A$ is a square matrix, what is the difference between these two regularized solutions: $x = (A + \alpha I)^{-1}b$ -- coressponding to eq.3 below $x = (A^TA + \...
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Tikhonov regularization in the context of deconvolution

I came across "Tikhonov regularization" and I have bare knowledge on it. It seems that it is a type of regularization that is important for deconvolution. Are there any good resources and examples? ...
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Is Tikhonov regularization the same as Ridge Regression?

Tikhonov regularization and ridge regression are terms often used as if they were identical. Is it possible to specify exactly what the difference is?