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Robustness of Nonlinear Representation Learning

Simon Buchholz, Bernhard Schölkopf

TL;DR

The paper tackles the robustness of unsupervised nonlinear representation learning when model assumptions are mildly violated, focusing on mixing functions that are close to local isometries. It develops a framework using a local-isometry distance $\Theta_p(f,\Omega)$ and a rigidity-based decomposition to show approximate identifiability of latent factors, first up to a linear transform and then for perturbed ICA in the presence of a small nonlinear component. It proves that, under near-isometric mixing, latent variables can be recovered with high fidelity (measured by MCC), and that for a perturbed linear ICA model $X=AS+\eta h(S)$, the linear part and the sources can be recovered approximately as $\eta\to 0$. These results collectively argue for approximate identifiability and robustness of nonlinear ICA and representation learning under realistic misspecifications, offering theoretical grounding for learning signals in real data that deviate only slightly from idealized models.

Abstract

We study the problem of unsupervised representation learning in slightly misspecified settings, and thus formalize the study of robustness of nonlinear representation learning. We focus on the case where the mixing is close to a local isometry in a suitable distance and show based on existing rigidity results that the mixing can be identified up to linear transformations and small errors. In a second step, we investigate Independent Component Analysis (ICA) with observations generated according to $x=f(s)=As+h(s)$ where $A$ is an invertible mixing matrix and $h$ a small perturbation. We show that we can approximately recover the matrix $A$ and the independent components. Together, these two results show approximate identifiability of nonlinear ICA with almost isometric mixing functions. Those results are a step towards identifiability results for unsupervised representation learning for real-world data that do not follow restrictive model classes.

Robustness of Nonlinear Representation Learning

TL;DR

The paper tackles the robustness of unsupervised nonlinear representation learning when model assumptions are mildly violated, focusing on mixing functions that are close to local isometries. It develops a framework using a local-isometry distance and a rigidity-based decomposition to show approximate identifiability of latent factors, first up to a linear transform and then for perturbed ICA in the presence of a small nonlinear component. It proves that, under near-isometric mixing, latent variables can be recovered with high fidelity (measured by MCC), and that for a perturbed linear ICA model , the linear part and the sources can be recovered approximately as . These results collectively argue for approximate identifiability and robustness of nonlinear ICA and representation learning under realistic misspecifications, offering theoretical grounding for learning signals in real data that deviate only slightly from idealized models.

Abstract

We study the problem of unsupervised representation learning in slightly misspecified settings, and thus formalize the study of robustness of nonlinear representation learning. We focus on the case where the mixing is close to a local isometry in a suitable distance and show based on existing rigidity results that the mixing can be identified up to linear transformations and small errors. In a second step, we investigate Independent Component Analysis (ICA) with observations generated according to where is an invertible mixing matrix and a small perturbation. We show that we can approximately recover the matrix and the independent components. Together, these two results show approximate identifiability of nonlinear ICA with almost isometric mixing functions. Those results are a step towards identifiability results for unsupervised representation learning for real-world data that do not follow restrictive model classes.

Paper Structure

This paper contains 22 sections, 25 theorems, 200 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Setting
  3. Robustness and approximate Identifiabilility
  4. Approximate linear Identifiability for approximate local Isometries
  5. Perturbed linear ICA
  6. Conclusion
  7. Notation and Mathematical Definitions
  8. Linear Algebra
  9. Probability Theory
  10. Sobolev spaces
  11. Differential Geometry
  12. Bounding $\mathrm{MCC}$ for almost Permutations
  13. Auxiliary result on distances to the orthogonal group
  14. Extension to the undercomplete case
  15. Proof and extension of approximate linear identifiability for approximate isometries
  16. ...and 7 more sections

Key Result

Theorem 2.3

We assume that $\mathcal{P}$ satisfies Assumption as:P2. Suppose $X=f(S)$ where $S\sim\mathbb{P}\in \mathcal{P}$ and $f\in \mathcal{F}_\mathrm{iso}$. If $X\stackrel{\mathcal{D}}{=} \tilde{f}(\tilde{S})$ for some $\tilde{f}\in \mathcal{F}_\mathrm{iso}$ and $\tilde{S}\sim\tilde{\mathbb{P}}\in\mathcal{

Figures (2)

  • Figure 1: (Left) Color map of Gaussian latent variable $Z$, (Center) Color map of the transformed data $X=f(Z)$ where $f$ is a piecewise linear approximation of a radius dependent rotation (i.e., $f(Z)\overset{\mathcal{D}}{\approx} Z$), (Right) Representation $X'=f'(Z)$ learned by a VAE with ReLU activation functions initialized with $f$ and $f^{-1}$ for decoder and encoder and small variance.
  • Figure 2: (Left) Distance of recovered linear unmixing $w$ from $\bar{w}$ (see \ref{['eq:def_barw']}) and $\tilde{w}$ (see \ref{['eq:tilde_wd']}) as a function of $\eta$. Plotted is the median over $5$ runs with $d=5$ (i.e., 25 recovered components). The shaded area shows the range from the $32\%$ to the $68\%$ quantile. (Right) Difference of the Mean $\mathrm{MCC}$ over 5 runs of $\hat{S}=W\Sigma^{-\frac{1}{2}}_XX$ from perfect recovery ($\mathrm{MCC}=1$). Regression lines are obtained by linear regression in log-log space for $0.01\leq \eta\leq 0.1$ and $\beta$ values indicate their slope.

Theorems & Definitions (53)

  • Theorem 2.3: Theorem 1, horan2021when
  • Theorem 3.1: Informal sketch
  • Theorem 4.1
  • Remark 4.2
  • Remark 5.1
  • Theorem 5.5
  • Remark 5.6
  • Theorem 5.7
  • Corollary 5.8
  • Theorem 5.9
  • ...and 43 more