autoregressive flows

**Autoregressive Flows** are a class of normalizing flow models that construct invertible transformations using autoregressive structure, where each output dimension depends only on the previous dimensions through a triangular Jacobian matrix. This autoregressive constraint enables exact and efficient computation of both the forward transformation and its log-determinant Jacobian, making density evaluation and sampling tractable while maintaining the expressiveness to model complex distributions. **Why Autoregressive Flows Matter in AI/ML:** Autoregressive flows provide **exact density evaluation with flexible, learnable transformations**, enabling precise likelihood computation for generative modeling, variational inference, and density estimation tasks where approximate methods are insufficient. • **Triangular Jacobian** — The autoregressive structure produces a lower-triangular Jacobian matrix whose determinant is simply the product of diagonal elements: log|det J| = Σ log|∂y_i/∂x_i|; this O(d) computation replaces the general O(d³) determinant, making flows practical for high dimensions • **Masked Autoregressive Flow (MAF)** — Each layer transforms x_i → y_i = x_i · exp(s_i(x_{