Attributes Shape the Embedding Space of Face Recognition Models
Authors: Pierrick Leroy
Antonio Mastropietro
Marco Nurisso
Francesco Vaccarino
Venue: ICML 2025 poster (2025)
Keywords: Face Recognition
Representation Learning
Interpretability
Embeddings
Contrastive Loss
Primary area: applications->computer_vision
TL;DR: We propose a metric to quantify how face recognition embeddings remain invariant to interpretable attributes.

Abstract: Face Recognition (FR) tasks have made significant progress with the advent of Deep Neural Networks, particularly through margin-based triplet losses that embed facial images into high-dimensional feature spaces. 
During training, these contrastive losses focus exclusively on identity information as labels. 
However, we observe a multiscale geometric structure emerging in the embedding space, influenced by interpretable facial (e.g., hair color) and image attributes (e.g., contrast).
We propose a geometric approach to describe the dependence or invariance of FR models to these attributes and introduce a physics-inspired alignment metric. 
We evaluate the proposed metric on controlled, simplified models and widely used FR models fine-tuned with synthetic data for targeted attribute augmentation. 
Our findings reveal that the models exhibit varying degrees of invariance across different attributes, providing insight into their strengths and weaknesses and enabling deeper interpretability.
Code available here: https://github.com/mantonios107/attrs-fr-embs.

Attributes Shape the Embedding Space of Face Recognition Models
Pierrick Leroy"! Antonio Mastropietro 2
Marco Nurisso" 13 Francesco Vaccarino"
Abstract
space representing images such that pictures of the same identity closer in the embedding domain (Chopra et al., 2005). Second, the usage of triplet-loss, in FaceNet (Schroff et al., 2015), where anchors and negative samples are compared to target image to obtain the gradient driving the embedding map training (Sankaranarayanan et al., 2016). The third consists in equipping the embedding space of an angular dissimilarity, so that it can be assimilated to hyper- sphere, e.g., AM-Softmax (Wang et al., 201 SphereFace (Liu et al., 2017), CosFace (Wang et al., 2018b), ArcFace (Deng et al., 2019a), AdaFace (Kim et al., 2022).
Face Recognition (FR) tasks have made signif- icant progress with the advent of Deep Neu- ral Networks, particularly through margin- based triplet losses that embed facial images into high- dimensional feature spaces. During training. these contrastive losses focus exclusively on identity information as labels. However, we observe a multiscale geometric structure emerging in the embedding space, influenced by interpretable fa- cial ge. hair color) and image attributes (e.g., contrast). We propose a geometric approach to describe the dependence or invariance of FR mod- els to these attributes and introduce physics- inspired alignment metric. We evaluate the pro- posed metric on controlled, simplified models and widely used FR models fine-tuned with synthetic data for targeted attribute augmentation. Our find- ings reveal that the models exhibit varying degrees of invariance across different attributes, provid- ing insight into their strengths and weaknesses and enabling deeper interpretability.. Code avail- able here:
However, our understanding and interpretation of the structure of the embedding space learned by an FR model remain significantly underdeveloped. Unlike inter- pretable attributes, deep face representations exist in a high- dimensional space, making it difficult to discern the specific properties they encode (O'Toole et al., 2018). The problem is further aggravated by the potential high risks associated with the social impact of FR technologies. The black-box nature of these embeddings raises concerns about bias, fair- ness, and security vulnerabilities (Mery, 2022; Gong et al., 2020). Previous studies have attempted to analyze these embeddings from different perspectives: Yin et al. (2019) introduced spatial activation diversity loss to encourage the network to learn structured face representations, ensuring that different filters respond to semantically meaningful fa- cial parts such as the eyes, nose, and jaw; On the other hand, Lin et al. (2021) proposed the xCos metric, which provides spatially interpretable similarity maps for face verification, whereas (Diniz& & Schwartz, 2020) investigates how well the attribute is implicitly learned by neural network layers.
watiiiioss0///tttee 107/attrs-
fr-embs.
1. Introduction
Deep Learning (DL) models have achieved state-of-the-ar performance for Face Recognition (FR) tasks, due to in- creased availability of curated datasets (Deng et al., 2019b) and improved recognition architectures and losses (Deng et al., 2022). In the challenging open-set scenario, novel face identities can appear at testing time, hence the problem is framed as metric learning task (Liu et al., 2017). There are three main innovation to reach outstanding FR results in this scenario. The first consists of learning an embedding
A more recent line of research focuses on reconstructing facial attributes from embeddings to understand what infor- mation is retained in the learned representations. Ren et al. (2024) developed two-stage attribute recovery framework, which reconstructs facial images from the embedding space and then quantifies the significance of facial attributes within the representation. Their analysis indicates that FR mod- els prioritize encoding shape-related attributes (i.e., jawline and facial proportions) over transient factors like expression and head pose. This aligns with findings that deep models tend to discard non-essential variations in favor of stable, identity-specific features (Yin et al., 2019). Other studies focus the best camera angle to avoid bias in facial im-
'Equal contribution 'Department of Mathematical Sciences, Politecnico di Torino, Turin, Italy 'Department of Computer Science, University of Pisa, Pisa. Italy *CENTAI Institute, Turin, Italy. Correspondence to: Antonio Mastropietro <anto- nio nio.mastropictroee di unipi.it>.
