Gaussian random projections for Euclidean membership problems

Ky Vu , Pierre-Louis Poirion (1) , Leo Liberti (2)

1 RIKEN Research Center
2 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]

We discuss the application of Gaussian random projections to the fundamental problem of deciding whether a given point in a Euclidean space belongs to a given set. In particular, we consider the two cases, when the target set is either at most countable or of low doubling dimension. We show that, under a number of different assumptions, the feasibility (or infeasibility) of this problem is preserved almost surely when the problem data is projected to a lower dimensional space. We also consider the threshold version of this problem, in which we require that the projected point and the projected set are separated by a certain distance error. As a consequence of these results, we are able to improve the bound of Indyk-Naor on the Nearest Neigbour preserving embeddings. Our results are applicable to any algorithmic setting which needs to solve Euclidean membership problems in a high-dimensional space.

Mots clés: Johnson-Lindenstrauss lemma; Machine Learning; Euclidean Distance Geometry; clustering

Domaines: Recherche opérationnelle [math.OC]

Ky Vu, Pierre-Louis Poirion, Leo Liberti. Gaussian random projections for Euclidean membership problems. Discrete Applied Mathematics, 2019, 253, pp.93-102. ⟨10.1016/j.dam.2018.08.025⟩. ⟨hal-02105075⟩