Perform PCA. Extract PCs. Can one then tell what the most important _original_ features were, from the PCs? Suppose that you have 1000 features, and a data set made up of say, 50,000 points. Suppose then that we perform PCA, and we extract the top 5 PCs, since they explain 99.99 percent of the variance, and thats all we care about.
From those top 5 PCs, can we 'go backwards', and be able to decipher, what the most 'important' features were from the original 1000? For example, can we answers the question, "What combination of my original 1000 features were responsible for my top PC?"
Thank you.
 A: Each of the principal components projects the whole original feature space onto several dimensions, which I will call the latent features. The more an original feature contributes to a latent feature, the more important it is for that feature.
Thus, look at the absolute values of the Eigenvectors' components corresponding to the $k$ largest Eigenvalues. The larger they are, the more a specific feature contributes to that principal component.
Mind, however, that these will typically be dense. If you want to find some kind of minimal feature space that explains most of the data, you might be interested in sparse pca.
A: You can answer the last question by looking at the loadings of that PC. You could also do this for each of the other 5 PCs. I am not sure what you could do about the overall importance of particular variables. 
A: Since eigen vectors are from the linear combination of the original variables, I don't think you can safely determine which of the original variables you can do away with from PCA.  I felt this is more like a Factor Analysis question (related to latent variables).
