Quantifying Structure in CLIP Embeddings: A Statistical Framework for Concept Interpretation
Authors: Jitian Zhao
Chenghui Li
Frederic Sala
Karl Rohe
Venue: Accepted by TMLR ()

Abstract: Concept-based approaches, which aim to identify human-understandable concepts within a model's internal representations, are promising for interpreting embeddings from deep neural network models, such as CLIP. While these approaches help explain model behavior, current methods lack statistical rigor, making it challenging to validate identified concepts and compare different techniques. To address this challenge, we introduce a hypothesis testing framework that quantifies rotation-sensitive structures within the CLIP embedding space.
Once such structures are identified, we propose a post-hoc concept decomposition method. Unlike existing approaches, it offers theoretical guarantees that discovered concepts represent robust, reproducible patterns (rather than method-specific artifacts) and outperforms other techniques in terms of reconstruction error. Empirically, we demonstrate that our concept-based decomposition algorithm effectively balances reconstruction accuracy with concept interpretability and helps mitigate spurious cues in data. Applied to a popular spurious correlation dataset, our method yields a 22.6% increase in worst-group accuracy after removing spurious background concepts.

Published in Transactions on Machine Learning Research (02/2026)
Quantifying Structure in CLIP Embeddings: A Statistical Framework for Concept Interpretation
Jitian Zhao
jahao8269wris.eedu
Deportment of Statistics, University of Visconsinn-Madion
Chenghui Li Department of Statistics, University of Wiscomsin-Mudiisnn
ck5399wissc.ett edu
Frederic Sala
fredsala Curisc. edu
Department of Computer Science, University of Wisconsin-Madisn
Karl Rohe
karl.rohe@wisc..o edu
Department of Statistics, University of Miscomsinn-Madison
Reviewed on OpenReview: https: // operrevee..........mm net/ pdfp id- D6KOWL3RRY
Abstract
Concept-based approaches, which aim to identify human- understandalle concepts within a model's internal representations, are promising for interpreting embeddings from deep neu- ral network models, such as CLIP. While these approaches help explain model behavior, current methods lack statistical rigor, making it challenging to validate identified concepts and compare different techniques. To address this challenge, we introduce hypothesis testing framework that quantifies rotation-sensitive structures within the CLIP embedding space. Once such structures are identified, we propose a post-hoc concept decomposition method. Unlike existing approaches, offers theoretical guarantecs that discovered con- cepts represent robust, reproducible palterns (rather than method-specific artifacts) and outperforms other techniques in terms of reconstruction error. Empirically, we demonstrate that our concept-based decomposition algorithm effectively balances reconstruction racy with concept interpretability and helps mitigate spurious cues in data. Applied to popular spurious correlation dataset, our method yields a 22.6% increase in worst-group accuracy after removing spurious background concepts.
Introduction
CLIP has become foundational model for a wide range of visual applications (Radford et al. 2021 Interpreting its high-dimensional embeddings is challenging due to the complex and entangled nature of the learned representations. Recent works address this via concept-based decomposition, aiming to identify interpretable semantic patterns within model components, embeddings, and neurons (Gandelsman et al. 2024a Balasubramanian et al. 2024).
Existing approaches broadly fall into two categories. The first category, exemplified by SpLiCE (Bhalla et al. 2024). decomposes CLIP embeddings into sparse, human-interpretable concepts such words. While this sparse decomposition improves interpretability, it introduces reconstruction errors, meaning that substantial portion of the original embedding's information is lost. This negatively impacts downstream tasks such as zero-shot classification, where preserving semantic information crucial. The second category uses Singular Value Decomposition (SVD) to decompose embeddings into linear combinations of concopt vectors (Fel et al. 22044 Graziani al. 2023 Zhang et al. (2021). These methods maintain high reconstruction fidelity (i.e., the reconstructed embedding closely approximates the original with minimal error by filtering

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out noise). However, they often struggle with concept interpretability, singular vector directions are not inherently aligned with human-interpretabbe concepts, making their meaning largely dependent on human intuition. This highlights a trade-off: methods that prioritize interpretability often sacrifice reconstruction fidelity, while those that preserve fidelity tend to lack meaningful concept alignment.
Beyond this seeming interpredabiltyy-rcoonnrru existing methods are ad-hoc rather than the result of a rigorous statistical framework. In real-world settings, embeddings are inherently noisy, and without statistical guarantees, it is unclear whether the concepts extracted by given method capture meaningful structure merely reflect artifacts of noise. A key challenge is that noise and meaningful structure can both produce scemingly interpretable components, leading to silent failures where methods extract spurious "concepts" that do not correspond to real semantic attribules.
fidelity trade-off, an even deeper issue remains:
To address both challenges, we first propose a hypoth- esis testing framework detect rotation-sensitive structure in the embedding subspace. Our key in- sight is that semantic concepts manifest direc- tional patterns in embedding space that are sensitive to rotation, unlike random noise, which remains sta- tistically unchanged under rotation. By testing for rotation-sensitive patterns, our method distinguishes noise from true underlying structure, ensuring that extracted concepts reflect meaningful, stable properties of the data rather than arbitrary artifacts.
Table 1: Sparsity and Fidelity Across Techniques
SVD
SPLICE Ours
Sparsity
Fidelity
Building the theoretical insights of Varimax rotation (Rohe& & Zeng 2020), we develop a post-hoc method that requires no additional training human annolation. Our approach achieves both the interpretabil-
ity benefits of sparse decomposition and the high reconstruction fidelity of SVD- based methods while offering statistical guarantees of concept recoverability.
The remainder of this paper is organized as follows:
In Section 3 we present a hypothesis testing framework to quantify rotation- concept structure in the embedding space. We provide the detailed test procedure, thcoretical guarantecs for test statistics, as well as empirical results.
In Section 4 we introduce a post-hoc concept-based decomposition method accompanied with an auto- matic concept interpretation algorithm.
In Section we formalize a statistical model that connects concepts with embeddings and prove concept identifiability under certain assumptions. We show that concept decomposition methods with a fixed, misspecified concept vocabulary can suffer from reconstruction errors.
5
In Section we show through qualitative analysis that our method learns interpretable concepts and maintains high reconstruction fidelity, evidenced by perrsstt-perooomance trade-off analysis. We also show that our method is effective in identifying and removing spurious concepts.
D
2 Background and Overview
We start with a brief discussion on statistical tests for rotation-sensitive structure and their implications for interpretability neural embeddings.
Hypothesis Test for Rotation-Sensitive Structure Early multivariate inference framed "no preferred direction" null hypothesis, leading to tests for departures from spherical symmetry. Classical methods such Mauchly's 's test for sphericity (Mauchly] 1940), John's test for identity covariance (John 197 and the Bingham test (Bingham 1974). all treat rotational invariance as the bascline, so significant rejections indicate directional (rotation-sensitivv)) structure. In high dimensions, random-matrix approaches extend this logic by comparing observed eigen-spectra to isotropic mulls (edoii & Wolf 2002 Chen et al. 2010). Building on these idcas, we adapt rotation-invariant bootstrap tests to neural embeddings, where coordinates are arbitrarily scaled or rotated. This allows us to assess deviations from isotropy before applying Varimax for interpretability.
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Rotation-sensitive Structure and Factor Interpretabilityy In classical factor analysis, raw factors are any orthogonal transformation of the loading matrix yields the same likelihood, so interpretability depends on selecting a "simple-structure rotation such as Varimax (Kaiser 1958). Recent work shows Varimax is not merely cosmetic-it it can consistently recover sparse structure under mild con- ditions (Rohe & Zeng 2023). Independent component analysis (ICA) addresses the samne indeterminacy by leveraging non- Gaussiantty to make the rotation identifiable, illustrating how probabílistic assumptions can anchor axes meaningfully (Hyvvrinen et al.) 2023). In modern representation learning, this symmetry objectives
persists: embeddings from contrastive or language-mood
only identifiable up to unknown linear map, making axis-aligned interpretations fragile unless the rotation is fixed (Roeder et al.; (021). Empirically, post-hoc rotations can sharpen semantics; for example, rotating word embeddings can align dimensions with human concepts without degrading performance PPark et al. 2017
3 Hypothesis Test for Rotation Sensitive Concepts
We develop hypothesis testing framework detect concepts in embedding spaces through rotational properties. We first characterize meaningful concepts through rotational sensitivity (Sec. 3.1.. validating our intuition on synthetic and real datasets. We then develop a resampling procedure (Sec. 3.2 test
statistics SSec. 3.3 and lest procedure with experimental results in (Sec. 3.4
3.1 Characterizing Meaningful Structure via Rotational Sensitivity
We first explain why a rotation-based approach is well-suited for statistically modeling concepts in embed- dings. Neural networks process embeddings through inner products with weight vectors, which measure how closely the embedding aligns with each weight vector's 's direction, making embeddings inherently direc- tional objects. When examining embedding spaces for meaningful structure, we essentially ask whether the disttibution of embeddings shows preferences for certain directions over others. A completely structureless embedding space samples points uniformly from all directions, making it rotationally invoriant. In contrast, mcaningful concepts manifest as preferred directions in the embodding distribution, breaking this invariance.
