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Abstract
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The factor analysis model is a statistical model where a certain number of hidden
random variables, called factors, affect linearly the behavior 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 a
-th
order factor analysis model, which 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
.
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.
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Keywords
factor analysis model, higher-order cumulants, symmetric
tensors
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Mathematical Subject Classification
Primary: 62R01
Secondary: 62H22, 62H25
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Milestones
Received: 30 September 2022
Revised: 11 May 2023
Accepted: 20 June 2023
Published: 28 November 2023
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Publishers). |
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