# The radius of statistical efficiency

arXiv: https://arxiv.org/abs/2405.09676 · HTML: https://mlsanity.com/paper/2405.09676/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 80 references that are in the map.
- First posted on arXiv 15 May 2024.

## Builds on (most cited first, 11 of 11)

- [Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview](https://mlsanity.com/paper/1809.09573/) (2018)
- [Fast low-rank estimation by projected gradient descent: General statistical and algorithmic guarantees](https://mlsanity.com/paper/1509.03025/) (2015)
- [Asymptotic Optimality in Stochastic Optimization](https://mlsanity.com/paper/1612.05612/) (2016)
- [Low-rank matrix recovery with composite optimization: good conditioning and rapid convergence](https://mlsanity.com/paper/1904.10020/) (2019)
- [Asymptotic normality and optimality in nonsmooth stochastic approximation](https://mlsanity.com/paper/2301.06632/) (2023)
- [Stochastic Approximation with Decision-Dependent Distributions: Asymptotic Normality and Optimality](https://mlsanity.com/paper/2207.04173/) (2022)
- [Orthogonal Invariance and Identifiability](https://mlsanity.com/paper/1304.1198/) (2013)
- [Variational analysis of spectral functions simplified](https://mlsanity.com/paper/1506.05170/) (2015)
- [A New Complexity Metric for Nonconvex Rank-one Generalized Matrix Completion](https://mlsanity.com/paper/2204.02364/) (2022)
- [The nonsmooth landscape of blind deconvolution](https://mlsanity.com/paper/1911.08526/) (2019)
- [When can you trust feature selection? -- I: A condition-based analysis of LASSO and generalised hardness of approximation](https://mlsanity.com/paper/2312.11425/) (2023)

## Built on by (most cited first, 1 of 1)

- [Instance-optimal stochastic convex optimization: Can we improve upon sample-average and robust stochastic approximation?](https://mlsanity.com/paper/2603.25657/) (2026)

## Abstract (arXiv)

Classical results in asymptotic statistics show that the Fisher information matrix controls the difficulty of estimating a statistical model from observed data. In this work, we introduce a companion measure of robustness of an estimation problem: the radius of statistical efficiency (RSE) is the size of the smallest perturbation to the problem data that renders the Fisher information matrix singular. We compute RSE up to numerical constants for a variety of testbed problems, including principal component analysis, generalized linear models, phase retrieval, bilinear sensing, and matrix completion. Interestingly, we observe a precise reciprocal relationship between RSE and the intrinsic complexity/sensitivity of the problem instance, paralleling the classical Eckart-Young theorem in numerical analysis. To establish our results, we develop theory for spectral functions of measures that extends well-known results from matrix analysis and eigenvalue optimization-a contribution that may be of interest beyond our immediate findings.