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REVIEW 4 major objections 5 minor 79 references

Does Data Scaling Lead to Visual Compositional Generalization?

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Compositional generalization in vision models comes from combinatorial data diversity, not data volume; an ideal linear structure needs only two examples per concept.

desk verdict Solid controlled evidence that diversity beats scale for compositional generalization, but the k=2 efficiency theorem is not proven as written and needs repair before the paper's central claim can stand. read the letter →

arxiv 2507.07102 v1 pith:LGIVQP7N submitted 2025-07-09 cs.LG

Arnas Uselis , Andrea Dittadi , Seong Joon Oh This is my paper
classification cs.LG MSC 68T0768T45
keywords compositionalgeneralizationdatadiversityscalinglinearlyfactoredrepresentationsvisualconceptsout-of-distributionrepresentationgeometryattribute-objectcombinations
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

Compositional generalization—recognizing an unseen attribute-object pairing by recombining known concepts—is a basic human cognitive ability that state-of-the-art vision models only partially exhibit. This paper asks whether the standard scaling recipe of ever-larger datasets and models will ever deliver it, and answers no: in controlled experiments that vary concept diversity and combination coverage independently of data volume, out-of-distribution accuracy on unseen pairings stays 60-80% below in-distribution accuracy even when training data is quadrupled. What closes the gap is diversity: more distinct concept values and more of their observed pairings push zero-shot accuracy above 90%, and only then do models organize concepts into linearly factored representations where a combined concept is the vector sum of its parts. The paper proves that under this structure, observing just two pairings per concept value is enough to generalize perfectly to all unseen pairings, and shows that pretrained models like CLIP and DINO have partial but imperfect versions of this structure. The practical reading is that dataset design should target combinatorial coverage, not just raw scale.

What carries the argument

Linearly factored embeddings (Definition 3.2) are the central object: for two concepts, the joint representation of value pair $(c_1,c_2)$ equals $u_{c_1}+u_{c_2}$, the sum of the two concept-value vectors. The load-bearing recovery mechanism is the conditional-mean estimator: average the feature vectors of all training images that share a concept value, centre by the global mean, and the result recovers that value's concept vector up to a shift; for pairwise combinations this yields the additivity identity $u'_{c_i,c_j}=u'_{c_i}+u'_{c_j}$. That identity turns the training set of $k=2$ pairings per value into a $2n$-equation linear system whose solution gives the factored vectors, and the separation of the two concept subspaces lets the paper construct projection-based classifiers that label every unseen pairing. This mechanism is what converts combinatorial coverage into representational geometry, and representational geometry into zero-shot generalization.

What would settle it

For $n=3$ in the linear system of equation (25), summing the three diagonal rows and the three shifted rows both give the all-ones vector, so the $6\times 6$ design matrix has rank at most 5 even though Part 2 of the proof of Proposition B.7 claims full rank; directly computing this rank for any $n$ settles whether two examples per concept value can recover the factored vectors.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the driver of visual compositional generalization is data diversity—the number of concept values and the fraction of their combinations seen during training—while data scale, measured as the number of images per combination, is not. In the paper's $(n,k)$ framework, models trained from scratch show three phases as combinatorial coverage grows: spurious features that cannot even discriminate the concepts, then discriminative features that are still not linearly factored, and finally, at high diversity, a geometry in which the representation of a combined concept is the sum of concept-value vectors, with near-perfect zero-shot accuracy on unseen pairings. Proposition 4.1 states the efficiency this buys: if representations are linearly factored and the concept subspaces are suitably separated, $k=2$ observed combinations per concept value suffice for a linear classifier to generalize perfectly to all $(n-2)n$ unseen combinations. The paper then tests pretrained models by recovering concept vectors from just two pairings per value and finds above-random performance that is concept-dependent—some attributes are nearly linearly accessible, others are not—so pretraining confers partial structure rather than full compositionality.

Load-bearing premise

The construction assumes that the small set of seen pairings pins down each concept's direction exactly, and that the two groups of concepts point into fully separate directions so that projections can separate them.

Editorial extensions

If this is right

  • Scaling up in-distribution training data will not by itself produce compositional generalization; closing the ID-OOD gap requires increasing the variety of concept values and the combinations of them that a model sees.
  • A reliable indicator of readiness for compositional generalization is the linearity and orthogonality of concept representations; decodability alone is insufficient.
  • Under a perfectly linearly factored representation, a model needs only two seen pairings per concept value to generalize to every unseen pairing, so data efficiency and representational geometry are linked.
  • Pretrained vision models will remain partially compositional—strong on some concept pairs, weak on others—unless they are trained or probed on diverse combination sets.
  • Sparse combinatorial coverage in large web-scale datasets is a structural obstacle that raw scale cannot overcome, because the number of possible combinations grows faster than any practical dataset.

