I'm not sure what you mean by "reliably". Your reconstruction is going to be lossy.

That being said if you've decomposed your original matrix as

$A=Q \Lambda Q^{-1}$

where $Q$ is the matrix of eigenvector columns and

$\Lambda$ is the diagonal matrix of eigenvalues

you should be able to just zero out the eigenvalues below some fidelity threshold, thus forming $\Lambda_{reduced}$.

Then

$A_{reduced}=Q \Lambda_{reduced} Q^{-1}$