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Attributes Shape the Embedding Space of Face Recognition Models
age analysis (Choithwani et al., 2023) or to develop an FR system which accounts for the difference between frontal or profile images (Yang et al., 2021).
attributes linked identities. Then, we focus on the mi- croscale, and we propose an invariance energy measure to quantify the invariance of the embedding model to each al- tribute. Since the embedding model at the microscale should filter out transformations of face images that preserve the identity, we hypothesize that the attribute invariance at the microscale level indicates the quality of the FR performance. We validate our hypothesis by fine-tuning a number of state- of-the-an FR models synthetic, generated data, varying set of interpretable attributes one by one in controllable manner through the GAN-Control model (Shoshan et al., 2021). The results show that the fine-tuning on a specific attribute increase the invariance metric on the same attribute.
We can observe that the training procedure and the con- trastive nature of the losses aim to achieve invariance of the embedding mapping over transformations of face images belonging to the same identity (Schroff et al., 2015: Chen et al., 2020b; Deng et al.,, 2022). Past works demonstrated how data augmentation impact feature learning for generic contrastive models (Wen & Li, 2021). In addition, (Chen et al.. 2020a) describes data augmentation as transforma- tions of image features that induce invariance in the learned representation. In theory, in the case of the FR task, we want dependency of the embedding map to the identity of the face image, while keeping invariance to every other image attribute (Schroff et al., 2015). For example, to ob- tain invariance of FR towards wearing a face-mask, Ge et al. (2024) makes use of a -iriatiiveddierrminaiieee approach to convert category-aware descriptors into identity-aware vec- tors. In addition, previous works have made public enriched datasets for training FR models by incorporating variations over a large range of attributes (Moschoglou et al., 2017: Cao et al., 2018).
2. Multi-Scale Geometry of the Latent Space
We establish a formal ground for the FR task as assumed by Deep FR models and losses. We assume an open set protocol, that is, the testing identities do not necessarily cover the training identities; hence, we need to map faces into a discriminative embedding space, so that the FR task gains similarities with the metric learning problem. For other FR protocols, we point the reader Liu et al. (2017). Furthermore, we assume a trained FR model ciently representative dataset, e.g., MSIMV3 (Deng et al., 2019b), and we inspect the embedding space to describe its geometric structure induced by the architecture and the loss.
suffi-
Taking the metric learning problem of open-set FR to the extreme, an optimal embedding model would map all the images of the same identity to a single point in the embed- ding space, with images of different identities mapped into distinct points. In practice, this behavior is not observed and might not even be desirable. Indeed, Dan et al. (2024) pro- poses a method to align the topological structure of the input and the embedding space to improve the generalization of FR models.
Let X = R² be the image space and D C XX be the space of face images, partitioned by the set of labeled identities I, D Use where Dj (x)(9) is the subset of face images associated with identity i. The dataset of face images D represents sample from D, and similarly, D; C D, Let f: X + E' be pretrained face embedding model, mapping images X to space of vector embeddings E, immersed in an outer space of dimension p, i.e. E = IR,, equipped with a distance (or dissimilarity) function d: E x - RR. For example, the embedding space of FaceNet (Schroff et al., 2015) is equipped with the Euclidean distance, whereas ArcFace (Deng et al., 2022) with cosine dissimilarity. We define the set of embeddings S: = f(D:) as an identity space. Likewise, we call P. = f(D)) as an identity point cloud, obtained by sampling from S, or by sampling from D; and embedding through / in F.
We begin our study of the embedding space by noting that there are attributes that orient the relation between identity point clouds. In addition, there is a structure emerging inside the regions corresponding to each embedded identity. Thus, suggest that the training procedure for FR determines at least two scales at which we can analyze the embedding space. First, the macroscale looks at relations and distances between point clouds of different identities. It is a global scale that looks at all the training images at once, clustered by identity classes. Second, the microscale inspects the inner structure of identity point clouds.
Furthermore, at inference time, we assume that there is a classification rule r: E + I U fu), such that if an embed- ding corresponding to an image close to a P., then r(e) i; we anticipate that x could be assigned to no identities, in which case the rule associates the unidentified identity u to the embedding e. Since the classification rule depends on the model and loss employed, we leave the rule as an abstract function. However, we imply that makes use of the distance (or dissimilarity) of F. In this setting, a i-decision region R, ri) is the subset
This paper aims to study how the embedding space geometry of FR models is shaped by human- -uman-interprettblle attributes. First, we examine the macroscale, obtaining insights into the models strategies to solve the FR task. In particular, we provide a procedure to check if attributes shaping the relations inter-identities are the ones most deterministically linked to an identity. Our results over the comparison be- tween FaceNet and ArcFace suggest that the embedding of the former has a higher structural dependency of the
f(x) is sufficiently
2

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Attributes Shape the Embedding Space of Face Recognition Models
fact, in the extreme case that model / is such that for each E I, the identity cloud trivially collapses f(D,) { and such that for distinct i, E I, e + e then it is easy to define rule to achieve the best recognition.
Table 1. Intra-class and inter-class distance on the L.FW dataset
Model
Architecture
d
db
ArcFace
R8
0.327
0.994
ArcFace
ResNet50
0.290
0.996
However, in real settings, the many losses proposed to tackle the FR task paired with the various architectures show non- trivial geometry, without ending in strong overfitting or mispredictions. This results in each identity point cloud possessing a structure that preserves some properties of the image space. Moreover, as far as the global scale is concerned, the triplet loss does not explicitly encourage any particular disposition of the identities in the embedding space, meaning that any random disposition of the point clouds, as long as they are sufficiently far away from each other, is equally valid.
