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Complete Decomposition of Symmetric Tensors in Linear Time and Polylogarithmic Precision

Pascal Koiran, Subhayan Saha

TL;DR

This work introduces a randomized, numerically stable approach for the complete symmetric tensor decomposition of order-3 tensors in $\mathbb{C}^n$ under the assumption of a diagonalisable, complete (rank $n$) representation $T = \sum_{i=1}^n u_i^{\otimes 3}$ with $\{u_i\}$ linearly independent. Central to the method is a finite-precision adaptation of Jennrich's simultaneous diagonalisation, augmented by a novel trace-based extraction step (TSCB) that recovers the weights $\alpha_i$ from a change of basis, enabling the reconstruction $u_i = (\alpha_i)^{1/3} v_i$ where $v_i$ come from the diagonalisation. The paper proves that, with high probability, one can obtain an $\varepsilon$-accurate decomposition in time $O(n^3 + T_{MM}(n)\log^2 \frac{nB}{\varepsilon})$ using polylogarithmic precision in $n$, $B$, and $1/\varepsilon$, and demonstrates robust finite-precision stability against roundoff and noise. A probabilistic condition-number analysis ensures the random pencil approach avoids the numerical instability highlighted in prior pencil-based studies, and the framework extends to undercomplete decompositions in follow-up work. Overall, the results yield a scalable, near-linear-time tensor decomposition algorithm with strong, quantifiable stability guarantees in floating-point arithmetic.

Abstract

We study symmetric tensor decompositions, i.e. decompositions of the input symmetric tensor T of order 3 as sum of r 3rd-order tensor powers of u_i where u_i are vectors in \C^n. In order to obtain efficient decomposition algorithms, it is necessary to require additional properties from the u_i. In this paper we assume that the u_i are linearly independent. This implies that r is at most n, i.e., the decomposition of T is undercomplete. We will moreover assume that r=n (we plan to extend this work to the case where r is strictly less than n in a forthcoming paper). We give a randomized algorithm for the following problem: given T, an accuracy parameter epsilon, and an upper bound B on the condition number of the tensor, output vectors u'_i such that u_i and u'_i differ by at most epsilon (in the l_2 norm and up to permutation and multiplication by phases) with high probability. The main novel features of our algorithm are: (1) We provide the first algorithm for this problem that works in the computation model of finite arithmetic and requires only poly-logarithmic (in n, B and 1/epsilon) many bits of precision. (2) Moreover, this is also the first algorithm that runs in linear time in the size of the input tensor. It requires O(n^3) arithmetic operations for all accuracy parameters epsilon = 1/poly(n).

Complete Decomposition of Symmetric Tensors in Linear Time and Polylogarithmic Precision

TL;DR

This work introduces a randomized, numerically stable approach for the complete symmetric tensor decomposition of order-3 tensors in under the assumption of a diagonalisable, complete (rank ) representation with linearly independent. Central to the method is a finite-precision adaptation of Jennrich's simultaneous diagonalisation, augmented by a novel trace-based extraction step (TSCB) that recovers the weights from a change of basis, enabling the reconstruction where come from the diagonalisation. The paper proves that, with high probability, one can obtain an -accurate decomposition in time using polylogarithmic precision in , , and , and demonstrates robust finite-precision stability against roundoff and noise. A probabilistic condition-number analysis ensures the random pencil approach avoids the numerical instability highlighted in prior pencil-based studies, and the framework extends to undercomplete decompositions in follow-up work. Overall, the results yield a scalable, near-linear-time tensor decomposition algorithm with strong, quantifiable stability guarantees in floating-point arithmetic.

Abstract

We study symmetric tensor decompositions, i.e. decompositions of the input symmetric tensor T of order 3 as sum of r 3rd-order tensor powers of u_i where u_i are vectors in \C^n. In order to obtain efficient decomposition algorithms, it is necessary to require additional properties from the u_i. In this paper we assume that the u_i are linearly independent. This implies that r is at most n, i.e., the decomposition of T is undercomplete. We will moreover assume that r=n (we plan to extend this work to the case where r is strictly less than n in a forthcoming paper). We give a randomized algorithm for the following problem: given T, an accuracy parameter epsilon, and an upper bound B on the condition number of the tensor, output vectors u'_i such that u_i and u'_i differ by at most epsilon (in the l_2 norm and up to permutation and multiplication by phases) with high probability. The main novel features of our algorithm are: (1) We provide the first algorithm for this problem that works in the computation model of finite arithmetic and requires only poly-logarithmic (in n, B and 1/epsilon) many bits of precision. (2) Moreover, this is also the first algorithm that runs in linear time in the size of the input tensor. It requires O(n^3) arithmetic operations for all accuracy parameters epsilon = 1/poly(n).
Paper Structure (34 sections, 55 theorems, 255 equations, 4 algorithms)

This paper contains 34 sections, 55 theorems, 255 equations, 4 algorithms.

Table of Contents

  1. Introduction
  2. Symmetric tensor decomposition
  3. Approximate tensor decomposition
  4. Model of Computation
  5. Results and Techniques
  6. Related work and discussion
  7. The Algorithm
  8. Proof overview and organization of the paper
  9. Condition numbers, numerical (in)stability and the negative result of BBV19
  10. Error Reduction by Repetition
  11. Preliminaries: Fast and Stable Linear Algebra
  12. Finite precision arithmetic
  13. Matrix Multiplication and Inversion
  14. Slices after a change of basis
  15. Diagonalisation algorithm for diagonalisable matrices
  16. ...and 19 more sections

Key Result

Theorem 1.2

There is an algorithm which, given a diagonalisable tensor $T$, a desired accuracy parameter $\varepsilon$ and some estimate $B \geq \kappa(T)$, outputs an $\varepsilon$-approximate solution to the tensor decomposition problem for $T$ in arithmetic operations on a floating point machine with bits of precision, with probability at least $(1- \frac{1}{n} - \frac{12}{n^2})(1 - \frac{1}{\sqrt{2n}} -

Theorems & Definitions (116)

  • Definition 1.1: Condition number of a diagonalisable symmetric tensor
  • Theorem 1.2: Main Theorem
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 3.1: Polynomial-Tensor Equivalence
  • Theorem 3.2
  • Corollary 3.2.1
  • Definition 3.3: Tensor Norm
  • Theorem 3.4
  • ...and 106 more