Addressing degeneracies in latent interpolation for diffusion models

Erik Landolsi; Fredrik Kahl

REVIEW 2 major objections 5 minor 31 references

Addressing degeneracies in latent interpolation for diffusion models

T0 review · 2 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A small bias in diffusion latents is amplified by sqrt(N) and makes centroid interpolation degenerate; the paper proposes a channel-wise mean adjustment that fixes it.

desk verdict A clean diagnosis of a real failure mode in many-input latent interpolation, with a simple fix that mostly works; the main soft spots are in the evaluation, not the theory. read the letter →

arxiv 2505.07481 v1 pith:PHHP7ARA submitted 2025-05-12 cs.CV

Erik Landolsi , Fredrik Kahl This is my paper · ORCID

classification cs.CV

keywords diffusionmodelslatentinterpolationDDIMinversionbiasamplificationnormalizationFIDCLIPdistanceimagemorphing

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Diffusion models are increasingly used to merge several images by interpolating between their inverted latent codes, but the paper argues that these operations quietly break down as the number of inputs grows. The cause is not simply the low norm of an averaged latent, the usual explanation, but a small per-channel bias that is common to the latents and gets amplified by roughly $\sqrt{N}$ when the centroid is normalized. The paper derives this amplification in a simplified noise-plus-bias model and verifies it experimentally on two Stable Diffusion versions. It then proposes a simple remedy: subtract each channel's mean from every latent, interpolate the residuals using a norm-adjusted blending method, and add the blended means back. Measured by FID and CLIP embedding distance, the fix reduces degeneration and improves quality even where the baseline images still look acceptable.

What carries the argument

The load-bearing object is the amplification identity $z'\approx\sqrt{N}\,d + e'$ for the normalized centroid of $N$ latents, derived in Section 4.2. It works through the sharply concentrated norm of an average of independent Gaussian variables, $\lVert \frac{1}{N}\sum_n e_n\rVert\approx\sqrt{L/N}$, which makes the normalization factor $\alpha$ grow like $\sqrt{N}$. That identity identifies channel-wise mean bias, not low norm alone, as the cause of degeneration, and it dictates the repair: remove the estimated per-channel means before norm-adjusting the residual, then reinsert the blended means.

What would settle it

Take latents with exactly zero channel-wise bias by drawing pure independent Gaussian noise, then compute normalized centroids for $N=2,8,\dots,96$ and generate images from them: if those images still degenerate, Eq. (9) is not the cause. In the opposite direction, add a known constant $d$ to each channel before normalization and check whether the image degeneration scales with $d\sqrt{N}$.

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Extended reading notes

Core claim

Concretely, the paper claims that if inverted latents obey $z_n = d + e_n$ with $d$ a small common deterministic bias and $e_n$ independent Gaussian noise, then the normalized centroid satisfies $z' \approx \sqrt{N}\,d + e'$. The normalization factor is boosted because the average of the noise terms has norm of order $\sqrt{L/N}$, so scaling the centroid back to the nominal norm $\sqrt{L}$ multiplies everything by about $\sqrt{N}$. The deterministic trace left in latents by imperfect schedulers and denoisers is therefore no longer negligible: it dominates the interpolated latent as $N$ grows, producing washed-out and degenerate images even though the norm is correct. The proposed fix decomposes each latent into a per-channel mean part $d$ and a residual $e$, interpolates the residuals with the norm-preserving schemes $f_{FIX}$ or $f_{NIN}$, and adds back linearly interpolated means; this preserves the input-reproduction property when $f_{NIN}$ is used.

Load-bearing premise

The load-bearing premise is that the amplified term is a common per-channel bias shared across latents and constant across spatial positions; if the bias varies from image to image or across space, the channel-wise mean fix stops removing the part that normalization amplifies.

