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

2026-01-27

AI Summary

This work addresses the challenge that nonlinear autoencoders often fail to learn ordered latent representations with interpretable variance, leading to inaccurate intrinsic dimensionality estimation. To overcome this limitation, the authors propose a novel autoencoder framework that naturally extends the ordered, variance-preserving properties of principal component analysis (PCA) to nonlinear settings by incorporating non-uniform ℓ² regularization and an isometry constraint. This approach jointly optimizes the structure of the latent space and the distribution of variance across its dimensions, thereby preserving the model’s capacity for nonlinear dimensionality reduction while yielding an ordered latent representation. As a result, the method significantly improves the accuracy of intrinsic dimensionality estimation compared to conventional nonlinear autoencoders.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Search and Optimization: Non-convex Optimization
• Computer Vision: Representation Learning for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

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.

Problem

• intrinsic dimension estimation
• ordered representations
• nonlinear autoencoder
• principal component analysis
• variance retention

Innovation

• Principal Component Autoencoder
• nonlinear dimensionality reduction
• ordered latent representations
• variance regularization
• isometric constraint