Cracking the Density Code: Why MAF Flows Where KDE Stalls

Kernel Density Estimation (KDE) is an effective method for one-dimensional density estimation, but its performance deteriorates significantly in higher dimensions due to the curse of dimensionality. Normalizing flows, specifically Masked Autoregressive Flows (MAF), provide a more robust solution for high-dimensional data by transforming a simple base distribution into a complex target distribution. This is achieved through a sequence of differentiable and invertible transformations, known as diffeomorphisms. The overall transformation is constructed by composing simpler functions, such as affine, elementwise, and coupling flows, which allows for modeling complex data structures while maintaining a tractable Jacobian determinant for computation.