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Can anyone help me understand the forward stagewise part in the LARS algorithm? I was reading the R code and could not figure out what is updateR and downdateR doing.

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Even without looking at the code, I bet I can guess.

The code is calculating what the R matrix in the QR decomposition is, from that for the previous model with one fewer or one additional predictor (X-) variable (update and downdate, respectively).

It can be computed from an adjacent model much more efficiently than computing it from scratch.

With many predictors, the speedup can be substantial.

For example the downdate probably uses either Givens rotations or Householder transforms to in effect reorder the variables, placing the to-be dropped variable last (and from where it can be ignored).

The book by Golub and Van Loan is probably a good place to get more information, but you could try something like this:

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.142.2571&rep=rep1&type=pdf

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