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REVIEW 2 major objections 5 minor 71 references

Towards an Explainable Comparison and Alignment of Feature Embeddings

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The principal eigenvectors of the normalized difference of two embeddings' kernel matrices mark the sample clusters one embedding captures and the other misses, computable in time linear in the number of samples.

desk verdict A useful spectral tool for embedding comparison that overstates its interpretability guarantee—worth reviewing, but the central claim needs a gap condition and the validation needs hardening. read the letter →

arxiv 2506.06231 v3 pith:ODH2OL5U submitted 2025-06-06 cs.LG cs.AIcs.CVmath.SP

Mohammad Jalali , Bahar Dibaei Nia , Farzan Farnia This is my paper
classification cs.LGcs.AIcs.CVmath.SP
keywords embeddingcomparisonkerneldifferencematrixspectralmethodPCArandomFourierfeaturesalignmentclustermismatchdetectionexplainableAI
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

This paper claims that the differences between two feature embeddings can be made explainable by eigendecomposing the normalized difference of their kernel matrices. The top eigenvectors of $\Lambda_{\psi_1,\psi_2} = \frac{1}{n}(K_{\psi_1} - K_{\psi_2})$ carry large entries precisely on samples that embedding $\psi_1$ groups into a cluster that embedding $\psi_2$ does not strongly group. Because the non-zero eigenspectrum of $\Lambda_{\psi_1,\psi_2}$ is shared by a $(d_1+d_2)\times(d_1+d_2)$ differential covariance matrix, the computation scales linearly in the sample size for bounded feature dimension, and random Fourier features extend the method to shift-invariant kernels such as the Gaussian RBF kernel. The same spectrum yields a distance (SPEC-diff) and a gradient-based alignment procedure (SPEC-align) that measurably reduces CLIP's mismatch with DINOv2 on ImageNet. A sympathetic reader would care because this replaces black-box accuracy comparisons with named, visualizable clusters of disagreement.

What carries the argument

The normalized kernel difference matrix $\Lambda_{\psi_1,\psi_2} = \frac{1}{n}(K_{\psi_1}-K_{\psi_2})$ is the object that carries the argument; its non-zero eigenspace is reinterpreted as a set of cluster indicators. The matching computational object is the differential covariance matrix $\Gamma_{\psi_1,\psi_2}$, a $(d_1+d_2)\times(d_1+d_2)$ block matrix built from the kernel covariance matrices of the two embeddings and their cross-covariance, which shares the non-zero eigenvalues of $\Lambda_{\psi_1,\psi_2}$ while having dimension independent of the sample size. For shift-invariant kernels, random Fourier features replace the infinite-dimensional feature map with a finite proxy and give the approximation error bound of Theorem 2.

What would settle it

Build a synthetic dataset with a well-separated cluster $I$ in embedding $\psi_1$ and make $\psi_2$ group inside $I$ but not across its boundary, so that the conditions of Theorem 1 hold with small $\epsilon_1$ and $\epsilon_2$; if the top eigenvector of $\Lambda_{\psi_1,\psi_2}$ then fails to concentrate on $I$, with its entries on $I^c$ exceeding $2\sqrt{\epsilon_1^2+\epsilon_2}/\gamma$ from Corollary 1, the central interpretation is false. On real data, the same check is available by computing the block norms for a SPEC-reported cluster and comparing the resulting $\xi$ with the measured eigenvalue gap.

Watch

Extended reading notes

Core claim

The paper's central claim is a spectral identification result: under two conditions on an index set $I$ — that embedding $\psi_1$ separates $I$ from its complement, $\|\frac{1}{n}K_{\psi_1}[I,I^c]\|_F \le \epsilon_1$, and that embedding $\psi_2$ groups $I$ only weakly, $\|\frac{1}{n}K_{\psi_2}[I,I]\|_2 \le \epsilon_2$ — the eigenvectors of $\Lambda_{\psi_1,\psi_2}=\frac{1}{n}(K_{\psi_1}-K_{\psi_2})$ concentrate their mass on $I$ or its complement, with total squared residual bounded by $\xi=4(\epsilon_1^2+\epsilon_2)$. Corollary 1 turns this into a per-eigenvector bound: if an eigenvalue is separated by $\gamma$ from the spectrum of the complement block, the off-cluster entries of the eigenvector have squared norm at most $4(\epsilon_1^2+\epsilon_2)/\gamma^2$. Proposition 1 supplies the scalable route: $\Lambda_{\psi_1,\psi_2}$ and the differential covariance matrix $\Gamma_{\psi_1,\psi_2}$ share their non-zero eigenvalues, so the eigendecomposition runs in $O(n(d_1+d_2)^2 + (d_1+d_2)^3)$ time, and Algorithm 1 computes the needed covariances in one streaming pass over the samples. The same eigenvalue spectrum defines the distance SPEC-diff, the spectral radius of $\Lambda_{\psi_1,\psi_2}$, and SPEC-align minimizes this distance by gradient descent using the top left and right eigenvectors of $\Gamma_{\psi_1,\psi_2}$.

