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Whitened CLIP as a Likelihood Surrogate of Images and Captions

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

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Pith's one-line read Whitening CLIP embeddings turns their squared norm into a direct log-likelihood estimate for images and captions.

desk verdict A useful and clearly-written heuristic for scoring CLIP embeddings, but the likelihood framing leans on an unvalidated joint-normality assumption that the paper's own norm statistics partially contradict. read the letter →

arxiv 2505.06934 v1 pith:VXRE7C6V submitted 2025-05-11 eess.IV cs.CV

Roy Betser , Meir Yossef Levi , Guy Gilboa This is my paper · ORCID
classification eess.IVcs.CV
keywords WhitenedCLIPembeddingslikelihoodestimationwhiteningtransformmultimodalrepresentationout-of-distributiondetectionimageartifactssphericalinterpolation
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 proposes Whitened CLIP (W-CLIP), a training-free linear whitening of the CLIP latent space, and argues that after this transform, image and caption embeddings behave approximately like standard normal vectors. If that approximation holds, the log-likelihood of an image or caption under CLIP's learned distribution is just $-\frac{1}{2}(d\log(2\pi)+\|y\|^2)$, where $y$ is the whitened embedding, so a meaningful probability score costs only one matrix multiplication. The authors support the normality assumption with Anderson-Darling and D'Agostino-Pearson tests and show the resulting scores separate real images from generated images with artifacts, detect domain shifts such as ImageNet-C and ImageNet-R, reveal generation bias, and make full-circle interpolation between images possible in the whitened space. A reader should care because explicit likelihoods for images have been hard to obtain, and this offers a fast, semantic alternative that plugs into existing CLIP-based pipelines.

What carries the argument

The carrying mechanism is PCA whitening: from a representative set of CLIP embeddings the paper builds the empirical covariance $\Sigma$, diagonalizes it as $\Sigma = V\Lambda V^\top$, and applies the invertible map $W = \Lambda^{-1/2}V^\top$ to centered embeddings, so $y = W(x-\mu)$ has zero mean and identity covariance. Under the paper's normality claim, this reduces the probability model to the standard normal density, making log-likelihood a function of the squared Euclidean norm. The companion object is the chi distribution of the norm, whose mean $\sqrt{d-\tfrac{1}{2}}$ drives the thin-shell intuition used to stabilize iterative generation and to explain why nearly all mass sits near a sphere of radius $\sqrt{d}$.

What would settle it

Take a held-out set of images never used to fit the whitening matrix, whiten their CLIP embeddings, and compare the empirical distribution of $\|y\|^2$ with the $\chi^2_d$ distribution predicted by Eq. (6). A large tail excess or a mean norm far from $\sqrt{d-\tfrac{1}{2}}$ would show the likelihood surrogate does not hold outside the fitting set.

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Extended reading notes

Core claim

The paper's central claim is that the CLIP embedding distribution, treated separately for images and for text, is approximately elliptical and becomes approximately isotropic standard normal after the invertible whitening map $y = \Lambda^{-1/2}V^\top(x-\mu)$. Under that model the paper defines the log-likelihood surrogate $\ell(x) = -\frac{1}{2}(d\log(2\pi)+\|y\|^2)$ and identifies the norm $\|y\|$ with a chi distribution whose typical radius is $\sqrt{d-\tfrac{1}{2}}$. The authors state that to their knowledge this is the first direct computation of likelihood functions for images and text prompts under the CLIP-learned distribution; the experiments are presented as preliminary demonstrations of the properties and applications of these scores.

Load-bearing premise

The load-bearing premise is that the whitened embeddings are jointly standard normal, not merely coordinate-by-coordinate normal and uncorrelated; the paper's checks are per-feature and are run on the same validation set used to fit the whitening transform.

