The framework of normalizing flows provides a general strategy for flexible vari-ational inference of posteriors over latent variables.

For time-series modeling we employ a Gaussian process with a nonlinear exogenous input structure.

As mentioned in section 3.1, we can use the matrix determinant lemma to efficiently compute the Proof: For k = 1,...,K 1, we have: P(H = k) = null As mentioned in the main manuscript, our V AE experiments closely follow Maddison et al. RELAX builds upon equation 16 to develop an estimator with reduced variance. Third, it should also be noted that the bias and variance of the gradient estimator of RELAX are central points of discussion by Grathwohl et al. We show results of running IGR and GS with and without RELAX in Table 2. Discrete Models MNIST IGR-I -94.18 GS -103.80 IGR-I + RELAX -81.95 GS + RELAX -83.41 Table 2: Test log-likelihood on MNIST for nonlinear architecture. Results are in Table 3 and we can see that again, IGR performs best.

Estimation of parameters in differential equation models can be achieved by applying learning algorithms to quantitative time-series data.

For time-series modeling we employ a Gaussian process with a nonlinear exogenous input structure.

Normalizing flows are deep generative models that enable efficient likelihood estimation and sampling through invertible transformations. A key challenge is to design linear layers that enhance expressiveness while maintaining efficient computation of the Jacobian determinant and inverse. We introduce a novel invertible linear layer based on the product of circulant and diagonal matrices. This decomposition reduces parameter complexity from $\mathcal{O}(n^2)$ to $\mathcal{O}(mn)$ using $m$ diagonal matrices and $m-1$ circulant matrices while still approximating general linear transformations. By leveraging the Fast Fourier Transform, our approach reduces the time complexity of matrix inversion from $\mathcal{O}(n^3)$ to $\mathcal{O}(mn\log n)$ and that of computing the log-determinant from $\mathcal{O}(n^3)$ to $\mathcal{O}(mn)$, where $n$ is the input dimension. We build upon this layer to develop Circulant-Diagonal Flow (CDFlow), which achieves strong density estimation on natural image datasets and effectively models data with inherent periodic structure. Furthermore, CDFlow significantly accelerates key operations in normalizing flows, providing practical benefits for scalable generative modeling.

In other words, this batch dependence is ignored in square flows, both in the log determinant computation, and when back-propagating through it.