In the case of CLIP, this directional structure is not merely empirical but is enforeed by the training objec- tive itself. CLIP's contrastive loss maximizes the dot product similarity between matched image and text embeddings (Radford et al. 22221), which imposes a linear readout property on the final representation layer: for the model to correctly associate a concept (e.g., "dog"), the image embedding must have a high projec- tion onto the corresponding direction in text embedding space. Unlike intermediate layers, where semantic content may be encoded nonlinearly, the final CLIP embedding space must therefore be linearly decodable by design, making singular vectors and linear decomposition the appropriate analytical tools for this space.
To accurately characterize these rotational preferences, focus on the rotational properties of singular vec- tors rather than raw embeddings. This choice is crucial because heterogeneous scaling in different directions (i.e,, elliptical patterns in data) can create apparent rotation sensitivity in the raw embeddings even when no meaningful structure exists. Singular vectors, being normalized to unit variance, allow us to identify true directional preferences while controlling for such scaling effects.
We now formalize these ideas, starting with a formal definition of rotational invariance:
Definition 1 (Probabilistic Rotational Invariance). A probability distribution with density function f on Rd is said to be rotationally invariant if for any rotation matrix R E Rdxd (i.e., R R = Ia and dct(R) 1), the distributtin of x is the same as the distribation of æR, f(x) = f(xR) for all x E Rd and all rolation matrices R.
Intuitively, this definition formalizes when distribution looks the same in all directions; there are no preferred directions patterns in the data. To illustrate this definition, we first examine classical examples of rotation invariance, both at the data distribution level and the concept level:
Example 1 (Standard Multivariate Normal is Rotationally Invariant). The multivariate Gaussian distribu- tion N (0, Ia) is rotationally invoriant.
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White noise images
ImageNet images have radial streaks.
Arogue Vecton
Pessuue Vesto
Singular Vector 2
Singular Vector 2
Figure 1: Visualization of singular vector loadings from CLIP embeddings from the ViT-B/32 backbone model, where loadings represent how much each singular vector contributes to an image's 's representation. Left: Projection onto 2nd and 3rd singular vectors (after Varimax rotation) emboddings from white noise images, showing rolation-invariant structure. Right: Same projection for ImageNet validation images, revealing distinct radial streak patterns that indicate rotation-sensitive structure. Each point represents image, with the first singular vector excluded to remove mean effects.
While Example T demonstrates rotation invariance at the data level, particularly interested in the rotational properties of concept-specific patterns, which we capture through singular vectors:
Example 2 (Singular Vectors of Gaussian Noise are Rotationally Invariant). Let A E Raxd be matrix with entries Aij drown i..dd from (00 1). Then the first k left singular vectors U E Baxk and right singular vectors V E gdxk from the truncated SVD of A are both rotationally invariant.
This serves as our null model, representing the absence meaningful concept structure. In contrast, em- beddings that encode meaningful concepts exhibit rotation sensitivity. For example,
Example 3 (Gaussian Mixture Model Shows Rotation Sensitivity). Consider data drawn from a Gaussian mixture model: WW,, Ta) WN((,,,T) where μ (1,0,0.....0) Rdd. This distribution is not rotationally invariant.
We can further validate these examples using real CLIP embeddings in Figure The left panel shows em- beddings of white noise images, displaying uniform distribution from all directions (i.e., rotation invariance) similar to Example 2 The right panel shows ImageNet embeddings, revealing clear directional structure through radial streaks, analogous to the structured distribution Example 3
To formalize these ideas, let A E Raxd be the data matrix. We obtain its truncated singular value decompo- sition, where U m contains the first & left singular vectors of A. Each column of U represents a principal direction of variation in the data. Our hypothesis test specifically examines the rotational properties of left singular vectors (matrix U).
3.2 Sampling from the Rotation invariant Distribution
Rotation sensitivity only manifests when embedding prefers certain directions. To mimic the absence of such structure, generate null sample by independently rotating each row of the embedding while preserving its length. We describe the details of this method in Algorithm 4 which is deferred to the Appendix. Comparing any test statistic to the distribution obtained from these rotated replicas yields a Monte-Carlo p-value for the presence of rotation-sensitive patterns.
The following proposition establishes the theoretical guarantees for our resampling procedure:

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Proposition (Statistical Properties of Resampling). Let { C IR be i.i.d. somples from a probability distribulion with densily f. For any measurable test. statistic T: Rdx= - R, define:
= T( Fn), x:at = R' R Rd Uniform(SOp T = T(1 20
If I is rolationally invariant, then T and Ti have the same distribulion and are conditionally independent. given the set of norms 111=:112)==1
This proposition guarantees that under the null hypothesis of rotational invariance, the resampled test statistic (T*) follows the same distribution as the original statistic (Ti).
3.3 Test Statistics
To detect and quantify rotation-sensitive structure in embedding spaces, propose two complementary test statistics that capture different aspects of rotation sensitivity: distributional non-Gaussianity through kurtosis and achievable sparsity under rotation through the Varimax objective function.
Kurtosis- based Statistic. Our first test statistic measures the non-Gaussian patterns in U, as meaningful concepts typically deviate from normal distributions. We define:
[ - 11)=1] kurtosis(U.:)), where kurtosis(X)- (E[ - 2))1)2 3, k
(U)
with / E[X]. Under the rotation invariance null hypothesis, we expect this statistic to be close to zero. Theorem 1. Under the null model of Ezample an equivalent rescaled version () follows standard normal distribution.
This normalization provides an efficient computational path for hypothesis testing under Gaussian assump- tions. We defer the detailed proof to the Appendix.
Varimax-based Statistic. Our second lest statistic optimizes over rotations to detect patterns that may be hidden in the original coordinate system. We define:
v(U R)=
UU R
- n
())))))))) TS²(U)= max v(U, R),
(1)
RESO,,
where e is the Varimax objective function, and SO; is the special orthogonal group of k x rotation matrices. This statistic measures the maximum achievable sparsity under rotation, making it particularly sensitive to structured patterns that may be hidden in the original coordinate system.
To apply these test statistics, we use a bootstrap approach. We first generate samples from the mill dis- tribution by applying random rotations to the original data matrix. We then compute both test statistics these null samples to form their empirical distributions. The p-values are calculated by comparing the observed test statistics against these null distributions.
3.4 Test Procedure and Results
We present hypothesis testing procedure for detecting rofation-sensitive structures in embedding spaces. Under the mull hypothesis of rotation invariance, we expect large p-values indicating meaningful structure (e.g., 2 0.05), while significantly small p-values (c.g., 0.05) suggest the presence of rotation-sensitive structure. Algorithm Sortthhnmuuu outlines our testing approach.
We evaluate our method using CLIP ViT-L/14 embeddings on three datasets: ImageNet validation set im- ages, white noise images (Gaussian noise fed into the CLIP model), and pure white noise emboddings. As shown in Figure g our test statistics reveal strong evidence of rotation-sensitive structure in CLIP embed- dings of TmageNet images, with both test statistics showing significant separation between their bootstrap
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Algorithm Hypothesis Test for Rolation-Sensitive Concepts
Require: Matrix U Ro*k resamples Nresample
Ensure: Varimax p-value pv: kurtosis p-value Pkur
1: for i = 1 to Nresample do
Upot f- Rolation Invariant Malrix(U)
Algorithm 4
3: 2 + -U x argmax o(U RR)
> Apply Varimax rotation in eq: equation
4:
Compute statistics TS,(Z; ) TS2 ( (2
5: end for
2 U argmax e(U, R)
Compute TS TS2
8: Compute p-values:
Pkur (TS (2)
Nimammpl
10: Pv
Neecomple
11: return Pkurs Pv
Image Loadings
Concept Dictionary
Image Embeddings
kin
yT
Images
Pretrained CLIP Model
Project
Text Embeddings
Zezz
Image taken in coral reef environment Image with vibrant underwater ecosystem
Texts
Picture taken coral reef setting Photo taken along coral reef
Text Loadings
Figure 2: Pipeline overview. CLIP emboddings from images and texts are processed extract interpretable concepts. Image embeddings factorized into sparse loadings (74mg) and concept dictionary (Y), while text embeddings are projected to obtain loadings (Ztat). Bottom: Example of a discovered concept with its representative images and text descriptions.
null distributions and obscrved values. Control experiments white noise confirm our method's validity. Both white noise embeddings (p-values: 0.55 for Kurtosis, 0.6 for Varimax) and white noise images (p-values: 0.62 for Kurtosis, 0.81 for Varimax) yield non-significant p-values, confirming that random noise contains meaningful structure.