Editorial extensions

If this is right

  • W-CLIP can rank real images above generated images with artifacts, giving a fast generated-image detector.
  • Corrupted ImageNet-C images and stylized ImageNet-R renditions receive lower likelihoods than clean ImageNet, with severity moving the norm histogram further away from the in-distribution set.
  • Iterative UnCLIP re-encoding drifts into noise, while normalizing each whitened embedding to radius $\sqrt{d}$ keeps outputs within the natural image space, exposing a likelihood bias of the generator.
  • Full-circle SLERP in W-CLIP keeps interpolated and extrapolated images natural and semantically meaningful, whereas raw-CLIP full-circle SLERP collapses into structured noise for off-segment angles.
  • Caption likelihoods decrease when captions become more specific or contain grammatical errors, and W-CLIP separates captions from general text more sharply than language-model likelihoods.

Reading between the lines

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

  • Editorial: the same whitening-plus-norm recipe should transfer to other contrastive vision-language embeddings; the paper tests two CLIP variants, so a held-out norm-distribution check on another model would test the generality.
  • Editorial: the thin-shell result implies a calibrated 'typical set' around radius $\sqrt{d}$; a threshold on normalized norm deviation might be more stable across datasets than raw likelihood.
  • Editorial: because image and text whitening matrices are fit separately, W-CLIP likelihoods are not directly comparable across modalities; a cross-modal calibration step would be needed before using the score to compare an image with its caption.
  • Editorial: the moderate correlation with language-model likelihoods (0.33-0.48) suggests W-CLIP carries complementary information, so a combined score could be more robust to text type and length than either measure alone.
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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 Whitened CLIP (W-CLIP), an invertible linear transformation of CLIP image and text embeddings via PCA whitening. Under the assumption that the whitened embeddings are jointly standard normal, the log-likelihood of an embedding is approximated as -1/2(d log(2π) + ||y||^2) (Eq. 6). The authors validate per-feature normality with Anderson-Darling and D'Agostino-Pearson tests, argue that whitening yields i.i.d. features, and demonstrate applications including artifact detection, domain-shift ranking, generation-bias measurement, and full-circle SLERP interpolation. They also compare their text likelihood scores with language-model NLLs and perform cross-dataset and cross-CLIP-model ablations.

Significance. If the central normality assumption is justified, W-CLIP offers a fast, training-free, and invertible likelihood surrogate for CLIP latents, with a clean derivation and a wide range of downstream uses. The paper's strengths include a reproducible code release, a principled derivation from the Gaussian assumption to Eq. (6), cross-dataset (Flickr8k) and cross-model (ViT-B/32) ablations, and a clear set of falsifiable experimental probes (artifact detection, ImageNet-C/R shifts, generation bias). The probabilistic claim is, however, the load-bearing component: Eq. (6) is only a likelihood if the whitened embeddings are jointly standard normal. The current validation establishes, at most, approximate marginal Gaussianity and enforced decorrelation in-sample; it does not establish joint normality, and an omitted Jacobian term further muddies the interpretation as a likelihood of the original embeddings. These gaps are fixable with additional multivariate tests and a clarifying statement, so the paper merits a major revision rather than rejection.

