# Universal Inference Meets Random Projections: A Scalable Test for Log-concavity

arXiv: https://arxiv.org/abs/2111.09254 · HTML: https://mlsanity.com/paper/2111.09254/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 45 references that are in the map.
- First posted on arXiv 17 Nov 2021.

## Builds on (most cited first, 4 of 4)

- [Universal Inference](https://mlsanity.com/paper/1912.11436/) (2019)
- [Recent progress in log-concave density estimation](https://mlsanity.com/paper/1709.03154/) (2017)
- [Optimality of Maximum Likelihood for Log-Concave Density Estimation and Bounded Convex Regression](https://mlsanity.com/paper/1903.05315/) (2019)
- [A new computational framework for log-concave density estimation](https://mlsanity.com/paper/2105.11387/) (2021)

## Built on by (most cited first, 3 of 3)

- [Online multiple testing with e-values](https://mlsanity.com/paper/2311.06412/) (2023)
- [A Multivariate Unimodality Test Harnessing the Dip Statistic of Mahalanobis Distances Over Random Projections](https://mlsanity.com/paper/2311.16614/) (2023)
- [On the Precise Asymptotics of Universal Inference](https://mlsanity.com/paper/2503.14717/) (2025)

## Abstract (arXiv)

Shape constraints yield flexible middle grounds between fully nonparametric and fully parametric approaches to modeling distributions of data. The specific assumption of log-concavity is motivated by applications across economics, survival modeling, and reliability theory. However, there do not currently exist valid tests for whether the underlying density of given data is log-concave. The recent universal inference methodology provides a valid test. The universal test relies on maximum likelihood estimation (MLE), and efficient methods already exist for finding the log-concave MLE. This yields the first test of log-concavity that is provably valid in finite samples in any dimension, for which we also establish asymptotic consistency results. Empirically, we find that a random projections approach that converts the d-dimensional testing problem into many one-dimensional problems can yield high power, leading to a simple procedure that is statistically and computationally efficient.