# On Spectral Learning for Odeco Tensors: Perturbation, Initialization, and Algorithms

arXiv: https://arxiv.org/abs/2509.25126 · HTML: https://mlsanity.com/paper/2509.25126/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 35 references that are in the map.
- First posted on arXiv 29 Sep 2025.

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

- [Escaping From Saddle Points --- Online Stochastic Gradient for Tensor Decomposition](https://mlsanity.com/paper/1503.02101/) (2015)
- [A statistical model for tensor PCA](https://mlsanity.com/paper/1411.1076/) (2014)
- [Rate-Optimal Perturbation Bounds for Singular Subspaces with Applications to High-Dimensional Statistics](https://mlsanity.com/paper/1605.00353/) (2016)
- [The landscape of the spiked tensor model](https://mlsanity.com/paper/1711.05424/) (2017)
- [Guaranteed Non-Orthogonal Tensor Decomposition via Alternating Rank-1 Updates](https://mlsanity.com/paper/1402.5180/) (2014)
- [Spectral Learning on Matrices and Tensors](https://mlsanity.com/paper/2004.07984/) (2020)
- [Large Dimensional Independent Component Analysis: Statistical Optimality and Computational Tractability](https://mlsanity.com/paper/2303.18156/) (2023)
- [On Estimating Rank-One Spiked Tensors in the Presence of Heavy Tailed Errors](https://mlsanity.com/paper/2107.09660/) (2021)

## Abstract (arXiv)

We study spectral learning for orthogonally decomposable (odeco) tensors, emphasizing the interplay between statistical limits, optimization geometry, and initialization. Unlike matrices, recovery for odeco tensors does not hinge on eigengaps, yielding improved robustness under noise. While iterative methods such as tensor power iterations can be statistically efficient, initialization emerges as the main computational bottleneck. We investigate perturbation bounds, non-convex optimization analysis, and initialization strategies, clarifying when efficient algorithms attain statistical limits and when fundamental barriers remain.