Deep neural networks have significantly advanced machine learning in recent decades.

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.

Withtheseconditions,adirectconsequenceisthat the singular values inon-manifold directions will not depend onσ2. Hence, if we fix the latent distribution to be standard Gaussian, wehavethat theNFused tolearnqσ2 must be f forall(u,v),i.e. However, these eigenvalues are exactly in direction of large variability, i.e. in on-manifolddirection. Thiswastobeshown. Let us assume thatσ21 = = σ2d in the following. B.1 Lolipop In [11], a manifold consisting of regions of different ID was considered - a 1 dimensional line segment, and atwodimensional disk such that theoverall manfiold resembles alolipop.

Usingourmethod weestablish anewbenchmark bycalculating the most accurate variational ground state energies ever published for a number of different atoms and molecules. Wesystematically break down and measure our improvements, focusing in particular on the effect of increasing physical prior knowledge.