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Learning Ordered Representations in Latent Space for Intrinsic Dimension Estimation via Principal Component Autoencoder

Qipeng Zhan, Zhuoping Zhou, Zexuan Wang, Li Shen

TL;DR

The paper tackles the challenge of preserving PCA-like interpretability in nonlinear dimensionality reduction by introducing PCAE, a nonlinear autoencoder that jointly optimizes reconstruction with a variance-ordering penalty and a soft isometry constraint. The key idea is to weight latent variances with strictly increasing coefficients and enforce distance-preserving behavior in latent space, enabling post-hoc intrinsic-dimension estimation by cumulative variance. Theoretical results show the weighted-variance objective recovers PCA components in the linear case, while the isometry constraint extends this ordering to nonlinear manifolds; empirical results on synthetic and real data demonstrate exact dimension recovery when known, alignment with MLE estimates when unknown, smoother interpolations, improved FID scores, and fast training relative to sequential or hierarchical baselines. Overall, PCAE yields interpretable, variance-ordered latent representations suitable for nonlinear DR, intrinsic-dimension estimation, and downstream tasks at scale.

Abstract

Autoencoders have long been considered a nonlinear extension of Principal Component Analysis (PCA). Prior studies have demonstrated that linear autoencoders (LAEs) can recover the ordered, axis-aligned principal components of PCA by incorporating non-uniform $\ell_2$ regularization or by adjusting the loss function. However, these approaches become insufficient in the nonlinear setting, as the remaining variance cannot be properly captured independently of the nonlinear mapping. In this work, we propose a novel autoencoder framework that integrates non-uniform variance regularization with an isometric constraint. This design serves as a natural generalization of PCA, enabling the model to preserve key advantages, such as ordered representations and variance retention, while remaining effective for nonlinear dimensionality reduction tasks.

Learning Ordered Representations in Latent Space for Intrinsic Dimension Estimation via Principal Component Autoencoder

TL;DR

The paper tackles the challenge of preserving PCA-like interpretability in nonlinear dimensionality reduction by introducing PCAE, a nonlinear autoencoder that jointly optimizes reconstruction with a variance-ordering penalty and a soft isometry constraint. The key idea is to weight latent variances with strictly increasing coefficients and enforce distance-preserving behavior in latent space, enabling post-hoc intrinsic-dimension estimation by cumulative variance. Theoretical results show the weighted-variance objective recovers PCA components in the linear case, while the isometry constraint extends this ordering to nonlinear manifolds; empirical results on synthetic and real data demonstrate exact dimension recovery when known, alignment with MLE estimates when unknown, smoother interpolations, improved FID scores, and fast training relative to sequential or hierarchical baselines. Overall, PCAE yields interpretable, variance-ordered latent representations suitable for nonlinear DR, intrinsic-dimension estimation, and downstream tasks at scale.

Abstract

Autoencoders have long been considered a nonlinear extension of Principal Component Analysis (PCA). Prior studies have demonstrated that linear autoencoders (LAEs) can recover the ordered, axis-aligned principal components of PCA by incorporating non-uniform regularization or by adjusting the loss function. However, these approaches become insufficient in the nonlinear setting, as the remaining variance cannot be properly captured independently of the nonlinear mapping. In this work, we propose a novel autoencoder framework that integrates non-uniform variance regularization with an isometric constraint. This design serves as a natural generalization of PCA, enabling the model to preserve key advantages, such as ordered representations and variance retention, while remaining effective for nonlinear dimensionality reduction tasks.
Paper Structure (41 sections, 4 theorems, 34 equations, 2 figures, 11 tables)

This paper contains 41 sections, 4 theorems, 34 equations, 2 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminary
  4. Principal Component Analysis
  5. Sequential optimization
  6. Single optimization
  7. Linear Autoencoder as PCA
  8. Isometric Mapping
  9. Methodology
  10. Failure of PCA-AE and Hierarchical Autoencoder
  11. PCA-AE
  12. Hierarchical Autoencoder
  13. Principal Component Autoencoder
  14. The choice of $\ell$ and $\gamma$.
  15. Determining the Intrinsic Dimension
  16. ...and 26 more sections

Key Result

Theorem 1

Let $\bm{\Sigma}$ be the covariance matrix of $\mathbf{X}$ with eigenvalues $\lambda_1 \ge \cdots \ge \lambda_p \ge 0$, and let $\bm{\Gamma}=\mathrm{diag}(\gamma_1,\ldots,\gamma_p)$ be diagonal with $0 \le \gamma_1 < \cdots < \gamma_p$. Then the minimum of is $\sum_{i=1}^p \lambda_i \gamma_i$. Moreover, $\mathbf{U}_*$ is optimal if and only if its $i^{\text{th}}$ column is a unit eigenvector of $

Figures (2)

  • Figure 1: Estimated variances of latent coordinates for dSprites and 3DShapes under different bottleneck sizes. In both cases, the recovered intrinsic dimension remains fixed at the ground-truth value, demonstrating that PCAE is robust to the choice of bottleneck dimension.
  • Figure 2: Linear interpolation between two randomly generated samples.

Theorems & Definitions (11)

  • Definition 1: Isometry
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Lemma 1: Von Neumann
  • proof
  • Lemma 2
  • proof
  • proof
  • ...and 1 more