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In lasso, we first standardize all variables to mean=0, var=1. As such, a beta is simply correlation, right? We want to keep the sum of absolute values of betas below certain value, so what do we do:

we keep adding variables, highest correlation first(in absolute value), until the sum of absolute values of their betas, i.e. of their correlations, is below the threshold.

Is this correct? If not, what's the correct explanation of lasso in terms of correlations(vs betas)?

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First, note the point that @gung made in a comment: betas for standardized variables are not quite the same as correlations. This issue holds true even "before returning to the original space," as you put it, as the answer from @glen_b on that page explains.

Your suggestion about a strict relation between correlations and the corresponding standardized regression coefficients would not (necessarily) hold, however, even in an application of the LASSO algorithm with a single predictor variable.

LASSO minimizes the residual sum of squares subject to a penalty on the sum of the magnitudes of the coefficients. With any non-zero penalty, the regression coefficient returned by LASSO will thus be smaller in magnitude than the standard linear regression coefficient. So LASSO is not simply adding one variable at a time at its "full" regression coefficient.

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