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Undercomplete Decomposition of Symmetric Tensors in Linear Time, and Smoothed Analysis of the Condition Number

Pascal Koiran, Subhayan Saha

TL;DR

This work tackles the problem of efficiently decomposing symmetric third-order tensors in the undercomplete regime ($r\le n$) under the assumption that the rank-one components are linearly independent. It introduces a randomized, robust algorithm that achieves $\varepsilon$-forward accuracy in linear time (in the tensor size $n^3$) within the exact arithmetic model, even under inverse-quasi-polynomial noise, by reducing to the complete case via a change of basis and performing the key steps implicitly to avoid dense tensor entries. A central contribution is the semi-unitary basis recovery (SUB) step, which stabilizes the subspace spanned by the decomposition vectors; robustness is enhanced through the DEFLATE subroutine and careful error/probability analyses. The paper also provides a smoothed analysis of the tensor-decomposition condition number, showing that favorable conditioning holds with high probability, which underpins the linear-time performance for most inputs. Collectively, these results advance practical, provably fast tensor decomposition methods with rigorous conditioning guarantees for undercomplete, diagonalisable tensors in the complex (and partially real) setting.

Abstract

We study symmetric tensor decompositions, i.e., decompositions of the form $T = \sum_{i=1}^r u_i^{\otimes 3}$ where $T$ is a symmetric tensor of order 3 and $u_i \in \mathbb{C}^n$.In order to obtain efficient decomposition algorithms, it is necessary to require additional properties from $u_i$. In this paper we assume that the $u_i$ are linearly independent.This implies $r \leq n$,that is, the decomposition of T is undercomplete. We give a randomized algorithm for the following problem in the exact arithmetic model of computation: Let $T$ be an order-3 symmetric tensor that has an undercomplete decomposition. Then given some $T'$ close to $T$, an accuracy parameter $\varepsilon$, and an upper bound B on the condition number of the tensor, output vectors $u'_i$ such that $||u_i - u'_i|| \leq \varepsilon$ (up to permutation and multiplication by cube roots of unity) with high probability. The main novel features of our algorithm are: 1) We provide the first algorithm for this problem that runs in linear time in the size of the input tensor. More specifically, it requires $O(n^3)$ arithmetic operations for all accuracy parameters $\varepsilon =$ 1/poly(n) and B = poly(n). 2) Our algorithm is robust, that is, it can handle inverse-quasi-polynomial noise (in $n$,B,$\frac{1}{\varepsilon}$) in the input tensor. 3) We present a smoothed analysis of the condition number of the tensor decomposition problem. This guarantees that the condition number is low with high probability and further shows that our algorithm runs in linear time, except for some rare badly conditioned inputs. Our main algorithm is a reduction to the complete case ($r=n$) treated in our previous work [Koiran,Saha,CIAC 2023]. For efficiency reasons we cannot use this algorithm as a blackbox. Instead, we show that it can be run on an implicitly represented tensor obtained from the input tensor by a change of basis.

Undercomplete Decomposition of Symmetric Tensors in Linear Time, and Smoothed Analysis of the Condition Number

TL;DR

This work tackles the problem of efficiently decomposing symmetric third-order tensors in the undercomplete regime () under the assumption that the rank-one components are linearly independent. It introduces a randomized, robust algorithm that achieves -forward accuracy in linear time (in the tensor size ) within the exact arithmetic model, even under inverse-quasi-polynomial noise, by reducing to the complete case via a change of basis and performing the key steps implicitly to avoid dense tensor entries. A central contribution is the semi-unitary basis recovery (SUB) step, which stabilizes the subspace spanned by the decomposition vectors; robustness is enhanced through the DEFLATE subroutine and careful error/probability analyses. The paper also provides a smoothed analysis of the tensor-decomposition condition number, showing that favorable conditioning holds with high probability, which underpins the linear-time performance for most inputs. Collectively, these results advance practical, provably fast tensor decomposition methods with rigorous conditioning guarantees for undercomplete, diagonalisable tensors in the complex (and partially real) setting.

Abstract

We study symmetric tensor decompositions, i.e., decompositions of the form where is a symmetric tensor of order 3 and .In order to obtain efficient decomposition algorithms, it is necessary to require additional properties from . In this paper we assume that the are linearly independent.This implies ,that is, the decomposition of T is undercomplete. We give a randomized algorithm for the following problem in the exact arithmetic model of computation: Let be an order-3 symmetric tensor that has an undercomplete decomposition. Then given some close to , an accuracy parameter , and an upper bound B on the condition number of the tensor, output vectors such that (up to permutation and multiplication by cube roots of unity) with high probability. The main novel features of our algorithm are: 1) We provide the first algorithm for this problem that runs in linear time in the size of the input tensor. More specifically, it requires arithmetic operations for all accuracy parameters 1/poly(n) and B = poly(n). 2) Our algorithm is robust, that is, it can handle inverse-quasi-polynomial noise (in ,B,) in the input tensor. 3) We present a smoothed analysis of the condition number of the tensor decomposition problem. This guarantees that the condition number is low with high probability and further shows that our algorithm runs in linear time, except for some rare badly conditioned inputs. Our main algorithm is a reduction to the complete case () treated in our previous work [Koiran,Saha,CIAC 2023]. For efficiency reasons we cannot use this algorithm as a blackbox. Instead, we show that it can be run on an implicitly represented tensor obtained from the input tensor by a change of basis.
Paper Structure (33 sections, 44 theorems, 122 equations, 11 algorithms)

This paper contains 33 sections, 44 theorems, 122 equations, 11 algorithms.

Table of Contents

  1. Introduction
  2. Symmetric tensor decomposition
  3. Approximate undercomplete tensor decomposition
  4. Results and Techniques
  5. Outline of the Algorithm
  6. The algorithm in more detail
  7. Organization of the paper
  8. Preliminaries
  9. Norms and Condition Numbers:
  10. Algebraic algorithm from KoiranSaha22
  11. Explanation for Step 6 of the algorithm
  12. Algorithm in finite precision from KoiranSaha22
  13. Moore-Penrose Inverse
  14. The condition number is well defined
  15. Change of Basis
  16. ...and 18 more sections

Key Result

Theorem 1.2

Let $T \in \mathbb{C}^n \otimes \mathbb{C}^n \otimes \mathbb{C}^n$ be an $r$-diagonalisable tensor for some $r \leq n$ and let $T' \in \mathbb{C}^n \otimes \mathbb{C}^n \otimes \mathbb{C}^n$ be such that $||T - T'|| \leq \delta \in~(0, \frac{1}{(nB)^C r^{C\log^4(\frac{rB}{\varepsilon})}})$ for some where $C_{CW}$ is some constant. The algorithm requires $O(n^3 + T_{MM}(r)\log^2(\frac{rB}{\varepsi

Theorems & Definitions (78)

  • Definition 1.1: Condition number of an $r$-diagonalisable symmetric tensor
  • Theorem 1.2
  • Definition 2.1: Tensor Norm
  • Definition 2.2: Condition number of a diagonalisable symmetric tensor
  • Theorem 2.3
  • Theorem 2.4
  • Definition 2.5
  • Lemma 2.6
  • Lemma 2.7
  • proof
  • ...and 68 more