# Statistical mechanics of low-rank tensor decomposition

arXiv: https://arxiv.org/abs/1810.10065 · HTML: https://mlsanity.com/paper/1810.10065/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Oct 2018.

## Built on by (most cited first, 4 of 4)

- [Computational Thresholds in Multi-Modal Learning via the Spiked Matrix-Tensor Model](https://mlsanity.com/paper/2506.02664/) (2025)
- [Optimal thresholds and algorithms for a model of multi-modal learning in high dimensions](https://mlsanity.com/paper/2407.03522/) (2024)
- [A Random Matrix Approach to Low-Multilinear-Rank Tensor Approximation](https://mlsanity.com/paper/2402.03169/) (2024)
- [Large-Scale Bayesian Tensor Reconstruction via Approximate Message Passing](https://mlsanity.com/paper/2505.16305/) (2025)

## Abstract (arXiv)

Often, large, high dimensional datasets collected across multiple modalities can be organized as a higher order tensor. Low-rank tensor decomposition then arises as a powerful and widely used tool to discover simple low dimensional structures underlying such data. However, we currently lack a theoretical understanding of the algorithmic behavior of low-rank tensor decompositions. We derive Bayesian approximate message passing (AMP) algorithms for recovering arbitrarily shaped low-rank tensors buried within noise, and we employ dynamic mean field theory to precisely characterize their performance. Our theory reveals the existence of phase transitions between easy, hard and impossible inference regimes, and displays an excellent match with simulations. Moreover, it reveals several qualitative surprises compared to the behavior of symmetric, cubic tensor decomposition. Finally, we compare our AMP algorithm to the most commonly used algorithm, alternating least squares (ALS), and demonstrate that AMP significantly outperforms ALS in the presence of noise.