# On maximum-likelihood estimation in the all-or-nothing regime

arXiv: https://arxiv.org/abs/2101.09994 · HTML: https://mlsanity.com/paper/2101.09994/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 21 references that are in the map.
- First posted on arXiv 25 Jan 2021.

## Builds on (most cited first, 8 of 8)

- [Statistical physics of inference: Thresholds and algorithms](https://mlsanity.com/paper/1511.02476/) (2015)
- [A statistical model for tensor PCA](https://mlsanity.com/paper/1411.1076/) (2014)
- [Tensor principal component analysis via sum-of-squares proofs](https://mlsanity.com/paper/1507.03269/) (2015)
- [Statistical limits of spiked tensor models](https://mlsanity.com/paper/1612.07728/) (2016)
- [Mutual information for symmetric rank-one matrix estimation: A proof of the replica formula](https://mlsanity.com/paper/1606.04142/) (2016)
- [All-or-nothing statistical and computational phase transitions in sparse spiked matrix estimation](https://mlsanity.com/paper/2006.07971/) (2020)
- [How to iron out rough landscapes and get optimal performances: Averaged Gradient Descent and its application to tensor PCA](https://mlsanity.com/paper/1905.12294/) (2019)
- [Statistical and computational thresholds for the planted k-densest sub-hypergraph problem](https://mlsanity.com/paper/2011.11500/) (2020)

## Abstract (arXiv)

We study the problem of estimating a rank-1 additive deformation of a Gaussian tensor according to the maximum-likelihood estimator (MLE). The analysis is carried out in the sparse setting, where the underlying signal has a support that scales sublinearly with the total number of dimensions. We show that for Bernoulli distributed signals, the MLE undergoes an all-or-nothing (AoN) phase transition, already established for the minimum mean-square-error estimator (MMSE) in the same problem. The result follows from two main technical points: (i) the connection established between the MLE and the MMSE, using the first and second-moment methods in the constrained signal space, (ii) a recovery regime for the MMSE stricter than the simple error vanishing characterization given in the standard AoN, that is here proved as a general result.