Multiway data analysis aims to uncover patterns in data structured as multi-indexed arrays, with multiway covariance playing a crucial role in many applications. However, the high dimensionality of multiway covariance presents significant computational challenges. To overcome these challenges, factorized covariance models have been proposed that rely on a separability assumption: the multiway covariance can be accurately expressed as a sum of Kronecker products of mode-wise covariances. This paper addresses the representability, certification, and approximation of such separable models, leaving statistical estimation or finite-sample properties aside. We reduce the question of whether a given covariance can be decomposed into a separable multiway form to an equivalent question about the separability of quantum states. Leveraging results from quantum information theory, we show that generic multiway covariances are typically \emph{not} separable and that determining the best separable approximation is NP-hard. These findings suggest that factorized covariance models can be overly restrictive and difficult to fit without additional structural assumptions. Nevertheless, our numerical experiments indicate that standard iterative algorithms, namely Frank-Wolfe and gradient descent, often converge close to the best separable approximation. As NP-hardness concerns worst-case computational complexity, Kronecker-separable approximations to multiway covariance could still be tractable to apply for analyzing many real-world datasets.

翻译：多通道数据分析旨在从多索引数组结构中揭示数据模式，其中多通道协方差在许多应用中发挥着关键作用。然而，多通道协方差的高维特性带来了显著的计算挑战。为克服这些挑战，研究者提出了基于可分离性假设的因式分解协方差模型：多通道协方差可以精确表示为各模态协方差克罗内克积之和。本文聚焦于此类可分离模型的可表示性、验证与逼近问题，暂不涉及统计估计或有限样本性质。我们将给定协方差是否可分解为可分离多通道形式的问题，等价转化为量子态可分离性问题。借助量子信息理论的结果，我们证明一般多通道协方差通常不可分离，且确定最佳可分离逼近是NP难问题。这些发现表明，若无额外结构假设，因式分解协方差模型可能过于严格且难以拟合。尽管如此，我们的数值实验表明，标准迭代算法（即弗兰克-沃尔夫算法与梯度下降法）通常能收敛至接近最佳可分离逼近解。由于NP难性针对的是最坏情况计算复杂度，针对多通道协方差的克罗内克可分离逼近仍可应用于分析许多真实世界数据集。