Manifold Decoders: A Framework for Generative Modeling from Nonlinear Embeddings

Classical nonlinear dimensionality reduction (NLDR) techniques like t-SNE, Isomap, and LLE excel at creating low-dimensional embeddings for data visualization but fundamentally lack the ability to map these embeddings back to the original high-dimensional space. This one-way transformation limits their use in generative applications. This paper addresses this critical gap by introducing a system- atic framework for constructing neural decoder architectures for prominent NLDR methods, enabling bidirectional mapping for the first time. We extend this framework by implementing a diffusion-based generative process that operates directly within these learned manifold spaces. Through experiments on the CelebA dataset, we evaluate the reconstruction and generative performance of our approach against autoencoder and standard diffusion model baselines. Our findings reveal a fundamental trade- off: while the decoders successfully reconstruct data, their quality is surpassed by end-to-end optimized autoencoders. Moreover, manifold-constrained diffusion yields poor-quality samples, suggesting that the discrete and sparse nature of classical NLDR embeddings is ill-suited for the continuous inter- polation required by generative models. This work highlights the inherent challenges in retrofitting generative capabilities onto NLDR methods designed primarily for visualization and analysis.