I use to explain the concept of the spectra reconstruction as trying to fix a puzzle. We have the pieces ( loadings) which once are multiplied by the scores have difference sizes, but the same pattern.
We fit all the pieces, but it can be that the puzzle does not fit correctly, that we have some gaps or spaces not filled,...., etc. is the concept of spectra reconstruction whe we have a error matrix which is the part of the puzzle not completed. It can be small, large,...
One application of this concept is used to define if the spectra belongs to a certain category.
In this blog you will find post about the spectra reconstruction.