4 Identifying Interpretable Concepts
We present our main algorithm for identifying rolation-sensitive concept structure the embedding space, as motivated by the notions in Section Our approach decomposes the embedding matrix into two components: sparse loading matrix that captures the relationships between individual images and concepts, and an orthogonal concept dictionary that maintains independence between the discovered concepts. This achieves both interpretability and high reconstruction fidelity.
4.1 Method Overview
As shown in Figure our pipcline processes image and text inputs through a pretrained CLIP model to obtain enbeddings. Given image embeddings A Raxd and text embeddings T E RMxd we learn k orthogonal concepts represented through three key matrices: concept dictionary matrix Y, imaae loadings Zumg: and

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Algorithm 2 Concept-based Embedding Decomposition
Input: Embedding matrix RRndd number of concepts
Output: Concept matrix Y [dd*,, image loadings 2 gaxk Á f (ormalization(A)
U, D,VT + (V)(A)
5: R < argmaxneso, R)
b Optimizes objective in Eq. equation |
2 -UUD
V+ VR
8: return 8
text loadings Zest The image and text loadings indicate how strongly each image text embedding aligns with the learned concept higher values represent stronger association with particular concept in the dictionary.
Algorithm@dee details our decomposition method, which has three key steps. First, we normalize the embedding matrix for better spectral estimation (see details in Appendix B Second, we perform SVD to identify the principal directions in the embedding space. However, these raw SVD components, while capturing the underlying structure, are not automatically aligned with human-interpretable concepts. This motivates our third step: Varimax rotation, which maximizes the variance of squared loadings for cach concept, naturally pushing individual loadings toward either high values zero and thus promoling sparsity. This sparsification is crucial for interpretability - it associates each data point primarily with its most relevant concepts and makes concepts more semantically distinct by connecting them only to related examples. The rotation step effectively aligns the rotation-sensitive structure we detected (Section 3) with interpretable axes in the embedding space. We empirically validate the necessity of this rotation Section G where we show that rotated sparse concepts exhibit more explicit semantics compared to raw SVD components.
While raw directions can be polysemantic, Varimax rotation directly addresses this entanglement by secking the simplest available linear basis. Our results on Waterbirds (Section 6.3) confirm that this linear disen- tanglement is sufficient to isolate and remove spurious concepts such as backgrounds, yielding substantial improvements in worst-group accuracy.
Our framework thus bridges rotation sensitivity and interpretability in two steps. First, the hypothesis test (Soction 3 uses rotation sensitivity to distinguish signal from noise-isotropic distributions are rotation- invariant, rotation sensitivity reveals "preferred directions" confirming non-random underlying structure. Second, Varimax rotation transforms these directions into sparse concept loadings, which is key to human interpretability: it associates data points only with their most relevant concepts and makes concepts seman- tically distinct. The connection between sparsity and interpretability is well-established in representation
learning (Subramanian et al. 2018 Guillot et al. 2023
Faruqui et al. 2015
4.2 Concept Interpretation
We propose two methods to interpret each concept. from the decomposition. The first method utilizes the image loading matrix 2 to identify represcntative examples for cach concept. For the j-th concept, we examine its corresponding column 2., and select the images with the highest loading scores. These typically share common semantic features, allowing to derive interpretable theme for the concept.
Our sccond method (Alg. 3 provides automatic concept interprctation through text descriptions without human intervention. This approach requires a pool of text descriptions for the image datasel, which can be obtained through LLMs or visual-language models (e.g., LLaVA proposed by Liu et al. (2023)). ). The algorithm projects these text descriptions onto our learned concept space and identifies the most relevant descriptions for each concept. In our experiments, we use the curated text description set from Gandelsman et al. 2024b), which provides general descriptions of ImageNet classes.
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Algorithm 3 Aulomatic Concept Interpretation
1: Input: Concept matrix y E IRd text descriptions {M) text embeddings 7 R.xxd descriptions
per concept r
2: Output: Text descriptions {Ti} lor each concept
L ---7 UTTY
Project text embeddings to concept space
4: for each concept j -= 1;.,,.., do
5:
Ij - indices of top r values in L.j
6:
f {M:
7: end for
return (Tg)
5 Theoretical Results: Identification and Recovery Bounds
In this section, we establish theoretical guarantees for our concept decomposition method. We first demon- strate that our method can reliably identify meaningful concepts under specific statistical assumptions, thereby extending previous work on Varimax rotation identification. We then analyze fundamental limita- tions of fixed-concept approaches, demonstrating why adaptíve concept learning is necessary.
5.1 Concept Identification
Our identification guarantees are based on a key insight: when embedding data exhibits sufficient statistical structure, a unique rotation (up to permutation) exists that aligns with interpretable concepts. We formalize this through the following assumptions:
(The identification assumptions for Varimax). The matriz Bank satisfies the identifica- tion assumptions for Varimax if all of the following conditions hold on the rows Zi E g* for = l....,... n: (t) ......... Zn are i.i.d., (ti) each vector Z; has k independent random variables (not necessarily identically distribuled), (iii) the elements of Zi have kurlosis 2 3.
Assumption 1
the vectors Z1, Zz......
The independence conditions (i) and (ii) ensure structural consistency across samples, while (iii) requires sufficient non-Gaussianity in the data. We relax the equal variance assumption from Rohe & Zeng (2020). allowing different concepts to have different strengths of expression. This is crucial. In the vintage sparse- PCA model, the data admit the factorization X = ZBY" where Z, Y are Varimax-rotated eigenvectors, and is the diagonal matrix eigenvalues, left and right multiplied by rotation matrices. Because B absorbs all scaling, Z and Y can be rescaled without losing orthogonality. Our model instead factors the data X ZY' for clear interprelation purposes, where Z is the data loading each concept, and Y is the concept dictionary. Hence, any attempt to transfer scale from Z to Y would break the orthogonality of Y, which makes the concept dictionary harder to interpret.
Theorem 2 (Varimax rotation identification). Suppose that Z Raxk satisfies Assumption 1 Define Ž Z - B(Z). For any rotation matrix e O(k), E (v(R,ZRT)) (RP: ( PE REO(K)
arg
where P(k) = (P E O(k): P.j (-1,0,1)) is the full set of matrices that allow for column reordering and sign changes, and is defined in equation equation
Under assumptions, this shows the Varimax objective identifies the correct concept rotation up to permutation.
5.2 Reconstruction Error Bounds for Fixed- Concept Methods
When the concept matrix is fixed, reconstruction errors arise from potential misalignment between predefined concepts and the ground-truth concept structure. To formalize this limitation, we denote the ground-truth

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latent concept matrix as C R and the fixed concept matrix (such as in SpLiCE) as Cw R where k S d <. m. We assume Cw may fail to capture some information present in C", which we quantify through the following condition: min ||CwP- - FF 2 6, where P is an arbitrary projection matrix, 8 > 0 represents the minimum possible misalignment between the fixed and true concepts, and || ||e represents the Frobenius norm.
Theorem 3 (Fixed concept-decompositton method reconstruction error lower bound). Given the misspec- ificalion condition above, consider A E RR such that A C*C*T with posilive k-th singular value, i.e. ox ox(Z*) > 0, then we have minzeRoxm ||A - ZC 2 oz(Z*)) (6, where oz(Z) = vox(Z (2(2222 2) is the absolute k-th largest singular value of Z.
This thcorem quantifies the risk of fixing the concept decomposition matrix in SpLiCE: when the prodefined concept vocabulary cannot, be aligned with the true concepts, reconstruction error is unavoidable. Our proposed method avoids this limitation by learning concepts from the data.
6 Experiment Results
We evaluate our method through qualitative and quantitative analyses. We assess the interpretability of learned concepts via visualizations and textual aligmment, analyze the trade-off between sparsity and recon- struction fidelity, and demonstrate the effectiveness of our method in removing spurious correlations across multiple datasets.
6.1 Qualitative Evaluation of Discovered Concepts
We evaluate the effectiveness of our concept decomposition method visually and textually
Setup. We apply our method to CLIP ViT-B/32 embeddings of the ImageNet validation set, using the curated text description set from Gandelsman et al. (2024a) that provides class-specific descriptions generated via ChatGPT.
