Transformation of box constraints after PCA

Question

I have a $n\times m$ data matrix where $n$ is the number of records and $m$ number of dimensions. Running principal component analysis (PCA) and selecting first $k<m$ components, I get an $n\times k$ matrix.

Consider I had a box constraints on the original data, say $x_{\min},x_{\max}\in\mathbb{R}^{m}$. How to construct new constraints for the first $k$ components?

The motivation is the following: I want to optimize an unknown function $f:X\to\mathbb R$ where $X$ is the hyperbox given by $x_{\min},x_{\max}$. I need to construct the function $\hat f$ from data. For practical reasons, I need PCA on the data. How to transform the set $X$?

Answer 1

It is a little bit hard to answer withouth further information : why would you like to impose box constraints after the PCA ? Did you impose them before the PCA or is it something you observed in the data ?

Indeed, there is no guarantee that the output of the PCA (i.e. the coordinates of the points on the principal axis) will follow the initial box constraints.

So you will have to rescale the output of your PCA to match the box constraints, just as you would do (did?) it before the PCA.