Reading between the lines

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

  • The paper studies pairs of concepts; if the additive mechanism is the reason diversity works, diversity should matter at least as much for three or more interacting concepts, since the number of combinations grows combinatorially.
  • A direct testable extension: explicitly regularizing a model's features toward an orthogonal, additive concept structure should yield OOD gains on unseen pairings even when $n$ is small, which would confirm that geometry rather than raw sample count is the causal channel.
  • Dataset construction for compositional competence could be guided by a coverage metric—how many distinct concept combinations appear—rather than by image count; comparing two datasets matched for diversity but differing in size would sharpen the headline claim.
  • The proof's reliance on disjoint concept subspaces suggests that concepts that are linearly factored but not orthogonal may need more than two examples; checking whether $k=3$ or $k=4$ restores perfect generalization in non-orthogonal settings would map the boundary of the result.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper studies whether scaling training data volume improves visual compositional generalization. Using a controlled (n,k) framework over synthetic datasets (dSprites, Shapes3D, FSprites, Colored-MNIST, PUG-Animal, plus MPI3D), the authors vary the number of concept values n and the number of observed combinations k per value. Their empirical findings are that in-distribution data scale alone does not close the ID-OOD accuracy gap, whereas increasing concept diversity and combination coverage improves zero-shot accuracy on unseen combinations. They also report a three-phase transition in learned representations: spurious features, decodable but non-linear features, and finally linearly factored, mutually orthogonal concept subspaces. Section 4.3 introduces Proposition 4.1 (proved as Proposition B.7) claiming that under linear factorization, observing k=2 combinations per concept value suffices to learn a linear classifier that perfectly generalizes to all unseen combinations. The paper then evaluates pretrained models (DINO, CLIP, ImageNet ResNet) through linear-factorization recovery and probing, finding partial but imperfect compositional structure. The central empirical diversity-over-scale claim is supported by controlled experiments across several datasets; however, the theoretical proposition as stated is not established by the proof in Appendix B.

Significance. If the empirical claims hold, the paper provides a useful, controlled counterpoint to the broad scaling narrative: for compositional generalization, combinatorial diversity of concept combinations can matter more than raw data volume. The dataset-grid methodology, the consistency of the three-phase feature-learning result across five synthetic datasets, and the public release of code are concrete strengths. The theoretical result, if repaired, would add a crisp mechanism for why linear factorization enables few-shot compositional generalization. The evaluation of pretrained models extends the findings beyond from-scratch training. The significance for the compositional generalization and representation learning community is moderate to high. However, the current proof of Proposition B.7 contains rank, orthogonality, and dimension issues that mean the stated theoretical claim is not yet supported.

major comments (4)
  1. [Appendix B, Proposition B.7, Eq. (25)] The proof claims that the 2n x 2n linear system in Eq. (25) is full rank because the diagonal and cyclic rows 'form distinct patterns that ensure linear independence.' This is false for the cyclic split used in the paper. For the diagonal pairs (i,i) and shifted pairs (i,i+1 mod n), the sum of the n diagonal rows equals the sum of the n shifted rows, so the row rank is at most 2n-1 and the system is underdetermined. The proof does not use the zero-sum constraints of Lemma B.4 to remove the null direction. Consequently, the k=2 sufficiency claim in Proposition 4.1 is not proven as written.
  2. [Appendix B, Proposition B.7, Part 3] Part 3 of the proof assumes that any vector in the joint span of the two concept subspaces decomposes uniquely as u1 + u2 with u1 perpendicular to u2, and that orthogonal projections onto U1 and U2 give optimal classifiers. Orthogonality of the two concept subspaces is not part of Definition 3.2 and does not follow from U1 intersecting U2 only at {0}; non-orthogonal subspaces can intersect trivially. Without a stated orthogonality or annihilator assumption, the existence of a linear classifier for concept 1 that ignores concept 2 is not guaranteed. The empirical orthogonality measurements in Section 4.2 motivate the assumption, but it is not stated in the proposition.
  3. [Proposition 4.1 / Proposition B.7, dimension statement] The proposition states that the joint span of {u_{c_1}} and {u_{c_2}} has dimension 2n-1, while the proof in Part 3 asserts dim(U1)=n-1, dim(U2)=n-1, and U1 ∩ U2 = {0}, which forces the joint span to have dimension 2n-2. These statements are arithmetically inconsistent unless the proposition refers to uncentered concept vectors and the proof to centered, zero-sum vectors, but that distinction is not made. The statement and proof need to be reconciled, for example by stating which representation (raw or centred) is used for each dimension count.
  4. [Proposition 4.1, generality of k=2 claim] The proposition does not specify which k=2 combinations per concept value are observed. The proof only goes through for a specific connected 2-regular split, namely the cyclic split in Eq. (15). For an arbitrary set of two combinations per value, the design matrix in Eq. (25) need not be identifiable even after adding zero-sum constraints. The statement should either restrict the observation pattern explicitly or justify that the result holds for any connected 2-regular split.
minor comments (5)
  1. [Appendix B, proof of Proposition B.7] There is a typo in the sentence introducing pairwise joint embeddings: 'contstructed' should be 'constructed.'
  2. [Section 5.1] The text refers to 'the detailed algorithm in Appendix 1'; the algorithm is actually in Appendix B.1, so the cross-reference should be corrected.
  3. [Figure 3 caption] The caption of Figure 3(b) says dataset size is kept fixed while increasing n with k=n-1, but the number of training combinations n(n-1) grows with n. Please clarify how total dataset size is controlled, or adjust the wording to describe what is actually held fixed.
  4. [Proposition 4.1 notation] The notation 'ci, cj ∈ {1,...,n}' overloads concept values with indices; the proposition should state that c_i ranges over the value set C_i and that combinations are indexed by i,j ∈ {1,...,n}.
  5. [Section 3.2 / Definition 3.2] The global mean is written as f in the appendix but as \bar{f} in the main text; unify the notation to avoid confusion between the feature extractor f and the mean embedding.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central conditional theorem and controlled empirical claims are self-contained; self-citations are peripheral.