major comments (2)
  1. [Sec. 3.3, Eq. (6)] The validation of the normal-distribution assumption is performed per feature on the same 5,000 MS-COCO validation embeddings used to compute the whitening matrix W. Since W is constructed to make the sample mean zero, the sample variance unity, and the sample covariance the identity on this same set, the covariance diagonality reported in Fig. 4(b) and the zero-mean/unit-variance claims in Fig. 4(d) are enforced by construction and do not constitute evidence about the population distribution. The Anderson-Darling and D'Agostino-Pearson tests are applied to individual coordinates and cannot detect non-Gaussian dependence among coordinates; the statement in Sec. 3.3 that 'in normal distribution, non-correlation is a sufficient condition for independence' assumes joint normality, which is precisely the property needing validation. As a result, the distribution of ||y||^2 need not be chi-squared, so Eq. (6) is not established as a log-likelihood. This is load-bearing because the entire probabilistic interpretation, the chi-distribution comparison in Tab. 2, and the use of the norm as a probability score depend on y being jointly N(0,I). The 13.24% relative deviation of the text norm standard deviation in Tab. 2 already signals a departure. Please add a direct test of joint normality or of the norm/chi-squared fit, and perform it on a held-out set with W computed on a separate training set (e.g., a 2k/3k split).
  2. [Sec. 3.4, Eq. (6)] Equation (6) is the log-density of the whitened vector y under a standard normal model, as Eq. (5) makes clear. If the paper claims to provide the likelihood of a raw CLIP embedding or of an image/caption, the invertible change of variables y = W(x - μ) introduces an additive log|det W| term that is missing. For a fixed W this constant cancels in relative comparisons, but it differs between the image and text whitening matrices, so the paper's image and text 'log-likelihood' scores are not directly comparable as likelihood values, and cross-modality comparisons (or any use in a setting with multiple W matrices) require the constant. Please state explicitly what random variable Eq. (6) describes and either include the Jacobian term or restrict the claims to within-space relative scores.
minor comments (5)
  1. [Sec. 3.3] The sentence 'in normal distribution, non-correlation is a sufficient condition for independence' is only valid under joint normality; please rephrase to avoid appearing to assume the very property being tested.
  2. [App. D.1, Alg. 1] The whitening procedure replaces highly correlated features with random noise r ~ N(0,0.1), and the correlation threshold τ is a free parameter. Please report the chosen τ and provide a sensitivity analysis; the current statement that the impact is minimal is not backed by quantitative evidence.
  3. [Tab. 3] Table 3 is difficult to parse because the column values for the two modalities are interleaved without clear separation. Please reformat so each row has explicit Data T, Data W, Avg. AD, Avg. DP, and Correlation columns, one row per modality.
  4. [Sec. 4.3] The text contains a typo: 'a dog with a bottle of bear' should read 'a dog with a bottle of beer'.
  5. [App. D.3, Eq. (12)] The Anderson-Darling formula is displayed with an awkward line break; please ensure the prefactor is typeset clearly as -n - (1/n) Σ ... so that the statistic is unambiguous.

Circularity Check

2 steps flagged · score 4.0 of 10

In-sample whitening makes part of the i.i.d. validation tautological, but cross-dataset and alternate-model checks keep the core claim from being fully circular.

  1. self definitional [Sec. 3.3, 'Whitened CLIP embeddings' (i.i.d. paragraph, Fig. 4)]
    "While the CLIP embeddings exhibit correlations between features, the covariance matrix of the whitened embeddings is almost exactly diagonal, indicating that the features are uncorrelated. This is expected, since the whitening transform is designed for exactly this purpose."

    By construction, W = Λ^(-1/2)V^T diagonalizes the empirical covariance of the same MS-COCO validation vectors used for Fig. 4, so the reported identity covariance and unit variances are not independent evidence; as the text admits, 'the whitening transform is designed for exactly this purpose.' The step then converts this forced uncorrelatedness into independence by invoking the claim that 'in normal distribution, non-correlation is a sufficient condition for independence,' which presupposes the joint normality that Eq. (6) needs and that the marginal Anderson-Darling and D'Agostino-Pearson tests do not establish. The load-bearing i.i.d. premise of Eqs. (5)-(6) is therefore partly a restatement of the whitening fit plus an assumed, rather than directly validated, joint Gaussianity.

  2. fitted input called prediction [Sec. 4 opening; Sec. 3.3, Tab. 1 and Tab. 2]
    "All the experiments in this section employ the CLIP ViT-L/14 model and utilize the MS-COCO validation set to compute the whitening matrix W."

    The normality statistics (Tab. 1), the covariance diagonality (Fig. 4), and the norm statistics (Tab. 2) are computed on the same 5,000 MS-COCO validation embeddings from which W is estimated. Since W is chosen to make this exact sample have zero mean and identity covariance, the covariance identity, the implied trace condition, and part of the 'i.i.d.' characterization of those same vectors are consequences of the fitting procedure rather than independent predictions. The paper partially mitigates this with cross-dataset (Flickr8k, Tab. 3) and alternate-backbone (ViT-B/32, Tab. 6) checks, but the in-sample validation used to justify Eq. (6) remains partly circular.

