[Paper Review] Generating Polynomials and Symmetric Tensor Decompositions

Jiawang Nie|arXiv (Cornell University)|Aug 25, 2014

Tensor decomposition and applications26 references4 citations

TL;DR

This paper introduces generating polynomials and generating matrices to characterize symmetric tensor decompositions, enabling efficient computation of tensor rank decompositions via algebraic geometry and polynomial systems. The key contribution is a method that leverages the apolar ideal structure and catalecticant matrices to compute symmetric tensor decompositions with high accuracy (error ≈ 10⁻¹²) in seconds to minutes, even for high-order tensors.

ABSTRACT

This paper studies symmetric tensor decompositions. For symmetric tensors, there exist linear relations of recursive patterns among their entries. Such a relation can be represented by a polynomial, which is called a generating polynomial. The homogenization of a generating polynomial belongs to the apolar ideal of the tensor. A symmetric tensor decomposition can be determined by a set of generating polynomials, which can be represented by a matrix. We call it a generating matrix. Generally, a symmetric tensor decomposition can be determined by a generating matrix satisfying certain conditions. We characterize the sets of such generating matrices and investigate their properties (e.g., the existence, dimensions, nondefectiveness). Using these properties, we propose methods for computing symmetric tensor decompositions. Extensive examples are shown to demonstrate the efficiency of proposed methods.

Motivation & Objective

• To develop a systematic algebraic framework for symmetric tensor decompositions using generating polynomials and matrices.
• To characterize the conditions under which a generating matrix determines a valid symmetric tensor decomposition.
• To provide efficient computational methods for computing symmetric tensor decompositions, especially for generic tensors.
• To analyze the existence, dimension, and nondefectiveness of generating matrices in relation to tensor rank.

Proposed method

• Represent symmetric tensor entries using generating polynomials, which encode recursive linear relations among tensor entries.
• Define a generating matrix as a matrix representation of a set of generating polynomials that determine a symmetric tensor decomposition.
• Use the apolar ideal and homogenization of generating polynomials to link tensor decompositions to algebraic geometry and polynomial systems.
• Apply catalecticant matrices to verify rank conditions and guide the decomposition process.
• Implement Algorithm 4.3 to compute symmetric tensor decompositions by solving polynomial systems derived from generating matrices.
• Leverage numerical algebraic geometry and homotopy continuation to solve the resulting systems efficiently, with error control.

Experimental results

Research questions

• RQ1 How can symmetric tensor decompositions be systematically characterized using polynomial relations among tensor entries?
• RQ2 What conditions must a generating matrix satisfy to yield a valid symmetric tensor decomposition?
• RQ3 How do the existence, dimension, and nondefectiveness of generating matrices relate to the rank of a symmetric tensor?
• RQ4 Can generating polynomials and matrices be used to compute symmetric tensor decompositions efficiently for generic tensors?
• RQ5 What is the computational performance and accuracy of the proposed method across different tensor orders and dimensions?

Key findings

• The method successfully computes symmetric tensor decompositions with residual errors on the order of 10⁻¹² for randomly generated tensors.
• For tensors in S⁵(ℂ⁴), decomposition times were a few seconds, with rank 15 decompositions computed accurately.
• In S⁶(ℂ⁴), the algorithm computed a rank-21 decomposition in a few minutes, with similar error levels.
• For (n+1,m) = (8,3), the average computation time was 1799.6 seconds, indicating scalability challenges for high-dimensional cases.
• The method achieved consistent success across 50 random instances per (n+1,m) pair, with all decompositions computed to desired length and high accuracy.
• The catalecticant matrix rank was found to be one less than the tensor rank in tested cases, supporting the correctness of the computed decompositions.