Improving OOD Generalization of Pre-trained Encoders via Aligned Embedding-Space Ensembles

Arash Khoeini; Martin Ester; Sharan Vaswani; Shuman Peng

REVIEW 4 major objections 4 minor 26 references

Improving OOD Generalization of Pre-trained Encoders via Aligned Embedding-Space Ensembles

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Aligned embedding-space ensembles improve OOD generalization of pre-trained encoders.

desk verdict A clean, honest MNIST-scale demonstration that aligned embedding-space ensembles beat single encoders and unaligned averages, but the theory is a corollary of the recovery assumption and the OOD transfer of the alignment is untested. read the letter →

arxiv 2411.13073 v1 pith:7M7Z3FKM submitted 2024-11-20 cs.LG cs.CV

Shuman Peng , Arash Khoeini , Sharan Vaswani , Martin Ester This is my paper · ORCID

classification cs.LGcs.CV

keywords ensemblelearningembedding-spacealignmentout-of-distributiongeneralizationself-supervisedcontrastivehypersphericalembeddingsKarchermeanorthogonaltransformation

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

The paper argues that the poor out-of-distribution (OOD) performance of self-supervised pre-trained encoders can be improved by ensembling several encoders in the embedding space, provided the individual hyperspherical spaces are first aligned. Its theoretical analysis extends an existing result that InfoNCE-trained encoders recover the true data latents up to an orthogonal transformation, showing that any two such encoders are related by an orthogonal map and that the mean of aligned embeddings recovers the correct latents. Based on this, the paper proposes Ensemble-InfoNCE, which learns an orthogonal alignment by minimizing geodesic distance with an orthogonality penalty, then aggregates aligned embeddings via the Karcher mean. Experiments on MNIST show that the aligned ensemble improves Recall@1 and MAP@R over single models in both in-distribution and out-of-distribution settings, with the largest gains on colored and cropped test inputs.

What carries the argument

The load-bearing identity is the orthogonal relationship between the embedding spaces of different InfoNCE-trained encoders: $f_1(x) = R f_2(x)$ for an orthogonal matrix $R$ (Proposition 1). The method learns this $R$ by minimizing the alignment loss $\mathcal{L}_{\mathrm{align}} = \frac{1}{N} \sum_{n=1}^{N} \arccos\big(f_i(x_n), R f_j(x_n)\big) + \lambda \| R^T R - I_D \|_F^2$ on in-distribution data, then aggregates the aligned embeddings with the Karcher mean on the hypersphere. This combination converts the ensemble mean into a quantity that provably recovers the true latent $z$ up to a rotation (Proposition 2).

What would settle it

Measure the geodesic alignment error between $f_1(x)$ and $R f_2(x)$ on OOD data: if the error does not stay as low as on the training data, or if the best-fitting $R$ on OOD data differs substantially from the learned $R$, the transferred alignment claim fails. Concretely, train on MNIST and evaluate on a genuinely shifted distribution (e.g., rotated or corrupted digits from a different dataset), and check whether the aligned ensemble mean still outperforms the better single encoder.

Watch

Extended reading notes

Core claim

The central claim is that taking an ensemble mean in the embedding space of self-supervised InfoNCE pre-trained encoders yields better embeddings on both in-distribution and out-of-distribution data than any single encoder, but only if the embedding spaces are aligned first. The paper proves (Proposition 2) that if each encoder $f_i$ recovers the ground-truth latents $z$ up to a per-encoder orthogonal transformation $R_i$, then the mean of $f_1(x)$ and $R f_2(x)$ equals $R_1 z$, so the ensemble recovers the true latents up to an orthogonal transformation. It further establishes (Proposition 1) that any two encoders trained on the same data are related by a single orthogonal map $f_1 = R f_2$. This justifies an unsupervised alignment procedure that learns $R$ by minimizing the geodesic distance between paired embeddings plus a soft orthogonality constraint, and the paper demonstrates on MNIST that the aligned ensemble outperforms single encoders and misaligned ensembles, with a $6.99\%$ improvement in Recall@1 and $17.38\%$ in MAP@R on colored OOD data.

