ReSi: A Comprehensive Benchmark for Representational Similarity Measures
Authors: Max Klabunde
Tassilo Wald
Tobias Schumacher
Klaus Maier-Hein
Markus Strohmaier
Florian Lemmerich
Venue: ICLR 2025 Poster (2025)
Keywords: representational similarity
benchmark
grounding
representations
Primary area: datasets and benchmarks
TL;DR: We present the ReSi benchmark -- A comprehensive benchmark for Representational Similarity Measures

Abstract: Measuring the similarity of different representations of neural architectures is a fundamental task and an open research challenge for the machine learning community. This paper presents the first comprehensive benchmark for evaluating representational similarity measures based on well-defined groundings of similarity. The representational similarity (ReSi) benchmark consists of (i) six carefully designed tests for similarity measures, (ii) 24 similarity measures, (iii) 14 neural network architectures, and (iv) seven datasets, spanning over the graph, language, and vision domains. The benchmark opens up several important avenues of research on representational similarity that enable novel explorations and applications of neural architectures. We demonstrate the utility of the ReSi benchmark by conducting experiments on various neural network architectures, real world datasets and similarity measures. All components of the benchmark are publicly available and thereby facilitate systematic reproduction and production of research results. The benchmark is extensible, future research can build on and further expand it. We believe that the ReSi benchmark can serve as a sound platform catalyzing future research that aims to systematically evaluate existing and explore novel ways of comparing representations of neural architectures. 
ReSi is available at https://github.com/mklabunde/resi.

Published as a conference paper at ICLR 2025
RESI: A COMPREHENSIVE BENCHMARK FOR REPRESENTATIONAL SIMILARITY MEASURES
Max Klabunde Tassilo Wald2.3,4+ Tobias Schumacher"..01
Klaus Maier-Hein 2,3,,,8 Markus Strohmaier Florian Lemmerich*
UUniversity Passau Medical Image Computing. German Cancer Research Center (DKFZ)
Hellmoltz Imaging, DKFZ University of Heidelberg University of Mannheim "RWTH Aachen University "Heidelberg University Hospital
National Center for Tumor Diseases (NCT) Heidelberg
GESSSS Leibniz Institute for the Social Sciences
"Complexity Science Hub
ABSTRACT
Measuring the similarity of different representations of neural architectures is fundamental task and an open research challenge for the machine learning com- munity. This paper presents the first comprehensive benchmark for evaluating representational similarity measures based on well-defined groundings of similar- ity. The representational similarity (ReSi) benchmark consists of (i) six carefully designed tests for similarity measures, (ii) 24 similarity measures, (iii) 14 neural network architectures, and (iv) seven datasets, spanning the graph, language, and vision domains. The benchmark opens up several important avenues of research on representational similarity that enable novel explorations and applications of neural architectures. We demonstrate the utility of the ReSi benchmark by con- ducting experiments on various neural network architectures, real-worke datasets, and similarity measures. All components of the benchmark are publicly available and thereby facilitate systematic reproduction and production of research results. The benchmark is extensible; future research can build on it and expand on it. We believe that the ReSi benchmark can serve as a sound platform catalyzing future research that aims to systematically evaluate existing and explore novel ways of comparing representations of neural architectures.
INTRODUCTION
Representations are fundamental concepts of deep learning that have garnered significant interest due to their ability to shed light on the opaque inner workings of neural networks. Studying and analyzing them has enabled insight into numerous problems, for example understanding learning dynamics (Morcos et al.) 2018 Mehrer et al.]20188, catastrophic forgetting (Ramasesh et al.) [201]) and language changes over time (Hamilton al.) 2016a). Such analyses commonly involve measur- ing similarity of representations, which resulted in plethora of similarity measures proposed the literature (Klabunde et al. 2023) Sucholutsky al. 2023). However, these similarity measures have often been proposed in an ad hoc manner, without comprehensive comparison to existing similar- ity measures. Moreover, they have often been proposed in conjunction with new quality criteria that were deemed desirable, with previously defined quality criteria being ignored. So far, only the few most popular measures have been compared (Ding al., [022]] Hayne al. (0244) or analyzed in more detail (Dujmović et al. 2023 Cui et al.) 2022 Davari et al. 2022
In this work, we present the first comprehensive benchmark for representational similarity measures. It comprises six tests that postulate different ground truth assumptions about the similarities between representations that measures could capture. We implemented these tests across several architectures and datasets in the graph, language, and vision domains. The ReSi benchmark enables tests for 24 similarity measures that have been proposed in the literature, and we illustrate how the results can
"Equal contribution. Author order among the co-first authors may be adjusted for individual use. "Corresponding author. towias..chumachhie www.uaccrr@uiimmmmmmhiiiii https::/// /i thub. abunde/resi

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Published as a conference paper at ICLR 2025
ReSi Benchmark Test
Desiign Modets
Define Similarity
Rank
Ground Truth Similarity
Representational Similarity Measures
7711
Model A
Model B
Bepresent Reprosentatinn Lafons
B
Model
Model D
Naprosettation
Reprosentatiess
Neprosefuutrr
Representationss
Decreasing Capture Truth
Figure I: Grounding similarity. all tests within the ReSi benchmark, we design set of models for which establish ground-truth about the similarity of their representations. The left heatmap illustrates the true similarity between a set of models, the other heatmaps the similarity values that different similarity measures assign to each model pair via their representations. We rank similarity measures by their ability to capture the ground truth. In practice, a ground-truth similarity between models is usually hard attain. For the ReSi benchmark. design tests where similarity is practically grounded.
provide insights into the properties and strengths and weaknesses of these measures. That way, ReSi can be useful as a test environment for new measures, reference that guides the choice of measures in an application hand, and a tool to obtain a deeper understanding of the differences between representations that are relevant to neural network behavior. All benchmark code and the corresponding models are openly accessible online.
GROUNDING REPRESENTATIONAL SIMILARITY
Before presenting the Resi benchmark, briefly introduce key terms and notations for represen- tational similarity, and discuss how ground-truths for representational similarity can be established.
2.1 REPRESENTATIONAL SIMILARITY
The ReSi benchmark is designed to evaluate the quality of measures that aim to quantify similarity of neural representations. Such representations can be derived by applying neural network model
f ....o f(),
where each function f() D' R denotes single layer, set of N inputs {X; N 1=1-for for
simplicity, we assume these inputs to be vectors in IR" even though, as the vision domain, these can also be multidimensional matrices. By stacking these inputs an input matrix X Nxp Xxp one can then slightly abuse notation and, at any layer I, extract the model's representation
= R (f() ((1-1) ..... (() (
where the rows R. = f(,) E R denote instance representations. Representational similarity measures compare full representation matrices R. R', and can thus be defined mappings
m R NxD NXD'
(3)
that assign scalar similarity score m R. R') to pair of representations R. R'. For brevity, throughout this work we will often denote these measures as similarity measures. Unless noted otherwise, we always consider representations from the final hidden layer of a neural network.
2.2 GROUNDING SIMILARITY
Approaches to measuring similarity of representations vary widely. For example, similarity can be related comparing pairwise distances in representation (Kornblith al. 2019), the ability
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Grounding by Prediction
Grounding by Design
Predictions
Rep. Matrix Space
Representation Matrix Spaze
1b
Regreseriatione Group B
A of
optimal
Sim
Rep. Sim within group
Rep. Sim across groupe
m
mi@
Desired:
Rep. Sim
Figure 2: Illustration of grounding approaches. We consider two approaches to establish ground- truths for representational similarity. When grounding by prediction, evaluate whether differ- ences in representation matrices correspond to differences in predictions of models, as, for instance, measured by Jensen-Shannon Shannon divergence (ISD). Ideally, representational similarity measure perfectly correlates with output similarity. When grounding by design, we design groups of models that are similar within and dissimilar across groups. A representational similarity measure should distinguish these groups accordingly.
to align two representations (Williams al., 2021 Li et al. 2015), or their topology (Barannikov
cet al. 2022) ffo broader overview of approaches, also the survey by Klabunde et al. (2023). While these approaches are usually justified by theoretical or practical desiderata, specific practical differences in models are not necessarily captured by these sapoaachs--modee difference is mul- tifaceted. One can distinguish models by their behavior like accuracy, robustness to augmentations or domain shifts, by their preference regarding texture shapes, or by aspects like differences in training data or their human-likeness, to name few examples. This multidimensionality leads to a crucial problem in measuring representational similarity, namely the lack of general ground truth that measures should reflect.
Due to this plurality in model behavior, number of possible targets to ground representational similarity have been proposed in the literature. For our purposes, we focus on the following two broad approaches to establish a ground truth for similarity, which we also illustrate in Figure 2 1. Grounding by Prediction. A straightforward way to obtain a ground truth for model similarity consider the differences in the predictive behaviors of a pair of models: when two models yield different predictions, they should also differ in their representations. This approach allows one to ignore where the source of difference between the models originates, but simultaneously implies that cannot be sure whether ground-truth similarities stem from differences in the classifier or the representations. Previous efforts that have used this way to ground similarity include the study by Ding al. (2021), who correlated representational similarity with accuracy difference, or the work by Barannikov et al. (2022), who grounded similarity to differences in individual predictions. 2. Grounding by Design. Through careful design, one can construct groups of representations for which one can impose a ground truth about similarity by relation. For instance, one can demand that representations from the same group be more similar to each other than representations from different groups. An example of such design is shown by Kornblith et al. (2019), who trained multiple models of the same architecture and formed groups of representations based on the depth of their producing layer. Then, they demanded that representations of the same layer should be more similar each other than to representations from other layers. These relative comparisons have the advantage that no deeper insights about the often opaque and nonlinear similarity measures are needed. However, the validity of the evaluation hinges on the validity the assumption that the representations actually follow the expected groups, which requires clear justification.
RESI: A BENCHMARK FOR REPRESENTATIONAL SIMILARITY MEASURES
We present the main components of the ReSi benchmark. It consists of (i) six carefully designed tests for similarity measures, (ii) 24 similarity measures, (iii) 14 neural network architectures, and
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Table I: Overview of the tests included in the ReSi benchmark. We design six tests that define
different ground truths for representational similarity.
Test
Intuition
Grounding Type
Correlation Accuracy Difference
Can variation performance of classifiers triined representaations cuptured?
Prediction
Correlation Ourput Difference
Can variation individual predictions of classtiers trained on representations be captured!
Prediction
Label Randomization
Can models that were trained on different labels be distinguished.
Design
Shortcur Affinity
Can models that rely different feutures distinguished??
Design
Augmentation
Can models different levels training sugmientation distinguished?
Design
Layer Monotonicity
Does layer similarity decrcase with increased layer distance within model?
Design
(iv) seven datasets, spanning the graph, language, and vision domains. In addition, we discuss how the results for our benchmark are evaluated.
3.1 TESTS
We provide descriptions of the six tests implemented within the ReSi benchmark. We begin with discussing two tests that are grounded by predictions, and then describe four tests that are grounded by design. An overview of these tests is given Table T In all tests, models were trained for classification tasks. Additional details of the test configurations can be found in Appendix B
Test 1: Correlation to Accuracy Difference. When two models are trained under similar condi- tions but differ in their performance, this can be seen as signal that the underlying representations are different. Following this intuition, we correlate the representational similarity of a pair of models with the absolute difference in their accuracies, thus similarity is grounded by predictions. Training Protocol. We roughly follow the protocol established by Ding et al. (2021). for which ten models are trained on each dataset, varying only by the training seeds. Afterward, we compute the accuracy of the models on the test set.
Test 2: Correlation to Output Difference. This test follows the same intuition as Test 1, with the difference that it focuses on differences in individual outputs of each model rather than their aggre- gated accuracy scores. Given that models with similar accuracy may still yield different predictions on individual instances, this provides more fine-graing grained signal which can serve as a more robust grounding for representational similarity. Thus, similarity again grounded predictions. Training Protocol. Due to the similarities to Test 1, we use the same models for this experiment.
Test 3: Label Randomization. In this experiment, we train models on the same input data, but with labels randomized to different degrees. The expectation is that models that learn to predict the true labels learn different representations compared to the models that learn to memorize random labels. These differently trained models are grouped by their degree of label randomization during training. We evaluate whether similarity within a group is greater than between groups. Therefore, representational similarity is grounded by design of these groups.
Training Protocol. For each domain, we create at least two groups of models. We always train one group fully correct labels and one group on fully random labels. Additional groups have partially randomized labels (25%) 2,5%556 50%, 75%). Across all domains, we always trained five models per group.
Test 4: Shortcut Affinity. The ability to identify whether two models use similar or different fea- tures can be a desirable property for a similarity measure. Hence, we create a scenario, in which we control feature usage by introducing artificial shorteut features the training data. Specific features correspond to each label, thus leaking them, but not necessarily perfectly-we can add incorrect shortcut features for some instances. Groups of models are then formed based on the degree to which the shortcut features in the training data match the true labels. Similarly to Test 3, we assess whether similarities within the same groups are greater than between groups. That way, representa- tional similarity is grounded by design of these groups.
Training Protocol. For each domain, we construct one group of models trained on data with short- cuts that always leak the correct label, one group trained on data with completely random shortcut features, and additional groups in which the labels were leaked on a fixed percentage of training samples. Representations been introduced. Each group consists of five models, in which varying training seeds affected model training and shortcut features.
then extracted from fixed test set, which random shortcuts have
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Table 2: Similarity measures included in the Resi benchmark. For each measure, we present their type of measure as specified in the survey by Klabunde et al. (2023), their abbreviations as used in our plots and tables, reference, and the preprocessing they require. RSM abbreviates Representational Similarity Matrix, i.e., matrix of pairwise similarities between the instances. CCA denotes Canonical Correlation Analysis.
Type
Measure
Proprscessigg
CCA
SVCCA
PWCCA
Singulaz Value CCA
Projection Weighied OCA
Monous Lll
Regal et
20
AligaCos
Aligned Cosine Similarity
Hamilton et al.
2016b
AngShupe
Orihagonal Angular Shape Metric
Wiramis
20/21
Conterol columns and unit matrix norm
HandCoer
Hand Correlation Mitch
LinRay
Regression
Ceniered columns and matrix norm
Alignment
OrthProc
Orthagmal Priemistss
PempPoee
Permitation Procrustss
Wilains Pl M
2021
ProdDist
Procrusies Size-an Shape Distance
Wir
2021
Cenierad columns
SofCCo
Solt Corelation Match
CKA
Centered Kemee Alignment
Korablith
Centredd columns
Correlation
BOS
Eigenspace Overbp Score
Max R K0D
RSM
GULP
GULP
Butt-Arbrri
2022
Conicred and nort to V
RSA
Representaionn Similarity Analysis
Norn Difference
110 SKl
2nd-Cos Cos
Sccond Order Cosine Similarity
[amilio el gl
20
Neighboss
Jaccard
Jaccard Sumlianit
Walls 2 3I
KankSim
Kank Similarity
Wang et 3l:
Topology
IMD
IMD Soore
Tsitulin
2020
Repesontaiion Topology Diverponce
ConcDiff
Concestriityy Difference
et
Statitic
MagDill
Magnitde Diflerence
Walls 3:
A05
UnilDiff
Unifommty Difference
Wang & bo0
20.0
Test 5: Augmentation. Augmentation is commonly used to "teach" models to become invariant to changes the input domain which do not affect the labels. In this test, we evaluate whether the sim- ilarity measures are able to capture robustness to such changes, by developing models with varying amounts of augmentation in their training data. Similar to before, we train groups of models with different degrees of augmentation. Models of the same group should yield more similar represen- tations than models from other groups trained on differently augmented data data-again, similarity is grounded by design of the models.
Training Protocol. In all domains, we trained one group of reference models on standard data, and additional groups of models with varying degree of augmentation the training data. Each group consists of five models, which only vary in their training seed. Representations are computed on the standard, ann-aueeeteed test data.
Test 6: Layer Monotonicity. As noted in Section 2.1 the individual layers (() of neural network f all yield representations R()) that can be compared. Given that these layers also represent a sequence transformations of the input data X, i.e. R² = ( (I((1))) it seems intuitive that representations of neighboring layers should be more similar than representations of layers that are further away, given the greater number of transformations between the further-apart layers] Thus, extract layer-wise representations of neural network and test whether
in this experiment, representational similarity measure can distinguish pairs of layers based on their distances from each other. Similarity of representations is, therefore, grounded by design.
Training Protocol. We reuse the trained models of Test I and Test 2, as no changes were made in the training scenario only for graph neural networks, we increased the number of layers five inner layers to enable sufficient number of comparisons.
3.2 REPRESENTATIONAL SIMILARITY MEASURES
ReSi evaluates 24 different similarity measures, for which provide brief overview in Table We used the reference implementations the measures where possible. Otherwise, we closely fol- lowed the given definitions and recommendations, also in the preprocessing of representations. Ex- plicit definitions of all measures and details on hyperparameter choices can be found in Appendix A
Thii could be violated if there were skip connections, but have controlled for this (see Appendix B.1
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3.3 MODELS
Overall, the ReSi benchmark utilizes a range of various graph, language, and vision models. De- tails on parameter choices can be found in Appendix [ All trained models publicly available. Graphs. As graph neural network architectures, we chose the classic GCN (Kipf & Welling (017)) GraphSAGE (Hamilton al. 2017), and GAT (Veličkovi 2018 models, using their respective
implementation from PyTorch ch Ge Geometro iic (Fey & Lenssen 2019). In addition, we considered
the desition-sensitiie P-GNN model (You et al. 2019), using the implementation of the authors. For each experiment and dataset, we trained these models from scratch.
Language. We chose the popular BERT architecture (Devlin et al. 2019) and its ALBERT variant (Lan et al. 2020)) well SmoILM2-I.7B (Allal et al. 2025) as LLM. For BERT, we use the 25 models pre-trained with different seeds from Sellam et al. (2022). For ALBERT and SmolLM2, a single pretrained model (see Appendix B.2 for details). In experiments,
fine-tuned these models on the given datasets to avoid computationally expensive pre- pre-training. We use the rep- resentation of the token that is passed into the final classifier, i.e., CLS for BERT and ALBERT, final token for SmolI.2, as well as mean-pooled representations over all tokens (also see Appendix I. Vision. We focus on prominent classification architecture families, namely ResNets (He et al. 2016). ViT' (Dosovitskiy et al. 2 and older VGG's (Simonyan & Zisserman 2013). To capture the ef- fect of scaling architecture sizes, we included ResNet18, ResNee33, ResNet101, VGG11, VGGI9, ViT-B/32 and the ViT-L/32 architectures. All models were trained from scratch, apart from the ViTs which were initialized with pre-trained weights from ImageNet21k (Deng et al. 2009).
3.4 DATASETS
We provide a brief overview of the datasets used within the ReSi benchmark. More detailed de- scriptions of the datasets can be found in Appendix [B.2.]]
Graphs. We focus on graph datasets that provide multiclass labels for node classification, and for which dataset splits into training, validation and test sets are already available. Specifically, we se- lect Cora (Yang et al., 2016). Flickr (Zeng et al., 2020), and OGBN-Arxin (Hu et al.) 2020). For the Cora graph, we extract representations from the complete test set of 1,000 instances, whereas for Flickr and OGBN-Arxiv, Language. We use two classification datasets: SST2 (Socher et al. 2013) is a collection of sen- tences extracted from movie reviews, labeled with their sentiment. MNLI (Williams et al.) 2018) is a dataset of premise-hypothesis pairs labeled with the true relation of these pairs. We used the vali- dation and validation-matched subsets extract representations for SST2 and MNLI, respectively. Vision. We use ImageNet1 100 (IN100), a random subsample of 100 classes of ImageNetlk (Rus-) sakovsky et al.] (015) and CIFAR- (Krizhevsky & Hinton 2009). IN100 reduces training time while keeping image resolution and content similar to ImageNet The image resolution was fixed to 224x224 on IN100 and 32x32 on CIFAR-100 except for ViT models, which used 224x224.
subsampled the test set 10,000 instances for computational reasons.
3.5 EVALUATION
Lastly, we describe how we evaluate and quantify the performance of representational similarity measures within the ReSi benchmark. Due to the different nature of the two approaches we use to ground representational similarity, we present the corresponding evaluation approaches separately.
When grounding similarity of representations with predictions of their corresponding classification models f. we specifically consider the outputs O == f(X) E NxC
Grounding by Prediction.
where we assume that each row O = f(X.) E IRC consists of the instance-witg wise class probability scores for C given classes.
In Test 1, we leverage these outputs to correlate representational similarity with absolute differences in accuracy. Thus, letting face (O) ;= qoce (O, y) denote the accuracy of an output O with respect to ground-truth labels y E IR we compute the absolute difference in accuracies
(4) Then, given a similarity measure m. and letting F denote the set of models trained for this test, for all model pairs f' F we collect the representational similarity scores ml RR, R') as well as the accuracy differences (O, O'), and report the Spearman correlation between these sets of values, together with the levels of statistical significance.
As, 1 ( () qaco (O'))