Results. Figure compares concept clusters discovered by our Varimax-rotated decomposition (left column) against those from raw SVD (right column). For two representative concepts, display the top nine images by loading score (2.z) alongside their automatically retrieved text descriptions from Algorithm 3 The top row shows a concept manually selected to demonstrate the effectiveness of our method, while the bottom row concept shows a randomly selected concept from a pool of 50. Our method consistently yields semantically coherent clusters (e.g. butterfly feeding scenes, screws) with concise, focused descriptions. contrast, raw SVD clusters mixed themes such as furniture and animals, screws and knitwears, accompanied with broader, less specific descriptions. These differences demonstrate that Varimax rotation effectively isolates meaningful concept directions in the CLIP embedding space, resulting in far more structured and interpretable concept representations than standard SVD decomposition.
6.2 perfrraeper Trade-off
We analyze how the number of concepts in our decomposition affects reconstruction fidelity, measured by the cosine similarity between original and reconstructed embeddings.
Setup. We evaluate our method on the ImageNet validation set using two CLIP models: VIT-B/32 (512 dimensions) and VIT-I./14 (768 dimensions). We compare our results against those of SpLiCE (Bhalla al. 2024) as baseline.
Results. Figure 4 shows that reconstruction fidelity improves with increasing number of concepts k for both models. ViT-L/14 consistently shows lower cosine similarity compared to ViT-B/32 at equal k. reflect- ing the challenge of capturing its richer 765-dimensional embedding space with the same concept budget. Our method achieves substantially higher reconstruction fidelity compared to SpLiCE when using comparable numbers of concepts.
The quality of reconstruction is crucial as it indicates how well our decomposition preserves the semantic information encoded in the original embeddings. While concept decomposition inherently involves trade-off
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Ours
Raw SVD
Image capturing butterfly
Image
suburban setting
Photo featuring Papilion butterfly feeding on nectar
Picture depicting modem room divider deeign
Close-up butterfly feeding on flower
Picture taken the Gulf of Mexico
Photo capturing butterfly's feedligg behavior
Picture of monarch butterfly in mid-flight
metbed netats
nobber
kitey
Picture with two drumeticks
Picture with two drumsticke
Close-up of bassoon reeds
Picture with bayonet
Picture taken in pencil manufacturing facility
Picture with papsicles arranged in pattern
depicting screwdriver
Photo with
Figure 3: Comparison of concept clusters obtained by our Varimax-rotated decomposition (left column) and raw SVD (right column) on CLIP image embeddings. Each cell shows the top nine images for given concept, annotated with their retrieved text descriptions. Our method produces tight, semantically coherent clusters and precise labels, whereas raw SVD yields mixed-semantics groups and more generic descriptions.
Table 2: Performance comparison across datasots. WGt: worst-group accuracy (higher better), Gap!: accuracy gap (lower better), mFl: micro-F1, MRec: recro- Best results in bold. Purple and green highlights indicate the best worst-group accuracy and the smallest accuracy gap.
Model
Waterbirds
iWildCam
CelebA
Avg
WGt
Gap
Ace.
mF1
MRec
Avg
WG
Gap
84.8
38.1
46.7
6.23
0.001
0.002
81.2
74.2
7.0
SVD-recon.
85.5
39.0
46.5
3.83
0.001
0.001
78.1
74.9
3.2
Spurious Removed
89.6
60.7
28.9
18.8
0.003
0.006
82.6
75.1
7.5
between interprotability and information preservation, our method offers flexible control through the number of concepts k, allowing users to balance these competing objectives. Unlike SpLiCE, which prioritizes inter- pretability at the cost of significant information loss, our approach maintains interpretable concepts while better preserving the original embedding structure. This preservation of semantic informa- tion essential for downstream applications and validates that our discovered concepts capture meaningful aspects of the data.
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Average Cosine Similarity vs Sparsity
1.0
Average Cosine Similarity
0.9
0.8
0.7
0.6
VIT-B/32
VIT-L/14
0.5
SpLiCE (VIT-B/32)
25
50
75
100
125
Rank
Figure 4: Reconstruction quality versus number of concepts. Higher cosine similarity indicates better preser- vation of the original embedding structure. Our method offers a better interpretabilit-fidllii trade-off than SpLiCE.
6.3 Removing Spurious Correlations
We evaluate our method's ability to identify and remove spurious correlations across three datasets: Water- birds (Sagawa et al.) 2019), WILDS--WildCam (Beery al. 2020), and CelebA (Liu et al. 2015 Dataset details are provided in Appendix B
Setup. For all experiments, we use CLIP VIT-B/32 embeddings (512 dimensions) and decompose them into = 50 concopts using Algorithm 2 Spurious concepts are identified using datasot-specific strategies (detailed Appendix BB, which analyze the correlation between concept embeddings and text descriptions emphasizing different attributes (e.g., target vs. background features). We defer the details of how to remove spurious concepts to the appendix.
We compare three embeddings: the full, original CLIP image embedding; the concept-based reconstruction, where embeddings are reconstructed from all learned concepts; and the spurious-removed reconstruction (Ours), where embeddings are reconstructed after removing spurious concepts. For classification, we follow the standard zero-shot classification setup.
Results. Our mothod consistently improves zero-shot prediction performance after removing spurious concepts (Table 2). On Waterbirds, removing spurious background concepts improves worst- group accuracy by 22.6% and reduces the accuracy gap by 17.8%. On iWildCam, the prediction accuracy triples from 6.23% to 18.8%; and CelebA, we achieve the highest average and porst-group accuracy while using only 5% of the original embedding dimensions. The SVD-rcconstructed embeddings maintain similar average accuracy to the original embeddings for the Waterbirds datasel, suggesting our method preserves task-relevant information while removing noise.
6.4 Additional Experiments
We present additional experiments in Appendix Specifically, Scction C.4 evaluates our models across
different backbones and datasets to demonstrate that our framework generalizes beyond CLIP. Section C.5 investigates model behavior when random rotations are applied to the CLIP embeddings. In Section C.6 we
use a concept coherence metric assess the semantic alignment of the discovered concepts. Finally, Section C.S analyzes the alignment between our learned concept directions and the attention heads of the model's
final layer.
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7 Conclusion
We introduced a hypothesis testing framework to quantify rotation-sensitive structures in the embedding spaces and proposed a seoepp--ecoomositiion method that achieves both high reconstruction fidelity and clear interpretability. We validated it through theoretical and empirical analyses. Applied to challenging dis- tribution shift benchmarks, our method consistently demonstrated significant improvements after identifying and removing spurious concepts.
Limitations. Our approach assumes linearity in the embedding decomposition, which may overlook complex non-linear structures potentially present in the embedding space. Additionally, note that the interpreta ity of discovered concepts depends partially on the quality and scope of the available text descriptions, which may introduce biases or limit generalization. Finally, while our hypothesis testing procedure is robust to rotationally invariant noise, it does not explicitly handle structured, non-rotational noise patterns, leaving room for further refinement in more nuanced settings.
Acknowledgements. The authors thank Kris Sankaran, Keith Levin, Yinqiu He, Changho Shin and Dyah Adila for the valuable discussions and feodback. Jitian Zhao and Chenghui Li gratefully acknowlodge support from the IFDS at UW-Madison and NSF through TRIPODS grant 2023239 for their support.
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Supplementary materials contain additional experiment details, results, and proofs. We provide glossary
table in Section A additional experiment details in Section B and results in Section technical lemmas and additional theoretical results in Section D and proof in Section E
A Glossary
The glossary is given in Table below.
Table 3: Glossary of Notation
Symbol
Delinition
U
Matrix containing singular vectors from SVD decomposition
Z
Image loading matrix, represents how images load onto concepts
SOs
Special orthogonal group (rotation matrices) in R** Input embedding matrix of size Rgxd
A
Y
Estimated concept matrix of size Rdxk
Number of concepts
Number of data points /images
d
Dimension of embeddings Rolation matrix Text embedding matrix The j-th concept
R
T
C
o2(Z)
The absolute d-th largest singular value of Z Ground-truth latent concept matrix Word concept matrix
C
Cw
(UU R)
Varimax objective function
(U)
Kurtosis test statistic
(U)
Varimax objective function test statistic Rescaled kurtosis test statistic
(U)
IIe kurtosis(U.))
Frobenius norm
Kurtosis the i-th column of U Set of permutation matrices in R
P(k)
Additional Experiment Details
All experiment results carried out using frozen pretrained weights from open-clip (VIT-B/32 and VIT-L/14), and additional model training is involved.
B.1 Concept Decomposition Algorithm Details
Scaling Data Matrix We scaled the data matrix A E Raxd before applying SVD in the concept decompo- sition algorithm. Define the row normalization vector as:
I 11 E I,
deg, A., T
D, = diag(deg, Tx E ga
Similarly, define the column quantities deg. = 1 E IRd, To Adeg E R, and D. diag(deg, + Tala) e dddd The scaled data matrix then defined as À = 1/2 1/2 which we use as input to the concept AD. dccomposition algorithm instcad of the original matrix A. This scaling step with regularization parameters Tr and Te helps stabilize the spectral estimation and prevents potential outliers in the singular vectors that could arise from noise in the data matrix (Le et al. 2017 Zhang & Rohe 2018).
