[Paper Review] Orthogonal tensor decomposition and orbit closures from a linear algebraic perspective

Pascal Koiran|arXiv (Cornell University)|May 13, 2019

Tensor decomposition and applications24 references6 citations

TL;DR

This paper provides a linear algebraic characterization of orthogonal tensor decompositions over real and complex fields, showing that symmetric and ordinary tensors admitting orthogonal Waring decompositions can be described by quadratic equations over ℝ, and by polynomial equalities and inequalities over ℂ. The key contribution is a complete characterization of the closure of orthogonal tensor decompositions using the approximate simultaneous diagonalization (ASD) property, with a counterexample proving that the ASD conditions are necessary but not sufficient for membership in the closure over ℂ.

ABSTRACT

We study orthogonal decompositions of symmetric and ordinary tensors using methods from linear algebra. For the field of real numbers we show that the sets of decomposable tensors can be defined be equations of degree 2. This gives a new proof of some of the results of Robeva and Boralevi et al. Orthogonal decompositions over the field of complex numbers had not been studied previously; we give an explicit description of the set of decomposable tensors using polynomial equalities and inequalities, and we begin a study of their closures. The main open problem that arises from this work is to obtain a complete description of the closures. This question is akin to that of characterizing border rank of tensors in algebraic complexity. We give partial results using in particular a connection with approximate simultaneous diagonalization (the so-called "ASD property").

Motivation & Objective

• To provide algebraic characterizations of orthogonal tensor decompositions using linear algebraic methods.
• To describe the set of decomposable tensors over ℝ and ℂ using polynomial equations and inequalities.
• To study the closure of orthogonal tensor decompositions, particularly over the complex numbers.
• To investigate the relationship between the ASD property and orbit closures in tensor decomposition.
• To resolve open questions about the structure of orbit closures by constructing counterexamples showing that ASD conditions are necessary but not sufficient.

Proposed method

• Uses the slice representation of order-3 tensors as n matrices to apply techniques from simultaneous matrix reduction.
• Applies results from quadratic forms and isotropic subspaces to analyze complex orthogonal decompositions.
• Employs the concept of approximate simultaneous diagonalization (ASD) to characterize limits of orthogonal decompositions.
• Leverages Witt’s extension theorem and properties of symmetric matrices to derive closure conditions.
• Constructs symmetric tensors from totally isotropic subspaces to generate counterexamples for closure membership.
• Uses dimension counting arguments to show that certain tensor families lie outside the closure of orthogonal decompositions.

Experimental results

Research questions

• RQ1 What algebraic conditions characterize the set of symmetric tensors admitting orthogonal Waring decompositions over ℝ and ℂ?
• RQ2 How can the closure of the set of orthogonal decomposable tensors be described over the complex numbers?
• RQ3 Are the approximate simultaneous diagonalization (ASD) conditions sufficient for a tensor to be in the closure of orthogonal decomposable tensors?
• RQ4 Can the orbit closure of orthogonal tensor decompositions be fully characterized using polynomial equations and inequalities?
• RQ5 What is the dimension of the variety of orthogonal decomposable tensors over ℂ, and how does it compare to other tensor families?

Key findings

• Over ℝ, the set of orthogonal Waring decomposable symmetric tensors is defined by quadratic equations, providing a new proof of results from Robeva and Boralevi et al.
• Over ℂ, the set of orthogonal decomposable tensors is a constructible set defined by polynomial equalities and inequalities, and its closure is characterized via the ASD property.
• The matrices formed by products of slices (e.g., X_k^T X_l) of tensors in the closure must be symmetric and approximately simultaneously diagonalizable.
• The closure of orthogonal decomposable tensors over ℂ is strictly larger than the set defined by the ASD property alone, as shown by a counterexample.
• For sufficiently large n (n ≥ 68), there exist symmetric tensors satisfying all ASD conditions but not belonging to the closure of orthogonal decomposable tensors.
• The dimension of the variety of orthogonal decomposable tensors over ℂ is at most 3n(n−1)/2 + n, which is quadratically bounded, while certain symmetric tensor families (e.g., those from totally isotropic subspaces) have cubic growth in dimension.