Editorial extensions

If this is right

Centroid-based augmentation and multi-image morphing pipelines should expect normalized averages of many inverted latents to drift away from valid image latents as $N$ grows, even when every individual latent looks normal.
Quality metrics such as FID and CLIP distance degrade with $N$ before degeneration is visually obvious, so latent interpolation methods should be evaluated at several $N$ values rather than by inspecting a few images.
Using channel-wise mean adjustment is claimed to improve FID and CLIP distance even at small $N$, so it can serve as a drop-in replacement for plain fixed normalization or normalization to interpolated norms.
The input-reproduction property survives the fix when residual interpolation uses $f_{NIN}$, because a one-hot weight vector returns the original latent exactly.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the same bias-plus-noise model describes any normalized averaging of features whose mean differs from zero, the $\sqrt{N}$ amplification may appear in other generative latent spaces, not only diffusion models; the remedy of per-dimension mean subtraction would then generalize to GAN inversion or embedding-space centroiding.
A low-pass-filtered estimate of $d$, which the paper lists as future work, should handle biases that vary across spatial positions; if spatial variation is strong, the channel-wise fix alone would still show residual quality loss at large $N$.
The paper's observation that $f_{FIX}$ slightly beats $f_{NIN}$ suggests the correct normalization target for far-from-input centroids is closer to the nominal $\sqrt{L}$ than to interpolated input norms; a closed-form norm schedule interpolating between the two regimes could combine both advantages without optimization.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper finds something real: when you average many inverted diffusion latents and renormalize, small per-channel biases get amplified by roughly sqrt(N), and the output degenerates. The derivation of Eq. (9) is simple, parameter-free, and consistent with the measured channel-mean growth in Figure 4. That is the genuinely new part, and it is solid. I also appreciate that the authors test the effect across two model families and several inversion schemes, including null-text inversion, which makes the phenomenon look robust rather than a quirk of one scheduler.

The proposed remedy, subtract per-channel means from each latent, interpolate the residuals with fFIX or fNIN, then add back blended means, is sensible and cheap. It resembles AdaIN, and the paper acknowledges that in the related work. The theoretical motivation makes the heuristic less ad hoc than in previous morphing papers, so credit is due there.

Where are the soft spots? The experimental evaluation would need error bars or at least multiple seeds to be fully convincing; the FID and CLIP curves in Figure 7 are presented without any spread. The closely related AdaIN baseline is mentioned but not directly compared, which leaves a bit of ambiguity about whether the specific decomposition matters or any instance normalization would do. And the stress-test concern stands: the d + e model assumes the bias is roughly constant per channel across spatial positions. If the bias comes from the terminal-SNR z0 component, it is image-specific and spatially structured, so channel-mean subtraction leaves a residual that still gets amplified by the Eq. (9) mechanism. The paper itself lists low-pass-filtered d as future work, so the authors are aware of the limitation, but the experiments do not measure how much spatial structure survives in the residual after chm. That is a genuine gap, though not a fatal one: the qualitative results show a large improvement, and the analysis correctly identifies the dominant bias.

I also see no circularity problem. The derivation starts from an explicit additive-bias model without fitting constants, and the single self-cited baseline, fNIN, is only used for comparison, not to support the main claim.

Who is this for? Anyone working on diffusion-based morphing, latent-space data augmentation, or centroid-based mixing with more than a handful of inputs. It is a subfield contribution with clear practical value, not a paradigm shift.

I would send it to peer review. The theoretical core deserves a serious referee, and the revisions needed are straightforward: add error bars, include an AdaIN-style baseline, and discuss or measure the residual spatial bias.

Referee Report

2 major / 5 minor

Summary. The paper studies degeneracies in latent-space interpolation for diffusion models when computing centroids of N inverted latents. It observes that even with norm normalization (fFIX or fNIN), the generated images degenerate as N grows, and it attributes this to a small deterministic bias d in the latents that is amplified by roughly sqrt(N) during normalization. The central theoretical statement is Eq. (9), derived in Section 4.2 for a toy model z_n = d + e_n with i.i.d. Gaussian e_n. The paper then proposes a channel-wise mean adjustment (chm): subtract each channel's mean from each latent, interpolate the residuals with norm normalization, and add back the interpolated means. Experiments on ImageNet classes with Stable Diffusion 1.5 and 3.5 measure FID and CLIP distance for N up to 64 and show that chm improves quality relative to the baselines. The authors also discuss the non-zero terminal SNR issue and alternative inversion procedures, and they explicitly leave spatially structured bias models (e.g., low-pass-filtered d) to future work.