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Figure 3: Aggregated ranks of measures across all models and lests, separated by domain. Lower is better. Measures are ordered by their median rank, and categorized according to the taxonomy by Klabunde et al. (2023). Tied measures all receive the best rank of their group. Boxplots indicate quartiles of rank distributions, the whiskers extend up to 1.5 times the inter-quartile range. No single measure or category stands out across all domains.
In Test 2, we consider differences in instance-wise predictions O, O rather than differences in aggregate performance scores. Thus, for all pairs of models f. f F, compute the disagreement
ADIS (
Σm= 1 (arg max; Oi, # arg maxj 0,g);
(5)
between their hard predictions, and the average Jensen-Shannon divergence (JSD)
AIsD(O, 2N Σm JSD(O.1(O))
(6)
of the class- wise probability scores, and report the Spearman correlation of both measures with the corresponding set of representational similarities m R, R').
Grounding by Design. For all tests of this category, we do not consider functional outputs of models as a ground truth for representational similarity anymore. Instead, we have created multiple groups of representations G, typically separated by differences in model training, through which we impose a ground truth about similarity by relation. In Test 3, 4 and 5, we postulate that for any similarity measure it should hold that representations R. R' G from the same group should be more similar to each other than representations R E 9, R" E g" from different groups, that is,
m(R, R²) S RR,, R²),
where we assume that for m. higher values indicate more similarity. Then, trivial performance measure is given by the conformity rate, i.e., the relative amount of times a similarity measure m satisfies (7) across all combinations of groups and representations. Given that this measure can, however, be biased by the ratio of intergroup vs. intragroup pairs, we additionally adapt the Area under the Precision-Recal curve (AUPRC) measure to our context. This is done by assigning a label y R. RR) == 1 [g G'} to each pair of representations R REG. G. R' E G' that can be compared within the given set groups, and then interpreting the representational similarities m (R, R2') as decision scores based which one should be able to "predict" the label y Rt, R').
Finally, in Test 6, we group representations R by the layers / they were extracted from. However, in contrast to the previous tests, we postulate that an ordinal relationship between the layers has to hold. Specifically, given model with L layers, for all tuples 1 iiisj k < L < L we require measure m to satisfy
m( R m(R RR())),
assuming that for m, higher values indicate more similarity. As in the previous tests, we report the corresponding conformity rate. Further, we report the Spearman correlation between the similarities m(R(), R(1) and the distance j of the corresponding layers over all tuples $ i < T..
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Table 3: Exemplary results for selected datasets and models. We show results of GraphSAGE on Flickr for graphs, BERT on SST2 (CLS token) for language, and ResNet18 on ImageNet100 for vision. For the first two tests, we report the Spearman correlation between representational simi- larity and the difference in accuracy and JSD, respectively. For Tests 3-5, we report the area under the precision-recall curve, quantifying if the corresponding groups of models can be separated by the similarity measures. For Layer Monotonicity (Test 6), we report the average Spearman correla- tion between representational similarity and layer distance. Higher values indicate that a similarity measure better reflects the ground truths from our tests. Our benchmark highlights that similarity measures have different strengths and weaknesses, and that there exists no measure that performs well across all tests and all domains. For example, PWCCA (first row) separates layers of Graph- SAGE and BERT well, but not the layers of ResNet18.
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PRECA
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0-47
0,13
0.21
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0.72
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08.76
0.81
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0.60
6.71
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0.45
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01.9
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0.37
0.55
06:3
0.45
01.5
0.64
4.35
Ind-Cie
-3.19
0.30
40.56
5u8m
10.33
16:2
03.37
1.00
1.86
0.64
1.50
0.52
13400
6.75
0..96
0.57
0,55
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0.25
0.17
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0.34
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14:33
0.15
1.10
11.03
0.64
1.10
11:35
0.39
6.29
0.51
0.16
6.35
5.16
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IMD
0.15
0.37
-316
5115
0:37
0129
-014
0.24
-8.11
4639
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0694
0131
01:47
1.50
1.40
1.80
0.39
0.34
0.67
1.00
0.57
1.80
0.38
0.30
0.57
0.56
0159
0183
0.99
0.58
-.10
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0.12
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0357
0.06
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300
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4.37
Statatical significance for prodiction grriiiionggddddddddd tests evaluated with cocrelation Indicated by (556) and (15).
4 BENCHMARK RESULTS
In the following, we illustrate selected benchmark results. Due to limited space, we focus on pre- senting (a) an aggregated overview and (b) an exemplary detailed result for a single dataset and a single architecture. We provide detailed results in Appendix [ We further present average runtimes
of the similarity measures, which can vary by multiple magnitudes, in Appendix D
For an aggregated result overview, we rank all measures for each combination of test, dataset, and architecture. The rank distribution across tests in each domain is shown in Figure 3 We observe that no measure outperforms the others consistently across the domains, with most measures having high variance in their rankings. However, some domain-specific trends can be identified, such as neighborhood-basee second-order cosine similarity (2nd-Cos), Jaccard similarity, or rank similarity measures that perform well in the graph domain. In the language domain, the popular linear CKA measure performs best. Further, in the vision domain, various types of measures perform well- particularly the less-used Jaccard similarity. Orthogonal Procrustes, which has overall performed well in the analysis by Ding et al. (202T). consistently ranks among the top 50% of measures. By contrast, other prominent measures, such as RSA or SVCCA, do not appear to stand out in general.
The ReSi benchmark can also provide more detailed results for analysis. To provide some exam- ples, Table presents the outcomes for a single dataset and architecture in each domain. Higher values always indicate better adherence of measures to similarity groundings. Again, we observe that, even for a single dataset, there no single measure that outperforms the others across all tests. Instead, most of the measures appear to perform well in some tests, but not in others. For instance, measures such as the eigenspace overlap score or GULP appear to be able to perfectly distinguish layers of GraphSAGE model but appear incapable of identifying whether they have learned spe- cific shortcuts, which most other measures appear to be capable of. One can also identify model- specific trends, such as neighborhood-bare based measures correlating particularly well with predictions from BERT models. In addition, considering the performance of descriptive sures may provide additional geometric insights representations: for instance, the concentricity of BERT representations appears to inform relatively well about the degree to which it was trained to memorize data.
srtissiis-bassd mea-
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5 TAKEAWAYS
No Free Lunch. Throughout our benchmark, we observe that ranks of individual measures vary sig- nificantly. For example, CKA (Kornblith et al. 2019) ranks first in Language, eighth in Vision, and 11th in Graphs when aggregating across all tests. This inconsistency extends beyond CKA, high- lighting limitations in the general applicability of established measures. This implies that measures should be chosen carefully for task at hand and that specific tasks may require the development of specialized similarity measures to appropriately capture relevant aspects of model behavior.
Preprocessing Matters. The similarity measures analyzed in this study include Orthogonal Pro- crustes (OrthProc) and Procrustes Size-and-Shape Distance (ProcDist), which differ only in the way the representations are preprocessed. Our results indicate that this seemingly small difference has a substantial impact on what properties of models similarity measures capture. This highlights av- enues for future work: preprocessing implicitly assumes that representations are equivalent under the preprocessing transformation. If, in the case of OrthProc, results are consistently worse after normalizing representations to unit norm, this could be an indicator that the absolute magnitude of each axis carries semantic meaning that similarity measures need to pick up on. Therefore, a bet- ter understanding of what kinds of representation are truly equivalent is crucial, and the tests from ReSi could be used to empirically investigate the impact of different preprocessing techniques. Need of Best Practices. Currently, similarity measures are often selected without deeper justifica- tion. Our results indicate that this practice leads to similarity scores that are difficult to interpret, it is unclear whether the score indicates similarity on an interpretable axis and whether the score is spe- cific to a measure. We argue that the community needs develop a holistic enables robust and reliable analyses with similarity measures, which was also recently advocated for by Soni et al.) (2024) in a neuroscience context. This further highlights potential for cross-domain pollination. With the following recommendations, we make a step towards such guidelines.
of best practices that
General Recommendations. Based on our results, from Figure 3] we can derive some general recommendations for applying similarity measures on specific domains. For graphs, neighborhood- based measures appear favorable, for language models, one should likely prefer CKA or distance correlation, and for vision models. Jaccard similarity appears to be the overall best choice. In gen- eral, the groups of alignment-, RSM- and neighborhood-bar form better than CCA-, topology- and descriptive crciistics-base measures. However, this does not imply that measures from these familics cannot be useful in specific applications. If one considers grounding to specific test as particularly important, the tables in Appendix Hapendix[[inoone inffrrm on the ability of similarity measures in that test. If, for instance, one considers correlation with output difference the best grounding, Jaccard similarity would be preferable for the vision domain, distance correla- tion for the language domain, and RSA for the graph domain.
measures also appear to overall per-
Unexpected Efficacy of Overlooked Measures. Within our benchmark, we implement a broad set of 24 measures, of which several measures have hardly been considered before in related literature and which sometimes performed surprisingly well. For example, Jaccard similarity emerged as the best-perforning measure in the vision domain. This finding challenges popular opinion and sug- gests that rarely used measures, despite their limited prior application in these contexts, may possess unique properties that make them particularly well suited for certain types of data. This unexpected performance highlights the potential for revisiting and rigorously testing lesser known measures. Potential for Deeper Insights. ReSi also be used in different ways to obtain deeper insights about neural representations. One way comes from direct analysis of the given results. For instance, the finding that neighborhood- based measures perform well in the graph domain may indicate that neighborhoods within GNN representations are driven by training objectives. Similarly, the results of Test 5 indicate that augmentation of BERT models particularly affects uniformity of their repre- sentations. Although these findings can only be considered preliminary signals, they illustrate how ReSi can provide valuable pointers for future research. Additionally, the groundings provided by ReS 1551's 's tests can be used to more systematically analyze the impact of preprocessing (as pointed out above), neural network design choices such as the type of normalization layer, or parameters of similarity measures such similarity functions for RSMs, on representational similarity scores.
RELATED WORK
Despite the large amount of representational similarity measures that have been proposed in the lit- erature (Klabunde et al. 2023), only few efforts have focused on comparative analysis of existing

