Gradient and Hessian of functions with non-independent variables

Matieyendou Lamboni

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arXiv · 2304.05835

Gradient and Hessian of functions with non-independent variables

Matieyendou Lamboni

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Abstract

Mathematical models are sometime given as functions of independent input variables and equations or inequations connecting the input variables. A probabilistic characterization of such models results in treating them as functions with non-independent variables. Using the distribution function or copula of such variables that comply with such equations or inequations, we derive two types of partial derivatives of functions with non-independent variables (i.e., actual and dependent derivatives) and argue in favor of the latter. The dependent partial derivatives of functions with non-independent variables rely on the dependent Jacobian matrix of dependent variables, which is also used to define a tensor metric. The differential geometric framework allows for deriving the gradient, Hessian and Taylor-type expansion of functions with non-independent variables.

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BibTeX RIS

Matieyendou Lamboni. 2023-04-10. Gradient and Hessian of functions with non-independent variables. https://arxiv.org/abs/2304.05835

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