Learning Tangent Bundles and Characteristic Classes with Autoencoder Atlases

Eduardo Paluzo-Hidalgo; Yuichi Ike

math.ATFeb 26, 2026

Learning Tangent Bundles and Characteristic Classes with Autoencoder Atlases

Authors: Eduardo Paluzo-Hidalgo, Yuichi Ike

Organizations: Department of Applied Mathematics I, University of Sevilla, Sevilla, Spain · Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo, Japan

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.

Figures & tables

Panel of Figure 1 — 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.

Panel of Figure 2 — Figure 2: Diagram of an atlas autoencoder. Given charts $(U_{i},E_{i},D_{i})$ and $(U_{j},E_{j},D_{j})$ with $U_{i}\cap U_{j}\neq\emptyset$ , the transition map $T_{ji}=E_{j}\circ D_{i}$ converts latent representations between charts.

Panel of Figure 3 — Figure 3: Diagram of an approximate atlas autoencoder. 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 $M$ , accommodating off-manifold points.

Metric	Value	Theoretical role
$\varepsilon$	$0.032\pm 0.008$	$\sup_{x}\\|D_{i}(E_{i}(x))-x\\|$
$\bar{\varepsilon}$	$0.007\pm 0.001$	$\mathbb{E}_{x}\\|D_{i}(E_{i}(x))-x\\|$
$\eta_{\mathrm{lat}}$	$0.54\pm 0.19$	$\sup_{x}\\|d(E_{i}\circ D_{i})_{E_{i}(x)}-I_{d}\\|_{\mathrm{op}}$
$\delta$	$0.101\pm 0.016$	$\min_{i,j,x}\|\det g_{ji}(x)\|$
Compatibility err	$0.008\pm 0.001$	$\\|T_{ji}(E_{i}(x))-E_{j}(x)\\|$
$\sigma_{\min}(dE)$	$0.66\pm 0.05$	$\min_{i}\sigma_{\min}(dE_{i})$

Table 1: Theoretical metrics for the

S^{2}

experiment, computed from a

4

-chart autoencoder atlas whose chart domains

U_{i}=\{x\in S^{2}:\langle x,v_{i}\rangle>-0.3\}

are induced by the vertices of an inscribed regular tetrahedron in

\mathbb{R}^{3}

(good cover; nerve

\partial\Delta^{3}\simeq S^{2}

). All quantities are reported as mean

\pm

standard deviation over

5

independent training runs. Symbols:

\varepsilon

, the maximum per-chart sup reconstruction error

\sup_{x\in U_{i}}\|D_{i}(E_{i}(x))-x\|

;

\bar{\varepsilon}

, the empirical mean reconstruction error

\mathbb{E}_{x\sim M}\|D_{i}(E_{i}(x))-x\|

;

\eta_{\mathrm{lat}}

, the latent-side differential proxy

\sup_{x}\|d(E_{i}\circ D_{i})_{E_{i}(x)}-I_{d}\|_{\mathrm{op}}

used as a practical estimator of

\eta

in Definition 4.1 (4);

\delta

, the global non-degeneracy gap

\min_{i,j,x}|\det g_{ji}(x)|

over all overlaps; “Compatibility err”, the pairwise transition residual

\|T_{ji}(E_{i}(x))-E_{j}(x)\|

; and

\sigma_{\min}(dE)

, the minimum singular value of the encoder Jacobian,

\min_{i}\sigma_{\min}(dE_{i})

. The bottom row reports the success rate of orientability detection via the coboundary test of Corollary 4.19 .

Metric	Value	Theoretical role
$\varepsilon$	$0.098\pm 0.018$	$\sup_{x}\\|D_{i}(E_{i}(x))-x\\|$
$\bar{\varepsilon}$	$0.021\pm 0.002$	$\mathbb{E}_{x}\\|D_{i}(E_{i}(x))-x\\|$
$\eta_{\mathrm{lat}}$	$0.47\pm 0.12$	$\sup_{x}\\|d(E_{i}\circ D_{i})_{E_{i}(x)}-I_{d}\\|_{\mathrm{op}}$
$\delta$	$0.36\pm 0.18$	$\min_{i,j,x}\|\det g_{ji}(x)\|$
Compatibility err	$0.027\pm 0.004$	$\\|T_{ji}(E_{i}(x))-E_{j}(x)\\|$
$\sigma_{\min}(dE)$	$0.55\pm 0.09$	$\min_{i}\sigma_{\min}(dE_{i})$