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similarity measures. In the following, we briefly provide overview of existing work. Grounding by Prediction. Ding et al. (202T) first proposed grounding similarity of representations to performance differences of probes trained them. They benchmarked three similarity measures, with tests focusing on identifying effects of seed variation or manually manipulating the represen- tations. Compared ReSi, they focus only a single test in our framework. Their work was extended by [Hanne et al.] (2024) with focus on measuring similarity of high-dimensional repre- sentations in CNNs. Similar Test 2 in ReSi, Barannikov al. (2022) tested the ability of their similarity measure to capture prediction disagreement.
Grounding by Design. Similar to Tests 3-5 in ReSi, some previous work has also tested whether similarity measures could distinguish representations based on differences in their underlying mod- els. Adix-A et al. (2022) considered representations of models from different architectures, and required similarity measures to distinguish representations by model family. [Tang et al. (2020) tested whether measures could distinguish models based on differences in their training data. In a setting similar to Test 6. Kornblith et al. (2019) grouped representations of models that only differ in their training seed by layer depth, and tested whether measures assign more similarity to represen- tations from corresponding layers. Finally, few analyses have tested whether similarity measures can detect model-independent changes in representations. For instance, it has been explored whether measures distinguished synthetic representations with different cluster structure (Barannikov al., 2022). levels of noise (Morcos et al. 2018). or subsampled dimensions (Shahbazi et al. 2021
7 DISCUSSION AND CONCLUSIONS
We briefly summarize our contributions and discuss limitations well potential avenues for fu- ture research to be built on ReSi
Contributions. In this paper, we have presented ReSi, the first comprehensive benchmark for rep- resentational similarity measures. By applying its six tests on a wide range of 24 similarity measures to evaluate their capabilities on graph, language, and vision representations, we have demonstrated that it (i can serve test environment for new measures, (ii) provide guidance regarding which measures to choose for specific domains, architectures or application scenarios, and (iii) can be leveraged to obtain deeper understanding of representational similarity. We provide all code pub- liely online, and invite the machine learning community to use it to test (potentially new) measures, or to extend it to include more models even additional tests.
Limitations. We have put considerable effort into properly grounding the benchmark tests. Specif- ically, when grounding by design, have not only carefully set up distinct training conditions, but also verified from the validation performance of the models under consideration that these in- deed follow the intended behavior and adapted the generation process if necessary. Still, we cannot fully control what models learn. Two behaviorally identical models could still differ in their internal process and, thereby, their representations, potentially confounding the resulting test scores. Never- theless, our tests would inform whether the grounding property can be inferred from representations without supervision. When grounding by predictions, we have deliberately restricted the tests to models that only vary in their training seed, to reduce the potential for additional confounders as much as possible. Furthermore, in our tests, we solely focus on last-layer representations. The va- lidity of the existing tests at other layers could be verified by evaluating whether the probes follow the same behavioral patterns as the full model.
Future Work. While ReSi implements 24 similarity measures, not all existing measures are in- cluded. Additionally, we did not explore the effects of different preprocessing approaches or pa- rameter choices of individual measures. Given that such factors can lead to different results (Boix- Adsera et al.] 2022] Timkey & van Schijndel (02T). ReSi opens the way for a systematic evaluation of such steps future work. Similarly, the six tests implemented in ReSi should be considered a comprehensive but extensible foundation of tests that users can build upon. We see great potential in i augmenting ReSi with additional tests, with the aim of a unified test suite that can inform the whole community of researchers and practitioners. For example, tests could be grounded via insights from interpretability, e.g., how similar reliance on specific input features is, as found via explana- tions (Eberle et al. 2022) aasuumng the explanation is faithful-or how similar internal features are, as discovered via sparse autoencoders (Lan et al. 2024). Finally, we acknowledge that ReSi currently evaluates similarity measures within single-modalit, settings. However, these measures are also crucial for measuring the alignment of representations in multimodal contexts. Extending ReSi to encompass multimodal settings represents highly promising future work.
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ACKNOWLEDGEMENTS
This work is supported by the Deutsche Forschungseemcinccaatt (DFG, German Research Foun- dation) under Grant No.: 453349072. The authors acknowledge support by the state of Baden- Württemberg through bwHPC and DFG through grant INST 35/1597-1 FUGG. This work was partly funded by Helmholtz Imaging (HI), a platform of the Helmholtz Incubator on Information and Data Science.
REPRODUCIBILITY STATEMENT
All our code and data well instructions how to run the benchmark are publicly available at https://githuue com/mk /k abunde/reee
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A REPRESENTATIONAL SIMILARITY MEASURES
In this section, first provide explicit descriptions of all similarity measures that included in ReSi, and then give details on the hyperparameter choices.
A.1 DEFINITIONS SIMILARITY MEASURES
In the following, we provide brief descriptions of all the representational similarity measures that consider in this study, where we follow the categorization proposed in the recent survey by Klabunde et al. (2023). For more detailed descriptions and broader overview of existing measures, also point to this survey.
CCA-based Measures. Canonical Correlation Analysis (CCA) Hotelling (936)) is based on the problem of finding weights WR E RD, WR' E RD' for the columns in the representations, such that the linear combinations Rw n and R' "ww nt' ER have maximal correlation. Assuming mean- centered representations, one can determine set of canonical correlations p: that satisfy
(Ra n Rt R'u(t)) Ro 1
P: max wR (
S.b. Ra (
Rww R
R'w LR'
Vj <i.
A single similarity score m(R, R') can then be obtained by aggregating the individual canonical correlations pi, e.g., via taking their mean:
MccA ( (R, R') D E.-P.
(10)
In Singular Value CCA (SVCCA) (Raghu al. 2017), this exact aggregation is used, but represen-
tations are first denoised by applying PCA on the mean-centered representations, removing those principal components, which explain less than fixed percentage (usually 1 1%) of the variance.
For their Projection-weighhee CCA (PWCCA) measure, Morcos et al. (2018) considered a weighted average the canonical correlations, where the weighting coefficients a; Σソ= (Rw (6) R y model the importance of each canonical correlation Pi-
Alignment-based Measures. Several measures from this category are based on solving the or-
thogonal Procrustes problem, which intuitively is based on finding the best orthogonal mapping of representations onto each other. Specifically, the Orthogonal Procrustes (OrthProc) measure defined via
morthProe R, R')
min QEO(D)
RRQ RRQ-R'I R (|
R -
(1I)
where it is assumed that columns the input representations are centered, and scaled to unit norm. The Procrustes Site-and-Shape--iiiie
only differs in preprocessing, as it does not assume that the matrix is scaled to unit norm. For the Permutation Procrustes measure, the minimization Equa- tion (TT) is restricted to permutation matrices. The Orthogonal Angular Shape Metric (Williams) ct al. 2021 follows a similar rationale, but optimizes minimizes a Frobenius norm:
M AngShape
R, R') = min arccos RQ, R')
(12)
QEG(D)
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The matrix Q+ that yields the solution of the Procrustes Problem in Equation (II is also used for the Aligned Cosine Similarity measure. Specifically, it considers the cosine similarities between all instance representations after alignment, and uses their average similarity score:
ClliguCos R, R') N Σ i=1 cos-sim ((RQ),, R.).
(13)
In a slightly different approach, (Li et al. 2015) proposed to align representations by matching neurons based on their correlation. For the Hard Correlation Match measure, neurons matched one-to-one a greedy fashion. Letting M denote the matrix that matches the neurons, which is a permutation matrix in this strict matching, the measure then yields the average correlation between the matched neurons:
Morr (R, RR) +
IR_ ; (R')) RR_ R)
(14)
For the Soft Correlation Match, this matching is relaxed, so that neuron from R can be matched to multiple neurons from R'.
Finally, the Linear Regression measure aligns representations by predicting one from the other via linear transformation. The resulting 1²2 score can then be used as similarity score:
MinnReg R. R') = min we I RWIl
(15)
[
[
RSM-based Measures. To avoid issues in finding optimal alignments, several methods consider
representational similarity matrices (RSMs), which describe the similarity of each instance R. to all other instances in representation R. Formally, the RSM S RNXN of representation R can be defined in terms of its elements via
(16) where s : IR x RD - R denotes a given instance- wise similarity function. Common choices for the similarity function include correlation Kriegeskorte et al. (2008) or kernel functions Kornblith jet al.] (2019). direct way obtain representational similarity measure is then to consider the norm of the difference between two RSMs, as in the RSM Norm Difference Yin & Shen (2018):
== = s===(< R,).
MRSMD:E R, R') |S S'
(17)
More broadly, Kriegeskorte et al. (0088 proposed Representational Similarity Analysis (RSA), which considers inner similarity function Sin compute RSMs, of which the lower triangles
are then vectorized to vector v(S) E NNN-11 and compared via an outer similarity function Sout
MASA RR R') Sour (v()) v(S)).
(18)
One of the most commonly used similarity measures in related literature is Centered Kernel Align- ment (CKA), which has been proposed by Kornblith al. (2019). CKA applies kernel functions as instance-wise similarity measures 8 to compute the RSMs, which are then compared via the Hilbert-
Schmidt Independence Criterion (HSIC) (Gretton et al. 2005). Specifically, the similarity score is computed via
McKA (R, R') v /HSICIS, HSSCCISSSSSSSSSS...
HSIC
(19)
Distance Correlation (DistCorr) Székely et al. 2007 is motivated from testing statistical depen- dence of two random variables, but can also be applied as representational similarity measure. As- suming that RSMs are mean-centered both rows and columns, one can compute their squared
sample distance covariance dCov?(S,)) (S,S") S') = N Σ Σ St and then derive the distance correlation via
MDDssCoo
CaaiioonnRRA (R.
dC.v"(S.s') /dCoaa (S,S) (S'
(20)
Eigenspace Overlap Score (Myy et al. 2019) compares RSMs by comparing the spaces spanned from their eigenvectors. Letting U E R XD U' = NDD denote the matrices of eigenvectors that correspond to the non-zero eigenvalues of S, S', respectively, the measure is defined as
mgos RR, R')
max(D.D")) UTU
(21)
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Finally, GULP Adser-A al. 2022 aims measure the extent of how differently linear (ridge) regression models that use either the representation R or the representation R' can gen- eralize. Letting the RSMs S & R R denote the matrix of covariance within representation, S SR, R Rt the cross-covariance matrix, and S (S + ALD) the inverse of a regular- ized covariance matrix, they provide a closed-fon form definition of the GULP measure
A (k)R)) 1/2
mGULP R') = Lir(S>SSS- S) + +r((" - 21r((S-ASnn -
(22)
where the hyperparameter 2 models.
corresponds to the regularization weight of the ridge regression
Neighborhood-basee Measures. The following set of measures is based on comparing the nearest neighbors of instances in the representation space. Each measure determines the sets of k nearest neighbors N (i) of each instance representation R; from the full representation matrix R with respect to given similarity function s, and then computes a vector of instance-wise neighborhood similarities (UNN( R, R'), NY. which are averaged over all instances to obtain similarity measures for the full representations R R':
mNN R, RR) = N Σ UNN R, R');
(23)
For the k-NN Jaccard Similarity, this vector simply contains the Jaccard similarities of the nearest neighbors of each pair of corresponding instance representations R, and Ry:
(Ffaccard R. RR),
(VR (+) (E [3]"
(24)
Second-Ore -Oderr Cosine Similarity (Hamilton et al. 2016a) considers the union of the instance- wise nearest neighbors in terms of cosine similarity ordered set (ji..... == (N) and then compares these cosine similarities to the nearest neighbors via
(wind-Cos R, RR)), == cos-sim ((Sij).... S.jK() (S S'ikk)))
where S denote the RSMs w.r.t. cosine similarity.
Rank Similarity (Wang et al. 2022) not only considers the cardinality of the overlap between instance-wise neighborhoods N (i). but also factors in the order of common neighbors with respect to cosine similarity. To increase the importance of close neighbors, this measure defines distance- based ranks TR, (j) for all j (t),,, where TR, (j) n if R, is the n-th closest neighbor of R, Based on these ranks, the instance- wise similarities are then defined as
(U (URankSim (R, R²), (Vennx)
EjEnHONNN (1+ G) Uillr 1i) G1)>
(25)
where )i is a normalization factor that limits the maximum of the ranking similarity to one.
Topology-based Measures. Measures of this category aim approximate and compare lower- dimensional data manifolds, which the high-dimensional representations are assumed to be con- centrated on. This approximation is often done using discrete structures such as graphs, and this approach has also been chosen by [sitsuuin et al. (2020) for their Multi-Scale Intrinsic Distance (IMD) measure. Specifically, they considered k-NN graphs G(R) for each representation, which are then compared through their heat kernel trace, which is defined as hkt R) X; denoting the eigenvalues of the normalized graph Laplacian of G(R). The IMD score is then defined as
(t) = ΣS e -tAs, with
MIMD R, R') = sup - 2(t+0 hhkgggn (t) hktg(re)
(26)
t>0
Similarly, the Representation Topology Divergence (RTD) (Barannikov et al., 2022) considers graphs (RZ), in which nodes are connected if the Euclidean distance of the corresponding in- stance representations is lower than a, and union graphs go (R. R'), where edges are formed if in
at least one of the two representations this condition is satisfied. Based on these graphs, collects a set (RR, R') of intervals (a) d22,, in which these graphs differ in the number of their connected components. The total length of these intervals, denoted b( R, R') E(a B 022 01,,
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then quantifies similarity between representations. Given that this similarity is not sym- metric, one, however, further takes the average
MRRD Rt Rt) -
+ R)).
(27)
Instead of computing one score for the full representation matrices R. R', Barannikov et al. (2022)
further recommend sampling multiple subsets of instances based on which this score can be com- puted, and again averaging the resulting scores in the end.
Descriptive Statistics. The final set of measures that we test within our benchmark considers statistical properties of individual representations R. Given a pair of representations R, R', and statistic Misstt, the individual statistics are then compared by taking their absolute difference:
Sift Matat (R) Mstat (R')).
(28)
Toward that end, the first statistic we consider is Magnitude (Wang et al. 2022). which corresponds to the length of the mean instance representation:
Ma [ == 1 N R
(29)
Concentricity (Wang et al. 2022). by contrast, considers the average distance of each instance rep- resentation to the mean representation:
McCon R = N Di=1 cos-sim R Σラ= R2).
(30)
Finally, Uniformity (Wang & Isola 2020) measures how close the distribution of instance represen- tations is to a uniform distribution the unit hypersphere, and defined as
MUni R log
N
Σラ= R
(31)
where t is hyperparameter Wang & Isola (2020)
A.2 HYPERPARAMETER CHOICES
Regarding the hyperparameters of the measures under study, we largely followed recommendations made in the original references. Specifically, made the following choices:
For SVCCA, we included as many principal components as necessary to explain 99% of the variance in the training data.
For RSMDiff, we computed the RSMs based on Euclidean distance
For RSA, we applied Pearson correlation as inner and Spearman correlation as outer simi-
larity function.
For CKA, we used the original, non- non-batched implementation, and chose a linear kernel as similarity function.
For the distance correlation (DistCorr), we computed RSMs based on Euclidean distance.
GULP, we set the regularization weight to
For all neighborhood-basee measures (Jaccard, RankSim, 2nd-Cos), set the neighbor- hood size to 10 and determined nearest neighbors based on cosine similarity.
For the IMD score, used 8,000 approximation steps and five repetitions.
For RTD, we always sampled 10 subsets of 500 instances.
For the uniformity difference (UnifDiff), we set 2.
All other measures did not provide any hyperparameters.
B EXPERIMENT DETAILS
Complementing the higher level experiment descriptions in the main manuscript, a more detailed description of them is provided in this section. This includes general information about the chosen model architectures, datasets and hyperparameters for each domain, with additional details that are unique individual experiments provided later.
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B.1 REPRESENTATION EXTRACTION
In order measure representational similarity, representations need be extracted from the trained architectures. As noted in the main manuscript, we use the representations of the last hidden layer for all experiments, except the monotonicity experiment, where we use additional hidden layers of the architectures as well. We chose to do so as we can verify our assumptions of a model learning shortcuts or learning to be more robust to augmentations with the predictions directly derived from these representations. In an earlier layer, for example, we would not be able to verify whether a model learned the shortcut or not. For the output correlation, similar argument can be made, as predictions are directly derived from these representations. Only for the layer monotonicity test, intermediate representations were used. Moreover, we carefully extract representations at locations where no residual connection bypasses the location of representation extraction to ensure that measure the entirety of the representation and not just a part of it, i.c., after transformer and residual blocks. This is especially important for the monotonicity experiment where disregard for this may invalidate the setting assumptions. Architectures, where this was important to consider, were all Transformer architectures, well the vision-specific ResNets.
Graphs. For the graph neural networks, the extracted representations naturally follow the desired matrix format R E NxD, where each row corresponds to the representation of specific node in the network. Thus, no additional processing was necessary.
Language. Transformers produce one representation per token of each input. As we focus on the last-layer representations, we focus the representation of the CLS token the representation for the whole input for BERT and ALBERT. As this token is used by the final classifier, we argue that all relevant information for that input will be contained in this representation making this token representation the most important one. However, we also compare models via representations mean- pooled over all tokens of an input to give representations of other tokens more direct influence. For SmoILM2, we use the final token of the input as it decides what the next prediction is, similar to the CLS token for BERT and ALBERT.
While it is possible to compare representations of all tokens, inputs with many tokens would take up more rows in the final representation matrix, which could bias similarity estimates towards sim- ilarity of long inputs. Additionally, the runtime of many representational similarity measures scale quadratically in the number of rows of the representations, which practically limits the size of the representations given the many comparisons that we had to run in our benchmark.
Vision. Vision models vary in their representations. CNNs generally conduct global average pooling operation before their classifier, leading to representations being in the desired R R format. Transformer-basee vision models yield CLS token in conjunction to additional tokens, hence we follow the language domain and only use the CLS token when comparing ViT representa- tions.
In the monotonicity experiment, CNN representations were extracted prior to the global average pooling layer, resulting in representations with varying spatial extents. This poses challenges for measures that leverage spatial extent to increase sample size N, as such measures often assume correspondence between features identical spatial positions. Furthermore, for spatial dimensions of w 112 and h = 112, the number of values in the representation exceeds 10,000, leading to potential memory issues. To address these concerns, we restricted representation extraction to layers with global stride 2 8, yielding = h = 28 in the ImageNet experiments. Additionally, we limited the extraction to the final six layers of the architecture. The extracted representations were subsequently scaled to uniform 7 x 7 spatial extent via average pooling, ensuring consistency across all layers except the classifier layer, which lacks spatial dimensions. For spatial representation comparisons, the spatial dimensions w and h were folded into the channel dimension c, converting representations of shape N x c x h x w to N x (chw), unless the measure required N > or exhibited prohibitively high computational complexity. In such cases, the spatial extent moved into the sample dimension, reshaping N x c x h x w to (Nhw) x c. To mitigate memory and time constraints, the sample size reduced from Nhw to N by subsampling every 49th value.
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For comparisons between spatial and non-spatial representations, spatial dimensions removed entirely through average pooling.
We note that this resampling and subsampling of the representations in the CNV case is suboptimal, yet this issue originates from the limitations of the similarity measure, namely requiring N or scaling prohibitively.
Moreover, in all vision experiments, representations were extracted from test images that models had not seen during training. For ImageNet100, the 50 validation cases per class used, resulting in N 5000 samples, and for CIFAR we use the full test dataset.
B.2 GENERAL EXPERIMENT DETAILS
Across all modalities and experiments, efforts were made keep architectures, training hyperpa- rameters or dataset choices as static as possible, while minimizing the overall compute effort that was necessary compare representations with all these measures. In the following, we describe general experimental settings which apply to all tests, separately for each domain.
B.2.1 GRAPH SETTINGS
Models and Parametrization. As graph neural network architectures, we chose the GCN (Kipf & Welling 2017), GraphSAGE (Hamilton al. 2017), and GAT (Veličković al. 2018) models due to their widespread prominence. our implementation, we used the respective model classes provided in the PyToreh Geometric (Fey & Lenssen 2019) package. We further applied the P-GNN (You et al.) 2019) architecture, since its position-aware approach to aggregation of node information provides contrast the other three models, which only aggregate information of neighboring nodes their convolutions. We applied the reference implementation by the authors, which was adapted the given node classification tasks. Due to its large memory consumption, was only feasible to apply this model on the Cora dataset. Further, we did not use this model for the augmentation test, because the DropEdge approach that position-avee aware embeddings dropping edges would alter the distances of nodes to the corresponding anchor sets, and these distances the key element the message passing of P-GNNs. Overall, also think that the consideration of datasets, where the corresponding task fits better to the strength of these position-ave posttion-aware GNNs, may be good extension for future studies that build on our benchmark.
for this test is not appropriate for
For each experiment and dataset, we trained these models from scratch. Regarding the hyperpa- rameters, we roughly followed values that were used for these models existing benchmarks. An overview of the hyperparameter choices for each algorithm and dataset are presented in Table The layer count also includes the classification layer, thus the number of inner layers is one less than the number given in the table. For the GAT model, used the Adam optimizer (Kingma & Ba 2015) as implemented PyTorch (Paszke et al. 2019) optimize cross-entropy loss the given multiclass datasets. To validate the performance, plot the training, validation and test accuracies the final models in Figure 4
always used h 8 attention heads. We always
All these parameter choices consistently applied in all our benchmark tests, except for Test 6 (layer monotonicity). where we increased the number of layers to six, yielding five inner layers.
Table Hyperparameters all architectures on the respective datasets in the graph domain.
Dataset
Architecture
Dimension
Layers
Activation
Dropout Rate
Learning Rate
Weight Decay
Epochs
GCN
relu
0.01
Cora
GraphSAGE
relu
0.5
0.005
0.0005
200
GAT
64
2
elu
0.6
0.005
0.0005
200
P-GNN
32
2
relu
0.01
200
relu
0.01
Flickr
GraphSAGE
256
3
relu
0.5
0.01
200
GAT
256
3
relu
0.5
0.01
0.0
200
GCN
256
3
relu
0.5
0.01
0.0
500
OGBN-Arxiv
GraphSAGE
256
0.005
0.0005
500
GAT
256
3
rolu
0.5
0.01
0.0
500
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Cora
Flickr
OGBN-Aniv
1.0
1.0
0.8
0.8
0.8
Acsuracy
0.6
0.4
0.2
0.2
0.2
0.0
GAT PONN
GCN
SAGE
SAGE
Train
Test
Figure 4: Validation accuracies of GNN models trained with standard parametrization. We used standard aainn/validationntesssspppp provided with the given datasets. Test accuracies largely corre- spond to those obtained in common benchmarks.
Datasets. We focus on graph datasets that provide multiclass labels for node classification, and for which dataset splits into training, validation and test sets already available. Thus, selected the following networks.
Cora (Yang al., 2016): in this citation network, nodes represent documents and edges represent citation links. Node features are given by bag-of-words representations of the corresponding documents, and node labels correspond to topical categories of the papers.
2. Flickr (Zeng et al. 2020j: in this network, nodes represent images uploaded Flickr, and edges are formed if two images share some common properties, such as geographic location, gallery, or users who have commented on Node features correspond to bag-of- word representations of the images, classes were formed based on image tags.
3. OGBN-Arxiv (Hu et al. 2020): this dataset represents citation network between Computer Science papers on arXiv. Each node corresponds to a paper, and directed edges indicate that one paper cites another one. Node features are given by aggregated word embeddings of paper titles and abstracts, labels correspond the subject area of each paper.
Statistics of the datasets can be found in Table
B.2.2 LANGUAGE SETTINGS.
Models. We BERT (Devlin al. 2019), ALBERT (Lan al. 2020 and SmolLM2 (Allal et al. 2025) models. BERT and ALBERT are encoder-only transformer models, whereas SmolL.22 a decoder-only transformer. Pretraining of these models expensive, so we rely on publicly avail- able models, which we will fine-tune for our tests. To make the tests challenging, the fine-tuning must induce nog-negligible differences behavior and representations. Otherwise, would be triv- ial to find models of the same group in the design-basee grounding tests, for instance. We generally fine-tune with different seeds, but BERT models in their whole are relatively little affected by this
Table 5: Statistics of the graph datasets used within the ReSi benchmark
Dataset
Nodes
Edges
Features
Classes
Cora
2,708
10,556
1,433
7
Flickr
89,250
899,756
500
7
OGBNVArXiv
169.343
1,166.243
128
40
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approach (Merchant et al. 2020). Thus, start fine-tuning of BERT from different pretrained models, provided by Sellam et al. (2022). For ALBERT, we fine-tune the same pretrained model the weights shared across layers, which should increase the changes induced in the model by fine-tuning. Finally, for SmolLM2, we finetune the 1.7B base mode
Datasets. We used two classification datasets: SST2 (Scher et al. 2013) a collection of sen- tences extracted from movie reviews labeled with positive or negative sentiment. MNLI (Williams et al. 2018) is dataset of premise-hypothesis pairs. They are labeled according to whether the hy- pothesis follows from the premise. We used the validation and validation-matched subsets extract representations for SST2 and MNLI, respectively.
As we used the pretrained models from Sellam et al. (0022) for BERT, only fine-tuned the models for experiments. While the number of epochs varies be- tween and 10 depending on the experiment, we always used the checkpoint with the best validation performance. We generally used linear learning rate schedule with 10% warm up to maximum of 5e-5, evaluate every 1000 steps, and used a batch size of 64. Otherwise, we used default hyperpa- rameters of the trans formers library For ALBERT, the training is identical with the exception of always training for 10 epochs. For SmoILM2, we use full finetuning for 500 steps with batch- size of 16. Otherwise, we use the default hyperparameters as released with the finetuning script of the Sm release.
Training Hyperparameters.
B.2.3 VISION SETTINGS
Architecture Choices. In the vision domain, plethora of architectures exist which could have been used in the scope of the benchmark. To narrow the scope and to keep computational overhead manageable, only architectures for classification were considered. Moreover, subset of architec- ture of prominent classification architecture families were evaluated, namely ResNets (He et al.] 2016), ViTs Dosovitskiy al. (202T) and older VGG's (Smmoyann & Zisserman (2015). To capture whether architecture size influences the measures, different sizes for the architectures were evalu- ated, resulting in ResNet18, 2. ResNer34, ResNet101, 4. VGG19, 5. VIT-B/32 and 6. ViT-L/32 the final architecture choice. Pre-trained checkpoints for ViT-B/32 and ViT-L/32 use taken from wegggingfaee provided under the Apache 2.0. Specific checkpoints are 'google/vit-bur 1at-422-22-in2111 and laalee/vi--arg--paachiit 2atthh32-224-i222k
In the vision domain, use the ImageNet100 (IN100) and the CIFAR- 100 Krizhevsky Hinton (009)) datasets. IN100 is derived from ImageNetlk (Russakovsky et al, 2015) by subsampling 100 randomly chosen classes. This reduces training time while keeping im- age resolution and content similar to ImageNetlk. When training the models on the IN100 dataset, the image resolution was fixed 224x224, while the resolution
Datasets.
32x32 when training
ht htteer//magiigaeee.ee///llllllee
http:///.....////aaaaaaea// /aann classes/
Table 6: Vision domain: Training hyperparameters for all architectures the ImageNet100 dataset.
Aside from the listed parameters, we note that all models shared more hyperparameters, namely label smoothing of 0.1 and cosine annealing learning rate schedule.
Dataset
Architecture
Batch Size
Learning Rate
Weight Decay
Optimizer
Epochs
ResNet18
128
0.1
4e-5
SGD
200
ResNet34
128
4e-5
SGD
200
ResNet101
128
0.1
4e-5
SGD
200
ImageNet1(0
VGG11
128
0.1
4e-5
SGD
200
VGG19
128
0.1
4e-5
SGD
200
VTT-B32
512
3e-3
0.1
Adam
300
VYT-L32
512
3e-3
0.1
AdamW
300
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Cora
Flickr
OGBN-ArXiv
0.8
0.7
0.7
0.7
0.5
0.5
0.5
Accuracy
Random Label
0%
0.4
0.4
0.4
25%
100%
0.3
0.2
0.2
0.2
0.1
0.0
0.0
0.0
GONSAGE GAT PGNN
GCN
SAGE
GAT
GCN
SAGE
GAT
Figure 5: Validation accuracies of GNN models trained for Test 3 (Label Randomization). Accu- racies were computed on test sets with regular labels. With increasing degree of randomization of target labels, performance degraded strongly. Clusters of accuracies per group are clearly separated.
models on the CIFAR-100 dataset. A notable exception the ViT models trained on CIFAR- 100, which used images CIFAR- 100 images that were upsampled from 32x32 to 224x224, as proposed by Touvron et al. (2021). This intervention was necessary remain compatibility the pre-trained weights of the patch encoder.
Training Hyperparameters. The ReSi benchmark requires models that can be differentiated by their training setting. Hence, it is mandatory to either train models from scratch or to fine-ture them accordingly. For all models of the vision domain, the choice was made to train the models from scratch for each respective dataset, except for ViTs whose encoders were initialized from pre-trained IN21k checkpoint. The chosen training hyperparameters are kept static across tests. A selection of hyperparameters is displayed in Table While the parameter choices worked well for ResNets and VGGs, the ViT performance their training settings. We assume this may originate from the fewer overall steps taken due to the 90% lower dataset size relative full ImageNet
lower than the ResNets, despite attempts to optimize
B.2.4 COMPUTE RESOURCES
To conduct the experiments, broad spectrum of hardware was used: For the model training, GPU nodes with up 80GB VRAM were employed. Depending on domain, representations were either extracted on GPU nodes and saved to disk for later processing, or extracted on demand on CPU nodes. Lastly, the representational similarity measures were calculated between representations on CPU nodes with 6-256 CPU cores and working memory between 80 and 1024 GB.
To extend the current tests with additional similarity measures, practitioners need have CPU compute resources to compute their own similarity measures on the existing model's representations. To introduce novel tests, practitioners would need to provide GPU compute resources for network training and subsequent CPU resources to evaluate all similarity measures on the new models.
B.3 TEST 3: LABEL RANDOMIZATION DETAILS
Graphs. trained groups five models each, for which r E {25%, 50%, 75%, 100%) of all labels are randomized during training. The models in each group were initialized with different training seeds, and different seeds also imply different randomization of labels. Given that we exclusively consider predictions datasets with L 2 classes, for all labels that were drawn to be randomized, we randomly drew new label from the set of all existing labels, excluding the actual ground-truth label, with uniform probability. After observing the performance of the models
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RandomLaaod Model Accuracies ImageNet 100
Randomiabbel Model Accuracies Citar 100
0.8
0.7
0.7
06-
4
0.4
0.4
0.3
0.2
0.2
Rondinn aielle 70%
0.1
Handan Laber 50%
Roessen
Hundan Lateds 50%
Pancon Label 0s
Rondom 0%
0.0
vaen
veir19
Rostle Rg
Rowele
Reater
VT VT-852
VIT-/22
voote
Reskters
Archilecuree
Figure 6: Validation accuracies of vision models trained for Test (Label Randomization) Training with varying degrees of random labels resulted in models of highly different accuracies. We high- light the three clusters used the Label Randomization test for the vision domain.
with randomized labels validation and test data, decided to only include models for which r E (25%, 100%] were altered during training for this test, along with five models trained on the regular data. This is mainly due the models already having shown behavior indicating that for higher degrees of randomization than 25%, there was hardly any relationship left between features and true labels that the models would have learned. We illustrate the performance of the models trained under varying degrees of label randomization in Figure where we can see that the models are strongly affected by such randomization.
Language. We create datasets where {25%, 50%, 75%, 100% of all labels are randomized so they can only be learned by memorization. an instance should get a randomized label, we pick uniformly at random from validating behavior, we use models without randomization of 0%, 75%, and 100% for BERT and ALBERT. For SmoILM2, we use 0% and 100%.
set of five labels that entirely unrelated to the data. After
Vision. We create datasets where E (25%. 50%, 75%, 100%) of all labels in the training set are randomized, forcing models to memorize. Whenever label chosen to be randomized, we randomly sample from all 100 existing labels, and replace the true label with the sampled one. Moreover, we re-use the first five models from Test and which represent dataset with r O. After validating behavior, use models with 0%, 50%, 75,, with clusters visualized in Figure
B.4 TEST 4: SHORTCUT AFFINITY DETAILS
Graphs. For all graph datasets, shortcuts were added by appending a one-hot encoding of the corresponding labels to the node features. Nodes which were not assigned a true level short- obtain a one-hot encoding of random label instead. We trained models with a ratio of E f0%. 25%, 50%, 75%. 100%} true shortcuts in the training data, but only considered the models trained on with p E (0%. 50% 100%) shorteuts in the training data for the benchmark. as otherwise there would not have been very clear separation of the models. We illustrate the performance of models influenced by the remaining degrees of shorteut likelihood in the training data Figure 7 where we can see that most models are indeed affected. In particular, GraphSAGE appears very sen- sitive to shortcuts, whereas other methods appear more robust this may also be due to differences in the way node information is updated in the convolutional layers.
Overall, on Cora the effect was the smallest, which is likely due to the models fitting easily to the training labels even without shortcut.
For all groups and seeds, we extracted representations based on the same fixed dataset, which con- tained one-hot encodings of random labels.
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Cora
Flickr
OGBN-Ardl
0.8
0.8
0.8
0.7
0.7
0.7
0.5
0.5
0.5
Shortcut
Accuracy
0.4
0.4
0.4
0%
50%
100%
0.3
0.3
0.2
0.2
0.2
0.1
0.0
0.0
0.0
GCN SAGE GAT PGNN
GCN
SAGE
GAT
GCN
SAGE
GAT
Figure 7: Validation accuracies of GNN models trained for Test 4 (Shortcut Affinity). Accuracies were computed on test sets with random shortcuts, but regular labels. With increasing degree of shortcut correlation with target labels, performance generally degraded strongly. Only for Cora, the effect was weaker, likely due the models not requiring the shortcuts for strong performance. Still, clusters of accuracies per group are mostly well-separated.
Language. We added shortcuts the data by prepending new special token each sen- tence for BERT and ALBERT. Their embedding matrix was extended with one token per class and trained during fine-tuning. We created datasets with five levels of shortcut strength, i.e. the rate at which the shortcut leaks the correct label. For SmoILM2, we added the correct answer token as shortcut during the task prompt. As completely incorrect shortcut tokens would per- fectly leak the label in binary problems like SST2, the lowest rate is the relative frequency of the majority class. Then we made four equally sized steps up to perfect shortcuts. This means that SST2 datasets have shorteut rates have r E {35.4%, 67.7%, 51.55%, 83.85%, 100%). J. We trained models with five different seeds for each shorteut dataset, but after validating sufficiently different behavior (see Figure 13 Figure 14 and Figure (5), we only used the models with the lowest and the two highest rates.
(55.8%, 66.8%, 77.9%, 88.9%, 100%) and for MNLI
Vision. In the vision domain, shortcuts were created through insertion of a colored dot. Each dot has static color, a diameter of 5, and is placed randomly in the image. To turn this into a shortcut, each class is assigned a unique color code, which is uniformly drawn once and shared across all experiments. By varying how often the pre-determined color appears on an image containing the corresponding class label, one control how useful the color dot feature is to the class in the image. A few exemplary images of the shortcuts are shown in Figure g
We trained models under five degrees of correlation p (0%, 25%, 50%, 75%, 100%), providing a more or less potent shortcut. To validate that these different ratios lead to sufficiently different behavior that our grouping assumption holds, measure the r == 0%. Models that learned to utilize the shortcut will show a stronger decrease in accuracy as the shortcut does not provide any meaningful signal anymore. In order to provide clearly distinguishable clusters, we discarded models trained with r E (25%, 50%]. The accuracy clusters after removing the two groups are shown Figure
accuracy of all models with
B.5 TEST 5: AUGMENTATION DETAILS
Graphs. To augment the graph data during training, we applied the DropEdge approach (Rong et al. 2020), which, at each training epoch, samples a fixed fraction of edges that are removed from the network for this specific epoch. Based on this approach, we trained four groups of augmented models, for which r E {20%, 40% 60%. 80%) of all edges were removed at each epoch. Again, after considering the performance of the models validation and test data, chose only include
25

