# Optimizing Orthogonalized Tensor Deflation via Random Tensor Theory

arXiv: https://arxiv.org/abs/2302.05798 · HTML: https://mlsanity.com/paper/2302.05798/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 35 references that are in the map.
- First posted on arXiv 11 Feb 2023.

## 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)
- [Introduction to Tensor Decompositions and their Applications in Machine Learning](https://mlsanity.com/paper/1711.10781/) (2017)
- [Statistical limits of spiked tensor models](https://mlsanity.com/paper/1612.07728/) (2016)
- [Reconstructing Training Data from Model Gradient, Provably](https://mlsanity.com/paper/2212.03714/) (2022)
- [Extension of PCA to Higher Order Data Structures: An Introduction to Tensors, Tensor Decompositions, and Tensor PCA](https://mlsanity.com/paper/1803.00704/) (2018)
- [A Random Matrix Perspective on Random Tensors](https://mlsanity.com/paper/2108.00774/) (2021)
- [On Estimating Rank-One Spiked Tensors in the Presence of Heavy Tailed Errors](https://mlsanity.com/paper/2107.09660/) (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)
- [On the Accuracy of Hotelling-Type Tensor Deflation: A Random Tensor Analysis](https://mlsanity.com/paper/2211.09004/) (2022)

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

- [Survey of Video Diffusion Models: Foundations, Implementations, and Applications](https://mlsanity.com/paper/2504.16081/) (2025)
- [On the Benefits of Weight Normalization for Overparameterized Matrix Sensing](https://mlsanity.com/paper/2510.01175/) (2025)
- [Hotelling Deflation on Large Symmetric Spiked Tensors](https://mlsanity.com/paper/2304.10248/) (2023)
- [Direction-Magnitude Decomposition for Low-Rank Matrix Optimization: Faster Convergence and Saddle-to-saddle Dynamics](https://mlsanity.com/paper/2606.31390/) (2026)

## Abstract (arXiv)

This paper tackles the problem of recovering a low-rank signal tensor with possibly correlated components from a random noisy tensor, or so-called spiked tensor model. When the underlying components are orthogonal, they can be recovered efficiently using tensor deflation which consists of successive rank-one approximations, while non-orthogonal components may alter the tensor deflation mechanism, thereby preventing efficient recovery. Relying on recently developed random tensor tools, this paper deals precisely with the non-orthogonal case by deriving an asymptotic analysis of a parameterized deflation procedure performed on an order-three and rank-two spiked tensor. Based on this analysis, an efficient tensor deflation algorithm is proposed by optimizing the parameter introduced in the deflation mechanism, which in turn is proven to be optimal by construction for the studied tensor model. The same ideas could be extended to more general low-rank tensor models, e.g., higher ranks and orders, leading to more efficient tensor methods with a broader impact on machine learning and beyond.