Load-bearing premise

The method assumes that the groups the two embeddings cluster differently are spectrally separable, showing up as a clear, well-gapped direction in the kernel difference matrix, and the paper gives no procedure for checking that this is true on a given real dataset.

Editorial extensions

If this is right

  • SPEC turns an embedding comparison into a checklist: each leading eigenvector names a concrete sample cluster, visualizable as images or sentences, that one model captures and the other misses.
  • Because the eigendecomposition runs in time linear in $n$ for bounded feature dimension, SPEC applies to datasets of 100K samples or more, such as MS-COCO and FFHQ, without subsampling.
  • SPEC-diff gives a symmetric, kernel-based distance between embeddings of different dimensions, and SPEC-align minimizes it by gradient descent; in the paper's setup, aligning CLIP to DINOv2 raised CLIP's frozen-feature ImageNet accuracy from 67.20% to 73.93%.
  • Random Fourier features extend the linear-time computation to shift-invariant kernels such as the Gaussian RBF kernel, with an error that decays as $O(1/\sqrt{m})$ in the number of features.
  • The comparison is directional: swapping the roles of the two embeddings reports clusters grouped by the second embedding but not the first, so the tool supports pairwise debugging of any two models.

Reading between the lines

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

  • One natural extension the paper leaves implicit is using SPEC as an unsupervised diagnostic for spurious correlations: a leading eigenvector that separates CLIP's text-overlaid clusters from DINOv2's content clusters is exactly the kind of label-free signal useful for auditing a model before deployment.
  • Because SPEC only needs kernel evaluations, it should apply to cross-modal and cross-architecture pairs beyond CLIP and DINOv2; the paper names cross-modal comparison as a future direction, and the same machinery would work for comparing image and text encoders with different dimensions.
  • The spectral-separability conditions suggest a practical validator: after a SPEC run, one could compute the measured block norms $\epsilon_1,\epsilon_2$ for each reported cluster and use the size of $\xi$ relative to the eigenvalue gap to flag which clusters are trustworthy.
  • A robustness check for real deployments would be to perturb the kernel matrices, by resampling the random Fourier features or subsampling the dataset, and ask whether the top clusters persist; the appendix reports RFF-count sensitivity, which supports this as a testable protocol.
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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

2 major / 5 minor

Summary. The paper proposes the Spectral Pairwise Embedding Comparison (SPEC) framework, which compares two feature embeddings by forming the normalized difference of their kernel matrices, Λ = (1/n)(K_ψ1 − K_ψ2), and interpreting the principal eigenvectors of Λ as clusters of samples grouped by ψ1 but not by ψ2. The paper proves Theorem 1, a residual bound over all eigenvectors under two conditions on a candidate index set I; Corollary 1, which bounds the entries of a single eigenvector on the complement set I^c under an additional spectral-gap condition; Proposition 1, showing that the nonzero spectrum of Λ can be obtained from a smaller differential covariance matrix with O(n) complexity for bounded feature dimension; Theorem 2, an RFF concentration bound for shift-invariant kernels; and Proposition 2, giving a gradient for a spectral-radius alignment objective. Experiments compare CLIP, DINOv2, Inception, SWAV, RoBERTa, and E5 embeddings on AFHQ, FFHQ, ImageNet, MS-COCO, and text datasets, and apply SPEC-align to finetune CLIP toward DINOv2, reporting improved ImageNet linear-probe accuracy.

