# Difference between PCA and regularisation

Currently, I am confusing about PCA and regularisation.

I wonder what is the difference between PCA and regularisation: particularly lasso (L1) regression?

Seems both of them can do the feature selection. I have to admit, I am not quiet familiar with the difference between dimensional reduction and feature selection.

Lasso does feature selection in the way that a penalty is added to the OLS loss function (see figure below). So you can say that features with low "impact" will be "shrunken" by the penalty term (you "regulate" the features). Because of the L1 penalty, the $$\beta_i$$ can become zero (which is not the case with Ridge, L2). In the Lasso case you would "eliminate" a feature when it is "shrunken" to zero, and you could call this feature selection. Lasso can be used in "high dimensions", i.e. when you have many features ("columns") but not so many observations ("rows").