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A Variational Manifold Embedding Framework for Nonlinear Dimensionality Reduction

John J. Vastola, Samuel J. Gershman, Kanaka Rajan

TL;DR

This work introduces a variational manifold-embedding framework for nonlinear dimensionality reduction that generalizes PCA while maintaining interpretability. By casting embeddings as solutions to an optimal embedding problem, it yields a KL-divergence-based objective with a geometric log-determinant term, connecting to score vectors and diffusion-model dynamics. The approach provides PDE-based characterizations of optimal embeddings, reveals conservation laws via Noether’s theorem, and shows PCA emerges as a special case under Gaussian priors and likelihoods. These insights offer a physics-inspired lens on inductive biases in nonlinear dimensionality reduction and highlight how symmetries constrain embedding geometry. Limitations include applicability to continuous data distributions and questions about scalability to high-dimensional datasets.

Abstract

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.

A Variational Manifold Embedding Framework for Nonlinear Dimensionality Reduction

TL;DR

This work introduces a variational manifold-embedding framework for nonlinear dimensionality reduction that generalizes PCA while maintaining interpretability. By casting embeddings as solutions to an optimal embedding problem, it yields a KL-divergence-based objective with a geometric log-determinant term, connecting to score vectors and diffusion-model dynamics. The approach provides PDE-based characterizations of optimal embeddings, reveals conservation laws via Noether’s theorem, and shows PCA emerges as a special case under Gaussian priors and likelihoods. These insights offer a physics-inspired lens on inductive biases in nonlinear dimensionality reduction and highlight how symmetries constrain embedding geometry. Limitations include applicability to continuous data distributions and questions about scalability to high-dimensional datasets.

Abstract

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.

Paper Structure

This paper contains 45 sections, 1 theorem, 99 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A tractable variational framework for dimensionality reduction
  3. When are two data manifolds the same?
  4. A variational objective for dimensionality reduction.
  5. A variational objective for dimensionality reduction.
  6. Examples of optimal one-dimensional embeddings.
  7. Interpreting optimal one-dimensional embeddings
  8. Optimal one-dimensional embeddings follow score vectors
  9. Connection to diffusion models
  10. Useful properties of the variational objective
  11. Flexibility-related properties
  12. Objective is geometry-sensitive but not coordinate-sensitive.
  13. Objective has expected trivial solutions.
  14. Interpretability-related properties
  15. Optimal embeddings satisfy EL equations.
  16. ...and 30 more sections

Key Result

theorem 1

Consider a transformation from coordinates $\vec{z}$ and fields $\vec{\phi}$ to where $\epsilon > 0$ is infinitesimally small. Note that the perturbations $\delta \vec{z}$ and $\delta \vec{\phi}$ are allowed to depend on $\vec{z}$ and $\vec{\phi}$. If this change of coordinates and fields changes the Lagrangian density by a total derivative, i.e., for some function $\vec{K}$, then we call such a

Figures (1)

  • Figure 1: a. The function $\vec{\phi}$ maps the latent space to the (higher-dimensional) ambient space. b. Optimal one-dimensional embeddings when $p_{\text{data}}$ is a mixture of Gaussians centered at points along a line (left), circle (middle), or sinusoid (right).

Theorems & Definitions (1)

  • theorem 1: Noether's theorem