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Learning Tangent Bundles and Characteristic Classes with Autoencoder Atlases

Eduardo Paluzo-Hidalgo, Yuichi Ike

TL;DR

It is shown that any reconstruction-consistent autoencoder atlas canonically defines transition maps satisfying the cocycle condition, and that linearising these transition maps yields a vector bundle coinciding with the tangent bundle when the latent dimension matches the intrinsic dimension of the manifold.

Abstract

We introduce a theoretical framework that connects multi-chart autoencoders in manifold learning with the classical theory of vector bundles and characteristic classes. Rather than viewing autoencoders as producing a single global Euclidean embedding, we treat a collection of locally trained encoder-decoder pairs as a learned atlas on a manifold. We show that any reconstruction-consistent autoencoder atlas canonically defines transition maps satisfying the cocycle condition, and that linearising these transition maps yields a vector bundle coinciding with the tangent bundle when the latent dimension matches the intrinsic dimension of the manifold. This construction provides direct access to differential-topological invariants of the data. In particular, we show that the first Stiefel-Whitney class can be computed from the signs of the Jacobians of learned transition maps, yielding an algorithmic criterion for detecting orientability. We also show that non-trivial characteristic classes provide obstructions to single-chart representations, and that the minimum number of autoencoder charts is determined by the good cover structure of the manifold. Finally, we apply our methodology to low-dimensional orientable and non-orientable manifolds, as well as to a non-orientable high-dimensional image dataset.

Learning Tangent Bundles and Characteristic Classes with Autoencoder Atlases

TL;DR

It is shown that any reconstruction-consistent autoencoder atlas canonically defines transition maps satisfying the cocycle condition, and that linearising these transition maps yields a vector bundle coinciding with the tangent bundle when the latent dimension matches the intrinsic dimension of the manifold.

Abstract

We introduce a theoretical framework that connects multi-chart autoencoders in manifold learning with the classical theory of vector bundles and characteristic classes. Rather than viewing autoencoders as producing a single global Euclidean embedding, we treat a collection of locally trained encoder-decoder pairs as a learned atlas on a manifold. We show that any reconstruction-consistent autoencoder atlas canonically defines transition maps satisfying the cocycle condition, and that linearising these transition maps yields a vector bundle coinciding with the tangent bundle when the latent dimension matches the intrinsic dimension of the manifold. This construction provides direct access to differential-topological invariants of the data. In particular, we show that the first Stiefel-Whitney class can be computed from the signs of the Jacobians of learned transition maps, yielding an algorithmic criterion for detecting orientability. We also show that non-trivial characteristic classes provide obstructions to single-chart representations, and that the minimum number of autoencoder charts is determined by the good cover structure of the manifold. Finally, we apply our methodology to low-dimensional orientable and non-orientable manifolds, as well as to a non-orientable high-dimensional image dataset.
Paper Structure (63 sections, 25 theorems, 98 equations, 6 figures, 7 tables)

This paper contains 63 sections, 25 theorems, 98 equations, 6 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Topological preliminaries
  4. Vector bundles
  5. Characteristic classes and orientability
  6. The first Stiefel--Whitney class
  7. Computing $w_1$ from transition functions
  8. Orientability via the sign cocycle
  9. Atlas-Based Formulation of the Orientability Obstruction
  10. Autoencoder atlases
  11. Transition maps and cocycle condition
  12. The tangent bundle of an autoencoder atlas
  13. Linearization
  14. Vector bundle construction
  15. Detecting orientability
  16. ...and 48 more sections

Key Result

Theorem 2.6

If $\mathcal{U}$ is a good cover of a paracompact Hausdorff space $X$, then there is a canonical isomorphism where $H^p(X; G)$ denotes the singular cohomology of $X$ with coefficients in $G$.

Figures (6)

  • Figure 1: Two different Möbius covers are shown. On the left, we have a good cover such that the intersections are contractible. However, on the right, we have that the intersection is another Möbius band and hence, not contractible.
  • Figure 2: Diagram of the Möbius band as a line bundle. The base space is $B$ that equals $S^1$, and given $b\in B$, the fiber $F_b$ is an interval coloured in dark blue.
  • Figure 3: Diagram of an atlas autoencoder.
  • Figure 4: Diagram of an approximate atlas autoencoder. We take into account that given a point $x\in U_i$, we need to extend the domain to $O_i$ because $y=D_i(E_i(x))$ may not lie on $U_i$. Also, $O_i$ may not be fully in $M$, taking into account off-manifold points.
  • Figure 5: Autoencoder atlas for the Möbius band (Section \ref{['sec:exp-mobius']}). The two-chart cover produces an overlap with two disconnected components on which the sign cocycle takes opposite values ($\omega_{10} = -1$ and $\omega_{10} = +1$), detecting non-orientability via the failure of the coboundary condition.
  • ...and 1 more figures

Theorems & Definitions (98)

  • Definition 2.1: Good cover
  • Remark 2.2: Nerve theorem for relaxed good covers
  • Definition 2.3: Čech cochain groups
  • Definition 2.4: Coboundary operator
  • Definition 2.5: Cocycles, coboundaries, and Čech cohomology
  • Theorem 2.6: Isomorphism for good covers Spanier1981
  • Definition 2.7: 1-cocycles and 1-coboundaries in $\mathbb{Z}/2$
  • Remark 2.8: Multiplicative notation
  • Definition 2.9: Real vector bundle milnor1974characteristic
  • Remark 2.10
  • ...and 88 more