# Spectral Methods from Tensor Networks

arXiv: https://arxiv.org/abs/1811.00944 · HTML: https://mlsanity.com/paper/1811.00944/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Nov 2018.

## Built on by (most cited first, 6 of 6)

- [Tensor cumulants for statistical inference on invariant distributions](https://mlsanity.com/paper/2404.18735/) (2024)
- [Average-Case Complexity of Tensor Decomposition for Low-Degree Polynomials](https://mlsanity.com/paper/2211.05274/) (2022)
- [Fast algorithm for overcomplete order-3 tensor decomposition](https://mlsanity.com/paper/2202.06442/) (2022)
- [Algorithms from Invariants: Smoothed Analysis of Orbit Recovery over SO(3)](https://mlsanity.com/paper/2106.02680/) (2021)
- [Concentration of polynomial random matrices via Efron-Stein inequalities](https://mlsanity.com/paper/2209.02655/) (2022)
- [Deep Neural-network Prior for Orbit Recovery from Method of Moments](https://mlsanity.com/paper/2304.14604/) (2023)

## Abstract (arXiv)

A tensor network is a diagram that specifies a way to "multiply" a collection of tensors together to produce another tensor (or matrix). Many existing algorithms for tensor problems (such as tensor decomposition and tensor PCA), although they are not presented this way, can be viewed as spectral methods on matrices built from simple tensor networks. In this work we leverage the full power of this abstraction to design new algorithms for certain continuous tensor decomposition problems. An important and challenging family of tensor problems comes from orbit recovery, a class of inference problems involving group actions (inspired by applications such as cryo-electron microscopy). Orbit recovery problems over finite groups can often be solved via standard tensor methods. However, for infinite groups, no general algorithms are known. We give a new spectral algorithm based on tensor networks for one such problem: continuous multi-reference alignment over the infinite group SO(2). Our algorithm extends to the more general heterogeneous case.