# L1 Estimation: On the Optimality of Linear Estimators

arXiv: https://arxiv.org/abs/2309.09129 · HTML: https://mlsanity.com/paper/2309.09129/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 46 references that are in the map.
- First posted on arXiv 17 Sep 2023.

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

- [Distribution-Free Conditional Median Inference](https://mlsanity.com/paper/2102.07967/) (2021)
- [Estimation in Poisson Noise: Properties of the Conditional Mean Estimator](https://mlsanity.com/paper/1911.03744/) (2019)
- [The Vector Poisson Channel: On the Linearity of the Conditional Mean Estimator](https://mlsanity.com/paper/2003.08967/) (2020)

## Abstract (arXiv)

Consider the problem of estimating a random variable X from noisy observations Y = X+ Z, where Z is standard normal, under the L1 fidelity criterion. It is well known that the optimal Bayesian estimator in this setting is the conditional median. This work shows that the only prior distribution on X that induces linearity in the conditional median is Gaussian. Along the way, several other results are presented. In particular, it is demonstrated that if the conditional distribution PX|Y=y is symmetric for all y, then X must follow a Gaussian distribution. Additionally, we consider other Lp losses and observe the following phenomenon: for p ∈ [1,2], Gaussian is the only prior distribution that induces a linear optimal Bayesian estimator, and for p ∈ (2,∞), infinitely many prior distributions on X can induce linearity. Finally, extensions are provided to encompass noise models leading to conditional distributions from certain exponential families.