Linked Questions

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1answer
1k views

Non-Singularity due to inclusion of non-zero lambda in ridge regression [duplicate]

There were many similar questions on this site , related to this but none were exactly to the point I wanted to ask So the question is relates to ridge regression and This link where there is a ...
2
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0answers
90 views

Is there difference between “spectral decomposition” and “singular value decomposition”? [duplicate]

Am I right that "spectral decomposition" for symmetric matrix and "singular value decomposition" for non square matrix? Any clarification would be appreciated.
2
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0answers
36 views

$L^2$ 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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0answers
23 views

How to prove biased estimator with SVD of X [duplicate]

Hi guys, I'm assuming that I am able to use SVD of X to solve this question. So, X = UΣV where U and V are nxn and pxp orthogonal matrices respectively and Σ is an nxp matrix containing the singular ...
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0answers
18 views

Penalized Regression: “ridge” RMSE and coefficients larger than those for plain “lm” [duplicate]

Working with the "prostate" dataset in "ElemStatLearn" package. ...
4
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3answers
5k views

Ridge Regression -Increase in $\lambda$ leads to a decrease in flexibilty

In Introduction to Statistical Learning, in the part where ridge regression is explained, the authors say that As $\lambda$ increases, the flexibility of the ridge regression fit decreases, ...
5
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3answers
2k views

Naive Ridge Regression in R?

I'm trying to learn some basic Machine Learning and some basic R. I have made a very naive implementation of $L_2$ regularization in R based on the formula: $\hat w^{ridge} = (X^TX +\lambda I)^{-1} X^...
16
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1answer
652 views

Reversing ridge regression: given response matrix and regression coefficients, find suitable predictors

Consider a standard OLS regression problem$\newcommand{\Y}{\mathbf Y}\newcommand{\X}{\mathbf X}\newcommand{\B}{\boldsymbol\beta}\DeclareMathOperator*{argmin}{argmin}$: I have matrices $\Y$ and $\X$ ...
7
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2answers
1k views

Maximum penalty for ridge regression

Consider a regression model $$ y = X \beta + \varepsilon. $$ I will use ridge regression to estimate $\beta$. Ridge regression contains a tuning parameter (the penalty intensity) $\lambda$. If I ...
3
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2answers
855 views

How to derive the covariance matrix of $\hat\beta$ in linear regression?

I just read this very insightful post about ridge regression, where the author stated that the variance of $\hat\beta$ is: $$\text{var}(\hat\beta) = \sigma^2(\textbf{X}^\prime \textbf{X})^{-1}.$$ I ...
6
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2answers
562 views

Is there a mathematical expression that shows how LASSO shrinks coefficients (including some to zero)?

By using singular value decomposition (SVD), I noticed from the derivation that ridge regression shrinks the coefficients by factor $\frac{D^2}{D^2+\lambda}$, where $D$ is the diagonal matrix of the ...
3
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1answer
199 views

Why does sklearn Ridge not accept warm start?

I am experimenting with some regularized linear regression methods using sklearn and noticed that Ridge does not accept warm start. I found it odd as many other methods do accept like Lasso, ...
1
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1answer
119 views

why ridge regression only decreases slope and not increases it?

I was following the below example from 'StatQuest with Josh Starmer' youtube channel. The example is pretty simple: red line is the usual 'least squares' (for the red points), and the blue one is ...
3
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1answer
64 views

What is the relationship between the sum of squares of all weights and lambda in the ridge regression [duplicate]

Currently I am reading chapter 8, regression. And I feel quite confused about the following paragraph(see picture below), does it mean in ridge algorithm, the sum of all weights will be less than ...
3
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0answers
87 views

How can $X^TX$ be decomposed?

Nikolaenko et al. claims that in ridge regression $A\beta=b$, where $A=X^TX+\lambda I \in R^{d\times d}$ and $b=X^Ty \in R^d$ (page 3), it can be decomposed into: $$A=\sum\limits_{i=1}^{n}A_i+\...

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