full rationale

The paper's central claims are not circular. The empirical result (diversity over scale) is established by controlled experiments that vary n, k, and dataset size; no fitted parameter is renamed as a prediction, and the reported zero-shot accuracies are measured on unseen combinations. The theoretical result (Proposition 4.1 / B.7) is explicitly conditional on the linearly factored embedding assumption of Definition 3.2, which is cited to external work (Trager et al., 2023). The proof derives identifiability of concept vectors from a linear system and then constructs projection classifiers; the conclusion that k=2 combinations suffice is not identical to the assumption, since it requires an invertibility/identifiability argument. Self-citations (Uselis & Oh 2025; Koishigarina et al. 2025; Sonthalia et al. 2025; Jeong et al. 2025) are contextual, supporting metrics or caveats, and are not load-bearing for the main derivation. The unproven full-rank assertion in Eq. (25) and the unstated orthogonality condition in the proof are genuine correctness concerns, but they are proof gaps rather than circularity: even if the proof is incomplete, the theorem's output is not equivalent to its input by construction. The derivation chain is therefore self-contained against the paper's own assumptions, and no circular step can be exhibited with a specific reduction.

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

The conditional theorem rests on exact linear factorization, a dimension assumption on the joint span, balanced sampling, and an unstated orthogonality condition. The empirical parts impose no fitted parameters; n and k are experimental controls. No new theoretical entities are introduced; the newly introduced FunnySprites dataset is a benchmark artifact rather than a conceptual entity.

assumptions (4)
  • domain assumption Exact linear factorization of feature embeddings, f(x) = sum over concepts of u_ci.
    Definition 3.2 and B.1 assume this additive structure; experiments only approximate it, so the theorem is conditional on a property models do not exactly satisfy.
  • ad hoc to paper The joint span of all concept vectors has dimension 2n-1.
    Stated in Proposition 4.1/B.7 to guarantee identifiability; the paper acknowledges this can fail when concept factors occupy low-dimensional subspaces.
  • domain assumption Balanced and uniform training data across observed combinations and unlabeled variations.
    Lemma B.4 and the mean-recovery formulas require equal counts per combination and per unlabeled variation; the (n,k) framework enforces this by construction.
  • ad hoc to paper Mutual orthogonality of the two concept subspaces.
    The projection part of the proof uses PU1 and PU2 as orthogonal projections with u1 perpendicular to u2, but this is not stated in Proposition B.7 and is only an empirical observation at high diversity.

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Pith. "Pith review of Does Data Scaling Lead to Visual Compositional Generalization?." pith.science (2026). https://pith.science/paper/LGIVQP7N

@misc{pith2026250707102,
  author       = {Pith},
  title        = {Pith review of: Does Data Scaling Lead to Visual Compositional Generalization?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LGIVQP7N}},
  note         = {Machine review of arXiv:2507.07102}
}
read the original abstract

Compositional understanding is crucial for human intelligence, yet it remains unclear whether contemporary vision models exhibit it. The dominant machine learning paradigm is built on the premise that scaling data and model sizes will improve out-of-distribution performance, including compositional generalization. We test this premise through controlled experiments that systematically vary data scale, concept diversity, and combination coverage. We find that compositional generalization is driven by data diversity, not mere data scale. Increased combinatorial coverage forces models to discover a linearly factored representational structure, where concepts decompose into additive components. We prove this structure is key to efficiency, enabling perfect generalization from few observed combinations. Evaluating pretrained models (DINO, CLIP), we find above-random yet imperfect performance, suggesting partial presence of this structure. Our work motivates stronger emphasis on constructing diverse datasets for compositional generalization, and considering the importance of representational structure that enables efficient compositional learning. Code available at https://github.com/oshapio/visual-compositional-generalization.

