# Rigorous Restricted Isometry Property of Low-Dimensional Subspaces

arXiv: https://arxiv.org/abs/1801.10058 · HTML: https://mlsanity.com/paper/1801.10058/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Jan 2018.

## Built on by (most cited first, 1 of 1)

- [Optimal Projections for Discriminative Dictionary Learning using the JL-lemma](https://mlsanity.com/paper/2308.13991/) (2023)

## Abstract (arXiv)

Dimensionality reduction is in demand to reduce the complexity of solving large-scale problems with data lying in latent low-dimensional structures in machine learning and computer version. Motivated by such need, in this work we study the Restricted Isometry Property (RIP) of Gaussian random projections for low-dimensional subspaces in ℝN, and rigorously prove that the projection Frobenius norm distance between any two subspaces spanned by the projected data in ℝn (n<N) remain almost the same as the distance between the original subspaces with probability no less than 1 - rm e-O(n). Previously the well-known Johnson-Lindenstrauss (JL) Lemma and RIP for sparse vectors have been the foundation of sparse signal processing including Compressed Sensing. As an analogy to JL Lemma and RIP for sparse vectors, this work allows the use of random projections to reduce the ambient dimension with the theoretical guarantee that the distance between subspaces after compression is well preserved.