# Statistical Limits in Random Tensors with Multiple Correlated Spikes

arXiv: https://arxiv.org/abs/2503.03356 · HTML: https://mlsanity.com/paper/2503.03356/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 44 references that are in the map.
- First posted on arXiv 5 Mar 2025.

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

- [Tensor Decomposition for Signal Processing and Machine Learning](https://mlsanity.com/paper/1607.01668/) (2016)
- [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)
- [The landscape of the spiked tensor model](https://mlsanity.com/paper/1711.05424/) (2017)
- [Optimality and Sub-optimality of PCA I: Spiked Random Matrix Models](https://mlsanity.com/paper/1807.00891/) (2018)
- [Statistical limits of spiked tensor models](https://mlsanity.com/paper/1612.07728/) (2016)
- [A Random Matrix Perspective on Random Tensors](https://mlsanity.com/paper/2108.00774/) (2021)
- [Long Random Matrices and Tensor Unfolding](https://mlsanity.com/paper/2110.10210/) (2021)
- [When Random Tensors meet Random Matrices](https://mlsanity.com/paper/2112.12348/) (2021)
- [Stochastic gradient descent in high dimensions for multi-spiked tensor PCA](https://mlsanity.com/paper/2410.18162/) (2024)
- [Hotelling Deflation on Large Symmetric Spiked Tensors](https://mlsanity.com/paper/2304.10248/) (2023)

## Abstract (arXiv)

We use tools from random matrix theory to study the multi-spiked tensor model, i.e., a rank-r deformation of a symmetric random Gaussian tensor. In particular, thanks to the nature of local optimization methods used to find the maximum likelihood estimator of this model, we propose to study the phase transition phenomenon for finding critical points of the corresponding optimization problem, i.e., those points defined by the Karush-Kuhn-Tucker (KKT) conditions. Moreover, we characterize the limiting alignments between the estimated signals corresponding to a critical point of the likelihood and the ground truth signals. With the help of these results, we propose a new estimator of the rank-r tensor weights by solving a system of polynomial equations, which is asymptotically unbiased contrary the maximum likelihood estimator.