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B.2 Hypothesis Test Experiment Details
To validate hypothesis testing framework, conducted experiments both real dataset- the Im- ageNet validation set-and two synthetic datasets: white-noise image embeddings and pure white-noise embeddings. The goal was to assess the framework's ability to detect non-random structures in different types of data.
Datasets.
ImageNet Validation Set: We used embeddings computed by a pretrained Vision Transformer (ViT-B/32) model on images from the ImageNet. validation set.
White-Noise Image Embeddings: We generated 10,000 white-noise images, each of size x24 224 pixels with 3 color channels. Each pixel value was drawn independently from a standard Gaussian distribution. These images were then processed by the pretrained VIT model to obtain embeddings of size 10, 000 x 512.
Pure White-Noise Embeddings: We directly generated random noise matrix of dimensions 10, 0000 512, with each entry sampled from a standard Gaussian distribution, without passing through the embedding model.
Experiment Details. For each dataset, obtained an embedding matrix A as described above. We then performed Singular Value Decomposition (SVD) on A, decomposing it into Á = U UDD. Here, U is matrix whose columns are the left singular vectors, representing orthogonal directions in the embedding space, and whose rows correspond to the images.
We observed that the first column of U (the first principal component) often captured mean or bias effects in the embeddings. The loadings on this component were concentrated around constant value, offering limited information about the latent structure of the data. Therefore, excluded the first column of U, defining Ü 2 :], to focus more informative components.
Next, we applied our hypothesis testing framework to Ü to compute p-values and test statistics, assessing the statistical significance of any non-random patterns present in the data.
Randomness in the Procedure
Our hypothesis testing procedure involves randomness in two key aspects:
Row-wise Random Rotations: To generate conditionally rotation-invariant data, we applied random rotations to each row of Ü. This step introduces randomness into the data transformation process.
2. Generation of Synthetic Data: The white-noise image and pure white-noise embeddings were generated using random sampling from standard Ganssian distributions.
To account for the variability introduced by these random processes, we performed additional tests using 5 different random seeds and varied the rank l of Ü.
Selection of Rank & We investigated how the choice of rank k, the number of singular vectors retained in Ü, affects the results of the hypothesis tests. We expect the p-values to increase with larger k indicating decreased ability to detect rotation-sensitive structure. As & increases, more columns of U are included, potentially introducing additional noise and reducing the statistical power to detect non-Gaussian signals. We report our results in Figure 5] We observed that for white noise image embeddings, p-values increase as increases, which aligns with our expectation. For white noise embedding, we observed p p-values oscillate around 0.5 and show no clear pattern l changes. This aligns with our thcoretical results from Example 2 which suggests U follows rotationally invariant distribution, and the p-value should follow an approximately uniform distribution between 0 and 1.
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P-values vs Rank
P-values vs Rank
Kurtosis vavalue
Kurtosis p-value
Var
Varimax p-value
0.8
0.6
Payalue
0.4
0.2
40
60
100
120
140
60
80
100
120
140
Rank (k)
Rank (k)
Figure 5: Illustraion of how p-values change with rank k. Left: white noise image embedding from a pretrained ViT-L/14 model. Right: white noise embedding of dimension 10, 000 768.
Dataset
Task
Spurious Attribute
#Images
#Caasses
Waterbirds
Bird Species Animal Species
Background
4,795
2
iWildCam
Location
42,791
182
CelebA
Hair Color
Eyeglasses
19,962
2
Table 4: Dataset characteristies and their corresponding spurious correlations.
B.3 Spurious Concept Removal Experiment
Datasets details
Waterbirds (Sagawa et al. 2019 Bird species (waterbird/landdird) with spurious background
correlations (waler/land)
WILDS-IWildCam BBerr et al.
2020 Koh et al. 2021
Animal species classification with spu-
rious location-specific features across camera traps
CelebA Liu al. 2015): Hair color classification (blonde/non-blonde) with spurious attribute
correlation (eyeglasses)
Dataset statistics can be found in table 4
Spurious Concept Detection Methods. We develop strategies to automatically identify spurious con- cepts, tailored to each dataset's characteristics:
For Waterbirds, we generate text descriptions following the template 'A {bird type) with a {back- ground type] background.' We identify spurious concepts as those where lop-ranking descriptions share common backgrounds but varied bird species.
For iWildCam, we employ a contrastive approach using two sets of descriptions: one focusing on animal features and another on location attributes. We compute cosine similarities between concept embeddings and these description embeddings to identify concepts that correlate strongly with location features.
For CelebA, we generate descriptions emphasizing either hair color or eyeglasses attributes, using similar contrastive approach to separate target concepts from spurious ones.
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Removing Spurious Concepts To remove spurious concepts, reconstruct image embeddings while setting the coefficients of identified spurious concepts to zero. Given the decomposition A, = E5 where Cj are learned concept vectors, we enforce O'epurious = 0 to filter out spurious information.
Waterbirds Experiment For the Waterbirds experiment, we use I'a landbird', waterbird"] as class prompts. the zero-shot prediction experiment, we first compute text embeddings for class prompts and compute cosine similarity between class prompt embeddings and image embeddings. Then, for each image, extract the class with the highest similarity the prediction.
For removing spurious concepts, we first decompose image embedding into linear combination of concepts with Algorithm 2 A, = E, a,Cj. Suppose we have identified spurious concepts with our proposed method, as explained in the main content. By removing spurious concepts, we set the coefficients for spurious concepts to 0. In other words, aspurious = O.
B.4 Algorithm to Generate Rotation-Invarian Matrix
See algorithm to generate Rotation-Invariant matrix in Algorithm
Algorithm Generate Rotation-Invarian, Matrix
1: Input: Matrix E Ra*k
Output: Rotation-invariant matrix Uroo e Raxk for each row u, E Rt 1.2....,nt do
Generale random rotation matrix R2 E Rixk
uot +- R.u. 6: end for
Rotate the row
7: UTot < [u]or uZ ..... "n T
b Matrix of rotated rows
8: return Thot
C Additional Experiment Results
C.1 Bootstrap simulation results
In Figure present bootstrap kurtosis and observed kurtosis distribution defined in Section 3.3
C.2 Additional Concept Results for ImageNet
We provide additional concept results for the ImageNet validation set in Figure 7 Embeddings computed by the V-T-B-32 model.
C.3 Concept Results for Waterbirds
We provide concept results for the Waterbirds dataset in Figure
C.4 Generalization Across Model Architectures
To demonstrate that our framework is not constrained to specific architecture, evaluate it on models with different backbones and training data: ResNet-50 (CNN-based, OpenAI weights) and OpenCLIP ViT- Across
B-32 trained on LAION-2B. Across all tested models, observe consistent pattern: concepts derived from raw SVD (without rotation) frequently appear as entangled mixtures of unrelated themes. However, after applying our proposed Varimax rolation, the directions consistently align into semantically unified themes. When applied to the same dataset (ImageNet)), different backbones (e.g., ResNet-50 vs. VIT) yield similar semantic concepts, identifying shared themes such as dogs, knitwear, birds, and ships. These results confirm that the rotation-sensitive structure is not an artifact of the ViT architecture but general property of CLIP-style contrastive embeddings.
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Bootstrap Varimax Objective
Bootstrap Kurtosis (log)
Vs Observed Varimax Objective
vs Observed Kurtosis (log)
250
250
200
Frequency
200
Frequency
150
150
100
100
Bootstrap Distribution Observed Statistic
Bootstrap Distribution Observed Statistic
50
50
0
0
1.5
2.5 Test Statistic Value
2.0
3.0
3.5
1.0
-0.5
0.0
0.5
1.0
1.5
1e-6
Test Statistic Value
(a) Bootstrap Varimax Objective Observed
(b) Bootstrap Kurtosis Observed Kurtosis
Figure 6: Comparison of bootstrap distributions and observed test statistics. The blue histograms show the distribution of test statisties computed from red dashed lines indicate the observed test statistics computed from CLIP embeddings of the ImageNet validation set.
resamples under the mull hypothesis. The
C.5 Robustness under Random Orthogonal Rotations.
To further validate our framework, we evaluate how different methods behave when the CLIP image embed- ding space randomly rotated. We sample a random orthogonal matrix Q ~ Uniform(SO.) and apply it to all image embeddings: À AQ. We then compare three methods on the rotated embeddings:
Our method maintains high interpretability and low reconstruction error on the rotated data, confirming that it successfully recovers the underlying structure even under arbitrary rotation.
