On the permanent of random tensors
Malihe Nobakht Kooshkghazi, Hamidreza Afshin
TL;DR
This work addresses approximating the permanent of random order-$d$ complex tensors when the mean magnitude $|\\mu|$ is at least $1/\\text{polylog}(n)$. It extends Barvinok-style additive approximation techniques to tensors by expressing $\\frac{per(X)}{(n!)^{d-1}}$ as a polynomial in $z$ with coefficients $a_k$, and approximating the first $t+1$ coefficients via computable statistics, connecting to a truncated exponential generating function. The core contributions are a deterministic quasi-polynomial time algorithm and a PTAS for such random tensors with $d=O(\\text{polylog}(n))$ and $|\\mu| \ge (\\frac{\\ln n}{(d-1)^6})^{-c}$, leveraging precise tail-control lemmas, moment bounds, and Hermite-polynomial techniques to show $\\sum_{k\\ge0}^{\\infty} V'_k z^k = e^{V_1 z - \\frac{\\xi z^2}{2}}$. These results generalize matrix permanents with vanishing mean to the tensor setting and provide practical deterministic algorithms with quasi-polynomial runtime for large $n$ and moderate $d$, including a PTAS.
Abstract
The exact computation of permanent for high-dimensional tensors is a hard problem. Having in mind the applications of permanents in other fields, providing an algorithm for the approximation of tensor permanents is an attractive subject. In this paper, we design a deterministic quasi-polynomial time algorithm and a PTAS that computes the permanent of complex random tensors that its module of the mean is at least 1/polylog(n).