The factor analysis model is a statistical model where a certain number of hidden random variables, called factors, affect linearly the behaviour of another set of observed random variables, with additional random noise. The main assumption of the model is that the factors and the noise are Gaussian random variables. This implies that the feasible set lies in the cone of positive semidefinite matrices. In this paper, we do not assume that the factors and the noise are Gaussian, hence the higher order moment and cumulant tensors of the observed variables are generally nonzero. This motivates the notion of kth-order factor analysis model, that is the family of all random vectors in a factor analysis model where the factors and the noise have finite and possibly nonzero moment and cumulant tensors up to order k. This subset may be described as the image of a polynomial map onto a Cartesian product of symmetric tensor spaces. Our goal is to compute its dimension and we provide conditions under which the image has positive codimension.

翻译：因子分析模型是一种统计模型，其中一定数量的隐藏随机变量（称为因子）以线性方式影响另一组观测随机变量的行为，并伴有额外的随机噪声。该模型的主要假设是因子和噪声均为高斯随机变量。这意味着可行域位于半正定矩阵的锥内。在本文中，我们不再假设因子和噪声服从高斯分布，因此观测变量的高阶矩和张量累积量通常非零。这引出了k阶因子分析模型的概念，即因子分析模型中所有随机向量的集合，其中因子和噪声具有有限且可能非零的k阶矩和张量累积量。该子集可描述为从多项式映射到对称张量空间笛卡尔积的像。我们的目标是计算该像的维数，并提供该像具有正余维数的条件。