# High-dimensional Location Estimation via Norm Concentration for Subgamma Vectors

arXiv: https://arxiv.org/abs/2302.02497 · HTML: https://mlsanity.com/paper/2302.02497/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 37 references that are in the map.
- First posted on arXiv 5 Feb 2023.

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

- [Outlier Robust Mean Estimation with Subgaussian Rates via Stability](https://mlsanity.com/paper/2007.15618/) (2020)
- [Sub-Gaussian estimators of the mean of a random vector](https://mlsanity.com/paper/1702.00482/) (2017)
- [Fast Mean Estimation with Sub-Gaussian Rates](https://mlsanity.com/paper/1902.01998/) (2019)
- [Finite-Sample Maximum Likelihood Estimation of Location](https://mlsanity.com/paper/2206.02348/) (2022)

## Built on by (most cited first, 5 of 5)

- [Large Language Models](https://mlsanity.com/paper/2307.05782/) (2023)
- [Faster Diffusion Sampling with Randomized Midpoints: Sequential and Parallel](https://mlsanity.com/paper/2406.00924/) (2024)
- [Improved Sample Complexity Bounds for Diffusion Model Training](https://mlsanity.com/paper/2311.13745/) (2023)
- [Finite-Sample Symmetric Mean Estimation with Fisher Information Rate](https://mlsanity.com/paper/2306.16573/) (2023)
- [Near-Optimal Mean Estimation with Unknown, Heteroskedastic Variances](https://mlsanity.com/paper/2312.02417/) (2023)

## Abstract (arXiv)

In location estimation, we are given n samples from a known distribution f shifted by an unknown translation λ, and want to estimate λ as precisely as possible. Asymptotically, the maximum likelihood estimate achieves the Cram'er-Rao bound of error mathcal N(0, frac1nmathcal I), where mathcal I is the Fisher information of f. However, the n required for convergence depends on f, and may be arbitrarily large. We build on the theory using smoothed estimators to bound the error for finite n in terms of mathcal Ir, the Fisher information of the r-smoothed distribution. As n → ∞, r → 0 at an explicit rate and this converges to the Cram'er-Rao bound. We (1) improve the prior work for 1-dimensional f to converge for constant failure probability in addition to high probability, and (2) extend the theory to high-dimensional distributions. In the process, we prove a new bound on the norm of a high-dimensional random variable whose 1-dimensional projections are subgamma, which may be of independent interest.