The tensor rank decomposition is a useful tool for the 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 $\Delta$ is to find vectors $\phi^i$ satisfying $\|Q-\sum_{i=1}^R \phi^i\otimes \phi^i\cdots \otimes \phi^i\|^2 \leq \Delta$. The volume of all possible such $\phi^i$ 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 $\Delta$ is given by the product of a hypergeometric function and a power function. We also extend the formula to generic decompositions of non-symmetric tensors. The derivation depends on the existence (convergence) of the partition function of a matrix model which appeared in the context of the CTM.

翻译：高压级分解是量重度的卡纳高压模型(CTM)中对高压体进行几何判读的有用工具。为了理解这种判读的稳定性,必须能够估计多少高压分解能接近给定的加压。更确切地说,找到一个对称高Q的近对称高压分解法的近似对称高调分解法和差错差差差差差差差差差差差差差值值,就是找到一个在量上达到$@-sum_i=1 ⁇ R\phi_i\otime=1 ⁇ R\phi\cdots\otimes\ otimes\ctime\ time\ cotime \ cotime \cleq\ Delta$。所有可能的美元等差错差差差差差值数量都用来测量在模型值内为 10Q$的可能分差差差差差差值的数值。虽然很难评估每Q$的数量,但我们发现一个类似数量的明确公式公式公式,通过将所有美元的基价内值整合值整合值整合值并入C-xxxxxxxxxxx值的公式, 的公式的表达法函数也显示。