# Transformation of box constraints after PCA

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 $X$?

• Please be more precise. What the new constraints should do? If X data is constrained to some range PCs are also constained. Do you want some other constraint for their range? – ttnphns Sep 11 '15 at 11:56