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planar flows

Planar flows are a class of invertible transformations used in normalizing flows and machine learning to map simple probability distributions into more complex, expressive distributions. In this framework, a planar flow modifies an input vector by adding a scaled non-linear activation of an affine projection, effectively expanding or contracting the distribution perpendicular to a hyperplane. Because of its particular algebraic structure, the determinant of the transformation's Jacobian can be computed in linear time using the matrix determinant lemma, enabling exact and computationally efficient probability density evaluation. By chaining multiple planar transformations together in sequence, generative models can warp elementary distributions, such as standard Gaussians, into multimodal and non-Gaussian densities while maintaining tractable sampling and inference.

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