# On Projections to Linear Subspaces

arXiv: https://arxiv.org/abs/2209.12485 · HTML: https://mlsanity.com/paper/2209.12485/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 15 references that are in the map.
- First posted on arXiv 26 Sep 2022.

## Builds on (most cited first, 3 of 3)

- [A Triangle Inequality for Cosine Similarity](https://mlsanity.com/paper/2107.04071/) (2021)
- [Accelerating Spherical k-Means](https://mlsanity.com/paper/2107.04074/) (2021)
- [ABID: Angle Based Intrinsic Dimensionality](https://mlsanity.com/paper/2006.12880/) (2020)

## Abstract (arXiv)

The merit of projecting data onto linear subspaces is well known from, e.g., dimension reduction. One key aspect of subspace projections, the maximum preservation of variance (principal component analysis), has been thoroughly researched and the effect of random linear projections on measures such as intrinsic dimensionality still is an ongoing effort. In this paper, we investigate the less explored depths of linear projections onto explicit subspaces of varying dimensionality and the expectations of variance that ensue. The result is a new family of bounds for Euclidean distances and inner products. We showcase the quality of these bounds as well as investigate the intimate relation to intrinsic dimensionality estimation.