Significance. If the central interpretation is justified, SPEC offers a scalable and genuinely interpretable tool for embedding comparison and alignment, complementing existing evaluation metrics that rely on downstream task accuracy. The O(n) differential-covariance implementation and the RFF concentration bound are concrete technical contributions; the large-scale experiments on 100K+ samples demonstrate practical scalability. The paper also makes an honest effort to validate its cluster interpretations beyond label-based metrics, notably the cosine-similarity ranking tests in Figures 16–17. However, the main theoretical claim is stated more strongly than what is proven: the eigenvector interpretation requires a spectral-separability condition that is neither stated as an explicit assumption nor verified empirically.

major comments (2)
  1. [Section 4 and Corollary 1] The paper states unconditionally that “the principal eigenvectors can be interpreted as the clusters of samples assigned by embedding ψ1 that are less strongly grouped by the second embedding ψ2,” but the proven support for this interpretation is conditional on a spectral gap: Corollary 1 requires λ − λmax(Λ[I^c, I^c]) ≥ γ > 0. Conditions 1 and 2 of Theorem 1 do not guarantee this gap; the block Λ[I^c, I^c] may itself have large positive eigenvalues, and the top eigenvalue may be degenerate, in which case an eigensolver returns an arbitrary mixture of cluster indicators. This gap between the stated claim and the proven theorem is load-bearing for the paper’s central contribution. Please restate the main claim as conditional on an explicit spectral-separability assumption, and add an empirical diagnostic (e.g., comparing the top eigenvalues of Λ with estimates of λmax(Λ[I^c, I^c]) for the identified clusters) to check this condition on real data.
  2. [Section 6, validation methodology] The AMI/NMI validation compares SPEC-derived cluster labels with K-Means labels computed on the same embedding that produced the eigenvectors. A high AMI for ψ1 only shows that SPEC labels are consistent with one possible clustering of ψ1’s space; it does not directly measure whether ψ2 fails to cluster those samples. The cosine-similarity ranking tests in Figures 16–17 are a much more direct validation of the “clustered by ψ1 but not ψ2” claim; please apply this type of check systematically to the reported clusters, or at least report a clustering-quality metric for ψ2 on the SPEC-identified sample groups.
minor comments (5)
  1. [Appendix A.1, final inequality] The proof uses the statement “for every PSD matrix A and vector v” to justify ∥Av − λv∥ ≥ |λ_j − λ| ∥v∥; the matrices Λ11 and Λ22 are not necessarily PSD. The inequality holds for all symmetric matrices, so replacing “PSD” with “symmetric” fixes the statement without changing the argument.
  2. [Definition 1] The notation “Λψ1,ψ2 ∈ Rn” is misleading; the matrix is n × n, so it should be R^{n×n}.
  3. [Algorithm 1, line 11] The mapped eigenvectors u_i are not normalized before being used for cluster identification; since clustering thresholds entries of u_i, normalization (or at least a stated convention) is needed for interpretability and comparability across eigenvectors.
  4. [Section 6, Experimental settings] The bandwidth selection rule “selected the embeddings bandwidths such that the difference between top eigenvalue is less than 0.01” is data-dependent and under-specified; please provide the exact procedure and report the chosen σ values for reproducibility.
  5. [Appendix A.3 and Remark 1] The Cholesky-based method for symmetric eigendecomposition of Θ is a useful contribution; however, the text should clarify that the correspondence holds only for nonzero eigenvalues and does not recover the null space of Λ, which is important when interpreting zero eigenvalues.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the SPEC eigenvector-cluster interpretation is supported by a self-contained perturbation bound; remaining issues are unverified spectral-gap assumptions and a self-referential sanity check, not circular reductions.

full rationale

The paper's central derivation is self-contained. Theorem 1 (Section 4, proof in Appendix A.1) treats the normalized kernel difference Λ = (1/n)(K_ψ1 − K_ψ2) as a perturbation of its block diagonal by the cross-block term Λ12, and uses Conditions 1–2 plus the PSD Schur complement to bound the sum of squared eigenvector residuals by ξ = 4(ε1² + ε2). Corollary 1 then converts this sum bound into an entrywise localization bound for an eigenvector whose eigenvalue clears the complement block by γ. This is a genuine mathematical argument with no fitted parameter and no appeal to the paper's own conclusions. The O(n) complexity (Proposition 1) is the standard nonzero-eigenvalue identity for AB and BA, proved in Appendix A.2; the RFF approximation (Theorem 2) is a separate concentration bound. SPEC-diff is defined as the spectral radius of Λ, and SPEC-align uses a gradient formula derived from the Jordan canonical form (Proposition 2), so the alignment objective does not presuppose the cluster interpretation. Two caveats are worth stating but they are not circularity. First, the paper's Section 4 wording, 'the principal eigenvectors can be interpreted as the clusters of samples assigned by embedding ψ1 that are less strongly grouped by the second embedding ψ2,' is stronger than what Conditions 1–2 alone prove; Corollary 1 requires an additional eigenvalue-gap condition, and the conclusion later admits 'the SPEC approach operates based on the assumption that the differently clustered samples can be detected by the spectral method.' This is an unverified assumption and a proof-to-claim gap, not a reduction of the claim to its inputs. Second, the KMeans-AMI sanity check in Section 6 compares SPEC labels to KMeans computed on the same embedding features whose kernel generated the eigenvectors, so high AMI is partly an internal-consistency check rather than independent ground truth; the bandwidth is also set using a criterion cited from the authors' prior generative-model papers. These choices affect interpretability of the experiments, but the theoretical result does not depend on them, and the ImageNet linear-probe result is an external benchmark. Overall, I find no step where a prediction is equivalent to a fitted parameter or to a self-citation by construction.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard linear algebra (AB/BA eigenvalue sharing), standard concentration (Vector Bernstein), and domain assumptions about normalized kernels and spectral separability of the target clusters. The main free parameters are the Gaussian bandwidth, selected by a data-dependent heuristic, the RFF count m=2000, which the paper's own ablation suggests is too small, and the alignment weight β in SPEC-align. No new entities are introduced.

