Symmetric tensor decomposition (0901.3706v2)

Abstract: We present an algorithm for decomposing a symmetric tensor, of dimension n and order d as a sum of rank-1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for binary forms. We recall the correspondence between the decomposition of a homogeneous polynomial in n variables of total degree d as a sum of powers of linear forms (Waring's problem), incidence properties on secant varieties of the Veronese Variety and the representation of linear forms as a linear combination of evaluations at distinct points. Then we reformulate Sylvester's approach from the dual point of view. Exploiting this duality, we propose necessary and sufficient conditions for the existence of such a decomposition of a given rank, using the properties of Hankel (and quasi-Hankel) matrices, derived from multivariate polynomials and normal form computations. This leads to the resolution of polynomial equations of small degree in non-generic cases. We propose a new algorithm for symmetric tensor decomposition, based on this characterization and on linear algebra computations with these Hankel matrices. The impact of this contribution is two-fold. First it permits an efficient computation of the decomposition of any tensor of sub-generic rank, as opposed to widely used iterative algorithms with unproved global convergence (e.g. Alternate Least Squares or gradient descents). Second, it gives tools for understanding uniqueness conditions, and for detecting the rank.

• The paper introduces a novel algorithm for decomposing symmetric tensors into sums of rank-1 tensors using an extended Sylvester’s method.
• It employs Hankel matrix properties and duality principles to establish conditions for uniqueness and accurate tensor rank detection.
• Numerical examples demonstrate the method's efficiency and potential applications in signal processing, data analysis, and telecommunications.

Analysis of Symmetric Tensor Decomposition

The paper "Symmetric Tensor Decomposition" articulates a comprehensive approach to decomposing symmetric tensors using an algorithm derived from Sylvester's method for binary forms developed in 1886. The authors, Jerome Brachat, Pierre Comon, Bernard Mourrain, and Elias P. Tsigaridas, extend this classical approach to tensors of higher dimensions and order, presenting a significant advancement in understanding tensor ranks and their decompositions.

Summary of Key Contributions

The primary contribution of this paper is the development of an algorithm for decomposing symmetric tensors into sums of rank-1 symmetric tensors. This approach builds on the classical Sylvester's method for binary forms, extending it to handle tensors of arbitrary dimensions and order. This advancement allows for:

• Efficient Decomposition: The proposed method efficiently computes the decomposition of any tensor of sub-generic rank, addressing limitations in current algorithms like Alternative Least Squares and gradient descent methods, which often lack global convergence guarantees.
• Understanding Uniqueness and Rank Detection: The paper provides tools to understand uniqueness conditions of polynomial tensor decompositions and to detect tensor rank accurately.

Theoretical Framework

The algorithm hinges on several theoretical concepts:

• Hankel and Quasi-Hankel Matrices: By employing Hankel matrices derived from multivariate polynomials, the authors establish necessary and sufficient conditions for the decomposition’s existence, even in non-generic scenarios.
• Duality and Apolarity: Reformulating Sylvester's approach from a dual perspective, the paper demonstrates that symmetric tensor decomposition problems connect to linear combinations of polynomial evaluations over distinct points.
• Polynomial Equations and Linear Algebra: The decomposition conditions lead to solving polynomial equations of small degree, significantly simplifying the problem relative to previous methods.

Numerical and Practical Implications

The paper also sheds light on numerical examples to illustrate the algorithm's effectiveness. These examples demonstrate its robustness and ability to deal with higher-order tensors while handling non-generic cases, providing key insights for practical applications in fields like signal processing, data analysis, and telecommunications.

Future Directions

The results of this research open several avenues for future exploration:

• Computational Complexity: Further analysis on the algorithm's computational complexity could optimize its implementation for large-scale tensors.
• Adaptations: The method holds potential for adaptation to decompositions involving asymmetrical tensors or tensors with different symmetry properties in specific modes.
• Applications in Data Science and AI: Given the dependence of many data science applications on tensor decomposition, this algorithm could be pivotal in advancing methods in areas like machine learning and AI.

Concluding Thoughts

The development of this algorithm signifies an important step in the field of multilinear algebra and its applications. By addressing both theoretical and practical challenges in symmetric tensor decomposition, the paper lays a foundation for future research and enhances the toolkit available for extracting meaningful insights from complex multivariate datasets.