Intrinsic dimensionality detection criterion based on Locally Linear Embedding

We revisit in this work the Locally Linear Embedding (LLE) algorithm which is a widely employed technique in dimensionality reduction. With a particular interest on the correspondences of nearest neighbors in the original and em- bedded spaces, we observe that, when prescribing low-dimensional embedding spaces, LLE remains merely a  weight preserving , rather than a neighborhood preserving algorithm. We propose thus a  ”neighborhood preserving ratio”  crite- rion to estimate a minimal intrinsic dimensionality required for neighbourhood preservation. We validate its efficiency on a set of synthetic data, including S-curve, swiss roll, as well as a dataset of grayscale images.