AdaFace
ResNet18
0.370
0.995
SphereFaceR
10e
0.276
0.993
FaceNet
iResNetvl
0.670
1.376
of E such that the projected embeddings by f are classified by r as identity i. Note that R, could be different from the identity space S,, and, in general, the decision regions of the training identities do not partition E.
Loosely speaking, a Deep FR model is trained by optimizing a loss function pushing the distance inter decision regions higher than the size of each region. The (inter-class) dis- tance between decision regions can be estimated through the corresponding distance between identity point clouds: (P,,,, P3) F.iE, Eeth e,.e)) d(e()) ,e())). Simi- larly for the (intra-class) distance within decision regions: d(P.)) |2//2
Interestingly, we find that real face recognition models ex- hibit strong global organization, such as the separation of male pictures from female pictures as shown for example in Figure 1. As it is not explicitly aimed at or due to the loss, this structure must be an emergent phenomenon due to the synergy between initialization, optimizer, architecture, and data. This fact is suggested also by Hill et al. (2019); the authors found that hierarchical structure emerges by analyzing the embedding space of a CNNs model for the FR task trained using triplet loss (Sankaranarayanan et al., 2016) and looking at interpretable attributes of face images.
1 Eety, ep, d(e(0), e()i) (Liu et al.. 2022:
Deng et al., 2022). Thus, after training, we can assume that the distance within is much smaller than the distance be- tween point clouds, i.e. d(ii= < j)( and d(j) < do(i, for i I. Indeed, Table 1 shows that there is at least factor of 2 between d and d, for FR models with various ar- chitectures, losses and distance metrics on the LFW dataset (Huang et al., 2008).
2.1. Attribute Invariance
We call image attribute a: - M. a function recognizing an interpretable property on face images; we name M. the modality space for a. We denote by A the set of attributes we want to recognize on each image. We can distinguish many attribute classes: for example, Hill et al. (2019) cites viewpoint, illumination, facial expression, or appearance. Some attribute may be strongly related to facial identity, such that a(x) is constant on each D;; on the opposite, others can be derived from the image properties, like contrast or brightness, or can vary between pictures of the same identity (e.g, face mask, eyeglasses). Depending on the attribute, Ma can be a manifold or just set. For instance, we can describe blonde hair a binary attribute that has value 1 if and only if the picture shows face with blonde hair a: e DD + Matonde = {0. 1}. Age is a continuous attribute that associates to picture the age of the person depicted at that moment. In this case, we can assume Mage = (0, 200). Finally, hair color is a continuous attribute that associates to a picture the person's 's hair color, belonging to a suitable color manifold Mootor-
From the assumptions made, if we look at the mapping of the dataset D to the embedding space E, we ought to see separate point clouds, each corresponding a single identity i. This geometry is expected, as it is direct effect of the contrastive loss used in training: embeddings belonging to the same identity are pulled together, while simultaneously being pushed away by the embeddings of other identities. Thus, the embedding space E of f. given the dataset D, has two natural levels of geometry corresponding to two (i) the microscale delves into the structure of each individual identity point cloud, whereas (ii) the macroscale pertains to the way various identity point clouds, now seen as single points, are arranged in E.
scales of observation:
It is important to recognize that, in terms of pure perfor- mance, the microscale is not crucial as long as all face images labeled end within the correct region RR, and R; does not contain elements of Dj. j A i. In particular, a model should possess an invariance property with respect to the attributes of facial images. Hence, if we transform face image I over a specific attribute through transformation t, while preserving the identity i, e.g., as of t(x) = x' E D;, then a performing model should satisfy an invariance prop- erty f((()))) = fx) Sec. 2.1 formalizes this property. In
In most cases, the attribute function may not have an explicit analytic definition, and we may have access to its values only through a dataset. For example, a widely used FR dataset as CelebA provides us with binary labels for a list
3

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Attributes Shape the Embedding Space of Face Recognition Models
CelebA embeddings
CelebA aubeddings
Farmll Maw
10-
dg dgrass
Facenet KS dislance
0.16
Bald Male
entrupy
0.1
08
0.12
P
06
deroms
0.4-
0.05
0.5
KS distance
02-
0.04
Mouth wwwghtliiyyyooooe Open 55mling
00
0.00
d.
1.35
1.50
0.12
0.16
Distance
Arcface KS distance
Figure 1. UMAP projection (McInnes et al., 2018) of the Facenet (a) and Areface (b) embeddings of CelebA. The colors show clear separation between images depicting males and females. Note that the UMAP projection does not accurately capture the distances in the original E space. c. For cach binary attribute, we compute the distributions of distances inside and across the two modalities, and compute their Kolmogorov-Sming Smirnov distance. d. Model comparisons: each point corresponds to an attribute whose coordinates encode the KS statistic for ArcFace and for FaceNet. The size of each point is proportional to the average intra-entropy of the corresponding attribute.
of 40 attributes (Liu et al., 2015), even if for some of them the definition is fuzzy (e.g. blonde hair). Moreover, we might not have direct access to the modality space Ma but, having fixed an image x, we could have a function to transform z while varying the value of ((x) in a controllable way. Inspired by the theory about data augmentation (Chen al.. 2020a), for a set of attributes a A we can define groups of transformations To {ta: D D), equipped with the composition operator o. For example, for the binary attribute having modalities Maluonde = {0, 1}, we can think of Ta = {z, w): is the zero element, =<x) = x, whereas w is a transformation from "blonde" to "not blonde" and vice versa, i.e. for each x DD (,((x)))) = a(xr), with the constraint (w w) (z) =(x) x. Furthermore, we define the action a₂ of group of transformations T₃ Moi
In general, note that a transformation of a could impact multiple attributes at the same time (e.g, age and hair color). Thus, by measuring the invariance over set of attributes we can observe their entanglement in E.