Load-bearing premise

The method assumes that a single orthogonal rotation $R$ learned on the in-distribution pre-training set also aligns the embedding spaces of the same encoders on out-of-distribution inputs; if distribution shift changes the relative geometry of the two embedding spaces in a way that is not a rigid rotation, the alignment and the ensemble mean will be incorrect.

Editorial extensions

If this is right

The method offers a label-free way to ensemble pre-trained encoders, preserving the interpretability of output-space ensembles and the flexibility of weight-space ensembles.
If the orthogonal alignment transfers, the same pipeline applies to any InfoNCE pre-trained encoder family, not just MNIST-scale models, potentially improving zero-shot OOD transfer.
The theoretical guarantee that the ensemble mean recovers the true latents up to rotation suggests that properly aligned ensembles should consistently improve embedding fidelity over single encoders.
Misaligned ensembles are shown to hurt performance, establishing alignment as a necessary step for embedding-space ensembling rather than an optional refinement.
Since the approach operates on embeddings rather than logits, it can be applied to retrieval, classification, and any downstream task that consumes frozen features.

Reading between the lines

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

The paper's own results show smaller gains on cropped OOD data than on colored OOD data, hinting that the transferred rotation assumption may be sensitive to the type of distribution shift; a global orthogonal map may not be the right model for shifts that alter local geometry.
A testable extension is to replace the single learned rotation with an input-dependent alignment map (e.g., a small neural network), which could handle shifts that change embedding geometry nonlinearly while preserving the interpretability of the ensemble.
The OOD evaluation in the paper keeps the same label set; a stronger test would be evaluating on unseen classes or a different domain, where the claim that the aligned ensemble recovers correct latents is more consequential and less likely to hold.

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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

Short version: the empirical claim is credible as far as it goes, the theory is mostly decoration, and the real load-bearing premise—that an alignment learned on ID data transfers to OOD inputs—is never tested. Worth engaging, but it needs more evidence.

What's new: the packaging of unsupervised orthogonal alignment plus Karcher mean for ensembling InfoNCE encoders. The objective in Eq. 2, with geodesic distance and a soft orthogonality constraint, is a reasonable, simple recipe. The controls are the right kind: the paper compares against unaligned embedding-space ensembles and weight-space ensembles, and it shows that aligned ensembling helps while unaligned ensembling hurts. The numbers in Tables 1 and 2 are internally consistent, and the appendix covers both supervised and unsupervised contrastive pre-training and two OOD shifts (color, crop). That is more than many workshop papers do.

The soft spots are real but not disqualifying. First, the theory is a direct corollary of the Zimmermann et al. recovery property. Proposition 1 is just composition of orthogonal maps; Proposition 2 says that under exact alignment the mean equals f1(x). So it does not explain why the ensemble should help—it only shows the ensemble does not destroy the recovered latents when alignment is perfect. Second, the OOD evaluation depends on the assumption that the orthogonal R learned on the training set remains valid on colored or cropped inputs. The paper gives no evidence for this: no OOD alignment loss, no comparison to re-learning R on OOD data. If the relative geometry of the two embedding spaces shifts nonlinearly, the method collapses to averaging misaligned vectors, which the paper's own results suggest is harmful. Third, the ensembles have no variance estimates, so we cannot tell whether the 1–7% gains are stable, and there is no significance testing. Fourth, no code is provided, which matters for a method whose value is mostly empirical. Fifth, it is MNIST only, and the gains are modest.

What holds up: the method is simple, clearly described, and the qualitative story—alignment matters, ensembling helps when spaces are aligned—is plausible and worth testing on larger data. The paper is honest about its preliminary scope.

This is for anyone working on ensembling or merging self-supervised encoders without labels. A workshop audience will get value. It deserves a serious referee, but the authors should be pushed to add variance bars, test the transfer of R across shifts, and ideally include one larger dataset. My recommendation: send it to review with that feedback.