Table 2: Theoretical metrics for the Möbius-band experiment, computed from a

2

-chart autoencoder atlas with chart domains

U_{0}=\{y(x)>-0.3\}

and

U_{1}=\{y(x)<0.3\}

on a sample of

n=1500

points from the standard immersion of the Möbius band in

\mathbb{R}^{3}

. The overlap

U_{0}\cap U_{1}

consists of two disconnected components related by the Möbius twist; each is treated as a separate

1

-simplex of the nerve as per Remark 2.5 . All quantities are reported as mean

\pm

standard deviation over

5

independent training runs. Symbols:

\varepsilon

, the maximum per-chart sup reconstruction error

\sup_{x\in U_{i}}\|D_{i}(E_{i}(x))-x\|

;

\bar{\varepsilon}

, the empirical mean reconstruction error;

\eta_{\mathrm{lat}}

, the latent-side differential proxy

\sup_{x}\|d(E_{i}\circ D_{i})_{E_{i}(x)}-I_{d}\|_{\mathrm{op}}

;

\delta

, the global non-degeneracy gap

\min_{i,j,x}|\det g_{ji}(x)|

; “Compatibility err”, the pairwise transition residual

\|T_{ji}(E_{i}(x))-E_{j}(x)\|

; and

\sigma_{\min}(dE)

, the minimum singular value of the encoder Jacobian. The bottom row reports the success rate of non-orientability detection by the coboundary test of Corollary 4.19 : no assignment of

\nu_{0},\nu_{1}\in\{\pm 1\}

can satisfy

\omega_{10}=\nu_{1}\cdot\nu_{0}

on both overlap components simultaneously, certifying

w_{1}\neq 0

Panel of Figure 4 — Figure 4: Autoencoder atlas for the Möbius band. The two-chart cover produces an overlap with two disconnected components on which the sign cocycle takes opposite values, detecting non-orientability.

Trial	Seed	$\varepsilon$	$\bar{\varepsilon}$	$\eta_{\mathrm{lat}}$	$\delta$	$\sigma_{\min}(dE)$	Detected	Correct
0	42	0.182	0.018	31.11	0.008	0.236	Orientable	✗
1	43	0.109	0.025	1.51	0.083	0.207	Non-orientable	✓
2	44	0.141	0.018	6.04	0.016	0.240	Orientable	✗
3	45	0.131	0.025	0.62	0.090	0.212	Non-orientable	✓
4	46	0.104	0.022	1.23	0.053	0.258	Non-orientable	✓

Table 3: Klein bottle: per-trial results over

5

independent runs of an

8

-chart autoencoder atlas. Charts are obtained by farthest-point sampling on the

100

-nearest-neighbour geodesic graph of

n=1000

points drawn from the standard immersion of the Klein bottle in

\mathbb{R}^{4}

(see Section 7.4.1 ); chart

U_{i}

collects sample points whose geodesic distance to landmark

i

falls below the

20

th percentile of the landmark distance matrix. Columns: “Seed” is the random seed of the training run;

\varepsilon

is the maximum over charts of the sup reconstruction error

\sup_{x\in U_{i}}\|D_{i}(E_{i}(x))-x\|

;

\bar{\varepsilon}

is the empirical mean reconstruction error;

\eta_{\mathrm{lat}}

is the maximum over charts of the latent-side differential proxy

\sup_{x}\|d(E_{i}\circ D_{i})_{E_{i}(x)}-I_{d}\|_{\mathrm{op}}

;

\delta

is the global non-degeneracy gap

\min_{i,j,x}|\det g_{ji}(x)|

;

\sigma_{\min}(dE)

is the minimum singular value of the encoder Jacobian across charts; “Detected” is the orientability verdict returned by the coboundary test of Corollary 4.19 ; and “Correct” (✓/✗) indicates agreement with the ground truth (non-orientable). Trials

0

and

2

are classified as non-converged by the post-training diagnostics (criteria (1)–(3) of Section 7.4.1 ): they exhibit the highest reconstruction error and overlap components with mixed signs in

\mathrm{sign}(\det g_{ji})

, indicating that the transition map Jacobian determinant passes through zero within the affected overlap.

