AInjectiveChange-of-VariableFormulaandStacking InjectiveFlows Wefirstderive(5)from(3). Bythechainrule,wehave: J[gφ ] g

We summarize our methods for computing/estimating the gradient of the log determinant arising inmaximum likelihood training ofrectangular flows. Algorithm 2showstheexactmethod, where jvp(f,z,)denotes computingJ[f](z) usingforward-mode AD,and i Rd isthei-thstandard basis vector, i.e. a one-hot vector with a1 on its i-th coordinate. Note that / θlogdetAθ is computed using backpropagation. Thefor loop is easily parallelized in practice.