Significance. If the diagnosis and remedy hold, the work is practically valuable for image morphing and data augmentation, where multi-input latent interpolation is increasingly common. The theoretical derivation in Section 4.2 is clean, parameter-free, and consistent with the measured channel-wise mean growth in Figure 4. The paper is also honest about the limitations of FID and about the restricted bias model it considers. The main strengths are the simplicity of the proposed fix and the breadth of the empirical evaluation across two model families and several inversion procedures. However, the significance of the claimed remedy is conditional on the assumption that the bias is dominated by per-channel constants, which the paper acknowledges but does not directly verify in the experiments.

major comments (2)

[Section 4.2 and Section 5] The derivation in Eq. (9) models the latents as z_n = d + e_n with a common deterministic bias d, and the proposed channel-wise mean adjustment in Section 5 removes only the per-channel spatial mean of d. If d also has a spatially varying component, or if the class-consistent part of the bias varies across examples as suggested by the terminal-SNR term sqrt(alpha_T) z_0 in Section 4.3, then that component survives the chm subtraction and is still amplified by the same sqrt(N) mechanism. The paper itself lists low-pass-filtered d as a possible decomposition in Section 5, indicating that a per-channel constant d is not guaranteed to capture the true bias. The experiments in Section 6 do not measure the residual bias after chm (e.g., the norm of the spatially varying part of the average latent), so the empirical support for the remedy is only established for biases that are approximately per-channel constant. To justify the broad claim that the method 'significantly reduces the degeneration effect', the authors should either directly measure the residual non-constant component or explicitly restrict the claim to the tested setting.
[Section 6, Figure 7] The primary quantitative evidence consists of FID and CLIP distance curves without error bars or per-class spread, despite being averaged over only 10 randomly drawn ImageNet classes. At moderate N, the differences between the chm methods and the baselines, and between fix/chm and nin/chm, may be within the natural variation across classes. Reporting only point estimates makes it difficult to assess whether the reported improvements are statistically significant, especially since the paper claims a 'significant' reduction in degeneration. Please report standard errors, confidence intervals, or per-class curves, and state in the caption or text that the curves are averages over 10 classes.

minor comments (5)

[Section 3.2] The notation 'Pwn = 1' should be typeset with a summation symbol and limits; currently it appears as an ambiguous 'P' rather than a sum.
[Section 3.3, Eq. (2)] The expression for fFIX(Z,{1,0,0,...}) has a missing delimiter; it should read sqrt(L)/||z1|| * z1 to avoid ambiguity.
[Section 4.1] The sentence 'we can conclude that inverted latents do not in general follow the statistics of random samples drawn from a normal distribution' is a strong conclusion drawn from a few qualitative examples; a more cautious phrasing would be appropriate.
[Section 6, Figure 7 caption] The caption says 'centroids of N noisy diffusion model latents' but does not specify that these latents come from inverting real ImageNet images (as described in Appendix A); please clarify.
[Section 4.4] The qualitative offset experiment in Figure 6 applies a constant offset only to the top quarter of one channel, which is a useful illustration but not a quantitative measurement of the spatial structure of the bias; a brief note acknowledging this limitation would strengthen the presentation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Eq. (9) follows from an explicit additive-bias model and the remedy is validated by external ImageNet metrics, not by construction.

full rationale

Eq. (9) is not circular: it is obtained by substituting the explicit decomposition z_n = d + e_n into fFIX and algebraically expanding the normalization factor alpha, with no constants fitted to the data being predicted. The observed sqrt(N) growth of channel-wise means in Fig. 4 is an independent test of that derivation. Section 5's channel-wise mean adjustment is a remedy targeted at the per-channel constant component of d; its usefulness is established by ImageNet FID/CLIP experiments, not by assumption. The paper's one self-citation ([12], for fNIN) is not load-bearing: fNIN is defined in Eq. (4), the bias-amplification derivation is performed for fFIX, and the fNIN variant is a baseline/option whose properties are analyzed in-paper. Section 5 also lists low-pass-filtered d as future work, so the paper does not claim its model covers all bias structures. No fitted parameter is renamed as a prediction and no uniqueness theorem or prior work is invoked to force the choice. Verdict: no significant circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The theoretical part is parameter-free: no constants are fitted to data. The cost is carried by modeling assumptions: additive decomposition of latents into common bias plus iid noise, and the sufficiency of channel-wise mean as an estimator for that bias. The paper is transparent that more advanced estimators, such as low-pass filtering, are left to future work.