SpLiCE, which relies projecting embeddings onto fixed concept vocabulary, catastrophically fails: the rotation breaks the aligmment between image and text embeddings, resulting in large increase in reconstruction error and a loss of semantic relevance.
Raw SVD is robust to the rotation transformation since singular vectors rotate with the data. However, the resulting concepts remain mixed and polysemantic, demonstrating that varianco-based decomposition alone is insufficient an explicit. search for the "simple structure" rolation required disentangle semantic concepts.
Specifically, sample random orthogonal matrices Q Uniform(SOa) and apply them to CLIP VIT- B/32 image embeddings: Ā = AQ. We then run each method (Ours, SpLiCE, Raw SVD) on the rotated embeddings. For our method and Raw SVD, we compute the SVD of the rotated data and apply Varimax rotation (or not, for Raw SVD). For SpLiCE, we project the rotated embeddings onto the original fixed text concept vocabulary. We track two metrics: (1) reconstruction error (mean squared error between original and reconstructed embeddings) and (2) concept coherence score (average pairwise cosine similarity among top-9 activating images per concept). The experiment was repeated with different random seeds to account for variability in the rotation sampling.
C.6 Quantitative Concept Coherence Results
To quantify the "semantic purity" of discovered concepts, we introduce a concept coherence score. For cach concept direction, we retrieve the top-9 most activating images and compute the average pairwise cosine similarity between their embeddings. A higher score indicates a tight, consistent semantic cluster, while a lower score indicates polysemantic concepts. As shown in Table our method achieves the highest coherence across all tested models and training configurations, confirming it minimizes concept entanglement better than both ICA and Raw SVD.
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Concept
Concept
Concept
Concest
Concest
Phataggas lopaicaliie
ee en sust wil
Proure
luage lotcr fada
Ficure
was with diee bel shutts
Mac dagaaa with Hadio Septh lald
Concept 13
Concept 11
Cuncept
Concept
Concept
rrgge descaddee tmbee Bevree
al
cra Pevvaiing bec' dedicd soap postare.
parko of musice infurenn beling slond
Thage toder eh
arndem 110 serting
Concept 15
Concopt 16
Cancest 17
Cancest
Concept 19
micars tave cothi holl forive
treie Poeer the sornd horee
patte Lated
1r300 merae in sedeet en cood deal
auide Recur crief Fubl
sitd qieerse loodiin Pres beaiit adlirne sod sor
tne
Aecre
marror
Concept 20
Concept 21
Concept 22
Concept
Concept 24
BUSTHEEE daral colir
pirk Bud Boriig pence caaig giver
Mail arrt uaee Pucrri daly biy
lilling
Figure 7: Top-24 concepts using our method with leading images and corresponding text descriptions. We observe that image and text concepts are well-aligned with similar semantic topics.
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LossL yyassher with banssce first background Lead Plycarches with natural reand
Calloriia bucagramt.
Goocsscys with aroud fersst buckrroudd r8g1thakk
Wermenn Out with ocean loakk (ooud
chack wiloo with troa deal fires belogrunn
A Paim WWrbber with a bembie forett baciground
A hee Crna with creen bacagmmnn
Aelling Out with a broacleat foree bacogreunc.
Pacfic Loon with naturel lake baciground. teckground
Arn crican Crow th bamboo forest bockground Fish Cros with hovvaco fureet bnckkrou Webbe with Cencooetyy Whll ART bamiboo forest backy
Purpie Finch with bemboo forest background
Horred Creae Amt or take tockyround Halard ribura
Ro Bunt bombice Strest bacermound Purple legaee kittheake naturall uko bookg sunnd.
tackground sund.
Pstaci recural like bockgiound.
CU
Figure Top-6 Waterbrids datasel concepts with text descriptions. We noticed there are bird-focused concepts (e.g., first row, left column) that specify the species more clearly and mention distinctive features. There background-focusee concepts (e.g., first row, middle column), that highlight the type of environ- ment. We also observed multiple birds concept (second row, left column).
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Table 5: Quantitative Concept Coherence Scores across different CLIP models. Higher scores indicate more
semantically coherent concepts. Best results in bold.
Model
Weights
ICA
Raw SVD
Ours
ViT-B-32
openai
0.649
0.621
0.764
RN50
openai
0.613
0.604
0.744
ViT-L-14
openal
0.643
0.601
0.725
ViT-B-32
laion2b s34b
b79k
0.578
0.499
0.677
Table 6: Spurious-Concept Removal Performance on Waterbirds. All concopt-based methods use = 50 concepts. "Recon" denotes reconstruction from all concepls; "Spurious Removed" denotes reconstruction after removing identified spurious concepts. Best unsupervised results in bold.
Recon Avg
Recon WG
Removed Avg
Removed WG
ERM (Supervised)
94.9%
29.2%
NMF
25.3%
0.0%
23.0%
0.1%
ICA
86.5%
40.8%
80.6%
52.1%
CRAFT
23.9%
0.0%
22.6%
0.0%
SVD
85.5%
39.0%
70.7%
60.4%
Ours
85.5%
39.0%
89.6%
60.7%
C.7 Additional Concept Removal Experiment Results for Waterbirds Dataset
To further contextualize method, compare against broader set of baselines on the Waterbirds dataset, including both supervised and unsupervised approaches.
Setup. We evaluate the following methods, all using k = 50 concepts where applicable: a supervised linear probe (ERM) trained with ground-truth labels, non-negative matrix factorization (NMF), independent component analysis (ICA), Sparse Autoencoder (Sparse-AE) (Cunningham et al. 2023 and CRAFT (Fel
ct al. 2023). Porformance is measured by zero-shot classification accuracy after reconstructing emboddings with and without spurious components removed.
Results. Table 6 confirms the intuition regarding the robustness-acot accuurra trade-off. The supervised ERM bascline achieves the highest average accuracy (94.9%) but is fragile, collapsing to 29.2% on the worst-group. NMF and CRAFT fail to produce meaningful reconstructions, yielding near-zero worst-grout accuracy. ICA achieves moderate performance but our method outperforms it on both average accuracy (89.6% vs. 80.6%) and worst-group accuracy (60.7% vs. 52.1%) after spurious concept removal. Our method sacrifices some average performance compared ERM gain substantial robustness, verifying that we successfully identify and remove the spurious background concepts that the supervised baseline relies
C.8 Connection to Internal Model Mechanisms.
To investigate whether global concepts reflect internal model mechanisms, analyze the alignment between our learned concept directions and the attention heads of the model's last layer. Following Gan- delsman et al. (2024a), who showed that CLIP's representation is primarily constructed in the late attention layers, we computed the cosine similarity between our concept directions and the outputs of the last-layer attention heads. The resulting alignment sparse and structured: specific global concepts align strongly with particular attention heads, indicating functional specialization. This suggests our concepts are not arbitrary directions but directly reflect internal model mechanisms.
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Prener buildoy
Boston
Brdbanon, geffee
Exgid Setter
Sainc Bomand
Amenian coot
oystorcatche
Iitl heren
Beston
wodbors
Rottweher
Toxtan
+
Chirunhus
Aeeercan Strfordentre
Votercicche
grous
megple
egiet
caee
Bick stark
and tiaateed mergganer
dentcher
vohore
Figure Demonstration of analogical reasoning with concepts. The equation Cgd (group of dogs) Ca (single dog) + C. (single bird) yields concept. that correctly identifies groups of birds in both image and text spaces.
C.9 Concept Learns Analogical Relations
Word embeddings are known to capture semantie relationships through vector arithmetie, famously demon- strated by analogies such as "king man + woman queen" (Mikolov et. al. 2013). We demonstrate that our learned concepts exhibit similar compositional properties with visual concepts.
To evaluate this, we identify three key concepts from our learned representation: Cyu (representing groups of dogs), Ca (single dog), and C, (bird). We then construct. new concept through vector arithmetic: C= Cgd Ca Cb. Intuitively, this operation should capture the transformation from "single entity" to "group" and apply it to birds. We evaluate this constructed concept in two ways: by projecting image embeddings (Score = AC where A contains image embeddings) and text embeddings onto this concept space. As shown in Figure 0 both the top-scoring images and their associated text descriptions align with our expectation, consistently returning groups of birds, demonstrating that our method successfully captures and transfers the concept of collectiveness across different semantic categories.
Technical Lemmas
In this section, we provide some technical results for convenience.
The following lemma is a generalization of Li et al. 2023
Lemma H.5) for a non-square matrix. We recall
(2) / / Z) as the k-th largest absolute singular value of Z.
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Lemma 1. For A Rd B E Rkxn where > d 2 k, we have
| Th (B) < AAB|e |A]l (0ll--o1BB) (B).
