# Subspace power method for symmetric tensor decomposition

arXiv: https://arxiv.org/abs/1912.04007 · HTML: https://mlsanity.com/paper/1912.04007/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Dec 2019.

## Built on by (most cited first, 14 of 14)

- [Symmetry Breaking in Symmetric Tensor Decomposition](https://mlsanity.com/paper/2103.06234/) (2021)
- [Stable Recovery of Entangled Weights: Towards Robust Identification of Deep Neural Networks from Minimal Samples](https://mlsanity.com/paper/2101.07150/) (2021)
- [Tensor decomposition for learning Gaussian mixtures from moments](https://mlsanity.com/paper/2106.00555/) (2021)
- [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)
- [Symmetry & Critical Points for Symmetric Tensor Decomposition Problems](https://mlsanity.com/paper/2306.07886/) (2023)
- [Contrastive independent component analysis](https://mlsanity.com/paper/2407.02357/) (2024)
- [Linear causal disentanglement via higher-order cumulants](https://mlsanity.com/paper/2407.04605/) (2024)
- [On a realization of motion and similarity group equivalence classes of labeled points in mathbb Rk with applications to computer vision](https://mlsanity.com/paper/2103.12980/) (2021)
- [Projected Tensor-Tensor Products for Efficient Computation of Optimal Multiway Data Representations](https://mlsanity.com/paper/2409.19402/) (2024)
- [Generalized Canonical Polyadic Tensor Decompositions with General Symmetry](https://mlsanity.com/paper/2601.05335/) (2026)
- [Multi-context principal component analysis](https://mlsanity.com/paper/2601.15239/) (2026)
- [Tensor-based second-order causal discovery](https://mlsanity.com/paper/2606.18074/) (2026)

## Abstract (arXiv)

We introduce the Subspace Power Method (SPM) for calculating the CP decomposition of low-rank real symmetric tensors. This algorithm calculates one new CP component at a time, alternating between applying the shifted symmetric higher-order power method (SS-HOPM) to a certain modified tensor, constructed from a matrix flattening of the original tensor; and using appropriate deflation steps. We obtain rigorous guarantees for SPM regarding convergence and global optima for input tensors of dimension d and order m of CP rank up to O(dlfloor m/2rfloor), via results in classical algebraic geometry and optimization theory. As a by-product of our analysis we prove that SS-HOPM converges unconditionally, settling a conjecture in [Kolda, T.G., Mayo, J.R.: Shifted power method for computing tensor eigenpairs. SIAM Journal on Matrix Analysis and Applications 32(4), 1095-1124 (2011)]. We present numerical experiments which demonstrate that SPM is efficient and robust to noise, being up to one order of magnitude faster than state-of-the-art CP decomposition algorithms in certain experiments. Furthermore, prior knowledge of the CP rank is not required by SPM.