Proof. Let W = Q and let fAR(x) = R̃h(Rx+ b). Indeed, any orthogonal matrix is invertible, so Q can be modelled by W. Also, note that R and R̃ are upper triangular and h is an elementwise function. The matrix product of the Jacobians is triangular, and thus R̃h(Rx+ b) has a triangular Jacobian and is therefore autoregressive. Hence, it can be modelled by fAR. Further, note that [10] bound RiiR̃ii > −1 ||h′||∞ , which ensures that the constraint ∂fAR(x)i ∂xi > −1 is satisfied. Hence, z = x +Q R̃h(RQx + b) can be written as Equation 1 when writing fAR(x) = R̃h(Rx + b) and M = Q without violating any constraints on M and fAR, and is therefore a special case. Remark 1: The increased expressitivity originates from fAR and not from W. To see why, suppose we replace Q and Q in the original formulation by W and W−1. Consider that any real square matrix W may be decomposed as QWRW. Hence, compositions W−1R̃ and RW can be written as QWR̃ ′ and RQW, where R̃′ = R−1 W R̃ and R ′ = RRW which are both still upper triangular. Hence, we have shown that even if the orthogonal matrix Q is replaced by an invertible matrix W, the transformation can still be written in terms of a shared orthogonal matrix QW and upper triangular matrices R̃′ and R′. Therefore, the source of the increased expressitivity is not the replacement of Q by W.
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