Deterministic sampling of autoencoder latent spaces via data-driven piecewise-linear expanding maps

Abstract

We propose a fully deterministic framework for generating new samples directly within learned latent spaces using a bank of data-driven piecewise-linear expanding maps constructed from empirical conditional density estimates. For each latent coordinate and conditioning context, an Inverse Frobenius–Perron formulation is used to synthesize a one-dimensional map whose prescribed invariant probability density locally approximates the corresponding empirical conditional distribution. New latent samples are then produced by a state-dependent deterministic switching rule that selects and iterates these local maps without requiring stochastic noise, learned vector fields, or additional neural training. The method can operate as a standalone latent sampler or as a complementary component within existing generative pipelines, providing a direct connection between empirical density estimation and deterministic map synthesis. Experiments on a synthetic Rössler dataset and on an end-to-end MNIST autoencoder show low latent-space distributional discrepancy and decoded-sample quality comparable to a KDE baseline in the reported configurations. The verified expansion properties apply to the constituent one-dimensional maps; no claim of a global invariant measure or global chaos is made for the complete high-dimensional switching system.