3. Macroscale Analysis
In this section, we analyse the structure of the embedding space at the macroscale, looking at the possible interaction of the identity point clouds P, at inter-identity distance dg. Given a model f. a dataset D, and an attribute a, it is interesting to determine whether a influences E across different P.. We propose a simple methodology to partially answer this question when a has discrete values. Using this approach, we analyze a high quality subset of CelebA consisting of [|| 1965 identities, with [D,] 30 and 40 binary attributes (e.g. "eyeglass", "smiling", "male") for five FR models: FaceNet, AdaFace, SphereFaceR and two versions of ArcFace
0o: : Ta x D D
(1,, + (x)
We recall that by the property of the action of group on a set: (1) for the group zero element 2 E Ta, aa(2, x) =(xr) = 2r, Vo E D: (2) the action is compatible with the composition of transformations, i.e. for every ta, 8a Ta. a. (fa Sa; x) ara (la, an (Sa, x)).
3.1. Attribute Dependence in Embedding Space
We present a simple procedure to test if an attribute has a structural impact on E relying only on dis- tunces/dissimilartiiss. More precisely, we propose to test whether conditioning a P; on the same attribute modality ma of an attribute a changes the distance distribution compared to conditioning on different modalities. In other words, test the hypothesis:
We can weaken the invariance property by asking that the embedding of the transformed images stay close to the base image (La (x)) xe fxx) As mentioned Sec. 2, the mi- croscale of an identity point cloud is measured by the within distance d(..). In particular, we state that the attribute sat- isfy approximate invariance property if, for all ta Ta. d (I((, ((x)) < d(:)), where å E [0, 1] is an hyper- parameter which determines the allowed displacement of t. (z). Hence, it is meaningful to measure the approximate invariance of an embedding model f relative to a set of attributes A and their respective group of transformations.
Ho: Fa(d(er, = Fa(d(er,
where
f(x;) are embeddings. F, denotes the cu- mulative distribution function (CDF) in the case when the images have the same modality for attribute a, i.e. axx1) = a(x2) = m, for some m E Ma. and Fg is the CDF when the modalities are different. We use the Kolmogorov-

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Attributes Shape the Embedding Space of Face Recognition Models
whereas smiling or mouth slightly open have high intra- entropy. This finding aligns with intuition, since the last two can easily vary for a fixed identity, while it is less likely for the first two, See Appendix C for the intra-entropy of each attributes and other summaries.
Table 2. Summary of Spearman correlation results.
Model
Architecture
Correlation
P-value
FaceNet
iResNetv1
-0.626
< 10
AreFace
ResNet50
-0.566
< 10-3 3
ArcFace
Re
-0.619
< 10-4
AdaFace
ResNet18
-0.656
< 10
Linking intra-entropy with KS statistics. For ArcFace and FaceNet on CelebA, we find significative negative Spear- man correlations between the intra-entropies Ha and the KS statistics SS, (Table 2). This confirms that the attributes shaping E the most are the ones most deterministically linked to an identity.
SphereFaceR
iResNet100
-0.606
S 10-4
Smirnov test from Massey Jr (1951) (KS-test) to check the equality of distributions as stated by Ho. We expect the attributes of CelebA to influence the embedding space of our FR models, precisely because these attributes are facial features, useful to solve the FR task.
Additional observations. In light of this analysis, we can make additional observations. On Figure 1d, we propose a comparative analysis by plotting two models against each other. The point cloud being above diagonal informs us that FaceNet has higher KS statistics, hinting bigger structural dependency on the attributes. While there slight variations, the two models tend to converge on which attribute have the bigger structural impact, as attribute are ranked similarly, as in (Li et al., 2015). All KS statistics are significantly lower than 1, meaning that no attribute is associated with distinet clusters of points, as it is the case for identities. Note also the correlation from Table 2 of intra- intra-entropy and KS statistics visualized as larger dots tend to concentrate towards the origin. Lastly, modalities are not necessarily symmetric and can have unbalanced KS statistics (example: bald, Figure 10 in Appendix).
To test Ho for an attribute we first sample modality point clouds Qm = {e E E e f(x), d(x) = m). Note that the Qm span embeddings across P,, with possi- bly multiple points per identity. Since we are only inter- ested the macroscale, we subsample each Qm to keep at most a single representative per identity. To compare between modalities of the same attribute, we further sub- sample such that, fixed all the Qm m E M, have the same cardinality q. Then we measure: the intra- distances dm = (d(en, e2) e1, 62 E Qm. e1 F e2), representing the pairwise distances between points in a Qm. and the inter- distances dm = (2()),e2) e, E Qm 62 & Qm repre- senting distances between points of different Qm Finally, for each modality, cal distributions of dmm and dmn, the statistic of the test be- ing the KS- disttnccee depicted on Figure 1c. For CelebA, m E 1 and we reject Ho for a large majority of at- tributes at the confidence level 0.001 (see detailed p- v-vllues on Figure 10 in Appendix C). Furthermore, to summarize the KS-statistics at the attribute level, we compute the mean over its modalities: KS,- me (KS)).
perform a KS-test on the empiri-
4. Microscale Analysis
After analyzing how identity attributes shape the macroscale geometry, we aim to see if and how much the microscale is influenced by the the same attributes, or if the embedding model is invariant to them.