Referee Report

4 major / 4 minor

Summary. The paper proposes Ensemble-InfoNCE, a method for ensembling self-supervised pre-trained encoders in the embedding space. The method learns an orthogonal transformation R that aligns the hyperspherical embedding spaces of M encoders on in-distribution pre-training data, then aggregates the aligned embeddings via the Karcher mean. The authors present two theoretical propositions claiming that encoder embedding spaces are related by orthogonal transformations and that the aligned ensemble recovers the ground-truth latents. They report experiments on MNIST in in-distribution, randomly colored, and randomly cropped settings, showing that the aligned ensemble improves Recall@1 and MAP@R over single encoders, whereas unaligned embedding-space ensembles degrade performance.

Significance. The paper identifies and addresses a real practical problem: naive averaging of misaligned embedding spaces from independently trained encoders degrades retrieval quality, and the comparison against unaligned embedding-space ensembles and weight-space ensembles is a useful sanity check. If the empirical gains are robust, the proposed method could offer a simple, unsupervised way to improve zero-shot OOD embedding quality. However, the theoretical contribution is largely a restatement of the paper's own assumption, the empirical evidence is restricted to a single dataset and a single ensemble that lacks variance estimates, and the transfer of the learned alignment to OOD inputs is an untested premise on which the OOD claim rests. The paper is honest in calling the experiments preliminary, but the central OOD claim needs stronger support.

major comments (4)

[Section 3.1, Prop. 2] The theoretical guarantee is a tautology under the paper's assumption. Prop. 1 follows immediately from f1(x)=R1z and f2(x)=R2z, and the proof of Prop. 2 (Eq. 5) substitutes f1(x)=Rf2(x) into the mean to obtain f1(x)=R1z. Thus the 'ensemble recovers the correct latents' result is simply the assumed single-encoder recovery property restated, and it provides no mechanism by which ensembling improves over a single encoder. The actual empirical gains must arise in the approximate-alignment regime, but the paper provides no analysis or bounds for that regime. This is load-bearing because the abstract and introduction present the theory as support for the method; I recommend either removing the theoretical-recovery claim or replacing it with an analysis of the mean under approximate alignment.
[Section 3.2 to Table 1] The OOD evaluation assumes that the orthogonal map R learned on the in-distribution training set (Eq. 2) remains a valid alignment for colored and cropped test inputs. The paper provides no argument or evidence for this transfer: there is no OOD alignment loss reported, no comparison between R learned on ID data and R re-estimated on OOD data, and no ablation varying the anchor encoder. If distribution shift changes the relative embedding geometry of the two encoders nonlinearly, the same R will misalign OOD embeddings and the Karcher mean will average semantically inconsistent directions. Because the OOD improvement is one of the paper's main claims, this untested transfer assumption needs to be addressed directly.
[Section 4, Table 1 and Figures 2, 4-8] The empirical claim that the aligned ensemble outperforms single models is not statistically established. The ensemble is a single deterministic combination of all M=5 models, so there is no variance estimate or confidence interval for the ensemble, and no significance test is reported. In the ID setting, the R@1 gain is 0.900 to 0.911, which is approximately 1.8 standard deviations of the single-model distribution; the OOD gains are larger, but without repeated ensembles or bootstrap resampling the reader cannot assess whether the differences are reliable. Please provide confidence intervals via bootstrapping over seeds or multiple independent ensembles, or otherwise quantify the variability of the ensemble performance.
[Section 4 and Section 5] The generality of the central claim is not supported by the evidence. All main experiments use MNIST with two synthetic shifts (color, crop), and the conclusion states that the method 'significantly enhances' OOD embedding quality. Given that the abstract and title refer to 'pre-trained encoders' generally, at least one additional dataset or a substantial tempering of the wording is needed before the claim can be accepted. The authors already acknowledge this in the future-work paragraph, but the abstract should reflect the MNIST-only scope.

minor comments (4)

