# A Variational Manifold Embedding Framework for Nonlinear Dimensionality Reduction

arXiv: https://arxiv.org/abs/2511.22128 · HTML: https://mlsanity.com/paper/2511.22128/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 34 references that are in the map.
- First posted on arXiv 27 Nov 2025.

## Builds on (most cited first, 10 of 10)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Neural population geometry: An approach for understanding biological and artificial neural networks](https://mlsanity.com/paper/2104.07059/) (2021)
- [Noether's Learning Dynamics: Role of Symmetry Breaking in Neural Networks](https://mlsanity.com/paper/2105.02716/) (2021)
- [The Unreasonable Effectiveness of Gaussian Score Approximation for Diffusion Models and its Applications](https://mlsanity.com/paper/2412.09726/) (2024)
- [Generalization through variance: how noise shapes inductive biases in diffusion models](https://mlsanity.com/paper/2504.12532/) (2025)
- [Capturing the dynamical repertoire of single neurons with generalized linear models](https://mlsanity.com/paper/1602.07389/) (2016)
- [Gradient Descent as Loss Landscape Navigation: a Normative Framework for Deriving Learning Rules](https://mlsanity.com/paper/2510.26997/) (2025)
- [Dynamical symmetries in the fluctuation-driven regime: an application of Noether's theorem to noisy dynamical systems](https://mlsanity.com/paper/2504.09761/) (2025)
- [Optimal packing of attractor states in neural representations](https://mlsanity.com/paper/2504.12429/) (2025)

## Abstract (arXiv)

Dimensionality reduction algorithms like principal component analysis (PCA) are workhorses of machine learning and neuroscience, but each has well-known limitations. Variants of PCA are simple and interpretable, but not flexible enough to capture nonlinear data manifold structure. More flexible approaches have other problems: autoencoders are generally difficult to interpret, and graph-embedding-based methods can produce pathological distortions in manifold geometry. Motivated by these shortcomings, we propose a variational framework that casts dimensionality reduction algorithms as solutions to an optimal manifold embedding problem. By construction, this framework permits nonlinear embeddings, allowing its solutions to be more flexible than PCA. Moreover, the variational nature of the framework has useful consequences for interpretability: each solution satisfies a set of partial differential equations, and can be shown to reflect symmetries of the embedding objective. We discuss these features in detail and show that solutions can be analytically characterized in some cases. Interestingly, one special case exactly recovers PCA.