# HOMER: Huber-of-Means for Efficient and Robust Estimation in Hilbert Spaces

arXiv: https://arxiv.org/abs/2607.27532 · HTML: https://mlsanity.com/paper/2607.27532/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 35 references that are in the map.
- First posted on arXiv 29 Jul 2026.

## Builds on (most cited first, 6 of 6)

- [Byzantine-Robust Distributed Learning: Towards Optimal Statistical Rates](https://mlsanity.com/paper/1803.01498/) (2018)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Adaptive Huber Regression](https://mlsanity.com/paper/1706.06991/) (2017)
- [Loss minimization and parameter estimation with heavy tails](https://mlsanity.com/paper/1307.1827/) (2013)
- [Distributed Statistical Estimation and Rates of Convergence in Normal Approximation](https://mlsanity.com/paper/1704.02658/) (2017)
- [MONK -- Outlier-Robust Mean Embedding Estimation by Median-of-Means](https://mlsanity.com/paper/1802.04784/) (2018)

## Abstract (arXiv)

Heavy tails weaken high-confidence control for the empirical mean. Geometric median-of-means (MOM) also lacks a threshold that moves toward mean efficiency. We propose HOMER, or Huber-of-Means for Efficient and Robust Estimation. HOMER aggregates block means through a radial Huber center. Its canonical and pseudo-Huber forms bound each block score and interpolate between median-like robustness and the empirical mean. We establish a Hilbert-space majority theorem and a MOM-order deviation bound under a finite second moment. Canonical HOMER recovers the sample mean inside its quadratic region. Pseudo-HOMER approaches the sample mean as the threshold grows. It also admits asymptotic linearity and consistent sandwich covariance estimation around the population block-Huber target. Under a finite third moment, fixed finite-dimensional projections support mean inference at the usual parametric rate. This result requires growing block sizes and counts, with block sizes increasing faster. Heavy-tailed simulations show that HOMER remains stable when a minority of block summaries is displaced. On clean Gaussian data, both versions closely approach the empirical mean's efficiency. Finite-block sandwich intervals undercovered, especially for skewed functional data. Further studies show failure when contamination affects most blocks or compromises ordinary within-block means.