[Equations (1) and (2)] Writing 'L_align = arg min_{R in R^{DxD}} ...' is formally incorrect: the left-hand side is a loss value but the right-hand side is a set of minimizers. Please write R* = argmin ... or define L_align as the objective function.
[Section 4, OOD data description] The description 'randomly colored' is underspecified; please state how colors are sampled (e.g., per-channel uniform draws) and whether the background as well as the digit is colored, so that the OOD construction is reproducible.
[Appendix A.5.4] The WSE* description contains an unclear sentence about hyperparameters 'which were only provided for ResNet50' and a typo 'from0.1'; please rephrase and list the exact hyperparameter search ranges used for the MNIST architecture.
[Section 1, related work] The claim of being the first embedding-space ensemble for self-supervised encoders would be strengthened by a brief discussion of prior unsupervised alignment approaches (e.g., Procrustes analysis or canonical correlation analysis) to make the novelty precise.

Circularity Check

2 steps flagged · score 6.0 of 10

Propositions 1 and 2 restate the assumption f1(x)=R1z, f2(x)=R2z: the proof of the 'ensemble recovers correct latents' result collapses to mean(f1,Rf2)=f1, so the theoretical guarantee is the input assumption rewritten; the empirical MNIST gains are independent.

self definitional [Section 3.1, Assumption and Proposition 1; Appendix A.2 proof]
"Assumption. ... we assume that f1(x) = R1z and f2(x) = R2z. ... Proposition 1 (Orthogonal transformation relationship). Under the above assumption, f1 and f2 learn the same latents up to an orthogonal transformation R, that is, f1(x) = Rf2(x)."

The 'relationship' is obtained by multiplying the assumed equations: since f1=R1z and f2=R2z, setting R=R1R2^{-1} gives f1=Rf2 by construction. The proof in Appendix A.2 does exactly this. No property of InfoNCE-trained encoders beyond the assumption is used, so the proposition is a restatement of the premise rather than an independently derived relationship between embedding spaces.
self definitional [Section 3.1, Proposition 2; proof in Appendix A.2, Eq. (5)]
"Proposition 2 (Ensemble recovers correct latents). The ensemble mean ¯f (x) of aligned embeddings f1(x) and Rf2(x) are the correct latents z up to orthogonal transformation R1, that is, ¯f (x) = R1z. ... Since f1(x) = Rf2(x), we have: ¯f (x) = 1 2 [f1(x) + f1(x)] = f1(x) = R1z (5)"

The proof shows the aligned mean is identically f1(x), so the claimed ensemble guarantee is exactly the assumption f1(x)=R1z. Under the assumption, averaging f1 and Rf2 contributes nothing: mean(f1,Rf2)=f1. Thus the 'theoretical result' that an ensemble recovers the correct latents is the single-encoder recovery assumption rewritten, not a derived property of ensembling. The abstract's claim that the theory demonstrates ensemble recovery is therefore a restatement of its input.

full rationale

The paper's two theoretical propositions are logically valid but vacuous: Proposition 1 is obtained from the assumption f1(x)=R1z and f2(x)=R2z by setting R=R1R2^{-1}, and Proposition 2's proof in Eq. (5) shows the aligned mean collapses to f1(x), so the claimed 'ensemble recovers correct latents' guarantee is exactly the assumed single-encoder recovery f1(x)=R1z. Thus the theoretical analysis does not establish any property of ensembling beyond its premise; it is a restatement of the input. The alignment matrix R is fit on in-distribution data (Eq. 2), and the theory says nothing about whether this R remains valid under the colored/cropped OOD shifts; that is an untested modeling assumption, a correctness risk rather than circularity. The experimental comparison (R@1, MAP@R on MNIST ID/OOD) is an actual empirical measurement, not forced by the theory, so the circularity is partial and confined to the theoretical claims. There is no load-bearing self-citation: the cited recovery guarantee [25] is external work by different authors. Score 6.

