# Random projections for Bayesian regression

arXiv: https://arxiv.org/abs/1504.06122 · HTML: https://mlsanity.com/paper/1504.06122/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Apr 2015.

## Built on by (most cited first, 4 of 4)

- [Oblivious sketching for logistic regression](https://mlsanity.com/paper/2107.06615/) (2021)
- [Scalable Learning of Multivariate Distributions via Coresets](https://mlsanity.com/paper/2603.19792/) (2026)
- [Joint Model and Data Sparsification via the Marginal Likelihood](https://mlsanity.com/paper/2605.29908/) (2026)
- [A Bayesian Vertical Federated Learning Framework for Multivariate Reduced-Rank High-Dimensional Regression](https://mlsanity.com/paper/2609.22654/) (2026)

## Abstract (arXiv)

This article deals with random projections applied as a data reduction technique for Bayesian regression analysis. We show sufficient conditions under which the entire d-dimensional distribution is approximately preserved under random projections by reducing the number of data points from n to k∈ O(operatornamepoly(d/varepsilon)) in the case ngg d. Under mild assumptions, we prove that evaluating a Gaussian likelihood function based on the projected data instead of the original data yields a (1+O(varepsilon))-approximation in terms of the ℓ2 Wasserstein distance. Our main result shows that the posterior distribution of Bayesian linear regression is approximated up to a small error depending on only an varepsilon-fraction of its defining parameters. This holds when using arbitrary Gaussian priors or the degenerate case of uniform distributions over ℝd for β. Our empirical evaluations involve different simulated settings of Bayesian linear regression. Our experiments underline that the proposed method is able to recover the regression model up to small error while considerably reducing the total running time.