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For learning the parameters of a manifold flow transformation, we need access to the differential volume ratio, <math>R_f</math>, or at least to its gradient w.r.t. the parameters. Moreover, for some inference tasks, we need access to <math>R_f</math> itself. Practical solutions include:
*Sorrenson et al.(2023)<ref name=manifold_flow/> give a solution for computationally efficient stochastic parameter gradient approximation for <math>\log R_f</math>.
*For some hand-designed flow transforms, <math>R_f</math> can be analytically derived in simple closed form, for example the above-mentioned '''simplex calibration transform''. Futher examples are given below in the section on simple spherical flows.
*On a software platform equipped with [[linear algebra]] and [[automatic differentiation]], <math>R_f(\mathbf x) = \left|\operatorname{det}(\mathbf{T_y}'\mathbf{F_xT_x})\right|</math> can be automatically evaluated, given access to only <math>\mathbf x, f, \pi</math>.<ref>With [[PyTorch]]:
<pre>
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