When working with categorical attributes, given that the distance in the embedding space E is used for the decision rule, is enough check the displacement induced by the attribute variation. Categorical attributes are then readily comparable because we can assess their associated changes in the embedding space when we change their modality in the attribute space. With continuous attributes, however, it is harder because there is no natural way to compare the amount of change for an attribute a, with another attribute 2g due to their possibly incomparable scales. For example, if one fixes change in illumination, it is impossible to choose "equivalent" change in hairstyle. To approach this issue, we propose an energy measure which is independent of the attribute scale. The general idea is to study the vector field in E induced by the infinitesimal variations of an attribute a, as we see in Figure 2. Every vector in the vector field is normalized to unit norm so that we can ignore the effect of
3.2. Geometry and Discriminative Power
Intra-entropy is proxy for discriminative power. In FR tasks, the discriminative power of an attribute reflects its contribution to prediction performance and information gain. For instance, smiling has likely low discriminative power since knowing whether someone is smiling provides little help in identifying them, whereas eye color is more distinc- tive and thus has higher discriminative power. We propose way to detect if the discriminative power of attributes is reflected by the geometry in embedding space. Given an attribute a, we approximate its discriminative power by its intru-entrupy Ha: first for each identity we com- pute H: = H{{((x) x e D.H), where H is the empiri- cal entropy. Then, the intra-entropy is simply an average over identities Ha = EiED Hi In CelebA, we observe that attributes such as male or bald have low intra-entropy,
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Attributes Shape the Embedding Space of Face Recognition Models
a
b
Image space
Embedding space
a +)) x 2()))2))
((x))
R Fad
31
B(c)
E(c)
[(c) 0.31
E(c) -
Allgued
Unaligned
Figure 2. a. Intuitive depiction of the computation of the invariance energy. The continuous variation of the hair color attribute results in a curve passing through cach embedding. Our energy measure computes the dissimilarity between the normalized tangent vectors to these curves at points which are closer than a threshold e. b. Visualization on how the model achieves approximate invariance w.r.t. dimension 22 by irregularly folding the space. c. Examples unit-norm vector fields on a simple 2-dimensional point cloud ordered by their E.
scale, leaving only direction and orientation.
In this case, the attribute's 's continuous variation will result in a locally constant displacement of the embeddings, that is, close embeddings are moved in the same direction.
In detail, let us consider a one- oeeoooneeeeee parameter group G (Varadara- jan, 2013) representing the action of the group of real num- bers f R varying the attribute o, Ora,t : Xa - Xa Vt TR This situation corresponds, for exam- ple, to the one provided by GAN-Contur if the attributes are changed through monodimensional sliders. Fixed an image x E D; we define the vector field at the embedding e= ==f(x) E P. as ve(e) with:
In detail, fixing scale parameter 2 O, define the invariance energy metric associated to the identity space S; and vector field U/ of a, as the expected cosine of the angle between vectors at points e, E E which have distance less or equal than E. (e) == Ee Be [1 [1- (a) vn ve(e)), (3)
with 0a,0 (z)
v. (e) = (e)
[e, (e)l (e(e)=
= d
f fo (x) E E.E,
where the first expectation is taken over uniform sampling of S,, and the second over uniform sampling of the closed ball with center and radius in the distance/dissimilarity function d, intersected with the identity space (e) n S, Since we have access to S, through the identity point cloud P., we compute an estimate of the invariance energy over the embedding of P.. The name "energy" comes from the fact that, when we approximate Equation (3) on given P., the energy (()) ) is closely reminiscent to the Hamiltonian of the nevector model (Stanley, 1968) describing the evolution of a system of pairwise-interacting n--dimensional spins.
dt t=
(2)
where Te E is the tangent space of E e, which we identify with Rd. If [e)l())]| = O, we write vo (e) O.
4.1. Energy Measure
Our conjecture is that, the FR model has learned to be ap- proximately invariant to collapse the directions associated to a in the input point cloud D, by folding and distorting the space in complex way. This folding, akin to crumpling a sheet of paper, will push together points which were further away in the input space. If this is the case, we can observe the behavior by computing the "roughness" of the attribute vector field Va- quantifying the lack of alignment of each vector v. (e) with the vectors close-by by points. An intuitive visualization of this fact is depicted in Figure 2b, where we see how the di- mension associated to attribute 22 is contracted non-lincarly, pushing closer vectors which were far away. On the other hand, the FR model will be sensitive to the attribute v, (e) aligned with the vectors associated with close-by points.
attribute a, it will do by trying
As we see in Figure 2c, if the vector field is perfectly aligned (a(e) = va (e") Ve, e' E P,, we will have minimum energy (e,((e) = Ve 0. will have low values when v. is locally aligned, while attains the highest values for disordered random-looking vector fields. The maximum energy value is &s,i 2, with &a, meaning that the vectors are on average orthogonal to one another.
Toy model. To validate the meaningfulness of our energy definition (3), we study a simple toy model where approx- imate invariance is achieved by a multi-layer perceptron
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Attributes Shape the Embedding Space of Face Recognition Models
b
Figure 2a. is important make sure the generated point cloud is perceived by FR models as single identity because we want to conduct microscale analysis. We do that by verifying that distances between embedded images and their single attribute variations correspond to the order of magni- tude of the intra distances reported on Table 1. Concretely, that assures us that the curves we are interested in do not represent traversals spanning more than a typical identity, While GAN-Control allows us to generate images varying high-level attributes, we also include 3 low-level data aug- mentations: brightness, hue and image quality, where the latter amounts to different levels of blurring and contrast enhancement.
bu
Endrgy Bi
04
Scale
Figure 3. a. The generated dataset for the toy model. The energies associated to each input dimension for different scales, over 30 neural network trainings.