assumptions (4)

standard math In high dimensions, the norm of the averaged iid Gaussian noise, ||(1/N) sum e_n||, concentrates at sqrt(L/N).
Used in Eq. (7) of Section 4.2 to replace the norm of the averaged noise by its expected value.
domain assumption Inverted latents can be written as z_n = d + e_n, where d is a small common deterministic bias and e_n are iid Gaussian noise.
Introduced in Section 4.2 and used in Section 5 to justify separating the interpolation of d and e.
ad hoc to paper The deterministic bias d is well approximated by a per-channel mean value that is constant across spatial dimensions.
Section 5 chooses option 3 among several estimators, leaving low-pass-filtered d as future work; this choice is justified mainly by the empirical success in Section 6, not by independent evidence.
domain assumption The target norm sqrt(L) is the appropriate normalization level for latent centroids.
Taken from prior work [24] and used in fFIX; the paper's own experiments show fFIX is slightly better than fNIN, but the assumption is not derived anew.

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Cite this review

Pith. "Pith review of Addressing degeneracies in latent interpolation for diffusion models." pith.science (2026). https://pith.science/paper/PHHP7ARA

@misc{pith2026250507481,
  author       = {Pith},
  title        = {Pith review of: Addressing degeneracies in latent interpolation for diffusion models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PHHP7ARA}},
  note         = {Machine review of arXiv:2505.07481}
}

read the original abstract

There is an increasing interest in using image-generating diffusion models for deep data augmentation and image morphing. In this context, it is useful to interpolate between latents produced by inverting a set of input images, in order to generate new images representing some mixture of the inputs. We observe that such interpolation can easily lead to degenerate results when the number of inputs is large. We analyze the cause of this effect theoretically and experimentally, and suggest a suitable remedy. The suggested approach is a relatively simple normalization scheme that is easy to use whenever interpolation between latents is needed. We measure image quality using FID and CLIP embedding distance and show experimentally that baseline interpolation methods lead to a drop in quality metrics long before the degeneration issue is clearly visible. In contrast, our method significantly reduces the degeneration effect and leads to improved quality metrics also in non-degenerate situations.

Figures

Figures reproduced from arXiv: 2505.07481 by the authors.

**Figure 1.** Images generated from centroids of N latents obtained from N input images of ImageNet class “tree frog” for N = 2, 8, 32, 48, 64, 96. Top row: Linear interpolation. Middle row: Linear interpolation with fixed normalization. Bottom row: channel-wise mean adjustment (our suggested method). {zn} and a corresponding set of mixing weights {wn : wn > 0, Pwn = 1}, finding a mixed latent representing a meaningful mixture of… view at source ↗

**Figure 2.** Interpolation paths produced by the methods from Section 3.3 for a 2D toy example. The gray circle represents latents with norm √ L, which is where randomly sampled latents are typically located. SLERP. Spherical linear interpolation (SLERP) [26] is a way of interpolating between two points on a unit sphere according to slerp([p1, p2], t) = sin (1 − t)θ sin θ p1 + sin (tθ) sin θ p2 (3) where θ = cos−1 (p T 1 p2) i… view at source ↗

**Figure 3.** Images generated from centroids created using the recipe in Section 4.1. ImageNet class “tree frog”, N = 2, 8, 32, 48, 64, 96, fixed normalization. 4.1 Initial investigation Let us first consider an ideal case of N i.i.d. latents zn ∼ N (0, I) and their normalized average z ′ = fFIX(Z, w). The weighted sum of i.i.d. normally distributed variables is also normally distributed with adjusted variance. After normaliza… view at source ↗

Figures from the paper (10 more)

**Figure 4.** Figure 4: Channel-wise mean of the first two channels of fFIX centroids computed from N inverted input images, plotted for 8 selected ImageNet classes. Left: SD 1.5, right: SD 3.5. The x-axes have a square-root scale to illustrate the linear dependence on √ N [PITH_FULL_IMAGE:f…

**Figure 5.** Figure 5: Images generated from centroids created using the recipe from Section 4.1, to ensure i.i.d. examples, using SD 3.5 to avoid the non-zero terminal SNR issue. N = 2, 8, 32, 48, 64, 96 (increasing to the right), fixed normalization. 4.4 The effect of latent bias To study …

**Figure 6.** Figure 6: Illustrating the effect of latent mean value offsets. Top row: offsets at timestep t = T. Bottom row: offsets at timestep t = 0. Columns: offsets b = 0.0, 0.1, 0.2, 0.4, 0.8. First, note that DDIM inversion is an approximation of the forward process that works best wit…

**Figure 7.** Figure 7: Image quality of images produced from centroids of N noisy diffusion model latents, measured using the FID metric and cos distance in CLIP embedding space [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