free parameters (3)
  • Gaussian kernel bandwidth σ (per embedding, per dataset) = not reported; selected so that top eigenvalues differ by < 0.01
    Section 6: bandwidths chosen by heuristic that the difference between top eigenvalues is less than 0.01. This data-dependent normalization affects the kernel matrices, eigenvalues, and identified clusters, and is not reported numerically.
  • Number of random Fourier features m = 2000 (main experiments)
    Section 6: m=2000 used for all main experiments, but Appendix B.1 shows clusters are inconsistent for m <= 3000 and clean at m=6000. This is a user-chosen approximation parameter that materially affects results.
  • SPEC-align alignment loss weight β = 0.1
    Table 2: alignment loss weight in the SPEC-align objective (Eq. 6). The reported alignment results depend on this hyperparameter; no sensitivity analysis is provided.
assumptions (5)
  • standard math AB and BA share the same non-zero eigenvalues for any matrices A, B
    Used in Proposition 1 and Appendix A.2 to justify that the eigenspace of the n x n kernel difference matrix can be obtained from the (d1+d2) x (d1+d2) differential covariance matrix.
  • standard math For a symmetric matrix A and vector v, ||Av - λv|| >= min_j |λ_j(A) - λ| ||v||
    Used in the proof of Theorem 1 (Appendix A.1). The proof text says 'for every PSD matrix A', but the blocks Λ11 and Λ22 need not be PSD; the inequality holds for any symmetric matrix via spectral decomposition.
  • domain assumption Kernel matrices are PSD and normalized kernels satisfy k(x,x)=1, giving trace((1/n)K)=1
    Used in the proof of Theorem 1 to derive the bound on the Frobenius norm of the off-diagonal block (the trace equality). This is stated in Section 3.2.
  • domain assumption The top eigenvalue of Γ, in absolute value, is simple (unique) so that the gradient formula in Proposition 2 holds
    Proposition 2 assumes a unique top eigenvalue in absolute value; if this fails, the spectral radius is not differentiable and the SPEC-align gradient is not defined.
  • standard math Vector Bernstein inequality applies to the sum of bounded random matrices
    Used in the proof of Theorem 2 (Appendix A.4) to bound the RFF approximation error. Cited to Gross (2011) and Kohler & Lucchi (2017).

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Pith. "Pith review of Towards an Explainable Comparison and Alignment of Feature Embeddings." pith.science (2026). https://pith.science/paper/ODH2OL5U

@misc{pith2026250606231,
  author       = {Pith},
  title        = {Pith review of: Towards an Explainable Comparison and Alignment of Feature Embeddings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ODH2OL5U}},
  note         = {Machine review of arXiv:2506.06231}
}
read the original abstract

While several feature embedding models have been developed in the literature, comparisons of these embeddings have largely focused on their numerical performance in classification-related downstream applications. However, an interpretable comparison of different embeddings requires identifying and analyzing mismatches between sample groups clustered within the embedding spaces. In this work, we propose the \emph{Spectral Pairwise Embedding Comparison (SPEC)} framework to compare embeddings and identify their differences in clustering a reference dataset. Our approach examines the kernel matrices derived from two embeddings and leverages the eigendecomposition of the difference kernel matrix to detect sample clusters that are captured differently by the two embeddings. We present a scalable implementation of this kernel-based approach, with computational complexity that grows linearly with the sample size. Furthermore, we introduce an optimization problem using this framework to align two embeddings, ensuring that clusters identified in one embedding are also captured in the other model. We provide numerical results demonstrating the SPEC's application to compare and align embeddings on large-scale datasets such as ImageNet and MS-COCO. The project page is available at https://mjalali.github.io/SPEC/.