We now take three different pretrained FR architectures, namely FaceNet, ArcFace, AdaFace, and compute the in- variant energies of the 8 controllable attributes described above for GAN-Control. We do this averaging over 40 synthetic identities D,. for multiple e-scales, to obtain a multiscale fingerprint of each model as a whole. Given that the embedding spaces of the three models are equipped with di different metrics (Euclidean and cosine dissimilarity), each scale & is chosen to be relative the average distance d between pairs of points in each P..
(MLP). We generate 3 6-dimensional blobs of 250 points each, such that their first two features 72, 42 are distributed according to three separated Gaussians as in Figure 3a. We name these dimensions useful because they can be employed by the MLP for classification. The values of the last use- less features Tz,.... 26, instead, are independently sampled from uniform distributions between- -5 and 5.
We train to convergence simple three-layered MLP to classify the blob each point belongs to. We see how the information about the blob separation is present only in the first two dimensions, meaning that the model will need to learn to be invariant to the last four. We pick a range of scales E E [0, 1] and compute the energy measure on the embeddings provided by the 16-dimensional penultimate layer. The results, shown in Figure 3b, clearly show how the energies are nicely divided into two groups, with the useless dimensions (attributes) having clearly larger energy than the useful ones at all scales.
In Figure 4a, we show the results of this analysis. First, we observe that the set of attributes we chose attains en- ergy values across a wide range, signifying the presence of different behaviors. In terms of the relative positioning of the attributes, we find that all models are most invariant to lower-level attributes like "contrast" and "illumination", while they achieve the lowest values complex attributes like "head angle" and "age", signifying more sensitivity, We note that this results align with the intuition of complex attributes being less "filtered out". Learning to be invariant to age, in particular, means that the model cannot rely on more lower-level identity features, like the hair color or the exact face proportions.
4.2. Energy of Pretrained FR Models
To compute the invariance energy for actual FR models, we need a way to sample face images with locally modified attributes, i.c. to mimic the action of group transformations over attributes. For this, we turn to generative modeling. GAN- GAN-Control (Shoshan et al., 2021) uses attribute classi- fiers during training to create a disentangled latent space. Then, it learns to map attribute vectors to their correspond- ing latent subspace. This allows users to generate images in a controllable manner, specified with interpretable at- tributes. We also experimented with ConfigNet (Kowalski et al., 2020) and Arc2Face (Papantoniou et al., 2024) but GAN-Control provided the best balance for our needs in terms of controllability, generation quality and efficiency. The disclosed pretrained GAN-Control model lets tin- ker with the following 5 attributes, fixing an identity: pose, age, hair color, illumination and expression. We use GAN- Control to generate vast point clouds of identities of a total number of over 120K embedded images sampled along curves where only one attribute changes, as portrayed on
From Figure 4b, moreover, it is clear how FaceNet achieves significantly lower energy across all attributes, at all scales. This. in line with the macroscale analysis of Section 3, sug- gests that the geometry of FaceNet's embedding space is more sensitive to the attributes also at the microscale of sin- gle identities. ArcFace and AdaFace, being similar models, have similar values, with AdaFace having slightly lower energy (more sensitivity) on the attributes related to color, i.e., "Hair color" and "Hue".
4.3. Invariance Energy Fine- Tunee Models
To assess the capability of our energy measure to capture the invariance of FR models, we ask if the invariance ob- served through the energy measure agrees with the effect of data augmentation on single attributes. In fact, by training a model with augmented data coming from the action &a of at-

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Attributes Shape the Embedding Space of Face Recognition Models
0.4 d
b
Adaasoe
Facerel
Energy
0.5
Energy
Head Hegle angle
Age
Hair oofor
Ilmiiaatoon
0.5
0.c
1-000
Energy
0.-
w
Expression
Brightness
Contrast
Hue
Head anglo
Age
Halr color
Bumiraatio
Expassion
Brightreas
Contraot
Hoe
0.
0.6
ca
0.4
0.
0.
0.8
0.6
0.8
E/
s/dd
s/dd
E/
Attributes
0.4
Auclisce I.s
Rei
0.4 d
d
Heed engle Age
Haado angle Energy
Arcbos
Burindtoon
Expresion
Bightrer
Dortno
Adaface
Ilumiiation Energy
+
enple
Har cole-
Busindion
Brgitnes
Head argle
Age
Numination
Expressiiy
Brighhress
Contrat
How
Head
Conter
Heat mas
Beat
Finetuning
Attributes
Altr ioules
Attributee
Figure 4. Energy distributions for each attribute and each FR model, computed across 40 synthetic identities at the scale 0.4 d. b. Average energy each attribute and FR model when we vary the scale from 0.4 d. e. The distribution of the Head angle (top) and Illumination energies after all the fine-tunings (x-axis) of AreFace and AdaFace. The highest energies, which attained ad the fine-tuned attribute, are highlighted. The post fine-tuning energy of every attribute and every fine-tuning under min-max rescaling over the line-tunings (rows).
tribute a, we can expect its internal representation become more invariant to (Chen et al., 2020a;b). Our goal is to fine-tune FR models with augmented data for each attribute, and observe the resulting change in the energy measure. In detail, the baseline ArcFace and AdaFace models are fine-tuned on monovariant datasets 10K identities created with C-N-Coltrol and with a single attribute variation (e.g. only variations illumination). We also generate validation and test sets containing all possible variations. However, since ArcFace and AdaFace have parametric losses specific to train labels, they can't 't be directly computed on a hold-out sets in the open-set protocol. We slightly adapt these losses to make them parameter-free (details in Appendix F.1) and so we are able to measure them on validation and test sets.
tively learned to be more invariant the variations of than the other models fa-- We observe that the specialization
induced by fine-tuning is not completely disentangled, as fine-tuning on one attribute (e.g., age or brightness) may increase performance on another, like hair color, more than on orientation. Fine-tuning metrics such as learning curves are presented in Appendix F.
