[Paper Review] On the number of Singular Vector Tuples of Hyper-Cubical Tensors

Shalosh B. Ekhad, Doron Zeilberger|arXiv (Cornell University)|Apr 30, 2016

Tensor decomposition and applications3 references3 citations

TL;DR

This paper derives a rational generating function for the number of simple singular vector tuples of generic hyper-cubical tensors, proving that the diagonal sequence $ C_d(n) = c(n,\dots,n) $ is holonomic and conjecturing an asymptotic formula $ C_d(n) \sim \alpha_d \cdot ((d-1)^d)^n \cdot n^{-(d-1)/2} $. The work combines generatingfunctionology, the MacMahon Master Theorem, and symbolic computation to establish recurrence relations and asymptotics for $ d=3,4,5,6 $, with a $ \$100 prize offered for a proof and $\$$25 for a closed-form expression of the constant $\alpha_d$. The results are implemented in a Maple package available online.

ABSTRACT

Shmuel Friedland and Giorgio Ottaviani's beautiful constant term expression for the number of singular vector tuples of generic tensors is used to derive a rational generating function for these numbers, that in turn, is used to obtain an asymptotic formula for the number of such tuples for n by n by n three-dimensional tensors, and to conjecture an asymptotic formula for the general d-dimensional case. A donation of 100 dollars, in honor of the first prover, will be made to the On-line Encyclopedia of Integer Sequences.

Motivation & Objective

• To determine the number of simple singular vector tuples in generic $ d $-dimensional hyper-cubical tensors.
• To derive a rational generating function for the sequence $ c(m_1,\dots,m_d) $, the number of such tuples.
• To prove that the diagonal sequence $ C_d(n) = c(n,\dots,n) $ is holonomic (P-recursive) for each $ d $.
• To conjecture an asymptotic formula for $ C_d(n) $, with a prize offered for its proof and for an explicit expression of the constant $ \alpha_d $.

Proposed method

• Using generatingfunctionology and the MacMahon Master Theorem with a $ d \times d $ matrix of ones off-diagonal, the authors derive a rational generating function for $ c(m_1,\dots,m_d) $.
• The generating function is expressed as $ \prod_{i=1}^d x_i (\prod_{i=1}^d (1 - x_i))^{-1} (1 - \sum_{i=2}^d (i-1)e_i(x_1,\dots,x_d))^{-1} $, where $ e_i $ are elementary symmetric functions.
• The holonomy of $ C_d(n) $ is established via algorithmic proof theory, leveraging the rationality of the generating function.
• For $ d=3 $, a fifth-order linear recurrence with polynomial coefficients is derived and verified numerically.
• Asymptotic expansions are computed using the WZ method, yielding $ C_3(n) \sim \frac{2}{\sqrt{3}\pi} 8^n n^{-1} (1 - \frac{13}{3}n^{-1} + \cdots) $.
• Numerical evidence supports the conjecture $ C_d(n) \sim \alpha_d \cdot ((d-1)^d)^n \cdot n^{-(d-1)/2} $ for $ d \geq 4 $, though a closed form for $ \alpha_d $ remains open.

Experimental results

Research questions

• RQ1 What is the exact number of simple singular vector tuples for a generic $ d $-dimensional hyper-cubical tensor of size $ n \times \cdots \times n $?
• RQ2 How can the generating function for $ c(m_1,\dots,m_d) $ be expressed in closed form, and is it rational?
• RQ3 Does the diagonal sequence $ C_d(n) = c(n,\dots,n) $ satisfy a linear recurrence with polynomial coefficients?
• RQ4 What is the asymptotic growth rate of $ C_d(n) $, and can it be expressed in terms of $ d $?
• RQ5 Can the constant $ \alpha_d $ in the asymptotic formula be explicitly expressed in terms of $ d $ and $ \pi $?

Key findings

• The generating function for $ c(m_1,\dots,m_d) $ is rational, as shown by transforming the singular vector tuple count into a constant term problem and applying the MacMahon Master Theorem.
• For $ d=3 $, the sequence $ C_3(n) $ satisfies a fifth-order linear recurrence with polynomial coefficients, verified using symbolic computation.
• The asymptotic expansion for $ C_3(n) $ is $ \sim \frac{2}{\sqrt{3}\pi} 8^n n^{-1} (1 - \frac{13}{3}n^{-1} + \frac{1477}{27}n^{-2} - \cdots) $, with the dominant base 8 being sub-dominant compared to 9.
• Numerical evidence strongly supports the conjecture $ C_d(n) \sim \alpha_d \cdot ((d-1)^d)^n \cdot n^{-(d-1)/2} $ for $ d \geq 4 $, though the constant $ \alpha_d $ remains unproven.
• The first 160 terms of $ C_4(n) $ were computed, but a recurrence could not be found due to insufficient data, suggesting high complexity.
• A $100 prize is offered by Doron Zeilberger for a proof of the asymptotic conjecture, and an additional $25 for an explicit formula of $ \alpha_d $.