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Bridge regression coefficient estimate $\hat{β}^{br}$ are the values that minimize the \begin{equation} \text{RSS} + \lambda \sum_{j=1}^p|\beta_j|^q , \end{equation} where $q \in \mathbb{R}$ and $q > 0 $.

My question is: why this kind of regression called BRIDGE regression?

I know that in 1993 Frank and Friedman proposed this in (1). However, at that time in that paper, there was no term like "bridge" nor "bridge regression". Confusingly, just 3 years later in 1996, Robert Tibshirani in the paper (2) cited the paper (1) using the term "bridge", viz., in section 11:

Frank and Friedman (1993) discuss a generalization of ridge regression and subset selection, through the addition of a penalty of the form $\lambda \sum_{j=1}^p|\beta_j|^q$ to the residual sum of squares. This is equivalent to a constraint of the form $\sum_{j}|\beta_j|^q \le t$; they called this the 'bridge'.

Emmm... They called? When the word "bridge" even do not occur in (1)?

I search on Google scholar and find no more paper before (2) citing (1), so where the word "bridge" come from? Do I miss something important?

I think my question might be related to Why is ridge regression called "ridge", why is it needed, and what happens when $\lambda$ goes to infinity?


References:

  1. A Statistical View of Some Chemometrics Regression Tool (pdf)
  2. Regression Shrinkage and Selection via the Lasso (pdf)
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  • $\begingroup$ Are you confusing "bridge" and "ridge"? $\endgroup$ – Stephan Kolassa May 21 '18 at 16:55
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    $\begingroup$ @StephanKolassa, "bridge" R is the generalization of ridge R. It seems to have been a play on words. $\endgroup$ – gung May 21 '18 at 17:02
  • $\begingroup$ @gung, Thanks for your edition. The link you added to paper (1) seems different from the Google scholar and i'm not sure whether the word 'bridge' occur in it $\endgroup$ – ming li May 21 '18 at 17:14
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    $\begingroup$ Hard to say, @mingli. This is the version on their website. A ctrl-f search doesn't work b/c it seems to be a scan of a typed paper. I skimmed it quickly & didn't see "bridge", FWTW. $\endgroup$ – gung May 21 '18 at 17:52
  • $\begingroup$ Maybe because it is a bridge between lasso and ridge? $\endgroup$ – kjetil b halvorsen Jan 16 at 15:24

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