Intrinsic dimension estimation: Advances and open problems

•The paper reviews state-of-the-art of the methods of Intrinsic Dimension (ID) Estimation.•The paper defines the properties that an ideal ID estimator should have.•The paper reviews, under the above mentioned framework, the major ID estimation methods underlining their advances and the open problems.

Dimensionality reduction methods are preprocessing techniques used for coping with high dimensionality. They have the aim of projecting the original data set of dimensionality N, without information loss, onto a lower M-dimensional submanifold. Since the value of M is unknown, techniques that allow knowing in advance the value of M, called intrinsic dimension (ID), are quite useful. The aim of the paper is to review state-of-the-art of the methods of ID estimation, underlining the recent advances and the open problems.