# Landscape analysis of an improved power method for tensor decomposition

arXiv: https://arxiv.org/abs/2110.15821 · HTML: https://mlsanity.com/paper/2110.15821/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 54 references that are in the map.
- First posted on arXiv 29 Oct 2021.

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

- [Tensor Decomposition for Signal Processing and Machine Learning](https://mlsanity.com/paper/1607.01668/) (2016)
- [Recovery Guarantees for One-hidden-layer Neural Networks](https://mlsanity.com/paper/1706.03175/) (2017)
- [Learning Mixtures of Gaussians in High Dimensions](https://mlsanity.com/paper/1503.00424/) (2015)
- [Notes on computational-to-statistical gaps: predictions using statistical physics](https://mlsanity.com/paper/1803.11132/) (2018)
- [On the Optimization Landscape of Tensor Decompositions](https://mlsanity.com/paper/1706.05598/) (2017)
- [Robust and Resource Efficient Identification of Two Hidden Layer Neural Networks](https://mlsanity.com/paper/1907.00485/) (2019)
- [Stable Recovery of Entangled Weights: Towards Robust Identification of Deep Neural Networks from Minimal Samples](https://mlsanity.com/paper/2101.07150/) (2021)
- [When Does Non-Orthogonal Tensor Decomposition Have No Spurious Local Minima?](https://mlsanity.com/paper/1911.09815/) (2019)

## Built on by (most cited first, 7 of 7)

- [Geometry and Optimization of Shallow Polynomial Networks](https://mlsanity.com/paper/2501.06074/) (2025)
- [Fast algorithm for overcomplete order-3 tensor decomposition](https://mlsanity.com/paper/2202.06442/) (2022)
- [Finite Sample Identification of Wide Shallow Neural Networks with Biases](https://mlsanity.com/paper/2211.04589/) (2022)
- [Lower Bounds for the Convergence of Tensor Power Iteration on Random Overcomplete Models](https://mlsanity.com/paper/2211.03827/) (2022)
- [Contrastive independent component analysis](https://mlsanity.com/paper/2407.02357/) (2024)
- [Approximate Real Symmetric Tensor Rank](https://mlsanity.com/paper/2207.12529/) (2022)
- [Multi-context principal component analysis](https://mlsanity.com/paper/2601.15239/) (2026)

## Abstract (arXiv)

In this work, we consider the optimization formulation for symmetric tensor decomposition recently introduced in the Subspace Power Method (SPM) of Kileel and Pereira. Unlike popular alternative functionals for tensor decomposition, the SPM objective function has the desirable properties that its maximal value is known in advance, and its global optima are exactly the rank-1 components of the tensor when the input is sufficiently low-rank. We analyze the non-convex optimization landscape associated with the SPM objective. Our analysis accounts for working with noisy tensors. We derive quantitative bounds such that any second-order critical point with SPM objective value exceeding the bound must equal a tensor component in the noiseless case, and must approximate a tensor component in the noisy case. For decomposing tensors of size D× m, we obtain a near-global guarantee up to rank widetildeo(Dlfloor m/2 rfloor) under a random tensor model, and a global guarantee up to rank O(D) assuming deterministic frame conditions. This implies that SPM with suitable initialization is a provable, efficient, robust algorithm for low-rank symmetric tensor decomposition. We conclude with numerics that show a practical preferability for using the SPM functional over a more established counterpart.