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Figure 8: Visualization of the shortcuts introduced the vision domain. We statically assign each class label color code representing the class. In our benchmark, we control the extent as to which the color dot correlates with the true label, thereby making the shortcut more or less useful. In the top row, the cases where color dots are placed randomly are displayed, while at the bottom the color labels of the dot always correspond to the class label.
Shortcut Model Acouracies ImageNet 100
Shortout Model Aocuracias Citor 100
5.
0.75
Acourayy
0.5
0.4
0.70
0.3
0.66
0.2
Shortout: 100%
Shortout: 100%
Shorlcut 75%
0.1
Shortout 78%
0.60
v0011
VG019
Resiee18
Revtlr34
Reeherrii
V1-8112
WT-L/32
Rovierl18
Rovibe04
Reaher ot1
yoor1
vas11
W7-832
Vinu222
Archilecture
Architecture
Figure 9: Validation accuracies of vision models trained for Test 4 (Shortcut Affinity). Models trained with highly correlated shortcuts show lower accuracy when the correlation of the shorteut feature with the image label is removed. After evaluating all trained models under this setting, removed groups that were not well separated from others.
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Published as a conference paper at ICLR 2025
Cora
Flickr
OGBN-Arxvv
0.8
0.8
0.7
0.7
0.7
0.5
0.5
0.5
Augmentation
Accuracy
0.4
0.4
20%
80%
0.3
0.2
0.2
0.2
0.1
0.1
0.0
0.0
GCN
SAGE
GAT
GCN
SAGE
GAT
GCN
SAGE
GAT
Figure 10: Validation accuracies of GNN models trained for Test 5 (Augmentation). Accuracies were computed unaltered test sets. In general, augmentation impacts the performance of models, though the these effect differ over models and datasets. In general, clusters of accuracies per group are identifiable.
the models with r E f20%, 80%} for this test along with a group models trained with standard parameters, which was due to the overall weak effect impact that the intermediate steps appeared to have. The performance these remaining models is shown in Figure T0 where we overall observe weak, but visible effect from augmentation.
Language. BERT has been trained on large-scale data of varying quality and, as preliminary experiments showed, highly robust to augmentation like word casing, synonyms, and typos There- fore, we chose more aggressive augmentation that can also scale to large amounts of data. We randomly replace words with synonyms, delete and swap words, and randomly insert synonym of word in the sentence in random position, as described by Wei & Zou (2099) and implemented in textattack library (Moriis et al. 2020). To create datasets with different level of augmen- (0%, 25%, 50%. 75%, 100%) } of all words per sentence. We train five models with different seeds per dataset. For SST2, we use the 0% and 100% models; for MNLI, 0%, 25%, and 100%for BERT models and 0% and 100% for ALBERT models. We do not implement this test for SmolLM2.
tation, augmented
Vision. For the augmentation test, we utilized increasing levels additive Gaussian noise with five different noise variance intervals [0, o). v {0, 1000, 2000, 3000, 4000). We refer to these levels as Off, S. M, and Max. Exemplary images of the augmented images are displayed in Figure 11] Aside from the noise, only RandomResizeCrop and HorizontalFlip augmentations used. After validating model behavior, we decided to use Max, and M for our benchmark, see Figure 12 for the resulting clusters.
B.6 TEST 6: LAYER MONOTONICITY DETAILS
Graphs. Since for the graph neural network models, we never used more than three layers in the previous experiments, for this test we trained a new set of five graph neural network models, each with five inner layers. That way, we can obtain a sufficient number comparisons. while still preserving decent performance- with higher number of layers, the performance began to strongly deteriorate. As all inner layer shared the same number channels, no preprocessing was necessary to compare the layer-wise representations.
'We tested most of the augmentations from http:///anggett.crr//cc///aaas robustness https: het.raaccc...rettecccccccc...
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Published as a conference paper at ICLR 2025
Figure 11: Visualization of the additive Gaussian noise used in the vision augmentation Test. We show an exemplary image to highlight the gradually increasing level of noise introduced. From left right (Off, S. M, L, Max).
GauseNNoie Model Accuracies ImageNet 100
GoussNoise Modil Accuracies Cilar 100
0.80
0.75
0.80
62
4
5
0.70
0.30
$
Acturaty
Assureoy
0.50
0.60
0.45
Nols:: Mix
Excluded
0.56
Noise: Medium
Noice: Small
0.00
Noice: Noice:Smal Small
voart
voo
Rastee
Rostetl
Rea
Restett
Restent4
Rewtertth
voori
vo019
vm-8222
WTAL22
Architacurre
Archilecturo
Figure 12: Validation accuracies of vision models trained for Test 5 (Augmentation) Models trained with varying degrees of additive Gaussian noise show lower accuracy on noise-free samples. Two groups were removed to improve separation. Further, ViT-B/32 and ViT-L/32 did not qualify the sepur-separation requirement and were excluded.
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Label Randomization
Shortcut Affinity
Augmentation
Rate
0.9
0.92
0.0
100.0
0.8
0.90
Validation Accuracy
0.6
Rate
0.7
Rate (SST2/MNLI)
0.88
55.8/35.4
0.4
75.0
100.0
0.6
68.9/83.9 9
100/100
0.86
0.84
0.5
0.2
0.82
0.4
0.0
0.80
SST2
MNLI
SST2
MNLI
SST2
MNLI
Dataset
Dataset
Figure 13: Validation accuracies of BERT models trained for tests 3-5. Across all tests, groups are clearly separated.
Label Randomization
Shortcut Affinity
Augmentation
0.9
Rate
0.0
++
0.90
0.8
Validatiio Accuracy
0.88
0.7
Rate
Rate (SST2/MNLI)
55.8/35.4
75.0
88.9/83.9
0.6
100.0
100/100
0.2
0.4
0.0
SST2
MNU
SST2
MNLI
SST2
MNLI
Dataset
Dataset
Dataset
Figure 14: Validation accuracies of ALBERT models trained for tests 3-5, For SST2, there is no separation between groups. Thus, we exclude the augmentation test on SST2 for ALBERT.
29

