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arxiv: 2606.21385 · v1 · pith:NAQ3O5UYnew · submitted 2026-06-19 · 💻 cs.LG · cs.AI

Unsupervised Disentanglement Without Compromises : How Functional Orthogonality Enforces Identifiability

Mathieu Cyrille Simon , Pascal Frossard , Christophe De Vleeschouwer This is my paper

Pith reviewed 2026-06-26 14:58 UTC · model grok-4.3

classification 💻 cs.LG cs.AI
keywords unsupervised disentanglementidentifiabilityJacobian orthogonalitygenerative modelsnormalizing flowslatent factorsrepresentation learningnonlinear models
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The pith

An orthogonality constraint on the Jacobian identifies general nonlinear generative factors without independence assumptions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper seeks to establish that latent factors can be uniquely recovered in unsupervised settings by requiring their influences on observations to be locally orthogonal, formalized as a constraint on the Jacobian of the generative function. This functional definition supports a proof of identifiability for arbitrary nonlinear models, bypassing the usual demands for factor independence or causal graphs. The argument holds only when every combination of factor values appears in the latent domain, which supplies the coverage needed for the orthogonality condition to pin down a single mapping. Regularized normalizing flows are used to test the idea and recover ground-truth factors in practice.

Core claim

We prove that this condition yields identifiability of general nonlinear generative models, without requiring statistical independence or causal assumptions, provided the latent domain admits all combinations of factor values.

What carries the argument

The orthogonality constraint on the Jacobian of the generative mapping, enforcing that distinct latent factors act through locally orthogonal directions.

If this is right

  • Identifiability holds for general nonlinear generative models under the stated condition.
  • Statistical independence between latent factors is not required.
  • Causal assumptions are unnecessary for the identifiability result.
  • Orthogonality-regularized normalizing flows recover ground-truth factors in experiments.
  • The observed success of VAEs can be explained by implicit satisfaction of the orthogonality condition.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Relaxing full combinatorial coverage would likely allow non-unique recoveries even when orthogonality holds.
  • The same constraint could be imposed on other generative architectures to test identifiability beyond flows.
  • Empirical checks on datasets whose factors do not span all combinations would directly test the necessity of the coverage assumption.

Load-bearing premise

The latent domain must admit every possible combination of factor values.

What would settle it

A concrete nonlinear generative model in which the Jacobian remains orthogonal everywhere yet two distinct factorizations produce identical observations when some factor combinations are missing from the latent domain.

Figures

Figures reproduced from arXiv: 2606.21385 by Christophe De Vleeschouwer, Mathieu Cyrille Simon, Pascal Frossard.

Figure 1
Figure 1. Figure 1: Grid Transformations: Visualizing Orthogonal Jacobians. form of disentanglement by requiring that the effect of each latent factor on the observations be decoupled from the oth￾ers. Importantly, this assumption applies to the entire model class M and is posited as a defining property of meaningful factors. In doing so, it shifts the inductive bias from statis￾tical independence of the latent variables to i… view at source ↗
Figure 2
Figure 2. Figure 2: Assume we aim to represent the position of a ball on a field as a two-dimensional latent variable. The most direct rep￾resentation uses Cartesian coordinates (x, y) (left), regardless of the distribution of positions. However, an equally valid alternative could be a polar representation (r, θ) (right). Without additional assumptions, there is no principled way to identify which repre￾sentation is the “true… view at source ↗
Figure 3
Figure 3. Figure 3: MCC and Amari distances quantitative results for d ∈ {3, 6, 9} in the independent/dependent cases [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Qualitative results under various sources domain. izations are provided in App.D, including comparison with VAEs across standard disentanglement metrics. Finally, Fig.4 presents some qualitative results illustrating the role of global domain information. We generate data with varying latent supports and train a Neural Spline Flow (Durkan et al., 2019) equipped with orthogonal Jacobian constraints. Consiste… view at source ↗
Figure 5
Figure 5. Figure 5: Geometric interpretation of Mobius transformations via stereographic projection. Points in ¨ R d are projected onto the sphere, transformed by a rigid motion of the sphere, and mapped back to R d via inverse stereographic projection. Plot generated using a visualization tool created by Juan Carlos Ponce Campuzano on Geogebra We now analyze which Mobius transformations can be automorphisms of the bounded do… view at source ↗
Figure 6
Figure 6. Figure 6: Method for constructing counterexamples. Any domain exhibiting inversion symmetry can be generated in this way. Crucially, aside from the points lying exactly on the inversion sphere, no point can remain fixed. Thus, for any other point in Ω, the existence of an automorphism involving inversion requires the existence of a corresponding point on the opposite side of the inversion sphere. It is precisely thi… view at source ↗
Figure 7
Figure 7. Figure 7: The condition in Prop.3 directly targets the defining geometric invariant of inversion. For a fixed center c, inversion about a sphere centered at c preserves, along every ray direction u, the product of the two intersection distances r1(u)r2(u) with the domain boundary. Domains that are invariant under inversion, such as disks or spheres, are characterized by the fact that this product is constant across … view at source ↗
Figure 8
Figure 8. Figure 8: Visualization of a disk automorphism induced by an inversion. The disk is mapped onto itself, confirming that the transformation is indeed an automorphism. The mapping is nonlinear and produces a visibly deformed geometry. B.2. The Two-Dimensional Case The two-dimensional case exhibits fundamentally different behavior from the higher-dimensional setting discussed above. This distinction arises from the mar… view at source ↗
Figure 9
Figure 9. Figure 9: Qualitative disentanglement results for crescent-shaped domains. From top to bottom: ground-truth sources sampled from the structured domain, corresponding observations, and reconstructions obtained using the orthogonally constrained model. Due to inversion symmetry, two equivalent solutions are observed. sharply with the rigidity imposed by Liouville’s theorem for d ≥ 3. In two dimensions, conformal mappi… view at source ↗
Figure 10
Figure 10. Figure 10: Example of transformations in 3D. This construction yields highly expressive nonlinear mixing functions with orthogonal Jacobians that go far beyond simple coordinate reparameterizations such as spherical or cylindrical coordinates. While Mobius transformations alone form a ¨ relatively restricted subclass, their block-wise and hierarchical composition produces a much larger family of admissible QD mappin… view at source ↗
Figure 11
Figure 11. Figure 11: Qualitative disentanglement results for independent sources. From left to right: ground-truth sources, observed variables, latent reconstructions obtained with the orthogonally constrained model, and latent reconstructions obtained with the unconstrained model. Reconstructed latents are visualized after a sigmoid transformation for clarity. 35 [PITH_FULL_IMAGE:figures/full_fig_p035_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Additional qualitative results for independent sources. Columns correspond to ground-truth sources, observations, recon￾structions from the constrained model, and reconstructions from the unconstrained model, respectively. Reconstructions are shown after sigmoid reparameterization. 36 [PITH_FULL_IMAGE:figures/full_fig_p036_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Qualitative disentanglement results for dependent sources. From left to right: ground-truth sources, source density, observed variables, latent reconstructions obtained with the orthogonally constrained model, and latent reconstructions obtained with the uncon￾strained model. Reconstructions are visualized after sigmoid transformation. 37 [PITH_FULL_IMAGE:figures/full_fig_p037_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Additional qualitative results for dependent sources. Columns show ground-truth sources, source density, observations, and latent reconstructions from the constrained and unconstrained models, respectively. 38 [PITH_FULL_IMAGE:figures/full_fig_p038_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Qualitative disentanglement results for non-cubical latent domains. From left to right: ground-truth sources sampled from a structured domain, observations, reconstructions obtained with the orthogonally constrained model, and reconstructions obtained with the unconstrained model. 39 [PITH_FULL_IMAGE:figures/full_fig_p039_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Additional results for non-cubical latent domains. The constrained model successfully recovers the sources up to rigid transformations of the domain, while the unconstrained model fails to preserve the latent geometry. 40 [PITH_FULL_IMAGE:figures/full_fig_p040_16.png] view at source ↗
read the original abstract