Metric	Value	Theoretical role
$\varepsilon$	$0.1147\pm 0.0116$	$\sup_{x}\\|D_{i}(E_{i}(x))-x\\|$
$\bar{\varepsilon}$	$0.0240\pm 0.0013$	$\mathbb{E}_{x}\\|D_{i}(E_{i}(x))-x\\|$
$\eta_{\mathrm{lat}}$	$1.1201\pm 0.3729$	$\sup_{x}\\|d(E_{i}\circ D_{i})_{E_{i}(x)}-I_{d}\\|_{\mathrm{op}}$
$\delta$	$0.0756\pm 0.0160$	$\min_{i,j,x}\|\det g_{ji}(x)\|$
Compatibility err	$0.0134\pm 0.0010$	$\\|T_{ji}(E_{i}(x))-E_{j}(x)\\|$
$\sigma_{\min}(dE)$	$0.2256\pm 0.0227$	$\min_{i}\sigma_{\min}(dE_{i})$

Table 4: Theoretical metrics for the Klein-bottle experiment, restricted to the

3

5

trials (seeds

43

45

46

) that satisfy the post-training convergence diagnostics. The atlas is the

8

-chart cover described in Section 7.4.1 ; values are mean

\pm

standard deviation over the

3

converged trials. Symbols:

\varepsilon

, the maximum per-chart sup reconstruction error

\sup_{x\in U_{i}}\|D_{i}(E_{i}(x))-x\|

;

\bar{\varepsilon}

, the empirical mean reconstruction error;

\eta_{\mathrm{lat}}

, the latent-side differential proxy

\sup_{x}\|d(E_{i}\circ D_{i})_{E_{i}(x)}-I_{d}\|_{\mathrm{op}}

for the differential reconstruction error

\eta

of Definition 4.1 (4);

\delta

, the global non-degeneracy gap

\min_{i,j,x}|\det g_{ji}(x)|

; “Compatibility err”, the pairwise transition residual

\|T_{ji}(E_{i}(x))-E_{j}(x)\|

; and

\sigma_{\min}(dE)

, the minimum singular value of the encoder Jacobian across charts. The coboundary test of Corollary 4.19 fails on all

3

converged trials, correctly certifying non-orientability.

Panel of Figure 5 — Figure 5: The $\mathbb{RP}^{2}$ line patches experiment. (a) Sample $10\times 10$ grayscale patches. (b) Selected transition maps $T_{ji}=E_{j}\circ D_{i}$ : source points (blue) and images (red) in $\mathbb{R}^{2}$ . The complete set of all pairwise transitions is in Figure 7 .

Metric	Value
$\varepsilon$ (sup reconstruction error)	$0.309\pm 0.081$
$\bar{\varepsilon}$ (mean reconstruction error)	$0.060\pm 0.003$
$\eta_{\mathrm{lat}}$ (differential error)	$12.90\pm 6.91$
$\delta$ (non-degeneracy gap)	$0.027\pm 0.010$
Compatibility error	$0.042\pm 0.008$
$\sigma_{\min}(dE)$ (encoder regularity)	$0.541\pm 0.064$

Table 5:

\mathbb{R}P^{2}

line patches: summary metrics for converged trials. Data are

5625

blurred

10\times 10

line-patch images (ambient dimension

N=100

) following [ 12 ] ; because a line at angle

\theta

is identical to a line at angle

\theta+\pi

, the resulting point cloud has the persistent cohomology of

\mathbb{R}P^{2}

(intrinsic dimension

d=2

). The atlas uses a

10

-chart cover from projective landmarks; training uses

\tanh

activations, no Jacobian regularisation (

\lambda_{\mathrm{jac}}=0

), and

1000

epochs. Of

5

training runs,

4

satisfy the post-training convergence diagnostics; each entry below is the per-trial value (e.g. for

\eta_{\mathrm{lat}}

, the maximum over charts), summarised as mean

\pm

standard deviation over the

4

converged trials. Symbols:

\varepsilon

, sup reconstruction error

\sup_{x\in U_{i}}\|D_{i}(E_{i}(x))-x\|

;

\bar{\varepsilon}

, empirical mean reconstruction error;

\eta_{\mathrm{lat}}

, latent-side differential proxy

\sup_{x}\|d(E_{i}\circ D_{i})_{E_{i}(x)}-I_{d}\|_{\mathrm{op}}

;

\delta

, global non-degeneracy gap

\min_{i,j,x}|\det g_{ji}(x)|

; “Compatibility error”, pairwise transition residual

\|T_{ji}(E_{i}(x))-E_{j}(x)\|

;

\sigma_{\min}(dE)

, minimum singular value of the encoder Jacobian across charts. The global hypotheses of Theorem 4.7 (

\eta_{\mathrm{lat}}<1

) are violated; nonetheless

\delta>0

holds and the coboundary test correctly fails on all

4

converged trials, certifying non-orientability. The discrepancy is explained by the codimension-driven gap between

\eta_{\mathrm{lat}}

and

\eta

discussed in Section 7.4.2 .