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Randomization
Affinity
0.9
0.0
Validation Accuracy
0.7
Rate (SST2/MNL)
0.7
88.9/83.9
0.6
0.6
100/000
0.5
0.5
Rate
0.4
75.0
100.0
0.3
SST2
MNLI
SST2
MNLI
Figure 15:
Validation accuracies of SmolLM2 models trained for lests 3 and 4. For the label random- ization test, we only use two groups as the three different training setups cannot be distinguished.
Language. We used models trained on standard data with varying seeds for each dataset. We used the CLS token representations or trean-pooled representations across all 12 transformer blocks and the embedding layer for BERT and ALBERT. For SmoILM2, we again use the final token of the prompt. As the skip-connections only skip layers inside the block, we argue that representations after the blocks should follow our assumptions.
Vision. For the layer monotonicity test, used the 10 models from Test I and 2. Differently to those tests, we extracted representations not only before the classifier, but we also needed to extract representations in intermediate layers.
C FULL BENCHMARK RESULTS
In the following, provide the full results of all experiments within our benchmark. For the tests grounded by prediction, we report statistical significance of the shown Spearman correlations on the 5% (*) and the 1% (**) level.
C.1 GRAPH RESULTS
We begin with presenting the results from the graph domain. Results are shown in Tables 12 with each table presenting results of single test. For the PWCCA measure, we report some NaN values-in these cases, the reference implementation consistently yields negative eigenvalues for covariance matrix, which then results undefined values when the square root of these cigenvalues taken to compute the pscudo-inverse of this matrix.
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Table 7: Results of Test (Correlation to Accuracy Difference) for the graph domain.
Eval Asch.
Speamua
OGBIN-Arch
GCN
SAGE
GAT
PCNN
GCN
GAT
GCN
GAT
CC1A
SVOCA
PACCA
-0.02
-005
-0.13
0.07
-0.19
-0.26"
-0.35'
-0.09
0.01
tih
-0.55
0.01
-0.18
-0.36
-0.27
-0.19
0.10
0.06
-4.10
-0,30
AlignCisc
2038"
0.38"
0.17
AngStupe
Aligamene
OenPax
Lanly
0.35
0.66
-0221
-0.9
-012
-0.11
-0.02
-0.14
4.0c
0.3M
6.17
0.28
-0.16
-0.35
-0.04
0.07
-0.01
0.59
-0.0M
-0.19
ParaPace
-0.2
0.18
-0.36
0.29
0.30
-0.9
0.15
-0.09
0.08
0.43"
Protilu
0006
0011
-011
0.29
0.00
-000
-0.17
0.07
0.43
SunCler
6.18
0.08
00.02
0.30
0.35
-007
0.35'
0.12
0.12
CKA
DisCCur
001
-0.7
0.88
0.17
041
0.42
-0.19
-0.10
0.15
-0.0s
RSM
GLLP
HOS
-034
-045"
0.08
0.08
-0.05
-0.2
-0.04
-002
0.35
0.04
-032
-0.27
-0.27
0.29
-0:21
-0.08
0.06
0.05
-..2
-0.12
KSA
-0.17
0.25
0:12
0.13
0.02
bral-Dax
Neighbors
Jaccard
0.32"
0.28
00.32*
RankSim
2000
-0.00
0.32
0.25
0.35"
0.31
-0:20
-0.28
0.06
40.99
Tegology
IMD
KTD
-0.0
0.06
0.00
-0.8
<0:46"
0.04
0..00
0.19
0.13
0.37
-0.09
-0.21
0.00
0.15
4.15
Stiiste
MiyDir
2.10
2004
40.1%
4.18
0.65
40.20
4038
Table 8: Results of Test 2 (Correlation to Output Difference) for the graph domain.
Tel:
IDD
Dduge Car.
Ditaet
Coca
Felk
OCINN Apil SAGE
OCEN banck
GCN
SAGE
PGKN
GEN
G4T
GEN
RONN
GCN
GCN
SAGE
GAT
0CA
SVOCA
400-
8:02
1.2h
-1.66
-014
0.25
6:33
0L19
1.35
4117
-006
104
8:37
0A2
+2.27
623
4.16
0.35
104
0.11
-0.33
000
01.5
830
1075
E.5
3.85
000
3:15
0130
E:17
Bug
8:36
0.23-
0813
8.5
0.30
8.89
0.45
1.38
8.00
1810
1335
0.27
8.80
0.30
860
0.36
6.34
00700
3.37
0:28
08.5
043
0.68
8:19
6.15
0-0m
663
3.5
0.98
3.99
0.03
810
633
Photlie
3.25
036
18.22
000
-018
3,35
9,30
010
042m
1624
-3.77
12.55
4013
-8.27
0.28
00.30
5.01
1000
60.36
3.55
0.68
+1.5
0.6""
0.4"
8.03
0.12
0.00
0,3"
0.45
0.35
0.407
1.33
00:0
2:00
01100
0.15
6-5-
8113
<0.7
0110
0.00
0.13
-3.22
6,33-
013
-0:11
0.15
23:35
0.35
08:11
4.00
0.16
6:32
-033
4.35
03.00
6.11
0.30
0.55
01.4
-3:5
0.4""
0330
-010
43.36
41.4
03.9
0.36
Neithee
-032
6.31
1311
0.30
2.39
644"
0.2
3.44
-.30
0.14
0.01
0.2
0030
8.54
0.50
0.27
3:4
10pelogy
0636"
3,4"
0:38
0.87
0.19
-114
6.64
2..6
1.38
4111
646"
1622
3.34
4.11
-2:1
4:12
0.01
40.1
0.77
46.9
Cow
000
3.53
000
0.38
0.17
0.3
6.33
013
040
0.3
0.10
333
0.03
321
000
0.33
0.55
0.07
6.16
0.1
339
011
0.30
0.5
0.0
0.6
0.0
011
600
0.6
6.15
000
0.5
0.1
0.1
0.6
Table Results of Test (Label Randomization) for the graph domain.
Bval
AUPRC:
Conformaty Rate Flatsr
Durnect
Cors
Tlukk SAGE
OGBIN-Ank
Com
OGBN Andi
Anch
SAGE
GAT
OCN
CCA
PRECCA
SVOCA
0.35
0.36
0.31
0.33
0.23
0.42
0.20
0.25
0.33
0.44
0.30
0.22
0.34
0.24
1,00
0.44
1.10
0.55
0.36
0.60
0.62
0.52
0156
0.59
0.64
6.58
0.30
0.69
0.91
0.64
0.53
0.54
6:43
1.00
0.61
1,00
0.9%
03.6
0.43
0:42
0.42
0.25
0.38
0.43
0.23
0.33
0.73
0.77
0.54
11.1
0.51
0.60
0195
0.60
0.33
6:12
0.54
0.28
Haddan
0:43
9/42
0.42
0.21
0.77
0.46
0.35
0.83
0.83
0.54
0.55
0151
0.51
0.54
0094
0.68
0.67
0.00
0.97
0.77
Aligstenn
Linhies
0.45
0.44
0.71
0.9
0.44
0.39
0.44
0.27
0.77
0.50
0.19
0.06
0.72
0.55
0.64
0.31
0.61
0.33
0:94
8:97
0.50
0.34
0.55
0.56
Pouttha
0.45
9:45
0.43
0.22
0.79
0.62
0.19
0.52
0.96
1.00
0.67
0U60
0.51
0.93
0.88
0.51
0.98
1.00
100
SuffCour
0.42
042
0.42
0.52
0.55
0.55
0.51
0.50
0.55
USA
0043
0:42
0.42
0.22
03.66
01.27
1,00
1,00
1.00
11.55
11.31
$3.31
0,54
0193
0.96
0.54
1.00
1
Dak..o orr
0.43
0.44
0.41
0.25
0.99
0.43
0.22
1,001
1.10
1.40
01359
16..5
11.52
0.69
0093
0.56
0.52
1
1,90
1.00
BOS
0.25
0.28
0.26
0.22
01455
1134
0.86
11.5
0065
0:25
0.58
0.30
BSA
0446
0.44
0.43
0.23
0:24
0142
10.35
0.59
0145
1140
0..9
1051
0.54
0.58
0.52
0.64
6:99
0.58
0..3
RSMbill
0.31
0.33
0.75
0.29
0.71
0.92
0.19
1,00
1.10
1.10
0.75
0.83
0.85
0,61
0.9
0.99
0.31
1.00
1.90
1100
Negghoes
Incard
0.42
0:41
0.41
0.24
0.56
0.45
0.333
0..5
1..1
11.55
11.1
0.55
0.37
6/47
0.53
0.95
Honlom
0445
9/41
0/41
00:22
0.:97
03:35
0.35
03364
11.5
10.0
1134
2064
0.54
0.65
0.53
0.:08
0.07
0.67
0.05
Topelogy
RTD
0.56
0.91
8.93
0.88
0.99
0.91
0.99
11:01
0.56
6.36
0.44
DoxiAM
0.20
0.39
0.39
0.18
0.37
0.57
0.1
0.90
1.00
1.00
0.56
0.18
0.74
0.66
0.73
0.85
0.48
0.95
1,00
1,00
Uniitdi
0.31
0.34
0.40
0.35
0.54
0.10
0111
0.7
0.39
0.56
31

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Published as a conference paper at ICLR 2025
Table 10: Results of Test (Shortcut Affinity) for the graph domain.
Bull
AUPRC
Confeenity
Diriot
Con SAGR
TTdle SAGIT
OGGRN--A Ands SATR
Cur
Fliter SAGP
OKTRNN Andiy
Anch
DCx
GAT
PENN
GEN
GAT
GUN
GAT
SAGRE
CIT
PYNN
CCX
GAT
CGEN
SAGRR
GRT
CCA
PRICCA
SVOCA
0.31
0.33
0.47
0.36
0.23
0:45
0.34
0.24
0.93
0.32
0.59
1.00
0.72
0.97
0.53
1.00
0.73
0.46
0:31
0.60
0:69
0.56
0.50
6.52
0.57
0.97
0.05
1.00
1.90
0.99
0.91
1.00
0.64
0/69
0.59
1,00
1,00
1.00
0.86
0.38
0.89
0.63
HadClnn
0.25
0.35
0.29
0.35
0.55
1,00
0.52
0.30
0.72
0.53
0.65
0.37
0.39
0,09
0.83
1.00
0.81
0195
0.83
0.50
Alyemann
Linkeg
0.30
0.24
0.90
0.61
1.811
1.10
1.80
0188
0.64
0.75
0.67
0.20
0.51
0.72
1.00
1.00
0.31
0.37
0.24
1.80
0.65
0.55
0.77
0.57
0.67
0.64
0.57
0.23
0880
0:91
0.64
1.22
0.80
1.08
11.5
1.00
1.00
1.00
11.77
11:1
2002
0.58
0382
1.00
041
1
130
1.00
SonK oer
0.31
0.40
0.93
0.39
0.06
1.001
0.32
0.72
0.85
01.06
0.79
2171
0.67
0.51
1.00
0.33
0.51
0.97
0.76
0.74
1.00
0.755
0:50
0/54
Dastoor
0.66
0.81
0.80
0.43
0.33
1.80
0.33
1.00
0.50
1.40
0.83
0:53
0:05
0.67
0.65
1.08
0.73
1.00
1
1.00
XMM
30
0.90
0.20
0.37
03:55
0.55
1.88
0.77
0.97
0.71
0176
0145
0.90
0.87
0.90
0.81
1.00
RSA
0:66
0:41
0.71
0.97
0.50
01:32
0.716
0/91
0.70
0.88
RSMmIr
0.46
0.50
0.69
0.37
0.48
0.99
01:66
1.301
1.00
1.40
0.70
0.776
2:41
0.64
0.75
0.47
0.59
1.90
1.89
1.50
0.81
Neighbos
Jaccar
0.74
0.76
1.00
1.50
1.60
0.91
0.53
6:97
0.53
1.00
Rinkdin
0.73
0.64
0.85
0.43
1.99
0.77
0.56
1.01
1.00
1.00
0.57
0.59
0:95
0.51
1.00
0.92
0.83
1.90
1.90
1.00
Topelogy
IMD
0.71
0.60
0.92
0.56
0:35
0.68
Daacihii
0.51
0.17
0.20
0.17
0.18
0.11
0.32
0.81
1..0
1.80
0.77
0.45
0.34
0.44
0.50
0.46
0.61
0.96
0.90
1.50
Statuiic
Maxtaa
0.37
811
0.18
0.11
0.35
0.53
1.00
0.99
[141
0.49
0.38
Table 11: Results of Test 5 (Augmentation) for the graph domain.
Fval.
AUPPCC Flickr SMGB
Confurmity Rate
Disaset
Coro SAGB
DGBN Artiv
Coca SAGE
Flickr SAGE
OGBNN Arsiv
Asch
GUN
GAT
GUN
GAT
GCN
SAGE
GAT
|CN
GXT
GCN
GXT
GUN
SAK3B
GAT
OUA
SVCCA
PWOCC
0.27
0.60
0.29
0.65
0.34
0.42
0.64
BAn
0.67
0.57
0.49
0.51
0.89
Daz
0.51
0.48
0.82
0.82
0.86
0.54
0.55
0.29
0.83
0.55
0.84
0.83
0.85
0.75
0.65
0.89
114
0.85
0.74
0.96
0.91
AligeCis
0.63
0.70
0.78
0.57
0.87
AusSSupe
0.78
0.72
0.55
0.83
0.59
Aligament
Orthihos
Linker
0176
0.83
0.70
0.72
0.95
0.55
0.96
0.55
0.35
0.53
0.94
0.85
0.81
0.89
0.91
Permihro
(161
0,60
0.755
0.89
0.87
01933
0.59
0.58
ProcDist
0.76
0.69
0.75
0.70
0,81
0.40
1.00
1.00
1.80
0.88
0.90
0.50
0.89
0.95
0.63
1.80
1,00
1.00
Softooe
0.63
0.70
0.50
0.75
0.58
0.55
0.43
0.46
0.45
0.89
0.92
0.80
0.87
0.85
0.54
0.51
0.71
0.66
CKA
0.70
0.75
0.91
0.57
0.75
0.51
1.00
1,00
0.59
0.00
0.92
0.97
0.53
0.91
0.74
1.80
1.00
1.00
DistCoo:
0.79
0.73
0.89
0.60
0.79
0.61
1.00
1,60
0.59
0.93
0.91
0.97
0.54
0.54
0.81
1.80
1,00
1.00
RSM
BOS
0:53
0.6
0.6
0.68
0.54
0.53
0:46
0130
0.56
0.48
0:44
1.401
1.411
0.995
0.75
0.90
0.30
0.75
0.76
0.58
0.52
0.81
0.90
0.84
0.90
0.96
0.75
RSA
0.57
0.72
0.49
0.68
0.33
0.56
0.92
0.99
0.55
RSMDM
0.79
0.93
1,60
0.93
0.57
0.97
1.60
1.00
2nd Cos
0.86
0,92
0.99
0.85
0.97
0.91
0.95
0.554
1,60
1.00
1.00
Neigtbors
Jncoard
0.81
0,55
0.59
0.55
0.59
1.00
RankSin
0.54
0.62
0.95
0.75
0.88
0.55
1.06
1.00
1.801
0.72
0.36
0.98
0.80
0.95
0.35
1.60
1.00
1,00
Tepelogy
RTD
IMD
0.91
0.74
0.70
0.34
0.20
0.98
0.72
0.59
0.57
1.00
0.54
0.28
1.00
1.00
1.00
1.00
1.80
1.801
0.98
0.93
0.64
0.95
8.99
0.46
0.87
0.90
0.95
1,00
0.75
0.53
1.80
1.80
1,00
1.00
1.00
1,00
CoueDME
0.17
0.50
0.62
0.41
0.35
0.53
0.43
0.53
0.51
0.69
0.78
0.87
0.78
0.65
0.74
0.76
0.80
0.54
Statitic
Mar
0.99
0.59
0.17
0.75
1.00
0.51
0.34
0.33
0.87
0.25
0.66
0.99
1,00
(191
UnittD
0.53
1.00
0.68
0.34
0.76
0.53
0.91
0.36
0.50
0.52
0.66
0.98
0.78
Table 12: Results of Test 6 (Layer Monotonicity) for the graph domain.
lovll
Diraact
Cors
TTudk SAUI
OKHIN-AAS
ilicor SAGG
OKIIN-AARY
Anch.
DCC
SAGEE
GAI
PENNN
GUNN
LAA
GUN
SALI:
LIAY
LCN
SALIE
CIAI"
JNNN
CON
GAL
GLNN
SAGI
CCC
SYOCA
0.51
0.50
0.51
1.31
13.64
0.:99
0.55
11:11
11:16
0.61
1.90
0.41
-2.10
0.91
1.48
1,40
08.64
0.51
0.77
01:65
0.53
0.95
0.55
01.55
1.80
1.40
0:45
0.33
0.68
0.04
0:54
13:54
SagSoge
01991
0.59
0.25
0.38
11.86
13.39
1.80
11.55
01.44
08.02
09:21
1.10
Aligsnes
OrcaPrss
Listfee
1,80
1.80
1,88
1.80
1,00
1
0.97
0.30
0.67
1.80
03.39
0.50
0.305
1.10
01:90
1.50
1.00
1,000
1.80
1.80
1.80
1.80
0:91
1
0.43
0.62
0.98
1.00
0:45
0.13
6.09
1
1.10
Pouztba
0399
1.90
1,00
0.99
0:75
1,80
08.75
0.50
0..96
1.60
0..90
1.80
1,80
0.01
0.62
6.83
0.85
1.00
SollC.o orr
0.95
0.99
0.75
0.52
0.30
0.35
0.66
0.511
0.75
1.40
1..96
11.59
01.50
0.24
0.79
0.99
0.11
0.91
0.62
110)
Dark. orr
1.60
1.30
1.30
0.59
0.99
11.5
0.385
11.0
01:22
1.00
1.50
1.30
0.03
0.99
0.47
0.95
R5M
BISS
ULLP
1.80
1,80
1,00
1,80
1,30
1,30
13.05
0.90
1,00
1.00
1/92
0.322
1.001
1.30
1.000
1.50
1.00
1,00
1.00
1,80
1.50
1.80
1.50
0.00
0.62
0.54
1.86
1.00
1.00
0.92
0.33
1,00
1,00
1100
RSMINI
0.99
1.90
1.30
0.85
0.33
1.80
0..99
1.50
1.30
0.91
1.00
1,80
1,48
1,30
0.90
13.96
11:07
0..95
1.50
1.50
1.30
1.40
1.30
6:41
0.81
0.46
0.51
1,30
Nrighton
Banksin
1.80
1.80
1.30
13:44
119%
13:90
11.96
1.50
1.30
1.50
1.30
0.94
0.99
1.00
Tapelogy
IMD
1,80
1,88
1.10
0.95
1,00
0.54
0.97
0.85
1.50
1.50
1.00
1.80
1.40
0.89
1,00
0.87
0.47
0.55
1,00
1.00
DaxiDiff
0.85
0.52
0.62
0.52
0.57
0.36
0.55
0.70
0.74
0.57
0.85
0.35
0.40
0.14
0.35
4.22
0.24
0.73
0.39
0.40
Strisie
MagDoT
0463
0.72
0.01
0.97
0.63
0.63
0.55
0:52
0.71
0.57
0.55
0.80
0.89
0.91
0.55
0.30
0.38
0.06
0.20
0,63
0.67
0.67
0.59
0.56
32

