Why fewer number of principal components give better results in clustering? I try to do an (unsupervised) clustering with sklearn, Python, by different algorithms (hierarchical, distance based, density based etc. ones). The data in question has few hundred original features and I do experiments with different dimension reduction algorithms before the actual clustering (PCA, IncrementalPCA, LocallyLinearEmbedding, Isomap, FactorAnalysis).
The strange tendency I observe is that the fewer component I keep with dimension reductions (more-less irrespective to the kind of the algorithm) the better result I get (by silhouette score). This tendency is true to the extreme: the best result seems to be the one with only 2 compressed final dimension.
This seems absurd, my question is what is the explanation here? I could imagine that either silhouette score is misleading in the evaluation (but then what to chose?), or that I may still happen to have a few key original explanatory variable which is enough to be kept?

The problem with clustering after dimensionality reduction is that the clustering can completely change by the projection. See e.g. this example:

Here, in higher dimensions, you have five perfect clusters and after PCA projection (to the red line) you end up with just one cluster. And even though the clustering after dimensionality reduction is nowhere near the proper clustering, the silhouette score can still be good, maybe even better than for the original data.
But clearly, this is not only a problem with the silhouette score, this follows from the projection which can severely alter the grouping of your data. So comparing any scores for clusterings after different dimensionality reductions is often not very informative.
Clustering metrics, especially in higher dimensions, are always problematic. None of them have ever worked for me except for toy examples. In my experience, methods like t-SNE and UMAP still work best.