full rationale

The paper's Eq. (6) is not derived from data; it is a proposal that whitened CLIP embeddings are approximately N(0, I_d), making the negative log-likelihood a function of the squared norm. The central circularity risk is therefore not in Eq. (6) itself, but in the evidence used to justify the N(0, I_d) premise. Two aspects of that evidence reduce to construction: the whitening transform W is fit to the MS-COCO validation set, and the same set is then used to display identity covariance and near-unit variances, which W enforces by design; and the independence claim is obtained by invoking the normal-distribution fact that uncorrelatedness implies independence, which assumes the very joint normality that is never directly tested. Those steps are partly self-definitional. However, the paper does not end there: Tab. 3 tests normality and likelihood correlations using Flickr8k as the test set with a COCO-fitted W, Tab. 6 repeats with a different CLIP backbone, and the ImageNet-A/C/R experiments provide external behavioral checks independent of the fitted values. Those checks give Eq. (6) genuinely independent content, so the circularity is partial rather than total. The self-citation to Levi and Gilboa 2025 for the modality gap is accompanied by Liang et al. 2022 and is not load-bearing. Overall score 4: some validation claims reduce by construction, but the central likelihood surrogate has independent empirical support.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central likelihood score rests on the assumption that whitened CLIP embeddings are jointly standard normal. The paper fits the whitening matrix (mean and covariance) to MS-COCO validation data, and validates normality only per-feature on the same data. The text modality additionally requires ad hoc removal of correlated features and injection of random noise. No new physical or conceptual entities are introduced.

free parameters (3)
  • Whitening matrix W and mean vector μ = 768x768 matrix and 768-dim vector (ViT-L/14); not reported in paper
    Computed from MS-COCO validation embeddings; every likelihood score depends on these fitted statistics.
  • Correlation threshold τ = not reported
    Algorithm 1 input for removing highly correlated text features; value is omitted, affecting the text whitening matrix.
  • Random noise for replaced features = N(0, 0.1)
    Ad hoc stochastic replacement for removed features in text whitening; introduces nondeterminism.
assumptions (4)
  • domain assumption Whitened CLIP embeddings are jointly standard normal.
    The log-likelihood formula (Eq. 6) is the Gaussian density; the paper checks only marginal normality and uncorrelatedness, not joint Gaussianity, on the fitting set (Sec. 3.3).
  • standard math Non-correlation implies independence for the whitened features.
    True for Gaussian vectors, but it is invoked to claim i.i.d. before joint Gaussianity is established (Sec. 3.3).
  • domain assumption Image and text CLIP embeddings are disjoint and each can be modeled as a single Gaussian after whitening.
    The paper treats the two modalities separately due to the modality gap, and assumes one whitened Gaussian per modality (Sec. 3.2).
  • domain assumption The empirical covariance of the validation set is non-singular and representative; for text, correlated features can be replaced with random noise without affecting results.
    The whitening algorithm requires invertible covariance; for text, features are dropped and replaced with N(0, 0.1) noise, an ad hoc choice (App. D.1, Alg. 1).

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Cite this review

Pith. "Pith review of Whitened CLIP as a Likelihood Surrogate of Images and Captions." pith.science (2026). https://pith.science/paper/VXRE7C6V

@misc{pith2026250506934,
  author       = {Pith},
  title        = {Pith review of: Whitened CLIP as a Likelihood Surrogate of Images and Captions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VXRE7C6V}},
  note         = {Machine review of arXiv:2505.06934}
}
read the original abstract

Likelihood approximations for images are not trivial to compute and can be useful in many applications. We examine the use of Contrastive Language-Image Pre-training (CLIP) to assess the likelihood of images and captions. We introduce \textit{Whitened CLIP}, a novel transformation of the CLIP latent space via an invertible linear operation. This transformation ensures that each feature in the embedding space has zero mean, unit standard deviation, and no correlation with all other features, resulting in an identity covariance matrix. We show that the whitened embeddings statistics can be well approximated as a standard normal distribution, thus, the log-likelihood is estimated simply by the square Euclidean norm in the whitened embedding space. The whitening procedure is completely training-free and performed using a pre-computed whitening matrix, hence, is very fast. We present several preliminary experiments demonstrating the properties and applicability of these likelihood scores to images and captions.