---

Published as a conference paper at ICLR 2025
C.2 LANGUAGE RESULTS
For the language domain, we present the results in Tables 13 Any missing values for RTD and IMD shown nan are due to excessive runtime. Missing values for other measures are due to numerical instability.
Table 13:
Results of Test (Correlation to Accuracy Difference) and Test 2 (Correlation to Output Difference) for the language domain using CLS token representations for BERT and ALBERT and final prompt token for Sm.1LM2.
Horr Spearas
Eail
Sairnia
Spuizau
MILJ
NE73
5013
MNLI
2312
SVETA
8.32
3.30
0.13
8.84
6.6
0.20"
01.3
10k
3.30
6:4mm
8:30
8:34"
048
6,38
0.00
E.33
0,377
6.6
0,777
-8:33
633
0.37
0.34
0.51
076
2.27
0.3-
0.33
0m
3.8m
0:35
0.35
0.38
1.amm
E.5m
0.23
0.25
0.13
0.34
4.55
1.34
0.40
0.23
1.2"
0.4"
2.4"
0.35
-8.25
-8.31
0.2
0.45
0.24
3.18
030
-.33
0.29
0.38
4033
2013
341
-001
0337"
6:37
-8.42
6:31
00207
0.30
0.397
0.13
0.2
0.96
0.69
0.48
0.327
3,27
0.12
0.18
0.22
0,5m
0,55
6114
1314
0:13
3.10
3.56
3:90
-151
8.31
4323
4.55
0.06
B5%
0.00
9.15
0.34
2.25
1.45
4.35
6.27
0.23
-2.91
0-4--
3.55
6:35
8.19
1.47
0046
1.57
6.55
0.1
Noghhn
18.31
-8.23
1040
8.57
1.55
0.51
0012
101
199
0347
332
6:34
811
-8.35
<00
0337
032
806
Tepelog
k1b
0.1
110
1.39
608
0.1
0.17
0.81
016
1.:0
3.0
096
0.29
600
000
000
0:11
0.35
0.9
300
816
000
600
6.0
001
Table 14: Results of Test (Correlation Accuracy Difference) and Test 2 (Correlation Output Difference) for the language domain using mean-pooled token representations.
Clr
Diugg Dee
Diract
JOILI
SST2
8.60"
0113
6.76
0119
8:14
6:11-
12:94
11.8
63377
40.12
0.25
10377
8.85
0112
0.33
0.364
8:37
8.39
0.79
3.27
4080
0.54
0.22
0.39
0.36
244
0.23
18.00
10.
11:4-
0.54
21.8
0.12
2.31
8:07
1000
0.37
9.15
8.527
RSM
11:0
0.35
25.0
0.2
-000
8.85
Zad
2,30
4018
0.3
0.25
-014
0.33
8.11
100
0.15
0.35
2-41-
6111
Tepelog
200
8:15
1113
0337
2004
0.27
8.30
000
3.91
000
800
0.0
011
1.00
606
Table 15: Results of Test 3 (Label Randomization) for the language domain.
(a) Results of Test (Label Randomization) using CLS//inal to- ken representations.
(b) Results of Test 3 (Label Randomization)
using mean- mean-pooled token representations.
19S7
FERT
S2RRET
109
116
1146
112
140
1113
0.00
DV172
1.44
1.25
171
6.7
140
109
0.80
1.4
140
477
330
6.30
3.11
8:10
1.40
100
1.80
140
144
1.22
1.8
8000
1.1
1:00
-110
115
1,00
1443
3.10
1.5
$210
0.50
3.5
4.5
132
600
Viddoon
135
5:31
04
522
6300
130
0.4
030
174
140
679
0:01
0.11
133
615
120
11
115
140
100
13
33

---

Published as a conference paper at ICLR 2025
Table 16: Results of Test 4 (Shortcut Affinity) for the language domain.
(a) Results of Test 4 (Shortcut Affinity) using CLS/final token (b) Results of Test 4 (Shortcut Affinity) using representations.
mean-pooled token representations.
ALPRC
Pawe
MELL
143
F1h
011
8.17
1:0
811
11
119
1.1
1.10
8.25
1:00
816
141
1:00
3.6
5.00
1.90
040
3.8
8.15
138
344
8:10
5.00
6.00
0:3
3.0
135
1.5
1014
1.20
638
344
6.55
133
637
Table 17: Results of Test 5 (Augmentation) for the language domain.
(a) Results of Test (Augmentation) using CLS to- ken representations.
(b) Results of Test (Augmentation) using mean-
pooled token representations.
VIVU SST2
5513
Bianst SENLI
5512
5313
SVOCA
0.64
0u4
0.51
0.38
6.71
1.60
6.5
C44
SYCCA
0:31
0.43
6.61
1:0
0.30
631
646
044
11:09
8:15
0:01
1:01
1:30
0148
14:38
1168
8:34
8.3
837
176
1:41
0.81
8.31
0.80
8:30
1331
130
1149
0.87
03.8
Algiracc
Torahre
0.31
0,33
1.50
1.44
131
0.17
2.51
1.27
1.9
0.45
04.3
0662
0.41
0.35
6.75
1.55
0.32
1.31
SulCier
0.21
0.30
0.3%
1.34
0.57
EA1
046
621
11:00
0.11
9:10
3:47
2162
0.90
0..3
3:00
1.30
045
19:79
0.30
8:32
0.76
1.30
1.7
039
010.7
1848
0.51
8.90
1.55
1236
0.38
0.30
1.5
0.34
0.42
1.35
0.22
044
Noightn
Seigher
Be Like
00:0
0.96
0:3
6.11
6:50
6:57
Reckher
0:40
0.36
6:10
1:51
019
055
647
Teyekor
140
01.3
0338
0.3
3.38
8:15
1.30
1.47
0.30
0.30
1.3
0:31
0.86
044
0.38
0.21
Statiic:
Madher
8:38
1.45
Mughe
6.6
844
Table 18: Results of Test 6 (Layer Monotonicity) for the language domain.
(a) Results of Test 6 (Layer Monotonicity) using C/f/rfiall token representations.
(b) Results Test (Layer Monotonicity)
using mean- pooledooeedddde token representations.
AbAC
ALBOCT
sact
ALBSCCT
ALBUKT
SoudaG
ALBSCCT
ALS.C
AGAEET
20.
6100
641
6.7
114
1.11
1.10
6010
1.10
3.35
5.00
8.00
130
140
1:99
3.6
5.00
6.300
1.80
1
140
1.0
1
1
3.00
1.00
670
0.44
646
0.6
5.6
1.00
340
4.00
Postien
140
1
4.00
170
170
520
1140
1.20
326
520
1750
150
1.50
1.6
110
500
0.4
1.2
1.4
134
344
045
1.0
600
100
1.4
149
344
509
1.00
0.31
600
1%:
113
2111
0.2
600
0021
10
131
110
332
800
1100
34

---

Published as a conference paper at ICLR 2025
C.3 VISION RESULTS
We present the results of the vision domain in Tables 19 30 Results are presented in test order, with each fest featuring one table. Some entries in the result tables feature NaN values. This was caused by various reasons: (i) Numerical instability due to operations like Singular Value Decompositions or negative values square roots and (ii) identical similarity values, leading to failure of correlation values. Whenever such failure occurred for an entire group of models, the entire measure was excluded for this case, as, for instance, removing the model group of Gaussian noise M may simplify the overall task of distinguishing models.
Table 19: Results of Test (Correlation to Accuracy Difference) for the vision domain on ImageNet- 100.
Spearman
Daraser
IN100
Arch.
RNetI8
RNe134
RNet101
VGG11
VGG19
VIT B32
VIT L32
CCA
SVCCA
0.29*
0.27
.0.00
.0.04
-0.30'
-0.01
-0.17
AlignCos
.0.08
.0.35
.0.01
-0.13
-0.12
0.07
0.05
AngShape
-0.02
HardCarr
0.21
0.13
00.01
.0.01
.0.03
0.35
.0.17
Aligunient
OnhProc
LinReg
0.21
0.19
41.11
.0.16
0.09
0.18
-0.02
-0.04
0.00
0.07
0.15
0.06
0.01
PermProc
0.07
0.09
0.08
0:14
-0.02
-0.06
-0.33
ProeDist
0.08
0,00
0.14
0.13
0.08
0.16
0.05
SofCoor
0.27
0.08
0.04
-0.03
-0.10
0.36'
-0.19
DistCor
0.31
40.08
0,08
0.03
-0.21
-0.26
0.03
EOS
0.05
-0.17
0.11
-0.22
0.08
0,47
0.03
-0.17
RSA
0.06
0..17
0.09
0.24
-0.35*
-0.12
-0.11
RSMD
0.09
40.10
0.11
-0.04
-0.08
0.01
-0.06
Neighborr
Jsccadd
-0.11
40.3
40.4
-0.22
-0.01
0.25
RankSim
0.07
0.04
0.13
-0.01
0.03
0.17
0.35
Topology
IMD
41.20
-0.17
-0.23
RTD
0.09
0.12
4.200
40.19
-0.11
0.10
ConcDiff
-0.11
0.34
4.04
40.11
.0.13
0.00
0.18
Statistic
40166
40.07
-0.18
-004
man
nan
Table 20: Results of Test (Correlation to Accuracy Difference) for the vision domain on CIFAR- 100.
Eval.
Spearman CIFARI0O
Archs
RNet18
RNe134
RNet101
VGG11
VGG19
VIT 852
VIT L33
PWCCA
-0.30"
0.07
0.23"
SVCCA
40.13
40.21
-0.03
AlignCos
-0.66
0.09
0.37
0.21
0.41*
-0.12
0.62
AngShape
-0.31*
0.14
0.44
-0.01
0.40"
-0.12
0.88
HardCarr
0.50
Alignment
LinReg
OrthProc
40.31*
-0.34*
0.14
0.12
0.44
0.42
.0.01
nan
0.40
nan
-0.12
-0.09
.0.24
0.88
PermProe
-0.66
-0.08
0.14
0.93
ProeDist
-0.37
-0.05
-0.05
0.44*
-0.16
0.83"
SoftCou
-0.66
0.05
0.37
0.07
0.48*
0.13
0.81
CKA
40.20
0.02
0.35
0.23
0.40
-0.17
0.90
-0.17
RSM
EOS
GULP
40.96
-0.8"
0.03
0.04
0.01
0.43*
-0.12
0.13
.0.24
0.46
-0.31
0.06
0.90
0.62
RSA
RSMDaff
.0.15
0.18
0.08
0.07
0.17
0.04
0.65
2nd-Coss
20.33
41.11
0.32
0.19
-0.07
-0.21
0.88
Neighbors
Jaccard
RankSim
20.15
40.20
0.10
0.08
-0.14
0.88
Topology
IMD
.0.12
0.06
40.00
.0.18
0.40"
-0.11
0.69*
40.03
0.93"
CoocDiff
.0.04
0.37
0.21
0.23
0.25
0..05
0.53*
Statistic
MagDii
40.17
0.24
0.01
.0.03
0.31
0002
0.33
UnifDitt
-0.00
4.15
4.12
-0.01
-0.88
35