Figures

Figures reproduced from arXiv: 2506.06231 by the authors.

Figure 1
Figure 1. Overview of the Spectral Pairwise Embedding Comparison (SPEC) framework: The SPEC performs an eigendecomposition of the difference of kernel matrices following the two compared embeddings (e.g., DINOv2 and CLIP image embeddings) on a given reference dataset. Every eigenvector can be interpreted as a differently captured sample cluster by the embeddings, and the corresponding eigenvalue quantifies the difference betw… view at source ↗
Figure 2
Figure 2. Comparison of different embeddings on 15K samples from the AFHQ dataset, consisting of 5K cats, 5K wildlife, and 5K dogs. The number at the top of each image represents the eigenvalue of the corresponding SPEC cluster. The last two images in each row show the UMAP representation of the SPEC clusters for each embedding individually. alignment of embeddings has also been explored in the con￾current work by Gong et al.… view at source ↗
Figure 3
Figure 3. Top 4 SPEC-identified clusters comparing CLIP and DINOv2 embeddings on 10 ImageNet classes with overlaid text labels. the SPEC-identified different-captured sample groups by the two embeddings. In the Appendix, we further provide the t-SNE (Van der Maaten & Hinton, 2008) and PaCMAP (Wang et al., 2021) plots of the FFHQ and AFHQ exper￾iments. Also, we have analyzed the found clusters using violin plots to visualize n… view at source ↗
Figures from the paper (23 more)
Figure 4
Figure 4. Figure 4: Top 4 SPEC-identified clusters by comparing CLIP and RoBERTa text embeddings on a dataset generated from GPT-4o. CLIP Kernel SPEC-align CLIP Kernel DINOv2 Kernel [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: Comparison of Kernel matrices after using SPEC-align to match the sample clusters of CLIP to DINOv2. also noted from the UMAP visualization of the samples that the top clusters of each embedding are not well clustered by the other, indicating that they focus on differe…
Figure 6
Figure 6. Figure 6: Top 4 SPEC-identified clusters comparing CLIP - DINOv2 with different Random Fourier features (RFF) r ranging from 1000 to 6000 using Cholesky decomposition on MS-COCO dataset. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]
Figure 7
Figure 7. Figure 7: Top 4 SPEC-identified clusters comparing DINOv2 - CLIP with different Random Fourier features (RFF) r ranging from 1000 to 6000 using Cholesky decomposition on MS-COCO dataset. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]
Figure 8
Figure 8. Figure 8: Top 4 SPEC-identified clusters comparing CLIP - DINOv2 embeddings on MS-COCO dataset using Cholesky decomposition with different Random Fourier Features (2000 and 6000). The second row shows the t-SNE and UMAP representation of the top 10 SPEC-identified clusters for e…
Figure 9
Figure 9. Figure 9: Top 4 SPEC-identified clusters comparing DINOv2 - CLIP embeddings on MS-COCO dataset using Cholesky decomposition with different Random Fourier Features (2000 and 6000). The second row shows the t-SNE and UMAP representation of the top 10 SPEC-identified clusters for e…
Figure 10
Figure 10. Figure 10: Comparison of SPEC-identified clusters across different visualization methods (PacMAP, t-SNE, UMAP) of [PITH_FULL_IMAGE:figures/full_fig_p024_10.png]
Figure 11
Figure 11. Figure 11: Top 4 SPEC-identified clusters by comparing CLIP and RoBERTa text embeddings on the WikiText-2 dataset with the visualization of the top 10 SPEC-identified clusters using t-SNE. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_11.png]
Figure 12
Figure 12. Figure 12: Top 4 SPEC-identified clusters by comparing CLIP and RoBERTa text embeddings on a dataset of 10K samples generated from GPT-4o with the visualization of the top 10 SPEC-identified clusters using t-SNE and UMAP. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_12.png]
