# Unraveling the Veil of Subspace RIP Through Near-Isometry on Subspaces

arXiv: https://arxiv.org/abs/1905.09608 · HTML: https://mlsanity.com/paper/1905.09608/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 May 2019.

## 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 a popular approach to tackle high-dimensional data with low-dimensional nature. Subspace Restricted Isometry Property, a newly-proposed concept, has proved to be a useful tool in analyzing the effect of dimensionality reduction algorithms on subspaces. In this paper, we provide a characterization of subspace Restricted Isometry Property, asserting that matrices which act as a near-isometry on low-dimensional subspaces possess subspace Restricted Isometry Property. This points out a unified approach to discuss subspace Restricted Isometry Property. Its power is further demonstrated by the possibility to prove with this result the subspace RIP for a large variety of random matrices encountered in theory and practice, including subgaussian matrices, partial Fourier matrices, partial Hadamard matrices, partial circulant/Toeplitz matrices, matrices with independent strongly regular rows (for instance, matrices with independent entries having uniformly bounded 4+ε moments), and log-concave ensembles. Thus our result could extend the applicability of random projections in subspace-based machine learning algorithms including subspace clustering and allow for the application of some useful random matrices which are easier to implement on hardware or are more efficient to compute.