---

Published as a conference paper at ICLR 2025
Table 21: Results of Test (Correlation to Output Difference) for the vision domain on ImageNet- 100.
Type Foll.
Grounding by Prodiction
ISD Con.
Desge Con:
IN100
IN
Arch
RNerl8
RNNec44
KNvt101
VOGU1
VOG119
KNet18
RNNrt4
KNett01
VOKI1
VO019
DCA
PWECA SVCCA
0.13
0.21
-0.00
0.15
0.35
0.15
-0.01
0.33
0.15
0.16
0.05
0.21
0.24"
0.18
0.27"
0.50'
0.07
0
0.14
0.06
0.12
0.03
00356
0.01
-0.06
0.02
1.15
0.07
AllenCo
-0:30
-0:32
-000
0:19
0.37
0.03
AngShape
0.19
0.26
0.40"
0.05
0.31
Allamnest
LinRer
OntiProc
0.24
0.31
0.23
0.30
0.19
0.25
40:26
0.25"
0.40
0.39
0.29
40.35"
-0.34
0.04
1005
PeraFio
0.18
0.16
0.27
-0.15
0.05
0.36'
-0.05
0.13
-0.25
-0.04
0.02
0.20
0.37
0.10
Proethet
0.10
00:44
0:30
-0.0%
0:27
40.05
0.03
0.08
-0.08
0:11
40:10
-0.07
40.07
DUS
SoftCoe
0.48"
0.37
0.31
-0.04
-0.16
0.01
-4..5
I.47"
-0.35
-0.07
40.0%
-0.79
0:37
-0.16
CKA
0.30"
008
0:30
-0.05
4106
0.06
0.29"
0..0
0:22
0.01
40:26
00:0
-0013
0.05
0.38
0.02
RSM
RDS
0.49
0.15
0.35
0.29
0.15
RSA
0.09
0.19
0:200
0.35"
0.30
0.04
0.00
0.04
RSMDai
-041"
-0.22
0.30
-0.21
0.07
0.02
-0.25
-0.17
-0.20
0.18
-0.01
-0.05
40:31
-0.17
2011
0:16
0:39
0.07
-0:70
0'4""
-0%
-0.21
0.88"
0.00
0.01
-0.07
0:19
-0.27
Neighbors
Iscard
0.36
0.26
0.32
0.05
0.33
0.47
00.30*
0.25
0,47
0.23
0.14
0.11
0.34
0.155
RankSin
2015
0.05
0:35
-0.00
-0:10
0.02
-0:32"
0:34
-0.33
Topslagy
Conclbit
4.2"
0.34
40.11
4.13
40.11
0.39"
-0.99
0.00
40.1
4006
4008
0.29
Statiati
MagDan
0.38
0.20
0.02
0.36
0.25
0.02
0.32
0.11
4.22
4.05
0.00
0.04
0.01
0.22
-0.5"
890
0.94
-0.17
man
-0.0
230
0.17
0.34
m0
Table 22: Results of Test 2 (Correlation to Output Difference) for the vision domain on CIFAR-106
Type
Groandng by Praliction
JSD Spcanuun
Desggr Con Spzamunn
Eval.
Dutaiet
CIFARI00
CIFARI00
Arch
RNer18
RNerl4
RNvr101
VCK11
VCK119
VIT 1932
WITE
RNe118
RNer34
RNar101
VXK1
V0K119
VIT 1812
VITT3
CCA
PWECA
SVCCA
-000
0.28
0.31
-0.43*
0,32"
1386
0.23
n38
nan
0.8
nat
-0.01
-1.14
.0.22
0
-0.35
0.19
0.17
830
0.07
0.78-
nan
0.00
-0.25
0.31
0.29
0.53
AngShope
0.41
0.69*
0.86"
0g8*
-0.36
0.39
0.30
0.38
0.29*
HardCoer
-00
0.31
0.11
0226
0.46"
-0.24
0.35
40.30
00225
0,30"
Alignneor
LisRex
OntiPi
0.17
0.00
044**
0.34
0.64
0.6"
0.7%
0.00
0386"
0.3)
-0.32
-0.95
0.19
0.39
0.30
0.10
008
0'36"
0.11
0,57*
-0.35"
0.79*
PoraFroo
0.06
0.11
0.30
0.35
0.42
0.51
0.5--
0.08
0.15
0.08
0.05
0.23
0.37
Bugg
ProcDist
0.18
0:23
0.61
0.05
0.4"*
0.57
0.8"
0002
0.20
0.32'
0.01
0'33"
0.50"
0.79"
SolCoa
-0.06
0:30
0.41
0.7
-0.21
0.35
0.16
-0.27
0.37
DinCon
0.48*
0.52
0.27
0.18
0.21
0.35'
0.54
0.30"
RSM
RDS
GULP
0.30
0122
0.28
0.27
0.00
0.57
0.19
0.21
0.22
0.25
0.57
0.73
0.94
0.5
-0.20
0.21
0.20
0.20
-0.09
0.26
0.09
6.38
40.34
0.53*
0.31
0,47*
0.91*
0.38
RSA
0.02
0'4""
0.00
0.29
0.79*
033"
-0.16
0.22
-0.1M
0.16
0.35*
0,50*
1.79"
RSMDAN
0.38
0.24
0.13
0.20
0.26
0.22
0.65
0.34
0.06
-0.06
0.15
-0.02
0.17
0.68
hul-Cos
0.38*
0.18
0:10
0.38
0.17
0:42*
Neighbors
Jocoad
0.47
0.20
0.17
0.43*
0.3
0.32
Topelogy
RTD
IMD
0:35
0.00
0.1
0.53
0.01
0.09
0.05
0.06
0.42*
0.54
0.24
0.01
0.00
0.96"
-0.0
0.36
-0.08
0.30
40.05
0.08
40.09
0:11
0.12
0.20
4000
0.09
0.50
0.92"
0.9
0.18
-013
0.16
0.7
0.34
0,
-0.05
0.32
40.19
0.30"
0:31
0.13
0.49*
Stststic
MayEKir
0.15
0.20
0.26
0.36
0.4**
0.20
0.30
0005
0.10
0.06
0.4"
0.14
0.07
0.29
Unifiii
2.18
40.3
0.33
0.19
man
0.05
40.16
0.17
-0.07
838
-0.16
Table 23: Results of Test 3 (Label Randomization) for the vision domain on ImageNet-1000
Eval.
AUPRC
Conformis Rute (NT10)
Donsat
IN/10
Anth.
RNe188
RNed4
RNa1101
VGGL1
VGG19
VIT B32
VIT 1.32
RNet18
RNe:4
RNec101
VGGL1
VOGL9
VIT B32
VIT L32
CCA
SVCCA
1,00
0.71
0.98
0.95
1930
1.00
0.33
0.99
0.99
AllynCos
0.45
0:43
0.46
0.59
0.69
AngShope
0.20
1
0.83
1,800
HadCcer
0.72
0.69
0.71
0.95
0.75
0.71
0.79
0.83
0.82
0.74
1.80
0.36
6:33
0.54
Alignmeor
Linkag
0.91
0.79
0:50
0.65
0.72
1.00
1,00
man
1.00
1.00
01:22
0..55
01889
0.88
0.63
0.96
0.83
0.40
0.55
1.00
1,00
730
1.00
1.00
0.30
0.7%
01899
0.89
PeraProc
0.70
0.42
0.42
0.73
0.70
0.45
0.94
0.67
0.50
0.51
0.34
0.67
0.66
0.95
Preethii
0:45
0:71
0.79
0:70
0.91
0.66
0.55
0.54
SonCon
0.72
0.45
0.70
0.55
0.53
0.68
0.75
0.95
0.73
CKA
1.00
1,00
1,00
1.00
1,00
DiaCon
1.80
1.00
1,00
1.00
1,00
RSM
GULP
BDS
0.64
0.59
0.71
0:31
6:56
0.56
0.30
0.35
0.77
0.76
0.7%
1.00
01.90
0.99
0.95
0.57
0.55
0.83
0.59
0.59
0.59
0.55
0.50
0.59
0:91
1.00
0.59
0..9
RSA
0.75
0.63
0:55
0.97
8.98
0.42
0.89
0.50
0.79
0.96
0.59
0.99
0.30
0.89
RSMDAR
1.00
1,00
1.00
0.96
1.06
1.00
0.85
1.00
1,80
1.00
0.59
1.00
1.00
0.93
0.95
1,00
Neighbors
Jocord
0.75
0.57
Topelogy
XTD
IMD
1.00
1,00
1,00
1.00
0.37
1,00
0.99
1,00
1.10
1.00
0.96
1.00
0.87
0.53
1.00
1.00
1,80
1.60
0.34
1,80
1,00
1,80
1.00
1,00
0.99
1.00
0.9
0.87
ConcDift
0.51
0.66
1.00
0.93
0.29
0.99
0.77
0.96
0.59
1.60
0.58
0.62
1.00
0.55
Staisiic
Maylanl
1,00
0.75
0.97
0.96
1,00
0.04
0.51
0:09
Unitbil
0.18
0.45
0.17
0.58
0.21
0.67
36

---

Published as a conference paper at ICLR 2025
Table 24: Results of Test 3 (Label Randomization) for the vision domain on CIFAR-100.
Roll
AUTPRC
Conforiit Roc CIFAR
Datest
CIFAR1ON
Arch
KNedi
RNedd4
Kivtto]
VGGI
VOG11
B322
VST
KNetS8
RNerd
KNect01
VVGS
VOGL1
B32
V3E
SVCEA
AligeCoe
0:42
0.72
0.99
0.32
0.82
AngShupe
0.42
0.43
0.72
0.47
1.00
0.75
0.51
0.50
0.61
0.55
0.70
1.00
0.85
0.74
Hank ver
0.45
01:4
0.70
0:44
0.61
03.81
0141
0.56
0.65
0.67
0.65
0.76
0.85
0..8
Aliemmnn
Ontaroo
Linkee
0.30
0.42
0:43
0.31
0.23
aan
0.47
man
1
114h
0.75
6.36
0.51
0.30
0.59
0.50
0.52
0.01
0.55
230
0.70
730
1.00
112
0:35
0:45
0.74
[1449
0.87
0.45
0.60
CKA
0:44
0.85
0.56
DalCon
0.45
0.44
0.70
0.94
1.06
0.72
0.43
0.67
0.65
0.67
0.98
1.00
0.33
0.71
RSM
GULP
BDS
0.72
0.72
0.55
0.58
0.51
1.00
01:33
0.37
0.41
0.74
0.74
01..5
0.34
0.433
0.83
0.83
0.06
0.66
0.58
1.80
0.61
0.55
0.79
0.82
0.7%
0.38
0.68
0.63
RSA
0.45
0:44
0.72
0:47
0.31
0.7.
0.52
0.96
0.63
0.50
0:71
0.93
0.79
0.73
0.36
0.99
0.95
1.00
Neighbor
h
0.50
KankSim
0.49
0.45
0.75
0.92
0.51
0.54
0.50
0.86
Tepology
IMD
0.58
0:85
0:8
0.76
0.83
0.61
0.46
0.81
0.92
1.80
0.85
095
0.87
0.65
KT1
1.00
088
1.00
097
0381
0.88
0.47
1.00
0.88
1.80
0.99
093
0.94
0.70
0.7
MapBam
1.00
0.78
0.88
Table 25: Results of Test 4 (Shortcut Affinity) for the vision domain on ImageNNet1000.
Eval.
Conformity
Arch
RNa188
RNc:4
RNa101
VOGLI
VOG19
RNc118
RN::34
RNc101
VOGLL
VOG19
CCA
SVCCA
PWOCA
0.99
0.55
0.68
1.00
0.51
0.65
0.68
0.99
0.29
0.25
0.60
0.82
0.21
0.93
0.51
1.00
0.54
1.40
0.51
0.71
0.91
1.80
0.62
0.90
0.42
0.97
0.57
0.97
AligeCos
1,00
1.00
1:00
1,00
1.00
1.00
0.59
1,00
1.80
1.80
1,00
1.00
1.00
1.00
AngShupe
1.00
1,00
1.00
1,00
1.00
1.00
0.99
1.00
1,80
1.40
1,00
1.00
1.00
1.09
0.92
0.37
LinReg
1.00
PemPPro
0.66
0.87
Preciber
1.00
1,00
1,00
1.00
0.80
1.00
1,00
1.80
1195
SofCoo
0.92
0.98
0.99
0.90
0.94
0.98
0.94
0.99
1,00
1.80
0.97
0.96
1.00
0.99
CKA
1.00
1,00
1.00
1,00
1,00
1.00
0.67
1.00
1.80
1.60
1,00
1.00
1.00
0.95
DuaUon
1.00
1,00
1,00
0.83
1.000
1,60
1,00
RON
1.00
0.95
1.000
1.40
0:98
2nd Cos
1,00
1,00
1.00
1,00
1.00
1.00
0.92
1.00
1,80
1.80
1,00
1.00
1.00
0.97
Nrighhors
hused
1.01
1,00
1,00
1,00
1.10
1.00
0190
1.00
1,80
1.80
1,80
1,00
1.00
0.94
RantSim
0.99
0.99
1.00
1.00
1.00
1.00
0.89
1.00
1.80
1.80
1.00
1.00
1.00
0.97
Topelagy
IMD
6:67
1,00
0.75
0.98
0.7%
1,00
0.67
1.00
0.48
0.9%
1,00
CuncDiff
0.76
0.95
Striatic
Maglir
0.37
0.37
0.5%
0.61
0.75
0.79
0.85
0.51
UaifEAr
0.75
0.73
0.87
0.60
0.55
0.61
0.17
0.90
0.91
0.96
0.83
0.83
0.34
0.00
Table Results of Test (Shortcut Affinity) for the vision domain on CIFAR-100.
Ival.
AUPRU CIFARIO0
Conformity Rate
Dutaiet
CIFAR 100
Arch.
RNer18
RNee34
RNct01
VO301
VGG19
VIT B32
VIT L.2
RNNr18
BNN:34
RNet101
VOGLL
VOG19
VIT B32
VIT L32
OCA
PR/OCA
SVCUA
0.56
1,00
0.1
1,00
0.1
1.00
0.87
1300
0.50
1186
0.79
1300
01:3
16:3
01.00
1,00
0.97
1,80
0.977
1.80
0.55
2340
0.7%
1980
0.90
880
0.65
AlignCos
1.00
1,00
1.60
HardCoer
1.00
0.35
1.00
0352
Aligmmn
LinRez
OxtlProx
1,00
1.00
1.00
1.00
1.00
1.00
1,00
1.00
0.99
0.20
0.77
0.38
1.00
1.00
1,00
1,50
1.80
1.50
1,00
1.00
0.43
1.00
0.76
0.50
0.93
0.34
1.00
0.36
0.37
(3:91
01.6
0.U95
0.55
1.80
0.25
0147
0.35
(1:7
ProcDist
1.00
1,00
1.00
1,00
1,00
0.92
0.70
3.00
1,50
1.00
1,00
1.00
0.97
0.72
SofCCor
1,00
0.71
1.00
0.39
RSM
EOS
GULP
1.00
1,00
1,00
1.00
0.54
0.71
0.65
0.64
0.46
3.00
1.00
1
1.80
0.55
0.58
0.39
6.51
0.71
RSA
1.00
1,00
1,00
1,90
1,00
13.91
0.09
1.00
1,80
1,80
1,00
1,00
0.55
18.74
RSMDzIf
1.00
0.96
1,00
0.45
0193
11.00
0.41
1.000
0.98
1,00
0.57
0.9%
0.96
0.65
2nd Cos
1,00
1,00
1,00
1,00
0.35
1,00
0.79
3.00
1,80
1,40
1,00
0.57
1,00
0.83
Neighhrre
hasedd
1.00
1
RantSin:
1.00
1,00
3,00
1,00
topology
RTD
IMD
1.00
0.54
1.00
1.00
0.99
1.00
0.99
0.99
0.79
0.59
1.00
0.5%
1,60
1.80
1,00
1.00
1.00
1.00
0.81
Condfaif
0.56
0.00
1.00
0.99
0.15
0.36
0.40
0.64
0.96
1.80
1.00
0.42
0.65
0.67
Statstic
MarDNi
0.99
0.65
0.29
1.00
0.51
0.22
0.42
1.00
0.91
0.97
1,00
0.31
0.52
0.71
UnitEter
0.47
0.47
0.30
0.21
0.50
0.17
0.18
0.70
0.50
0.68
0.41
0.71
0.00
0.20
37

---

Published as a conference paper at ICLR 2025
Table 27: Results of Test 5 (Augmentation) for the vision domain ImageNet-100..
Eval.
AUPRC
Conformit Rize
Dataut
IN:10
IN
RNirt4
RNar001
RNer1
VOK11
VC001
SVCCA
AllynCio
0.71
0.84
0.90
1,00
0.81
0.76
0.75
0.94
0.96
0.97
1,00
0.95
0.92
0.94
AngShupe
0.71
0.52
0.56
0159
0.24
0.74
0.90
0.90
0.51
0.53
1.80
0194
0.91
0.95
BarCcer
0.66
0:44
0.50
0:59
0.43
0.42
0.64
0.71
0.68
0.76
0.34
0.82
0.78
0.86
Alignme
LidRez
0.94
0.64
0.75
1,00
0.75
0.37
0.85
0.98
0.90
0.92
1,00
0.94
0.65
0.94
PernFroo
SunCCon
CKA
0.90
0:73
0.67
0.99
0.75
0.37
0.71
0.97
0.92
0.59
1,00
0.95
0.73
0.94
DurCon
0.83
0.68
0.62
0.99
0.75
0.39
0.70
0.96
0.90
0.57
1,00
0.96
0.74
0.14
RSM
GULP
EOS
0.93
0.92
0.59
0.00
0.69
0.09
0.96
0.9%
ILS8
0.88
0.80
0.45
0.72
0.72
0.98
0.995
0.36
0.56
0.89
0.89
1.00
1/00
0.96
0.96
0.85
0.85
0.88
0.5
RSA
And-Cos
Neighbors
Jocoard
0.55
0.59
RunkSin
0.71
0.57
0.89
0.83
0.80
0.51
0.63
0.92
0.57
0.96
0.95
0.92
0.82
0.57
Topelogy
KTD
IMD
0.56
0.5
0.65
0.59
0.76
0.64
0.78
0.39
0.75
0.75
0.30
0.23
0.39
0.21
0.82
0.56
0.90
0.88
0.92
0.59
0.94
0.58
0.92
091
0.32
0.66
0.50
UnitESi
Table 28: Results of Test 5 (Augmentation) for the vision domain on CIFAR-100.
Eval.
AUPRC
Conformity Rate
Dataset
CIFARI00
CIFARIOD RNer101
Arch.
RNet18
RNer34
RNet101
VGGI
VGG19
RNet18
RNer34
VGGH
VGG19
CCA
PWCCA
0.47
nan
nan
nan
SVCCA
0.41
0.55
0.61
0.42
0.25
0.62
0.81
0.80
0.65
0.49
AlignCos
0.98
1.00
1.00
0.98
0.50
0.99
1.00
1.00
0.99
0.82
AngShape
0.43
0.54
0.98
0.43
0.49
0.58
0.77
1.00
0.57
0.81
HardCorr
0.49
0.54
0.99
0.41
0.38
0.73
0.82
1.00
0.57
0.71
Alignment
LinRey
0.45
0.50
0.98
0.67
0.76
0.99
OrthProe
0.43
0.54
0.98
0.43
0.49
0.58
0.77
0.99
0.57
0.81
PemmProc
0.51
0.66
0.52
0.44
0.36
0.74
ProdDist
0.42
0.76
1.00
0.43
0.43
0.50
0.92
1.00
0.58
0.78
SoftCor
0.51
0.53
1.00
0.41
0.42
0.74
CKA
0.45
0.73
1.00
0.69
0.49
0.68
0.86
1.00
0.92
0.83
DisiCon
0.63
0.74
1.00
0.79
0.52
1.00
0.94
RSM
EOS
0.45
0.46
0.84
0.52
0.54
0.66
0.71
0.94
0.81
0.80
GULP
0.45
0.46
1.00
0.34
0.66
0,70
RSA
0.66
0.71
1.00
0.66
0.42
0.91
0.88
1.00
0.88
0.77
RSMDiff
0.92
0.89
1.00
0.28
0.98
1.00
0.52
2nd-Cos
0.46
0.45
0.63
0.47
0.55
0.70
Neighbors
Jaccard
0811
0.91
1.00
1.00
0.74
0.94
0.98
1.00
1.00
0.92
RankSim
0.96
0.99
1.00
0.89
0.69
1.00
Topology
IMD
1.00
0.87
0.89
1.00
0.29
1.00
0.96
0.97
1.00
0.56
RTD
1.00
1.00
1.00
1.00
0.88
1.00
1.00
ConcDiff
1.00
1.00
1.00
1.00
0.39
1.00
1.00
Statistic
MagDiff'
1.00
0.81
0.63
1.00
0.25
1.00
0.96
0.89
1.00
0.64
UnitDift
0.89
0.98
1.00
0.20
0.63
38