We expect the invariance energy capture the shift in inter- nal representation caused by the fine-tuning. Figure 4c, shows two examples ("head angle" and "illumination") where, fixing an attribute the fine-tuning on is the one increasing the energy value the most. This holds for both Ar- cFace and AdaFace in the same way. Figure 4d summarizes the results by showing the per-attribute energies min-max rescaled across the tine-tunings. Notice how, for both Are- Face and AdaFace, the maximum value of each column is achieved at the diagonal, meaning that E, is increased the most when fine-tuning on the same a.
We fine-tune multiple times our baselines on monovariant datasets and use the carameter-free loss determine the best checkpoint. During fine-tuning, we track key met- rics on the validation and test sets and on CelebA (cf Ap- pendix F). After the process, we obtain a family of special- ized models, one for each attribute. Let fa be the FR model fine-tuned on attribute a and L. denote the parameter-free loss measured on a hold-out monovariant set D. contain- ing only variations of attribute Then, we observe that La(fa) < ),,r,, Va # a', i.e. each model fa has effec-
5. Conclusion
In this paper, we explored the geometric properties of the embedding space of FR models, emphasizing how human- interpretable attributes influence both macroscale and mi-
8

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Attributes Shape the Embedding Space of Face Recognition Models
croscale structures. For macroscale, we focused on categor- ical attributes, and proposed a simple approach to assess their influence and discriminative power using distribution distance. For the microscale, we focused on continuous attributes and introduced an energy measure to quantify the degree of invariance of a model to an attribute, Our pro- posal effectively quantifies the sensitivity of FR models to different attributes, highlighting variations across models, and demonstrating how they encode attributes with vary- ing degrees of invariance. The measurements agree with fine-tuned models, confirming that targeted augmentation enhances attribute-ppecffic invariance. Our findings provide deeper interpretability of FR embeddings and suggest new avenues for improving the robustness of FR systems. Al- though we focus on FR, our ideas could be applied in other metric learning contexts, provided attributes are available.
age, pose, illumination, which may influence recognition performance in unintended ways. Greater transparency in embedding learning can contribute to fairer and more ac- countable biometric technologies, ensuring that face recog- nition systems function reliably across diverse populations while mitigating potential misuse.
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Impact Statement
The risks associated with FR models extend beyond techni- cal inaccuracies. Biases in the embedding space can lead to disparities in recognition accuracy across demographic groups, raising ethical concerns in applications such surveillance, border control, and authentication systems. The reliance on black-box embeddings also creates vulnera- bilities, where adversarial manipulations or synthetic alter- ations can exploit weaknesses in the learned representation. These issues highlight the need for interpretability- approaches that can assess and guide the structure of FR embeddings. A clearer understanding of the embedding space of FR models is essential for ensuring their reliability, fairness, and security. Given that these models learn high- dimensional feature representations, their decision-making process remains largely opaque, making it difficult to as- sess whether embeddings capture identity-related features robustly or if they encode unwanted biases. Investigating the structure and invariance properties of embeddings can help uncover hidden dependencies on attributes such
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Attributes Shape the Embedding Space of Face Recognition Models
A. Datasets
LFW(Huang et al., 2008) LFW is dataset of approximately 13K images with no additional attributes beyond identity in its standard version. The number of images associated with each identity varies between handful and several hundred. We use this dataset as a sanity check dataset for the models. Indeed, we expect the performance of all models to be almost perfect on the face verification (Huang & Leamed-MMller 2014) task (classifying matching and non-matching pairs), which is saturated on this dataset. We show examples of distance distributions in Figure 5 and metrics in Table 3
1400
1700
Orgaes
Figure 5. Distribution of distances on the face verification task on random pairs extracted from LFW for AdaFace, ArcFace, SphereFaceR and FaceNet. The models where the embedding space equipped with cosine distance (AdaFace, ArcFace, SphereFaceR) show the distance in degrees.
model
best accuracy
mean intra distance
mean inter distance
ArcFace (r18)
0.9974
0.3270
0.9944
ArcFace (r50)
0.9991
0,2896
0.9960
AdaFace (r18)
0.9953
0.3697
0.9950
SphereFaceR (iresnet100)
0.9983
0.2760
0.9933
FaceNet (iresnet v!)
0.9851
0.6702
1.3763
Table 3. Models metrics. The best accuracy corresponds to the estimation of accuracy at the optimal threshold.