(2)
Proof. Assume the SVD of B is UDDDVE VB. Then,
|AB|| F |AUBDBVBlle |BBDDBll
(3)
By applying similar induction proof in Li et al 2023 Lemma H.5) AUBDB where AUB Bd and DB E g&xk, we obtain
|AnDDBlF |AUB||e on (D())) -All=-O((DD)
(4)
and
|AUB ||UB|| (DB) = -|A||F (DB).
(5)
This concludes the proof.
E Missing proofs
E.1 Proof of Example I
Proof.
The density of the standard multivariate normal distribution is given by
1 exp (-11))
f(x)
(2T)
which depends only on the L2 norm || For any rotation matrix RG we have ||=|||| ||>||| because rotations preserve norms. Therefore,
f(2R) (2rr)d/2 exp (-111aR11)
[111)_1))] = f(x).
(2.r)d/2 exp
This shows that the density (and thus the distribution) of x is unchanged under rotations, proving rotational invariance.
E.2 Proof of Example 2
Proof. It suffices to show the result when & = min m}. Due to the nature of normally distributed random variables, for any orthogonal matrices G E Rmxm and H E IRaxn, the entries of GAH are i.i.d. and follow a standard normal distribution. Therefore, this guarantees spherical symmotry for the left and right singular matrices U and VT of A, implying that both U and VT follow a uniform distribution with respect to the Haar measure on the Stiefel manifold.
For any rotation matrix R R have (UR) (UR)URR- Iq. Consider fı(UU) UR, which defines to-one map from the Stiefel manifold to itself. By the one-to-one properly, UR has the same distribution U, namely following the uniform distribution to the Haar measure on the Stiefel manifold. A similar result holds for VT.
This guarantees rotation invariance.
E.3 Proof of Example 3
Proof. To show that this distribution is not rotationally invariant, we need to find a rotation matrix R such that the distribution of zR differs from the distribution æ.
Let R be a rotation matrix that rotates the vector μ to another direction. For simplicity, consider a rotation R that maps μ to Ry v, where (0, 1, 0).
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Published in Transactions on Machine Learning Research (02/2026)
The original distribution of Æ has two components centered at μ and -p. After rotation, the distribution of x R has components centered at Ru= v and R(- -v.
However, the probability density function (pdf) of æ before rotation is
1 1 exp (-21e= (-21x-a1²)) + 2 (2m)a/2 exp (-2 1 + m1l?)
f(x) 2 (2T))/22
After rotation, the PDF becomes
1 f f(xRt
(-51LeR 1112) + (2 exp
(-1 10l?))
2 (2x)
exp
But since | - 1 || ul|² in general, the pdf fR(Z) is not equal to f(z). Specifically, the locations of the mixture components have changed, resulting in a different distribution.
Moreover, consider evaluating the probability at specific point. For example, at 14, we have
1(4) 2(2 exp( -Žllu - -wl?)
1 exp - -1lllu u1l?)
2(2x)a/2
d/2 exp( (xxp(-211e11²]]
After rotation, at ue, we have
(R(U) f(ult)- 2 (27) exp (-5llue- - u11
(-5llae= (aIle++11? m1())
2 (2x)
exp
Since pR + n. the values of f(x) and fR(H) are different, confirming that the distribution is not rotationally invariant.
E.4 Proof of Proposition 1
Under Ho, to show that Ao is rotation invariant, we need to prove that for any fixed rotation matrix R E SO(d), the distribution of Arot R is the same as that of Arot.
Since conditional on the same ||A: l2 = || 2|,, for any A there must exist R, such that
A A,R,,
where R2 uniformly sampled from SO(d). For any e SO(d), multiplying ro on the right by R:
1 R = (A,R,) (RRRR) = A,R.,
where we define R, R.R. Since R, are uniformly distributed over SO(d) and independent, and R is fixed element of (O(d), the products R, = R, R are also uniformly distributed over SO(d), independent from cach other, and independent from A, Therefore, the distribution of A, is the same as that of A,
A, d A.R.
This implics that the rows of Aro R have the same joint distribution the rows of A
(A R d 1A
This completes the proof.
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Published in Transactions on Machine Learning Research (02/2026)
E.5 Proof of Theorem 1
For theoretical analysis, we derive equivalent standardized form:
TSa(U)= = Vnk
3n
|ur)osis(U.))-
(6)
v33
n +
Recall that U =I, therefore we have プド=1 Uz
1 for any i. We simplify the rescaled kurtosis
(U.( Vkk
71
k
U
3n n+2
v33
k
n
i=1
j=1
We recall from the proof of Example Denote
2 that for fixed i, U. follows normal distribution on Haar measure.
3n 1 + 2
X; n
U
==1
We compute
I (s + 2 I + ()
E
When s = 2, we have
E
T²nn + 2)
Therefore, by plugging this formula into our computation, we obtain
3n
3n
E
0,
72 2
0 2
and
7531 1
E
2222
( + () 2) + 1)
we obtain (Uij, Uvr) and U,
Un-U /2
On the other hand, by rewriting Uly as distributed. Therefore, we have
are identically
Σ, 57
v2
EE BUgUUZ =E E (U
Uii + Uvj (Ui++Uu))
(Ui (Ug- Up!
( v2
V2
This can be reduced to On the other hand, we have Combining equation g and equation
E [UN]]] = 4E [U - 5E [U}U}}]
(8)
15 (n + 6)(n + 4) + 2)n
EE
(B [U E [19,])
(9)
n
g we obtain,
9 E [ (n + 6) + 4)(n + 2)n
By plugging in the computations of moments, obtain
E [ = n [y] + - 1
n
105 x n²
9n² - 1)
9n(n -
+
(n (n 4)(n + 2)
(n + 6)(n 4) + 2)
(n + 2)
The leading order term of this variance is 33 random variables, we conclude the result.
By the central limit theorem and the fact that X? are i.id. "
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Published in Transactions on Machine Learning Research (02/2026)
E.6 Proof of Theorem 2
Recall Assumption that E/ [[Zzy] = 0, (B(2))) = o3. B((T))) ni 2 30
n (15 SiRRRR )
v (R, (R,2R))
[2RR]
8=1
To simplify notation, we denote 0 RR E O(k). We want to optimize vl R, 2T R(T) over O. We analyze two terms, respectively. For the fourth moment term
1 k Σ
k
E
2012
=
1=1
1=1
8=1
1=1
j=1
k n Σ
2z, 2h 08
=
2 03 +3 Σ
n
1=1
t=1
i=1
Fegh-!
O +3 Σ
i=
/=1
high!
00 +3
Σ
o
hg
because the expectation of all other cross terms in the computation contains at least one moment of entry, which is 0 according to the independence and [[ 0. For the second moment term,
2002) 27
Zn0 On)
E
n
4=1
q=1
E
Σ
Σ
Z Omi )
q=1 highte
43 0
cross terms
Term 1
Zyh Ont
highth
Term
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Published in Transactions on Machine Learning Research (02/2026)
Since B(Z,,) = 0, the expectation of all the cross terms should be 0 and can be removed from the equation. Now we compute terms 1 and 2, respectively. For term 1,
1 1)=
E (Term
n
Σ
E
03
43
11
1=1
9=1
j=1
1
k
Var
Σ
22j03
+
ΣΣΣ
&
2300
n
t=1
x=15=1
g=1
n
n
Σ
Oj.Var (Zzg)
+
ΣΣ
03,31
t=1
9=1
1=1
k
Σ
Σ
O3(01 -09) +
Σ 03003
12
k
k
k
Σ
Σ
o3
Σ
(n) - -3
t=1
E=1
For term 2,
n
E (Term 2) = n
Σ
E
Ond Om
1=1
9=1 Reggh
(E gh 1(22
n²
Pm 9=1 hygh
k
(
2
- Eonohh
n
t=l
h=1
Combining the computation for the second and fourth moments, we obtain
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Published in Transactions on Machine Learning Research (02/2026)
k
(
Th
O +3
Σ
of o
R 0%
ligh!
1
J1
nh 03
12 8=1
j=1
2
oh 0
Σ
o Oll
h=1
3n of) Of + (2- Σ
Σ
0
7
/=11=1
1-1 7
3n 03 + 2 Σ
11
8= 1=
b
k
2-1
3 + (2- (2- 2
Σ Σ
ojj
1=1 1=
Equality can and can only be achieved when O is a permutation matrix, where each row and each column have and only have exactly one 1. This completes the proof.
E.7 Proof of Theorem
Denote the generalized inverse of C. Cw as Cw)t. Then, by projecting from the column space of Cw to C", we can rewrite the condition min |[ P- - 1r 2 8 as
CW - C" 2.