---

Published as a conference paper at ICLR 2025
Table 29: Results of Test 6 aaaer Monotonicity) for the vision domain on ImageNet-100..
Confermti
Speammun
CCA
PALCA SVCCA
0.72
0.58
0.39
0.72
0.75
0.75
0.69
0.35
0.78
0.20
0.27
0.30
0.45
0:42
01:32
0.40
0.47
0.61
AliyeCos
0.64
0.56
0.80
0.90
0.65
1.00
1.09
0.52
0.63
0.52
0:93
0.11
1,00
1.08
AngShupe
0.55
0.00
0.00
0.97
0.9%
0.55
0.05
0.05
0.56
1.00
HardCoer
LinRex
Alignmeo
PeinFroo
0.71
0.69
Precibial
0.85
0.30
0.79
0.50
0.81
0,70
0.55
0.42
0.39
0:48
0.67
0.71
SofCoe
0.71
0.85
0.58
0.30
0.89
1.00
1.00
0.11
0.50
0.70
0.52
0.64
1,00
1.00
CKA
0.93
0.97
0.99
0.90
0.92
1.00
1.00
0.57
0.52
0:97
0.35
0.93
1.00
1.09
DunCon
0.95
0.98
0.99
0:50
0.92
1.00
1.10
0.97
0.79
0.97
0.35
0.93
1.00
1.00
ROS
1.00
RSMDiff
0.85
Dul-Cos
0.85
0.91
6:55
1,00
1.09
0.55
0.77
0:92
1,00
1,00
1,30
Neighbors
Jocoard
0.85
0.90
0.1
1,90
1.10
1.00
1.09
0.55
0.65
0.75
1,00
1.00
1.00
1.09
RankSin
0.85
0.90
0:90
1.00
1.00
1.00
1.00
0.55
0.65
0.67
1.00
1.00
1.00
1.00
Tapslogy
IMD
092
0.46
0.83
0.52
CorcDif
0.32
Statstic
MayDam
UailNNi
0.6
0.68
631
0.39
0.49
1.00
0.18
0.30
0.55
40.30
4.06
Table 30: Results of Test 6 (Layer Monotonicity) for the vision domain on CIFAR-100.
Speattnntt VOGLL
CIPAR VOGLL
RNa188
RNN:34
RNa101
RNc:34
RNst01
V3T
CCA
PWDCA
SVCCA
0.88
0.33
0.68
0.88
6:50
0.58
0.65
1,00
0.70
1.00
0.50
0.93
1.09
-0.28
3/10
0.27
0.65
-0.36
0.40
1,00
0.31
0:10
0.66
0.57
AliguCos
0.77
0.76
0.68
0.82
0.27
0.56
1.00
0.19
0.41
0.18
0:53
0.41
0.32
1.08
AngShupe
0.87
1,00
0.51
1,00
HardCoer
Alignnoor
OnthPhos
1.80
PermiProc
0.74
0.61
Predibal
0.70
0.92
0:54
0.70
0.76
0.97
0.90
0.10
0.76
0.35
0.60
0.66
6.98
0.92
SolCoor
0.76
0.50
0.59
0.31
0.91
1.00
0.97
0.14
0.31
-0.10
0.52
0.66
1.00
0.86
CKA
0.65
0.77
0.87
0.95
8.99
0.97
0.97
0.00
0.26
0.62
0:94
0.99
0:38
0.87
DunCon
0.55
0.30
0.38
0.99
0.97
0.97
0.17
0.30
0.54
0.59
0.99
0.9%
0.55
RSMDur
0.25
Inl-Cos
0.90
0.89
0.74
1,00
0.88
1.00
1.08
0.58
0.62
0.20
1,80
0.74
1.00
1.08
Nrightors
hocird
0.85
0:35
0.65
0:55
0.39
1.000
1.19
0.55
0.52
0.02
0337
0.73
1.00
1.19
RaniSin
0.85
0.85
0.64
0.95
0.90
1.00
1.00
0.55
0.57
0.02
0.97
0.75
1.00
1.00
Topology
0.62
Concbift
0.47
0.30
0.23
Statinic
MagDin
0.55
0.39
0.30
Usifbir
0.78
0.34
0.76
0.40
0.63
0.98
1.00
0.48
0.16
0.16
0.41
0.30
0.79
0.95
39

---

Published as a conference paper at ICLR 2025
D RUNTIME EXPERIMENTS
While running our experiments, we have also recorded computational runtimes of all comparisons per measure. In 3]] we provide an overview of runtimes averaged across all experiments, sep- arated by architecture. We note that these similarities were not all computed on the same machine, and also, there may be variations over different datasets that were aggregated here, given that these often varied in size.
Yet, one can identify clear differences in the runtimes of different measures, which even span differ- ent orders of magnitudes. Specifically, the copology-based IMD and RTD scores were consistently among the slowest measures, and further, LSM-based measures such as RSA, HOS and in particular RSMDiff are also significantly slower than, for instance, most alignment- measures. Given that the computational cost RSMs is generally quadratic the number inputs, is also no surprise that these algorithms do not scale that well.
Table 31: Runtimes of the similarity measures in seconds, averaged over all comparisons. Since embedding dimensions and the number of inputs vary for different datasets, and we aggregate across experiments and datasets within each domain, the runtime should be interpreted broad estimate of the runtime to be expected.
Modality
SACIR
SmalI.M.
RNa11
RNN134
RNee101
VIT-R/30
0.073
1,991
2.237
2.635
3.748
5.224
Allances
0.051
0,014
0.424
01998
2.604
AngShape
0.001
0.794
0.224
2,409
0.350
0.292
1.062
2.656
0.032
0.04
0.001
0.355
0.415
5.195
110.375
7.335
10.254
0.333
0.743
Concliff
0.10
0.010
0.011
0.085
0.003
0.083
1.290
0.034
0.055
2:272
2:281
0.018
1.182
1:302
2.955
6.058
5.256
52:490
10:243
10.77
3:215
4.283
EOS
23.954
22.0128
24.517
0,029
5.210
4.963
14.502
11.345
12:435
32.257
23.746
16.018
20.656
30,714
GULP
0.042
0.143
0.944
0.001
0.990
0.293
3,527
0.400
0,361
9.583
0.512
0.538
0.254
1.945
HandCor:
0,225
0.023
0.025
0,001
0.251
0.120
1.427
0.170
0.168
2.505
0.254
0.258
0.185
0.393
IMD
248.439
250.130
26,
$4,068
153.065
154.229
113:272
142.055
920.261
4231,700
549.156
282.879
517.934
587.420
Joccard
$:156
3.248
3.576
0,060
1:715
1,351
3.891
1,371
1.731
3,822
2.415
2.557
2.454
5,30
Linkog
0084
0.043
0.050
0.002
0.709
0.545
14:355
1.885
0.880
15:250
0.621
0.003
2,666
5,078
Magbbfff
0.00
0.001
0.01
0.000
0.005
0.005
0.012
01009
0.017
0:995
0.018
0.033
0.00
0.00
Onthitone
8002
0.03
0.04
0.001
0.792
0.215
2.007
0.251
0.223
5.495
0.134
0.288
0.300
196
PWCCA
0.182
0.265
0.272
0.003
3,575
1.532
19,657
1.155
1.154
26.363
1,398
3.031
2.474
5,806
PemmProe
0023
0.031
0.029
0.000
0.341
0.280
4.177
0.213
01199
5:224
0.319
0.300
0.308
0.599
ProeDie
0.000
0.011
0:011
0.001
0.771
0.211
2.215
0.229
5.421
0.301
0.252
0353
2.163
RSA
19315
21.201
21.284
0.145
3.518
10,629
60:452
17:195
17.672
12.687
RSMDaff
0.177
59.77%
151165
21.79
20.058
214.915
2.693
40


Reviews:
[{"review_id": "HE0q8fmRSi", "reviewer": "Reviewer_vg5P", "summary": "This work presents a comprehensive benchmark for evaluating 23 representational similarity measures across six similarity tests, twelve different architectures, and six datasets from diverse domains. Additionally, the paper offers insights for future research and aims to democratize a unified framework that ensures fair comparisons across different studies.", "questions": "* What happens with different architectures for the text modality (e.g. some LLMs)?\n* What is \"IN21k\" in line 282? Please write out the full name instead of using the acronym, as it’s unclear without a reference to the actual name.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "* The paper is well-written and easy to follow, with experiments clearly presented and tasks well-defined. Figures and tables are self-explanatory, significantly enhancing the manuscript's readability and accessibility.\n\n* The authors have provided the code, ensuring full reproducibility of results and facilitating the community's ability to expand and build upon the benchmark. This openness supports transparency and encourages collaborative advancement in the field.\n\n* The problem addressed is highly relevant and important for the community, as it remains an open question which metric is best suited for different scenarios.", "weaknesses": "* It’s unclear whether the similarities are calculated between two latent spaces (as suggested in formula 3, line 094) or between two individual representations (a single point), as written in line 095. Clarifying this distinction (and how each metric is adapted accordingly) could improve the reader's understanding. It seems to be between two latent spaces, but in Figure 2 (left), the metric is calculated between two points.\n\n* When evaluating the text architectures:\n  * Could you clarify the choice to test the same model with 25 different seeds, rather than exploring architectural variations like RoBERTa or ALBERT, which are available pretrained on Hugging Face? Including a wider variety of architectures might enhance the robustness and generalizability of the findings.\n  * It would be helpful to either move the explanation of the choice to use the CLS token to the main manuscript or to reference the appendix. Additionally, you might consider testing alternative aggregation modalities, such as the mean or the last token, since these can yield different representations and may impact the results.\n\n* The vision task may benefit from broader testing since only ImageNet with 100 subclasses is used. Expanding the evaluation to include the full ImageNet dataset or other widely-used datasets, such as ImageNet-1k, CIFAR-10, or CIFAR-100, could provide a more comprehensive assessment.\n\n* The analysis could benefit from stronger takeaways. For instance, in lines 438-439, you suggest that the choice of measures should be task-specific, and in lines 453-455, you advocate for the development of best practices for using similarity metrics. However, these insights feel somewhat broad, and more concrete guidance could help readers understand how to apply the benchmark effectively.\n\n* In line 413, it’s mentioned that some domain-specific trends can be identified, yet later (line 424), you note that no single measure consistently outperforms others across all tests, even within a single dataset. The first claim—that some metrics appear more suitable for specific modalities—is interesting and would benefit from a more in-depth discussion. Also, adding a summary table of these findings of the paper could help readers better understand the details.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "g4RHmiZuzl", "reviewer": "Reviewer_KrQ9", "summary": "This paper presents the ReSi benchmark, a framework for assessing representational similarity measures across neural architectures from different domains. It features 23 similarity measures, twelve neural network architectures, and various datasets from graph, language, and vision. ReSi is proposed to support research on representation learning, reproducibility, and exploration of new approaches for comparing neural representations.", "questions": "- Should Eq. 3 map to R^N?\n- It did not become clear initially what type of accuracy measures are considered; e.g. what task the model is trained to solved?\n- I think coloring the labels of the similarity measures in Table 3 may be helpful for the reader (similar to the coloring in Fig. 3)", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 2, "strengths": "- The challenge of understanding learned representations better, especially in an unsupervised way is highly relevant and of practical importance.\n- Reproducible and extensive experiments covering a wide range of metrics, domains and tasks.\n- The paper is overall well-written and easy to follow. The figures and findings are clearly presented.", "weaknesses": "- I think the paper misses the opportunity to give the reader a better understanding of what drives the differences and inconsistent results of measuring similarity. Which types of measures perform well together and what feature of the representation is this driven by. Relevant aspects could be due to the way representations are normalized, what arithmetic function is used for computing the distance (e.g. dot-product, summed differences, etc.)\n- Another angle to better understand similarity computations is from the point of interpretability, i.e. what features do the respective models pick up on and how do these change across models. I think this is an important and complementary direction to measuring different similarity scores and should be discussed (cf. [1,2,3]). \n- ReSi is presented as the “the first comprehensive benchmark for representational similarity” and can “yield rich insights” and a “theoretical understanding of representational similarity”. While I did find the general guidelines interesting and helpful, I think a deeper understanding of what causes these differences across metrics and domains is missing, and thus the above statements overstate the level of achievable insights from ReSi for me.\n\n\n[1] Eberle et al. “Building and interpreting deep similarity models”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 44(3), 2020.\n\n[2] Janizek et al. \"Explaining explanations: Axiomatic feature interactions for deep networks\", CoRR, 2020.\n\n[3] Vasileiou et al. “Explaining Text Similarity in Transformer Models”, NAACL 2024.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "tyhAnKT6Ec", "reviewer": "Reviewer_jBQa", "summary": "This paper proposes a suite of benchmarking tasks to test the quality of a representational similarity metric. The tasks are designed around training models such that they are different in one specific way that leads to a certain expectation for similarity. Specifically, the authors propose 6 tasks:\n\n(T1): Correlating similarity vs accuracy. Models are trained with only varied seeds.\n(T2): Correlating similarity to actual model outputs.  Models are trained with only varied seeds.\n(T3): Label Randomization: Randomizing different percentages of the labels. Models are expected to be more similar within the same percentage of randomization.\n(T4): Shortcut affinity: Models are trained with random \"shortcuts\" that leak the correct label. The shortcut that always leaks the correct label, sometimes leaks the correct label and never leaks the correct label. Models should be more similar within group. \n(T5): Augmentation: models are trained with different kinds of augmentations adn models are expected to be more similar within the same type of augmentations. \n(T6): Layer Monotonicity: The similarity between layers is expected to be related to the distance between the layers. \n\nThey apply their benchmarking tasks to three domains, graphs, language and vision and test 23 different representational similarity methods. \n\nThe authors find that no similarity metric is always the best according to these metrics, however, there are some that tend to perform better than others, especially within a specific sub-domain (graph, language, or vision).", "questions": "Q1. Which layers are being used in the vision model experiments? I find that this information is not clearly provided in Sec. A.5 Vision.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "(S1) The goal of the paper is well-motivated and could be a valuable contribution to the community, since it is not clear how to compare representational similarity methods. \n\n(S2) The problem is quite challenging as it is hard to know exactly how two models differ, therefore the authors take reasonable steps to develop tasks in which they can approximately group models by differences in their performance or training method. \n\n(S3) The authors apply their benchmarking method on a comprehensive set of representational similarity methods in three popular domains in machine learning. \n\n(S4) The authors provide some interesting insights into differences between similarity methods and show that no single similarity method always performs the best.", "weaknesses": "(W1) Task 6 - I find that there are some careful considerations to use task 6 (see Q1). If the layers are chosen incorrectly, task 6 would have some issues for at least some vision models since skip connections could make neighboring layers dissimilar. Correcting/explaining this issue would shift my score to a 6. \n\n(W2) I found the depth of analysis to be somewhat limited. The authors could significantly strengthen the paper by providing some deeper analysis into a few similarity methods that explains how the similarity methods arrive at different similarity scores when comparing the same representations. For example, the authors show the impact of preprocessing by comparing two similarity methods that differ only in preprocessing: Orthogonal Procrustes (OrthProc) and Procrustes Size-and-Shape Distance (ProcDist). Can the authors analyze what the preprocessing does to the representation and how it changes the results of the method on the benchmarks? If this analysis and/or others are provided, I would consider improving the rating further. \n\nMinor ----\n(W3) Section 3.1 should indicate more details of the tasks will be provided in Sec 3.5.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "UMwUlEwcCO", "reviewer": "Reviewer_5jzy", "summary": "The paper introduces ReSi, a comprehensive bechmark for comparing neural representations.\nThe benchmark consists of (i) six carefully designed tests for similarity measures, (ii) 23 similarity measures, (iii) twelvecneural network architectures, and (iv) six datasets, spanning over the graph, language, and vision domains.\nThe benchmark is extensible; future research can build on it and further expand it.\nThe source code is publicly available.", "questions": "1. Can you please clarify notations at Figure 3? Does boxes and bars mean quantiles?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "1. Novelty. To the best of my knowledge, authors have introduced the first representational similarity bechmark.\n2. Inside the benchmark, several rarely used similarity measures were tested (like Jaccard similarity), however, they showed promising results.\n3. An in-depth analysis is provided; cases where some similarity measures tend to work well as studied.\n4. The benchmark is quite diverse: in includes datasets from spanning graph, language, and vision domains.\n5. The language of the paper is fine and it's easy to follow.\n6. Impact. The comparison of neural representational similarity measures is a long-standing problem in deep learning.", "weaknesses": "1. Authors cite (Barannikov et. al, 2022.) several times as an example of representation comparison measure and experimental methodology, but don't include the proposed measure (RTD) into comparisons.\n2. Stitching (Bansal, 2021) is not included intro the study.\n3. Some important details of experiments are not disclosed: for example, CKA could be evaluated on the whole dataset or batches. Other similarity measures probably include hyperparameters as well.\n4. The study might benefit from an Appendix with a brief description of all similarity measures under evaluation.\n\nBansal, Y., Nakkiran, P., & Barak, B. (2021). Revisiting model stitching to compare neural representations. Advances in neural information processing systems, 34, 225-236.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)