# Successive Rank-One Approximations for Nearly Orthogonally Decomposable Symmetric Tensors

@article{Mu2015SuccessiveRA, title={Successive Rank-One Approximations for Nearly Orthogonally Decomposable Symmetric Tensors}, author={Cun Mu and Daniel J. Hsu and Donald Goldfarb}, journal={ArXiv}, year={2015}, volume={abs/1705.10404} }

Many idealized problems in signal processing, machine learning, and statistics can be reduced to the problem of finding the symmetric canonical decomposition of an underlying symmetric and orthogonally decomposable (SOD) tensor. Drawing inspiration from the matrix case, the successive rank-one approximation (SROA) scheme has been proposed and shown to yield this tensor decomposition exactly, and a plethora of numerical methods have thus been developed for the tensor rank-one approximation…

## 20 Citations

Recovering orthogonal tensors under arbitrarily strong, but locally correlated, noise

- Computer Science, MathematicsArXiv
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The problem of recovering an orthogonally decomposable tensor with a subset of elements distorted by noise with arbitrarily large magnitude can be solved through a system of coupled Sylvester-like equations and how to accelerate their solution by an alternating solver is shown.

Perturbation Bounds for (Nearly) Orthogonally Decomposable Tensors

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We develop deterministic perturbation bounds for singular values and vectors of orthogonally decomposable tensors, in a spirit similar to classical results for matrices such as those due to Weyl,…

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- Computer ScienceAISTATS
- 2017

This new method built on Kruskal's uniqueness theorem to decompose symmetric, nearly orthogonally decomposable tensors provably handles a greater level of noise compared to previous methods and achieves a high estimation accuracy.

Perturbation Bounds for Orthogonally Decomposable Tensors and Their Applications in High Dimensional Data Analysis

- Computer Science, MathematicsArXiv
- 2020

The implications of deterministic perturbation bounds for singular values and vectors of orthogonally decomposable tensors are illustrated through three connected yet seemingly different high dimensional data analysis tasks: tensor SVD, tensor regression and estimation of latent variable models, leading to new insights in each of these settings.

Linear convergence of an alternating polar decomposition method for low rank orthogonal tensor approximations

- Mathematics, Computer ScienceMathematical Programming
- 2022

An improved version iAPD of the classical APD is proposed, which exhibits an overall sublinear convergence with an explicit rate which is sharper than the usual $O(1/k)$ for first order methods in optimization.

Optimal orthogonal approximations to symmetric tensors cannot always be chosen symmetric

- Mathematics, Computer ScienceArXiv
- 2019

It is shown that optimal orthogonal approximations of rank greater than one cannot always be chosen to be symmetric.

Successive Partial-Symmetric Rank-One Algorithms for Almost Unitarily Decomposable Conjugate Partial-Symmetric Tensors

- Mathematics
- 2019

In this paper, we introduce the almost unitarily decomposable conjugate partial-symmetric tensors, which are different from the commonly studied orthogonally decomposable tensors by involving the…

Robust Eigenvectors of Symmetric Tensors

- Computer Science, MathematicsArXiv
- 2021

This paper shows that whenever an eigenvector is a generator of the symmetric decomposition of a symmetric tensor, then (if the order of the tensor is sufficiently high) this eigen vector is robust, i.e., it is an attracting fixed point of the Tensor power method.

O C ] 9 D ec 2 01 9 LINEAR CONVERGENCE OF AN ALTERNATING POLAR DECOMPOSITION METHOD FOR LOW RANK ORTHOGONAL TENSOR APPROXIMATIONS

- Mathematics, Computer Science
- 2019

An improved version of the classical APD, iAPD, of the alternating polar decomposition method is proposed, which exhibits an overall sublinear convergence with an explicit rate which is sharper than the usual Op1{kq for first order methods in optimization.

Greedy Approaches to Symmetric Orthogonal Tensor Decomposition

- Computer ScienceSIAM J. Matrix Anal. Appl.
- 2017

This paper review, establish, and compare the perturbation bounds for two natural types of incremental rank-one approximation approaches for finding the symmetric and orthogonal decomposition of a tensor.

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