(10)
Similarly, by plugging A = T
min ZeRaxA
||A 2C
min ZeRaxt
1 -CC lF
1 - T'0 CWll
(e1l=-ox("")
where the inequality follows from Lemma 1 We conclude the result by plugging equation 10 in the above equation.
29


Reviews:
[{"review_id": "G3F6jnMN1T", "reviewer": "Reviewer_Njv9", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes a statistically grounded framework for interpreting CLIP embeddings by explicitly testing whether the embedding space contains recoverable structure before attempting concept attribution. The paper appears more statistically rigorous than most concurrent papers aiming to attributing semantic concepts to embeddings, although a tradeoff might be lower benchmark performances.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The paper's writting is fairly carful, not claims are well supported.", "audience_yn": "Yes", "audience_explanation": "people who work in vision and embedding will be interested to read the paper, i would also like to argue that everyone in the ML field should read it.", "requested_changes": "1. While the paper deliberately does not optimize for benchmark performance, including comparisons to stronger but potentially fragile baselines would help contextualize the tradeoffs. Even if performance is lower, such comparisons would clarify what is gained in terms of robustness and what is sacrificed.\n\n2. Given that rotation sensitivity is central to the paper’s thesis, it would strengthen the claims to evaluate how competing methods behave under random orthogonal rotations of the embedding space. Demonstrating degradation in interpretability or performance for baselines would provide direct empirical support for the proposed framework.\n\n3. I wonder if the authors can come up with some quantitative measures that are more reliable than the previous comparisons. \n\n4. It seems the method is not strictly constrained to CLIP, some demonstration of its usage in other embeddings could be helpful.", "broader_impact_concerns": "none noted.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "BZe49cfNFn", "reviewer": "Reviewer_MqaU", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "1. This paper clearly points out an issue underlying the trade‑off between interpretability and reconstruction fidelity — that existing methods lack a rigorous statistical framework and are inherently noisy.\n2. Establishing a connection between rotational invariance and the interpretability and correctness of concepts is an interesting attempt.\n3. The method for detecting rotational invariance is clear and has solid theoretical guarantees (although such conclusions may rely on certain assumptions).\n4. The effectiveness to remove spurious correlations is impressing.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The paper is generally well written, and the proposed method is supported by some mathematical analysis. The drawback is that the empirical evaluation is not very sufficient — the extracted concepts should be evaluated across a more diverse set of tasks.", "audience_yn": "Yes", "audience_explanation": "As what I stated above.", "requested_changes": "1. More theoretical or empirical evidence of the correlations is needed: An important concern in this paper is why the authors can claim that the interpretable concepts or semantics are sensitive to rotation, unlike other noise patterns or artifacts. Although Section 3.1 provides some explanations, it is not sufficient to substantiate the paper’s most critical assumption.  \n(a) The paper implicitly assumes a one-to-one correspondence between concepts and directions. However, in real representation spaces, semantics often exhibit nonlinear or multidirectional distributions. For example, a concept may arise from a combination of multiple sub-directions or dimensions rather than a single vector direction. As a result, defining concepts purely through “directionality” may oversimplify the semantic structure.  \n(b) Furthermore, the paper only argues that “breaking rotational invariance” indicates the existence of some structure, but such structure may not necessarily be related to specific semantic concepts or particular tasks.  \n(c) The paper proposes replacing raw embeddings with singular vectors to mitigate scaling effects. However, linear decompositions such as SVD assume that the data follow linearly separable principal component structures; moreover, in high-dimensional neural representations, complex nonlinear relationships may not be fully captured through singular directions.  \n\n2. The experimental comparison provided in Sec.6.1 is not convincing enough, it seems that such vsualization is only provided with raw SVD, more comparisons with advanced methods are recommended.", "broader_impact_concerns": "See in previous sections.", "additional_comments": "N/A", "extra_scores": {}, "extra_text": {}}, {"review_id": "AvEKAR7QlK", "reviewer": "Reviewer_gnQX", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes a concept-based framework for analyzing the CLIP embedding space by identifying directions that are sensitive to rotation and interpreting them as human-understandable concepts. \nThe proposed approach achieves a favorable trade-off between interpretability and reconstruction fidelity compared to prior methods that either prioritize interpretability through predefined semantic concepts (e.g., word-based decompositions) at the cost of reconstruction fidelity, or preserve embedding fidelity through SVD-based techniques while sacrificing interpretability.\n\nThe main contribution of the paper is the introduction of a statistical hypothesis testing framework to detect rotation-sensitive structure in embedding subspaces. \nThe key insight is that meaningful structure (i.e., concepts) should break rotational invariance, whereas noise remains statistically invariant under rotation. \nBuilding on this idea, the paper applies Varimax rotation to align principal components with sparse, axis-aligned directions that are argued to be more interpretable, while maintaining low reconstruction error. \nThe authors further demonstrate that removing certain learned directions, identified as spurious, can improve accuracy on standard robustness benchmarks such as Waterbirds and iWildCam.\n\nWhile the idea of rotation-sensitive structure is appealing, the paper does not carry the analysis far enough to meaningfully interpret the CLIP embedding space or the internal behavior of the model in the way earlier work does (e.g., Gandelsman et al., 2024a). \nThe proposed “concepts” are global and linear, and are not tied to model components, training dynamics, or representational mechanisms such as attention. \nIn contrast to earlier work that directly analyzes how CLIP representations are formed and used, the interpretation step here remains largely illustrative. \nAs a result, it is unclear how the extracted concepts advance understanding of CLIP’s internal dynamics, embedding geometry, or interactions between concepts beyond providing sparse decompositions with favorable reconstruction properties.", "claims_and_evidence_yn": "Yes", "claims_explanation": "In general, the paper provides sufficient evidence to support its core contribution on rotation-sensitive structure, including theoretical motivation and empirical validation on both synthetic noise and real CLIP embeddings. \nThe results convincingly show that CLIP embeddings from natural images are significantly more rotation-sensitive than white noise. \nThe reconstruction experiments and comparisons to existing methods are clearly presented and sufficient to substantiate claims about the interpretability/fidelity trade-off.\n\nHowever, broader claims regarding interpretability are primarily supported by qualitative and illustrative examples. \nThe interpretation of rotation-sensitive directions as meaningful concepts relies on post-hoc alignment with images and text descriptions, without quantitative validation of semantic coherence or stability across datasets and models. \nSimilarly, while spurious correlation removal improves worst-group accuracy on benchmark datasets, it remains unclear whether these gains reflect deeper interpretability or simply the removal of dataset-specific nuisance features (e.g., background-related features in the Waterbirds dataset that are already known to be spurious).\n\nOverall, the evidence is sufficient for the statistical methodology, but more limited for claims about the interpretability of the learned CLIP representations or concepts.", "audience_yn": "Yes", "audience_explanation": "While the paper introduces an interesting statistical perspective on rotation-sensitive structure in embedding spaces, it remains unclear how the proposed analysis meaningfully advances understanding of the CLIP embedding space itself. \nThe experimental evaluations rely on largely dataset-specific and ad-hoc setups, making it difficult to assess how the findings generalize or provide broader insight into CLIP representations. \nAlthough the rotation-sensitivity analysis is potentially useful, the paper does not follow through with a deeper investigation of how the identified concepts arise within the model or how they relate to CLIP’s internal representational mechanisms.", "requested_changes": "1- While the paper claims improved interpretability of the CLIP embedding space, it remains unclear how the proposed analysis leads to a deeper understanding of CLIP representations. Providing insights into how the identified rotation-sensitive directions relate to CLIP’s internal dynamics would help close this gap. Although not required for the current scope, connecting learned concept directions to architectural components, training signals, or representational mechanisms (e.g., attention patterns or layers) would significantly strengthen the interpretability narrative.\n\n2- While the statistical detection of rotation-sensitive structure is convincing, the paper should more clearly explain why such structure should correspond to human-meaningful semantic concepts as claimed in the introduction section.\n\n3- The spurious correlation experiments would benefit from a clearer discussion of whether the observed performance gains arise from meaningful disentanglement of semantic concepts or simply from removing known dataset-specific nuisance features (e.g. the background in Waterbirds dataset). \n\n4- It would be great to get some insights on whether the proposed rotation-sensitivity analysis is expected to generalize to other vision language models. Clarifying the extent to which the approach is CLIP-specific versus model-agnostic would improve the positioning and relevance of the work.", "broader_impact_concerns": "There are not ethical implications of that work that needs to be addressed.", "additional_comments": "Minor comments:\n- The language at some places sound informal such as  P.1 \"CLIP is a powerful tool\".\n- The introduction of dimensionality $k$ in Example 2 in P.4 is abrupt without clarifying it to be the $k$ for truncated SVD.", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)