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A Statistical Analysis of Wasserstein Autoencoders for Intrinsically Low-dimensional Data

Saptarshi Chakraborty, Peter L. Bartlett

TL;DR

This work analyzes statistical guarantees for Wasserstein Autoencoders when data lie on intrinsically low-dimensional structures within high-dimensional spaces. By deriving an oracle-based excess-risk bound and carefully balancing misspecification and generalization errors, the authors show that convergence rates depend on the intrinsic Minkowski dimension $d_\mu$ rather than ambient dimensionality, with a rate of $\tilde{O}\left(n^{-\frac{1}{2+d_\mu}}\right)$ under Lipschitz smoothness. The results establish encoding/decoding guarantees and data-generation guarantees, and provide explicit network-size scaling with sample size and smoothness, for both Wasserstein-1 and MMD-based dissimilarities. The findings bridge theory and practice for WAEs, enabling reliable learning and generation on data with low-dimensional structure, and suggest pathways for extending the analysis to regularized networks and sharper intrinsic-dimension notions.

Abstract

Variational Autoencoders (VAEs) have gained significant popularity among researchers as a powerful tool for understanding unknown distributions based on limited samples. This popularity stems partly from their impressive performance and partly from their ability to provide meaningful feature representations in the latent space. Wasserstein Autoencoders (WAEs), a variant of VAEs, aim to not only improve model efficiency but also interpretability. However, there has been limited focus on analyzing their statistical guarantees. The matter is further complicated by the fact that the data distributions to which WAEs are applied - such as natural images - are often presumed to possess an underlying low-dimensional structure within a high-dimensional feature space, which current theory does not adequately account for, rendering known bounds inefficient. To bridge the gap between the theory and practice of WAEs, in this paper, we show that WAEs can learn the data distributions when the network architectures are properly chosen. We show that the convergence rates of the expected excess risk in the number of samples for WAEs are independent of the high feature dimension, instead relying only on the intrinsic dimension of the data distribution.

A Statistical Analysis of Wasserstein Autoencoders for Intrinsically Low-dimensional Data

TL;DR

This work analyzes statistical guarantees for Wasserstein Autoencoders when data lie on intrinsically low-dimensional structures within high-dimensional spaces. By deriving an oracle-based excess-risk bound and carefully balancing misspecification and generalization errors, the authors show that convergence rates depend on the intrinsic Minkowski dimension rather than ambient dimensionality, with a rate of under Lipschitz smoothness. The results establish encoding/decoding guarantees and data-generation guarantees, and provide explicit network-size scaling with sample size and smoothness, for both Wasserstein-1 and MMD-based dissimilarities. The findings bridge theory and practice for WAEs, enabling reliable learning and generation on data with low-dimensional structure, and suggest pathways for extending the analysis to regularized networks and sharper intrinsic-dimension notions.

Abstract

Variational Autoencoders (VAEs) have gained significant popularity among researchers as a powerful tool for understanding unknown distributions based on limited samples. This popularity stems partly from their impressive performance and partly from their ability to provide meaningful feature representations in the latent space. Wasserstein Autoencoders (WAEs), a variant of VAEs, aim to not only improve model efficiency but also interpretability. However, there has been limited focus on analyzing their statistical guarantees. The matter is further complicated by the fact that the data distributions to which WAEs are applied - such as natural images - are often presumed to possess an underlying low-dimensional structure within a high-dimensional feature space, which current theory does not adequately account for, rendering known bounds inefficient. To bridge the gap between the theory and practice of WAEs, in this paper, we show that WAEs can learn the data distributions when the network architectures are properly chosen. We show that the convergence rates of the expected excess risk in the number of samples for WAEs are independent of the high feature dimension, instead relying only on the intrinsic dimension of the data distribution.
Paper Structure (48 sections, 47 theorems, 93 equations, 3 figures)

This paper contains 48 sections, 47 theorems, 93 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A Proof of Concept
  3. Background
  4. Notations and some Preliminary Concepts
  5. Notation
  6. Wasserstein Autoencoders
  7. Intrinsic Dimension of Data Distribution
  8. Theoretical Analyses
  9. Assumptions and Error Decomposition
  10. Main Result
  11. Related work on GANs
  12. Proof Overview
  13. Implications of the Theoretical Results
  14. Encoding Guarantee
  15. Decoding Guarantee
  16. ...and 33 more sections

Key Result

Lemma 4

Let, $f \in \mathscr{H}^\gamma\left( \mathscr{A} , [0,1]^{d_2}, C\right)$, with $\mathscr{A} \subseteq [0,1]^{d_1}$. Then, $\overline{\text{dim}}_M \left(f\left( \mathscr{A} \right)\right) \le \overline{\text{dim}}_M( \mathscr{A})/(\gamma \wedge 1)$.

Figures (3)

  • Figure 1: Average generalization error (in terms of FID scores) and reconstruction test errors for different values of $n$ for GAN and MMD variants of WAE. The error bars denote the standard deviation out of $10$ replications.
  • Figure 2: Plot of $\xi_{a, b}(\cdot)$
  • Figure 3: A representation of the network $h(\cdot, \cdot)$. The magenta lines represent $d^\prime$ weights of value $1$. Similarly, cyan lines represent $d^\prime$ weights of value $-1$. Finally, the orange and teal lines represent $d^\prime$ weights (each) with values $+0.25$ and $-0.25$, respectively. The identity map takes $2 d^\prime \mathcal{L}(f)$ many weights (see remark 15 (iv) of JMLR:v21:20-002). The magenta, cyan, orange and teal connections take $6d^\prime$ many weights. All activations are taken to be ReLU, except the output of the yellow nodes, whose activation is $\sigma(x) = x^2$.

Theorems & Definitions (81)

  • Definition 1: Neural networks
  • Definition 2: Hölder functions
  • Definition 3: Maximum Mean Discrepancy (MMD)
  • Definition 4: Upper Minkowski dimension
  • Lemma 4
  • Proposition 4
  • Lemma 4: Oracle Inequality
  • Theorem 4
  • Remark 5: Number of Weights
  • Remark 6: Rates for Lipschitz models
  • ...and 71 more