**Figure 8.** Figure 8: Images generated from centroids of latents obtained from N input images of ImageNet class “monarch butterfly” for N = 2, 8, 32, 48, 64, 96, SD 1.5. Top row: Fixed normalization. Middle row: fix/chm. Bottom row: nin/chm. methods as N grows sufficiently large. Qualitativ…

**Figure 9.** Figure 9: Images generated from centroids of latents obtained from N = 12 input images of ImageNet classes “boa constrictor”, “crate”, “hourglass”, “oxygen mask”, “shark” using SD 3.5. Top row: Fixed normalization. Bottom row: nin/chm. 7 Discussion We have identified and dissect…

**Figure 10.** Figure 10: As expected, the image quality is worse than when using guidance, [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]

**Figure 10.** Figure 10: Images generated from centroids using 500 diffusion timesteps without classifier-free guidance, showing that the degeneration issue persists also under more ideal DDIM inversion conditions. N = 2, 4, 8, 16, 32, 64 (increasing to the right), fixed normalization, ImageN…

**Figure 11.** Figure 11: Images generated from centroids computed using null-text inversion, showing that the degeneration issue still persists. N = 2, 4, 8, 16, 32, 64 (increasing to the right), fixed normalization, ImageNet class “goldfish” [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]

**Figure 12.** Figure 12: Images generated from centroids of latents obtained from N input images of ImageNet class “baboon” using SD 3.5. N = 2, 8, 32, 48, 64, 96. Top row: Fixed normalization. Middle row: fix/chm. Bottom row: nin/chm [PITH_FULL_IMAGE:figures/full_fig_p016_12.png]

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Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

arXiv preprint arXiv:2304.08466 (2023)

Azizi, S., Kornblith, S., Saharia, C., Norouzi, M., Fleet, D.J.: Synthetic data from diffusion models improves imagenet classification. arXiv preprint arXiv:2304.08466 (2023)

arXiv 2023

[2] [2]

arXiv preprint arXiv:2211.01324 (2022)

Balaji, Y., Nah, S., Huang, X., Vahdat, A., Song, J., Kreis, K., Aittala, M., Aila, T., Laine, S., Catanzaro, B., et al.: ediffi: Text-to-image diffusion models with an ensemble of expert denoisers. arXiv preprint arXiv:2211.01324 (2022)

arXiv 2022

[3] [3]

ACM Transactions on Graphics (TOG)20(2), 95–126 (2001)

Buss, S.R., Fillmore, J.P.: Spherical averages and applications to spherical splines and interpolation. ACM Transactions on Graphics (TOG)20(2), 95–126 (2001)

work page 2001

[4] [4]

In: Proceedings of the IEEE/CVF conference on computer vision and pattern recognition

Chong, M.J., Forsyth, D.: Effectively unbiased fid and inception score and where to find them. In: Proceedings of the IEEE/CVF conference on computer vision and pattern recognition. pp. 6070–6079 (2020)

work page 2020

[5] [5]

In: Forty-first International Conference on Machine Learning (2024)

Esser, P., Kulal, S., Blattmann, A., Entezari, R., Müller, J., Saini, H., Levi, Y., Lorenz, D., Sauer, A., Boesel, F., et al.: Scaling rectified flow transformers for high- resolution image synthesis. In: Forty-first International Conference on Machine Learning (2024)

2024

[6] [6]

arXiv preprint arXiv:2208.01618 (2022)

Gal, R., Alaluf, Y., Atzmon, Y., Patashnik, O., Bermano, A.H., Chechik, G., Cohen-Or, D.: An image is worth one word: Personalizing text-to-image gener- ation using textual inversion. arXiv preprint arXiv:2208.01618 (2022)

arXiv 2022

[7] [7]

arXiv preprint arXiv:2403.14602 (2024)

Garibi, D., Patashnik, O., Voynov, A., Averbuch-Elor, H., Cohen-Or, D.: Renoise: Real image inversion through iterative noising. arXiv preprint arXiv:2403.14602 (2024)

arXiv 2024

[8] [8]

Advances in neural information processing systems30 (2017)

Heusel,M.,Ramsauer,H.,Unterthiner,T.,Nessler,B.,Hochreiter,S.:Ganstrained by a two time-scale update rule converge to a local nash equilibrium. Advances in neural information processing systems30 (2017)

2017