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

The central claim rests on the Zimmermann et al. recovery theorem (cited, not rederived), the paper's own near-tautological orthogonality assumption, and the unstated transfer of the learned alignment to OOD data. There are no new physical entities; the free parameters are hyperparameters and the alignment matrix, which is fitted on the training set.

free parameters (4)

Orthogonality regularization coefficient lambda = 0.5 (supervised); 0.1/0.3/0.5 (unsupervised)
Controls the soft orthogonality penalty in Eq (2); chosen by hand per training mode, no validation procedure described, and the reported OOD gains depend on this choice.
Embedding dimension D = 8
Projection head dimension; chosen by hand, shapes the hyperspherical geometry and the capacity of the learned alignment R; no sensitivity analysis is given.
Alignment training epochs and learning rate = 20 epochs, LR 0.1
Hyperparameters for fitting R on the pre-training set; chosen by hand, they affect the estimated alignment matrix.
Number of InfoNCE negative samples = 16
Standard contrastive hyperparameter; the theoretical recovery guarantee from [25] depends on the asymptotic contrastive objective, so this finite-sample choice is part of the setup.

assumptions (4)

domain assumption InfoNCE-trained encoders recover ground-truth latents up to orthogonal transformations (Theorem 2 of [25]).
Used as the Assumption in Section 3.1 and inherited from prior work; the paper's theoretical results are conditional on it.
domain assumption The contrastive objective is minimized in the limit of infinite data and the generative process is invertible (conditions from [25]).
Required for the recovery guarantee and for the approximate orthogonality claim on real data; not restated or proved in this paper.
ad hoc to paper For a pair of encoders f1, f2, there exists an orthogonal R such that f1(x)=R f2(x) across all x (Section 3.1 Assumption).
This is exactly the relationship the paper claims to 'reveal'; introducing it as an assumption makes Propositions 1 and 2 tautological.
ad hoc to paper The alignment matrix R learned on the training set transfers to OOD test inputs.
Unstated premise; Eq (2) is minimized on pre-training data, and OOD evaluation assumes the same R aligns shifted embeddings.

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Cite this review

Pith. "Pith review of Improving OOD Generalization of Pre-trained Encoders via Aligned Embedding-Space Ensembles." pith.science (2026). https://pith.science/paper/7M7Z3FKM

@misc{pith2026241113073,
  author       = {Pith},
  title        = {Pith review of: Improving OOD Generalization of Pre-trained Encoders via Aligned Embedding-Space Ensembles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7M7Z3FKM}},
  note         = {Machine review of arXiv:2411.13073}
}

read the original abstract

The quality of self-supervised pre-trained embeddings on out-of-distribution (OOD) data is poor without fine-tuning. A straightforward and simple approach to improving the generalization of pre-trained representation to OOD data is the use of deep ensembles. However, obtaining an effective ensemble in the embedding space with only unlabeled data remains an unsolved problem. We first perform a theoretical analysis that reveals the relationship between individual hyperspherical embedding spaces in an ensemble. We then design a principled method to align these embedding spaces in an unsupervised manner. Experimental results on the MNIST dataset show that our embedding-space ensemble method improves pre-trained embedding quality on in-distribution and OOD data compared to single encoders.

Figures

Figures reproduced from arXiv: 2411.13073 by the authors.

**Figure 2.** Comparing embedding qualities of single models (blue), an ensemble of unaligned embed [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** For in-distribution (ID) evaluation, images like those in (a) were used. For out-of [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Supervised contrastive pre-training with Colored MNIST as OOD evaluation data. Com [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: Supervised contrastive pre-training with Cropped MNIST as OOD evaluation data. Com [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: Unsupervised contrastive pre-training with Cropped MNIST as OOD evaluation data. [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

**Figure 7.** Figure 7: Comparing embedding qualities of single models (blue), an ensemble of unaligned em [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

**Figure 8.** Figure 8: Comparing embedding qualities of single models (blue), an ensemble of unaligned em [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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work page 2023

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3d object representations for fine- grained categorization

Jonathan Krause, Michael Stark, Jia Deng, and Li Fei-Fei. 3d object representations for fine- grained categorization. In Proceedings of the IEEE international conference on computer vision workshops, pages 554–561, 2013

2013