This paper explores unsupervised disentangled representation learning from a functional perspective. We define latent concepts as factors that influence observations through locally orthogonal directions, formalized as an orthogonality constraint on the Jacobian of the generative mapping. We prove that this condition yields identifiability of general nonlinear generative models, without requiring statistical independence or causal assumptions, provided the latent domain admits all combinations of factor values. Experiments with orthogonality-regularized normalizing flows empirically confirm the theory, demonstrate reliable recovery of ground-truth factors, and shed light on the success of VAEs. These findings challenge the prevailing impossibility claims for unsupervised disentanglement and provide a principled alternative foundation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper claims that defining latent factors via locally orthogonal directions (formalized as an orthogonality constraint on the Jacobian of the generative mapping) yields identifiability for general nonlinear generative models without statistical independence or causal assumptions, provided the latent domain has full combinatorial coverage of factor values. It supports the claim with a proof and with experiments on orthogonality-regularized normalizing flows that recover ground-truth factors on synthetic data and offer insight into VAE behavior.

Significance. If the conditional identifiability result holds, the work is significant: it supplies a functional-orthogonality route to identifiability that avoids the independence and causal assumptions common in the literature and directly challenges impossibility theorems for unsupervised disentanglement. The machine-checked or explicit derivation (if present) and the reproducible flow experiments constitute concrete strengths that could guide new regularization strategies.

minor comments (2)
  1. [§3] §3 (or wherever the main theorem is stated): the precise statement of the domain-coverage assumption should be repeated verbatim in the theorem box so readers can immediately see the exact premise under which the Jacobian-orthogonality condition implies unique recovery.
  2. [Experiments] The experimental section would benefit from an explicit ablation that isolates the orthogonality regularizer from other flow hyperparameters to confirm that the reported factor recovery is attributable to the Jacobian constraint rather than the flow architecture alone.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the work, recognition of its significance in providing a functional-orthogonality route to identifiability, and recommendation for minor revision. We will prepare a revised manuscript addressing any minor points.

Circularity Check

0 steps flagged

No significant circularity; conditional identifiability theorem is self-contained

full rationale

The paper states a conditional mathematical result: Jacobian orthogonality of the generative mapping implies identifiability of general nonlinear models when the latent domain has full combinatorial coverage, without independence or causal assumptions. This follows directly from the stated definitions and the explicit domain-coverage premise; the abstract and reader's summary indicate no reduction of the claim to fitted parameters, self-referential equations, or load-bearing self-citations. The derivation chain is a proof under premises that are declared upfront rather than smuggled in, making the result independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof depends on the domain-coverage assumption and the definition of latent concepts via locally orthogonal Jacobian directions; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption The latent domain admits all combinations of factor values
    Explicitly required for the identifiability theorem to hold.

pith-pipeline@v0.9.1-grok · 5641 in / 1268 out tokens · 23758 ms · 2026-06-26T14:58:29.366952+00:00 · methodology

discussion (0)

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Reference graph

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