Figures

Figures reproduced from arXiv: 2505.06934 by the authors.

Figure 1
Figure 1. Raw, centered and whitened CLIP geometry. The whitened CLIP space is isotropic, transforming the original ellip￾soid shaped space into an hypersphere. In this space, the embed￾ding norm reflects likelihood level. Higher norms correspond to lower probabilities. dual text-image semantics. However, its potential as a likeli￾hood surrogate remains unexplored. This paper introduces Whitened CLIP (W-CLIP), a linear whiten… view at source ↗
Figure 2
Figure 2. Log-likelihood of real and generated images with artifacts. Real images of a hand and a dog (left) and three similar AI generated images with artifacts. Real images have higher log-likelihood than generated images with artifacts. 4. We show W-CLIP can be used to estimate probability drifts in generative models, discover artifacts in im￾age generation and rank statistical deviation of out-of￾distribution (OOD) benchm… view at source ↗
Figure 3
Figure 3. Norm histograms of ImageNet variations. Top left: ImageNet-A, comprising of natural adversarial examples, closely aligns with clean ImageNet due to their natural origins. Top right: ImageNet-C histograms under varying impulse noise levels of severity display significantly larger norms than clean ImageNet, indicating distributional deviations. Bottom: ImageNet-R comparison shows that different styles cause varying li… view at source ↗
Figures from the paper (23 more)
Figure 4
Figure 4. Figure 4: Raw CLIP and W-CLIP analytic comparison. The covariance matrices of raw CLIP (a) and W-CLIP (b) demonstrate the effectiveness of the whitening transformation in achieving unit variance and zero correlation among features. Histograms of four CLIP features (c) vary in me…
Figure 5
Figure 5. Figure 5: Likelihood variation for different levels of details. The original MS-COCO caption is marked with an arrow, with deviations underlined. Left: Removing details, such as character names or locations, increases likelihood. Right: Adding specificity, such as replacing “wom…
Figure 6
Figure 6. Figure 6: Bias in image generation. Left: Using CLIP-encoded images in MS-COCO validation set as a condition for generating new images. The histogram shows a bias towards lower likelihoods in generated images. Right: Iteratively using UnCLIP to generate images encoded by CLIP wi…
Figure 7
Figure 7. Figure 7: Differences between likelihood functions. Our proposed likelihood estimation is highly sensitive to grammatical errors (a), demonstrated by the removal of all nouns from the captions, and text type (b), where Text Data refers to a general text dataset (OpenWebText) and…
Figure 8
Figure 8. Figure 8: Full circle SLERP example. The full circle SLERP is performed in both the raw CLIP space (a) and in the W-CLIP space (b). The different angle between embeddings in both space is presented. In the raw CLIP space the full circle SLERP results with noise for most of the d…
Figure 9
Figure 9. Figure 9: Log-likelihood of real and generated images with artifacts. Real images of zebras and a surfer (left) and three similar AI generated images with artifacts. Real images have higher log-likelihoods than AI generated images with artifacts. 13 [PITH_FULL_IMAGE:figures/ful…
Figure 10
Figure 10. Figure 10: Log-likelihood of real captions and captions with artifacts. Real captions, framed with a blue frame and atrifacted captions, where we removed the first or last words from a caption, or one of the middle words. In all cases the original caption has the highest log-lik…
Figure 11
Figure 11. Figure 11: ImageNet-C histograms on all corruptions. All corruptions have a significantly higher norm (lower log-likelihood) than ImageNet. For most corruptions, as level of corruption increases the norm increases. Some corruptions do not show this monotonic behavior (motion/gla…
Figure 12
Figure 12. Figure 12: Likelihood for different text types. Comparing likelihood values computed for MS-COCO captions and OpenWebText general text sentences. The sentences from OpenWebText are filtered to have similar lengths to MS-COCO captions. LLMs (OPT, NEO, GPT2) treat captions similar…
Figure 13
Figure 13. Figure 13: Likelihood drift when removing nouns Comparing likelihood values computed for MS-COCO captions with and without nouns. None of the models show a drift like our likelihood (Tab. 5). 16 [PITH_FULL_IMAGE:figures/full_fig_p016_13.png]