Manifold	Dim	Ambient	Ground truth	$\bar{\varepsilon}$	$\delta$	Converged	Accuracy
$S^{2}$	2	$\mathbb{R}^{3}$	Orientable	0.007	0.101	5/5	100%
Möbius band	2	$\mathbb{R}^{3}$	Non-orientable	0.021	0.36	5/5	100%
Klein bottle	2	$\mathbb{R}^{4}$	Non-orientable	0.024	0.076	3/5	100%
$\mathbb{R}P^{2}$ (patches)	2	$\mathbb{R}^{100}$	Non-orientable	0.060	0.027	4/5	100%

Table 6: Summary of all four orientability-detection experiments reported in Section 7 , restricted to runs that satisfy the post-training convergence diagnostics. Columns: “Manifold” lists the test manifold; “Dim” is its intrinsic dimension

d

; “Ambient” is the ambient space

\mathbb{R}^{N}

used for sampling (or, for

\mathbb{R}P^{2}

, the line-patch image space); “Ground truth” is the known orientability class of the manifold;

\bar{\varepsilon}=\mathbb{E}_{x}\|D_{i}(E_{i}(x))-x\|

is the empirical mean reconstruction error;

\delta=\min_{i,j,x}|\det g_{ji}(x)|

is the global non-degeneracy gap; “Converged” is the number of converged runs out of

5

; and “Accuracy” is the proportion of converged runs on which the coboundary test of Corollary 4.19 returns the correct orientability verdict. Across the four manifolds,

17

20

runs converge and the coboundary test is correct on all

17

of them.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 6: Transition maps $z\mapsto T_{ji}(z)=E_{j}(D_{i}(z))$ for the Klein bottle experiment (Section 7.4.1 ). Each panel shows the learned transition map between a pair of charts $(i,j)$ , with source points in blue and their images under $T_{ji}$ in red (or green for triple-overlap regions). The title of each panel reports the percentage of overlap points with $\det g_{ji}>0$ ; panels showing $0\%$ or $100\%$ indicate sign-consistent overlaps. On of the overlap display 99% of consistency as it displays two points from another connected component.

Figure 7: Transition maps $z\mapsto T_{ji}(z)=E_{j}(D_{i}(z))$ for the $\mathbb{RP}^{2}$ line patches experiment (Section 7.4.2 ). Each panel shows the learned transition map between a pair of charts $(i,j)$ , with source points in blue and their images under $T_{ji}$ in red (or green for triple-overlap regions). The title of each panel reports the percentage of overlap points with $\det g_{ji}>0$ ; panels showing $0\%$ or $100\%$ indicate sign-consistent overlaps. The balanced distribution of positive and negative signs across the 25 overlap components confirms that the sign cocycle $[\omega]\in\check{H}^{1}(\mathcal{U};\mathbb{Z}/2)$ is non-trivial, detecting the non-orientability of $\mathbb{RP}^{2}$ .

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Autoencoders are widely used for nonlinear dimensionality reduction and manifold learning. While most common implementations rely on both nonlinear encoders and decoders, we investigate the specific role of the encoder and the extent to which it can be constrained to be linear without reducing accuracy. We conduct a comparative study on four autoencoder architectures: standard fully nonlinear autoencoders (AE), linear-encoder autoencoders (Lenc-AE), linear-decoder autoencoders (Ldec-AE), and fully linear autoencoders (LAE), evaluated on synthetic manifolds, computational mechanics data sets, and real-world image data sets including MNIST. We demonstrate that imposing a linear encoder preserves most of the representational capacity of the autoencoder, provided the decoder remains nonlinear. In particular, Lenc-AE consistently outperforms both Ldec-AE and LAE, and achieves reconstruction quality comparable to fully nonlinear AE, while offering advantages in terms of parsimony and interpretability of the latent representation. These results suggest that the nonlinear decoder is the critical component for manifold learning, rather than the encoder. A geometric interpretation of this finding is developed, which identifies the precise conditions under which a linear encoder is sufficient, and the specific manifold configurations that expose its limitations.

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