# Estimating Certain Integral Probability Metric (IPM) is as Hard as Estimating under the IPM

arXiv: https://arxiv.org/abs/1911.00730 · HTML: https://mlsanity.com/paper/1911.00730/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Nov 2019.

## Built on by (most cited first, 11 of 11)

- [Neural Estimation of Statistical Divergences](https://mlsanity.com/paper/2110.03652/) (2021)
- [Non-Asymptotic Performance Guarantees for Neural Estimation of mathsff-Divergences](https://mlsanity.com/paper/2103.06923/) (2021)
- [Sample Complexity Bounds for Estimating Probability Divergences under Invariances](https://mlsanity.com/paper/2311.02868/) (2023)
- [Reversible Gromov-Monge Sampler for Simulation-Based Inference](https://mlsanity.com/paper/2109.14090/) (2021)
- [Adversarially Balanced Representation for Continuous Treatment Effect Estimation](https://mlsanity.com/paper/2312.10570/) (2023)
- [The Representation Jensen-Shannon Divergence](https://mlsanity.com/paper/2305.16446/) (2023)
- [Online Learning to Transport via the Minimal Selection Principle](https://mlsanity.com/paper/2202.04732/) (2022)
- [Scalable Sobolev IPM for Probability Measures on a Graph](https://mlsanity.com/paper/2502.00737/) (2025)
- [Unified Perspective on Probability Divergence via Maximum Likelihood Density Ratio Estimation: Bridging KL-Divergence and Integral Probability Metrics](https://mlsanity.com/paper/2201.13127/) (2022)
- [Generalized Sobolev IPM for Graph-Based Measures](https://mlsanity.com/paper/2510.25591/) (2025)
- [Distributional Shrinkage I: Universal Denoiser Beyond Tweedie's Formula](https://mlsanity.com/paper/2511.09500/) (2025)

## Abstract (arXiv)

We study the minimax optimal rates for estimating a range of Integral Probability Metrics (IPMs) between two unknown probability measures, based on n independent samples from them. Curiously, we show that estimating the IPM itself between probability measures, is not significantly easier than estimating the probability measures under the IPM. We prove that the minimax optimal rates for these two problems are multiplicatively equivalent, up to a log log (n)/log (n) factor.