Figure 13
Figure 13. Figure 13: Top 4 SPEC-identified clusters by comparing CLIP and E5-Large-V2 text embeddings on a dataset of 10K samples generated from GPT-4o with the visualization of the top 10 SPEC-identified clusters using t-SNE. 28 [PITH_FULL_IMAGE:figures/full_fig_p028_13.png]
Figure 14
Figure 14. Figure 14: Top 4 SPEC-identified clusters by comparing RoBERTa and E5-Large-V2 text embeddings on MS-COCO 2017 train captions ( 120K prompts) with the visualization of the top 10 SPEC-identified clusters using t-SNE. 29 [PITH_FULL_IMAGE:figures/full_fig_p029_14.png]
Figure 15
Figure 15. Figure 15: Top 4 SPEC-identified clusters comparing CLIP and DINOv2 embeddings on ImageNet-1k dog breeds. The last row shows the UMAP representation of the top 10 SPEC-identified clusters for each embedding. 30 [PITH_FULL_IMAGE:figures/full_fig_p030_15.png]
Figure 16
Figure 16. Figure 16: Comparing similarity ranking for SPEC clusters in DINOv2-CLIP on the AFHQ dataset. The leftmost images show the top 4 samples of two SPEC-identified clusters. Cosine similarity is computed with 4 cluster members (green-bordered) and 4 random images (red-bordered), sor…
Figure 17
Figure 17. Figure 17: Comparing similarity ranking for SPEC clusters in DINOv2-CLIP on the FFHQ dataset. The leftmost images show the top 4 samples of two SPEC-identified clusters. Cosine similarity is computed with 4 cluster members (green-bordered) and 4 random images (red-bordered), sor…
Figure 18
Figure 18. Figure 18: Comparing Different embeddings on the 120K samples from MS-COCO 2017 dataset. 5 0 15 10 5 10 5 0 5 10 15 20 8 6 4 2 10 2 4 6 8 10 12 14 9 8 7 6 5 4 3 2 1 0 12.5 10.0 7.5 5.0 2.5 0.0 2.5 5.0 7.5 10.0 7.5 5.0 2.5 0.0 2.5 5.0 7.5 10.0 8 6 4 12 10 4 2 0 2 4 6 9 8 7 6 5 4 …
Figure 19
Figure 19. Figure 19: Comparing embeddings on 70K FFHQ samples. Top numbers show SPEC cluster eigenvalues. Last two images per row display UMAP representations of SPEC clusters for each embedding. 32 [PITH_FULL_IMAGE:figures/full_fig_p032_19.png]
Figure 20
Figure 20. Figure 20: Comparing embeddings on 70K FFHQ samples. Top numbers show SPEC cluster eigenvalues. Last two images per row display UMAP representations of SPEC clusters for each embedding. B.6. SPEC-align Experiments SPEC-align using ℓ2-Norm of Kernel Difference Matrix. As mentione…
Figure 21
Figure 21. Figure 21: Comparison of different embeddings on 15K samples from the AFHQ dataset, consisting of 5K cats, 5K wildlife, and 5K dogs. The number at the top of each image represents the eigenvalue of the corresponding SPEC cluster. The last two images in each row show the UMAP rep…
Figure 22
Figure 22. Figure 22: SPEC-align using the ℓ2-norm of the eigenvalues of the kernel difference matrix. Left: original CLIP kernel. Center: aligned CLIP kernel using ℓ2-norm-based SPEC-align. Right: target DINOv2 kernel. Bottom: SPEC-diff score (ℓ2-Norm) over alignment iterations. 35 [PITH…
Figure 23
Figure 23. Figure 23: Top 8 Kernel-PCA (Gaussian RBF kernel) clusters for CLIP, DINOv2, and CLIP aligned with DINOv2, trained on the ImageNet dataset. 36 [PITH_FULL_IMAGE:figures/full_fig_p036_23.png]
Figure 24
Figure 24. Figure 24: Top 8 Kernel-PCA (Gaussian RBF kernel) clusters for CLIP, DINOv2, and CLIP aligned with DINOv2, trained on the ImageNet dataset. 37 [PITH_FULL_IMAGE:figures/full_fig_p037_24.png]
Figure 25
Figure 25. Figure 25: Comparison of Kernel matrices after using SPEC-align to match the sample clusters of CLIP to T5-XL with measuring SPEC-diff during the training. 39 [PITH_FULL_IMAGE:figures/full_fig_p039_25.png]
Figure 26
Figure 26. Figure 26: Comparison of Kernel matrices after using SPEC-align to align CLIP to DINOv2. 40 [PITH_FULL_IMAGE:figures/full_fig_p040_26.png]

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.