Figure 14
Figure 14. Figure 14: Log-likelihood values and correlations. Log-likelihoods are computed on 5,000 captions from the MS-COCO validation set, with correlations measuring the alignment of each model’s log-likelihood and ours [PITH_FULL_IMAGE:figures/full_fig_p017_14.png]
Figure 15
Figure 15. Figure 15: Examples of differences between language models and our method. The sorted index represents the position out of 5000 captions from MS-COCO, ranked from low to high likelihood values. The relative likelihood index is compared among GPT2 (LLM), BLIP (VLM), and our metho…
Figure 16
Figure 16. Figure 16: Normal distribution tests on image embeddings. Top row - mean value and standard deviation per feature, over all groups of embeddings. Bottom - histogram of mean values of each feature. In all plots the red line represents the test threshold. Left - Anderson-Darling t…
Figure 17
Figure 17. Figure 17: Normal distribution tests on text embeddings. Top row - mean value and standard deviation per feature, over all groups of embeddings. Bottom - histogram of mean values of each feature. In all plots the red line represents the test threshold. Left - Anderson-Darling te…
Figure 18
Figure 18. Figure 18: Mean and variance of all whitened features. We show the mean and variance of all the 768 features of the whitened embeddings. There are minor deviations from 0 (for mean) and 1 (for variance). For mean values the deviation is up to 0.0015% and in the case of the varia…
Figure 19
Figure 19. Figure 19: Normal distribution tests with different data sizes Anderson-Darling average scores and standard deviation (a) and D’Agostino-Pearson p-value average scores and standard deviation (b). Threshold is under 0.752 (marked with a black line) and above 0.05, respectively. C…
Figure 20
Figure 20. Figure 20: 2D full circle SLERP example. The SLERP points are in blue and the circle perimeter is in orange. Examples of a simple 2D case of full circle SLERP. When both points are on the circle (a) the SLERP points follow the circle perimeter perfectly. If one of the points dev…
Figure 21
Figure 21. Figure 21: Full circle SLERP example. The full circle SLERP is performed in both the raw CLIP space (a) and in the W-CLIP space (b). The different angle between embeddings in both space is presented. In the raw CLIP space the full circle SLERP results with noise for most of the …
Figure 22
Figure 22. Figure 22: Opposite image generated in the raw CLIP space. The structured noise produced by CLIP exhibits 4×4 pixel blocks and a restricted color palette (black (’0’ in all color channels), white (’1’ in all color channels), red, green, blue, magenta (’1’ in red and blue channel…
Figure 23
Figure 23. Figure 23: Generation bias. Iteratively using UnCLIP to generate images encoded by CLIP with two fixed seeds. The raw process gradually becomes noisy, whereas with normalization (to √ d at each encoding step), the content drifts but remains within a natural and reasonable image …
Figure 24
Figure 24. Figure 24: Generation bias. Iteratively using UnCLIP to generate images encoded by CLIP with two fixed seeds. The raw process gradually becomes noisy, whereas with normalization (to √ d at each encoding step), the content drifts but remains within a natural and reasonable image …
Figure 25
Figure 25. Figure 25: Adding details to concepts. The original caption from MS-COCO is framed in blue. Adding details decreases the likelihood. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_25.png]
Figure 26
Figure 26. Figure 26: Removing details from concepts. The original caption from MS-COCO is framed with a blue frame. Removing different details increases the likelihood. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_26.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. The Hyperspherical Geometry of CLIP Latent Space: A Semantic Mixture Model

    cs.LG 2026-07 conditional novelty 5.0 of 10

    CLIP embeddings are modeled as a mixture of von Mises-Fisher distributions on the unit sphere, improving out-of-distribution detection and semantic decomposition over single-Gaussian baselines.

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Pith tools

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