Encoder-Decoder Manifold Alignment for Idempotent Generation

The gist: The authors propose a new training framework that explicitly closes the gap between encoder and decoder manifolds by enforcing latent-space consistency, which they argue is necessary for idempotent generation and stable editing.

Why it matters

This work addresses a key failure mode in generative models where repeated application of a model leads to instability and drift because the encoder projects inputs onto one latent manifold while the decoder implicitly learns to reconstruct data from a different one This geometric mismatch prevents the model from learning truly idempotent mappings By aligning these geometries, the method encourages stable projections and achieves significantly lower idempotency error compared to existing methods.

How it works

The proposed training framework introduces a simple idempotency regularization term added to the standard reconstruction objective L = Lbase + λLidem. This loss penalizes drift in the latent representation by computing z = Eϕ(x) and zˆ = Eϕ(Dθ(sg(z))) and defining Lidem as Ex∼pX h∥zˆ − sg(z)∥2. The stop-gradient on z serves to keep the target fixed during optimization while preventing degenerate solutions.

The framework is based on the idea that idempotency is a necessary condition for an encoder–decoder composition to act as an optimal projection onto the data manifold M, meaning D(E(D(E(x)))) = D(E(x)). A sufficient condition for output-space idempotency is latent-space consistency, which is enforced by the condition Eϕ(Dθ(Eϕ(x))) = Eϕ(x). This ensures that both decoding paths operate on identical representations and yield the same output.

Key theoretical findings

The authors provide formal proofs for their claims, showing that idempotency is a necessary condition for projection optimality, as D(E(D(E(x)))) = D(E(x)). They also prove that if the encoder satisfies the latent consistency condition Eϕ(Dθ(Eϕ(x))) = Eϕ(x) ∀ x ∈ X, then f is idempotent, meaning Dθ(Eϕ(Dθ(Eϕ(x)))) = Dθ(Eϕ(x)). Furthermore, they show that the output-space idempotency error is bounded by the latent-space idempotency error, justifying the use of the latent-space loss as a practical surrogate for enforcing output-space idempotency.

Empirical validation

The method was empirically validated on image generation and editing tasks, showing consistent improvements in reconstruction quality and edit success rate. For image generation on MNIST, the VAE + Ours method achieved the best FID (1.12) compared to VAE + IGN (30.50), demonstrating that the alignment framework prevents progressive drift under repeated application. In editing tasks, qualitative results show that our idempotent VAE better follows editing instructions while preserving source image structure and fine-grained details.

Conclusion

The authors conclude that encoder–decoder alignment achieves lower idempotency error and higher edit success rate across all evaluated architectures and datasets. They suggest that encoder–decoder alignment deserves to be treated as a first-class design objective in generative modeling, on par with reconstruction quality and disentanglement. The work shows that latent-space consistency is sufficient to enforce output-space idempotency. The research opens directions for scaling this alignment to large-scale diffusion and flow-matching models, and extending the framework to speech and video modalities. All these results underscore the effectiveness of aligning the geometry of encoder and decoder components. >

NeurIPS Paper Checklist

1. Claims

Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope?

Answer: [Yes]

Justification: The abstract and introduction clearly state the three main claims: (1) idempotency is a necessary condition for projection optimality in encoder-decoder models, (2) our regularization reduces drift under repeated application, and (3) enforcing encoder-decoder alignment improves editing quality. All three are supported by theoretical proofs in Appendix B and empirical results in Section 4.

Guidelines:

• The answer [N/A] means that the abstract and introduction do not include the claims made in the paper.

• The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A [No] or [N/A] answer to this question will not be perceived well by the reviewers.

• The claims made should match theoretical and experimental results, and reflect how much the results can generalize to other settings.

• It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.

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Question: Does the paper discuss the limitations of the work performed by the authors?

Answer: [Yes]

Justification: The limitations are in section 6

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• The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.

• The authors should reflect on the factors that influence the performance of the approach.

For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.

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While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren’t acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.

3. Theory assumptions and proofs

Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?

Answer: [Yes]

Justification: All assumptions are stated explicitly in the proposition statements. Formal proofs for all theoretical claims are provided in Appendix B, including the necessity of idempotency for projection optimality (Proposition 1), latent consistency as a sufficient condition (Proposition 2), and the Lipschitz error propagation bound (Proposition 3). Informal arguments in the main text are complemented by formal proofs in appendix.

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4. Experimental result reproducibility

Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?

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Justification: Full implementation details are provided in Appendix C

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1. Latent space stability during repeated generation will be significantly enhanced by enforcing latent space consistency, as this directly controls output-space drift. This allows generative models to produce stable results when applied iteratively, which is crucial for tasks like iterative image editing or inpainting.

2. Editing models will achieve higher identity preservation and fine-grained detail retention because the proposed method preserves finer details, including object positions, colors, and scene structure across edited scenes. This means edits will be more accurate without introducing color drift or structural distortion during repeated application.

3. The model's learned representation will act as an optimal projection onto the data manifold because enforcing idempotency is a necessary requirement for optimality when the encoder–decoder composition acts as a projection operator. This ensures that latent edits remain localized and do not unintentionally affect unrelated attributes like identity or fine acoustic details.

Recently, several learning paradigms have been introduced to enforce idempotency in generative models. The goal of this property is to ensure that repeated application of a model leaves samples unchanged once they lie on the target data manifold. In this work, we explore enforcing idempotency in encoder--decoder models. We argue that a key reason for the failure of generative models on some tasks, such as editing, is a geometric mismatch between the manifolds learned by the encoder and decoder. The encoder projects inputs onto one latent manifold, while the decoder implicitly learns to reconstruct data from a different manifold. This discrepancy prevents the model from learning truly idempotent mappings. To address this issue, we propose a new training framework that explicitly closes this gap by forcing the encoder and decoder to learn consistent representations of the same underlying data manifold. By aligning the geometry of these components, our method encourages stable projections. Empirically, we show that our approach achieves significantly lower idempotency error and consistently regenerates identical outputs under repeated application, compared to existing methods. We demonstrate the effectiveness of the proposed framework on both image generation and image editing tasks. Finally, we show that enforcing idempotency in this manner improves identity preservation and information stability, leading to more realistic and controllable generative editing models.