Counting Tensor Rank Decompositions

Abstract

Tensor rank decomposition is a useful tool for geometric interpretation of the tensors in the canonical tensor model (CTM) of quantum gravity. In order to understand the stability of this interpretation, it is important to be able to estimate how many tensor rank decompositions can approximate a given tensor. More precisely, finding an approximate symmetric tensor rank decomposition of a symmetric tensor Q with an error allowance Δ is to find vectors ϕi satisfying ∥Q−∑i=1Rϕi⊗ϕi⋯⊗ϕi∥2≤Δ. The volume of all such possible ϕi is an interesting quantity which measures the amount of possible decompositions for a tensor Q within an allowance. While it would be difficult to evaluate this quantity for each Q, we find an explicit formula for a similar quantity by integrating over all Q of unit norm. The expression as a function of Δ is given by the product of a hypergeometric function and a power function. By combining new numerical analysis and previous results, we conjecture a formula for the critical rank, yielding an estimate for the spacetime degrees of freedom of the CTM. We also extend the formula to generic decompositions of non-symmetric tensors in order to make our results more broadly applicable. Interestingly, the derivation depends on the existence (convergence) of the partition function of a matrix model which previously appeared in the context of the CTM.

Emergence of Lie group symmetric classical spacetimes in canonical tensor model

“…Therefore it is generally a non-trivial matter whether the integral (6) converges or not. The convergence for R N 2 /2 was first noticed in [33], and was systematically analyzed in [34]. Since our value of R in (5) is roughly smaller by a factor of 2, the current analysis does not suffer from the divergence, which indeed was checked in our actual Monte Carlo simulations.…”

“…This implies that, if R is sufficiently large, there will always be such a solution to the equation (34).…”

Tensors and Algebras: An algebraic spacetime interpretation for tensor models

“…This was done using the unperturbed tensor rank decomposition P abc = 7 i=1 β i p i a p i b p i c with 7 points as discussed above. The first point p 1 , which should be noted is arbitrarily picked, was taken and shifted with a vector q p 1 a → p 1 a + q a , Figure 8: A plot of the functions f 2 (x) of the circle with a shifted point as described in (49), where the function values are defined as in (26). On the left using = 0.1, ∆ = 8.1 • 10 −31 , in the middle = 0.25, ∆ = 5.6 • 10 −30 and on the right = 0.75, ∆ = 2.3 • 10 −28 .…”

“…Note that we assume a tensor rank decomposition to be any decomposition of P abc such that (29) is satisfied as in[49]. Oftentimes it is defined instead to be the decomposition for the lowest R possible, which we call a minimal tensor rank decomposition.…”

Splitting-merging transitions in tensor-vectors systems in exact large-$N$ limits

“…Though the exponent in ( 1) is semi-definite in our case, it is a non-trivial question whether (1) is finite or not, because the exponent contains flat directions, such as φ i a = −φ j a , which extends to infinity. This question about the finiteness was systematically studied mainly by numerical methods in [23] 7 , and it was checked/conjectured that the system is finite only for R (N + 1)(N + 2)/2. In this paper, we are free from this instability, because we consider large-N limits with finite R. This paper is organized as follows.…”