Figures

Figures reproduced from arXiv: 2507.07102 by the authors.

Figure 1
Figure 1. Sparse concept combinations in large-scale datasets. Left: An indicator matrix of noun-adjective co-occurrences in LAION-400M shows significant sparsity in concept combinations; the majority of cells are unobserved (zoomed-in view), demonstrat￾ing that even common concepts rarely combine in the dataset. This sparsity biases models toward memorizing frequent combinations rather than learning compositional structure. … view at source ↗
Figure 2
Figure 2. Investigating compositional learning through concept scaling. The figure illustrates our two main experimental settings. Left (Data setting): Training data consisting of images with corresponding concept combinations shown in the grid, where blue cells indicate observed combinations during training. Right (Model setting): Two approaches—training models from scratch (Section 4) where we systematically increase the nu… view at source ↗
Figure 3
Figure 3. Compositional generalization emerges through different forms of concept diversity. (a) In basic settings with limited diversity, models show substantial accuracy drops on unseen combinations (brown) compared to seen combinations (yellow), demonstrating the inherent difficulty of compositional generalization. (b) When increasing the number of target classes (n) while keeping dataset size and diagonal training combina… view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: Increasing ID training data quantity does not solve compositional generalization. Despite training with significantly more in-distribution samples, models still struggle to generalize to unseen concept combinations. The gap between ID and OOD performance remains large …
Figure 5
Figure 5. Figure 5: Linearity emerges with data diversity, while feature discriminability alone does not imply linear structure. (a) Feature discriminability emerges early but does not imply compositional structure, (b) Linear concept representations only emerge with increased training di…
Figure 6
Figure 6. Figure 6: Compositional generalization capabilities of pre-trained models under assumed linear factorization. Bar plots show both training (transparent) and testing (solid) accuracy across different datasets (DSPRITES, SHAPES3D, CMNIST, PUG-ANIMAL) when using minimal training da…
Figure 7
Figure 7. Figure 7: Importance of linear feature structure for compo￾sitional generalization. We illustrate a schematic for shape and color classification using linear models in a 2-dimensional feature space, comparing zero-shot and adapted cases with frozen feature extractor. (1) If the …
Figure 8
Figure 8. Figure 8: Even with pretraining, models struggle with composi￾tional generalization. Despite the benefits of pretraining, models still face challenges in generalizing to unseen concept combina￾tions. While larger models like CLIP and DINO VIT-L show the strongest performance, th…
Figure 9
Figure 9. Figure 9: Performance scaling with concept diversity. OOD accuracies across four datasets: Shapes3D, dSprites, FSprites, and Colored￾MNIST. Each heatmap shows performance for different combinations of concept values (n) and seen combinations (k) per concept value. Increasing con…
Figure 10
Figure 10. Figure 10: Compositional generalization in pre-trained models. Heatmaps show out-of-distribution accuracy for different combinations of n (concept values) and k (training combinations) across datasets. Darker colors indicate higher accuracy. Pre-trained models exhibit improved g…
Figure 11
Figure 11. Figure 11: Sample images from the MPI3D dataset (Gondal et al., 2019). The dataset contains real-world images of objects with varying properties like color, shape, size and camera viewpoint. Examples from the testing set of n = 6, k = 5 are shown. MPI3D 0.0 0.2 0.4 0.6 0.8 1.0 A…
Figure 12
Figure 12. Figure 12: Accuracy comparison for n = 3, k = 2 using ResNet-50. As shown in the main text, compositional generalization is difficult: the model struggles to generalize to the object-shape concept. MPI3D Random 30 60 120 Dataset Size (×10³) 0.35 0.40 0.45 0.50 OOD Accuracy 0.5 0…
Figure 13
Figure 13. Figure 13: Compositional generalization only improves with data diversity, not data quantity. Top left: Under few training combinations (n [PITH_FULL_IMAGE:figures/full_fig_p022_13.png]
Figure 14
Figure 14. Figure 14: Evaluating pre-trained vision models on MPI3D. Left: Accuracy comparison for classifiers constructed under linear factorization. [PITH_FULL_IMAGE:figures/full_fig_p023_14.png]
Figure 15
Figure 15. Figure 15: FunnySprites dataset examples. Shape and orientation variations for n = 14 concept values with k = 2 training combinations. Each sprite is generated by connecting traced points to form unique geometric shapes, providing a challenging test for compositional generalizat…
Figure 16
Figure 16. Figure 16: Colored-MNIST examples. Digit and color combinations for n = 10 values with k = 3 training combinations. This dataset combines the MNIST digits with color variations to test compositional understanding of shape and color attributes. We introduce the Funny Sprites data…

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.