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Black Box Absolute Reconstruction for Sums of Powers of Linear Forms

Authors Pascal Koiran, Subhayan Saha

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Author Details

Pascal Koiran
  • Univ Lyon, EnsL, UCBL, CNRS, LIP, F-69342, LYON Cedex 07, France
Subhayan Saha
  • Univ Lyon, EnsL, UCBL, CNRS, LIP, F-69342, LYON Cedex 07, France

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Pascal Koiran and Subhayan Saha. Black Box Absolute Reconstruction for Sums of Powers of Linear Forms. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 24:1-24:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.FSTTCS.2022.24

Abstract

We study the decomposition of multivariate polynomials as sums of powers of linear forms. We give a randomized algorithm for the following problem: If a homogeneous polynomial f ∈ K[x_1 , . . . , x_n] (where K ⊆ ℂ) of degree d is given as a blackbox, decide whether it can be written as a linear combination of d-th powers of linearly independent complex linear forms. The main novel features of the algorithm are:  
- For d = 3, we improve by a factor of n on the running time from the algorithm in [Pascal Koiran and Mateusz Skomra, 2021]. The price to be paid for this improvement is that the algorithm now has two-sided error. 
- For d > 3, we provide the first randomized blackbox algorithm for this problem that runs in time poly(n,d) (in an algebraic model where only arithmetic operations and equality tests are allowed). Previous algorithms for this problem [Kayal, 2011] as well as most of the existing reconstruction algorithms for other classes appeal to a polynomial factorization subroutine. This requires extraction of complex polynomial roots at unit cost and in standard models such as the unit-cost RAM or the Turing machine this approach does not yield polynomial time algorithms. 
- For d > 3, when f has rational coefficients (i.e. K = ℚ), the running time of the blackbox algorithm is polynomial in n,d and the maximal bit size of any coefficient of f. This yields the first algorithm for this problem over ℂ with polynomial running time in the bit model of computation.  These results are true even when we replace ℂ by ℝ. We view the problem as a tensor decomposition problem and use linear algebraic methods such as checking the simultaneous diagonalisability of the slices of a tensor. The number of such slices is exponential in d. But surprisingly, we show that after a random change of variables, computing just 3 special slices is enough. We also show that our approach can be extended to the computation of the actual decomposition. In forthcoming work we plan to extend these results to overcomplete decompositions, i.e., decompositions in more than n powers of linear forms.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Computing methodologies → Algebraic algorithms
  • Computing methodologies → Linear algebra algorithms
Keywords
  • reconstruction algorithms
  • tensor decomposition
  • sums of powers of linear forms
  • simultaneous diagonalisation
  • algebraic algorithm
  • black box

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