CelebA(Liu et al., 2015) We use the CelebA dataset to conduct an analysis of the attributes at the global scale, that is, across identities. Since the finetuning is done on generated data, we also track few metrics on CelebA during our finetuning process to keep track of the dynamics real data. This dataset contains over 200K images and 10K identities, as well as 40 binary attributes (e.g., "eyeglass", "smiling", "male"). To extract faces, we run the Retina Face(Deng et al., 2020) face detector. We encountered common data quality and processing issues:
1. Multiple faces detected on an image.
2. No face detected on an image.
3. False negatives: for given identity, there are sometimes mislabels i.c. images of diffcrent persons.
We took some steps to manage these problems and create higher-quality subset since we don't need all the data. We address issues 1) and 2) above by discarding images with not exactly 1 face detected. We also restrict ourselves to the identities with more than 30 images. Then, we apply statistical filtering to remove obvious outliers (cf. Section 3.2 (Deng al., 2019 2019))): first, we project images into the embedding space using face recognition models, and then we remove those points that are far from all other points. We verify the improvement of statistical filtering by comparing simple clustering with the identity point clouds. We do this statistical filtering using ArcFace and FaceNet.
After these steps, we reduce CelebA to subset containing around 55K images of <00 images by identity. All analyses are made on the higher-quality subsets.
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Attributes Shape the Embedding Space of Face Recognition Models
B. Models
Model
Architecture
Metric
Train set
Images (M)
Source repository
FaceNet
iResNetvl
euclidean
VGGFace2
3.31
davi faviddannbberrffcceett
ArcFace
ResNet50
cosine
MSIMV3
5.18
deepinsigght/lisighhffacc
ArcFace
ResNet18
cosine
MSIMV3
5.18
deepinssiihhh//nsiighhfaaae
AdaFace
ResNet18
cosine
VGGFace2
3.31
mk-minchul/AdaFace
SphereFaceR
iResNet100
cosine
MSI
ydwen/openspperee
Table 4. Main characteristics of models used in this work.
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Attributes Shape the Embedding Space of Face Recognition Models
C. Macroscale analysis details
C.I. Summaries: global average, inter entropy, intra entropy
The main quantity of interest is the intra-entropy, as defined in the main text. We have also computed the global average of each attribute and another entropic summary, the inter-entropy, which quantifies the variability of an attribute across identities. If m |.. ExeD, a(x) is the attribute a's average over identity
1. 1 Eier ma is the attribute global average;
2. (mm)). the attribute inter-entropy is the differential entropy of a Gaussian kernel density estimation of (m²) ke
3. H. is the attribute intra-entropy, as defined in the main text
Summary
global average
Baid
entropy
inter entropy
Blurry
Wearing Hat-
Double Chin
Eyeglasses
Chubby
Mustache
Wearing Nocktie
Goatee
Pale_Skin
Sidebums
Receding Hairline
Receding_Harrliee
Rosy Cheeks
5_o_Clock _ohlook_Shdoww Shadow
Oval_ Face
Narrow, Eyes
Bushy Eyebrows
Bland_ Hair
Bangs
Wearing NNecklaeeeee
Brown Hair
Straight Hair
Black Hair
Wearing Wearing_Earring Earrings
Bags Under_ Eyes
BlgeNoce Nose
Male
Pointy_ Nose
Big Lips
Wavy Hair-
Arched_ Arched_Eyebrowe
High. Cheekbones
Smiling
Mouth Souuth_Slgghly__00 Openn
Heavy Makeup
Wearing Lipstick
Attractive
Young
No, Beard
2
value
Figure 6. CelebA attribute summarics. The global average is measured on all images. The intra-entropy first computes the entropy at the identity level, then averages over all identities, and conversely for the inter-entropy.
14

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Attributes Shape the Embedding Space of Face Recognition Models
Summary min-max rescaled
global average
Bald
intra entropy
Gray Hair
inter entropy
Bluryy
Wearing Hat
Dauble Chin
Eyeglasses
Chubby
Mustache
Wearing, Nedctie
Goates
Sidebums
Receding, Hairtine
Rosy.
5_0, Clock Shadow
Oval_ Face
Narrow Eyes
Bushy Eyebrows
Blond_Haa Hair
Bangs
Wearinng_Ncklace
Wearing Necklace
Straight, Hair
Black Hair
Wearing, Earrings
Bags, Under Eyes
Big Nose
Male
Pointy Nose
Big
Wavy Hair
Arched Eyebrows
High. Cheekbanes
Smiling
Mouth Slightly Open
Heavy Makeup
Wearing Lipstick
Attractive
Yaung
No, Beard
value
Figure 7. CelebA attribute summaries, rescaled. This figure min rescaled version of Figure to enhance between summary comparison.
15

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Attributes Shape the Embedding Space of Face Recognition Models
C.2. Distributions of mean and intra entopy for each attribute
In Figure and Figure we show the estimated distributions of attribute mean and intra entropies for each attribute over cleaned subset of CelebA.
Desritutionn AE Bado mosn,, Wlmitiy
26
40 18
26
10
26 20
26
20
28
20
20
20
20
12
20 00
00
20 m
20
Figure Distribution attribute mean over identities of CelebA. In parenthesis entropy distribution that correspond the inter-entropy
16

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Attributes Shape the Embedding Space of Face Recognition Models
D-SE-tAtIOn entrerriges, derdit
4
1 Aomesmm
Figure Distribution of intra entropies identities of CelchA for each attribute. parenthesis is the of this distribution that correspond to the intra-entropy.
17

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Attributes Shape the Embedding Space of Face Recognition Models
C.3. p-values associated with the KS-test of Ho
In Section 3.1 we perform statistical test at the attributes-by attribute--y--modaiii by-modality level, so in total Nattributes X Temodalities 40 x 2 80 statistical tests for each models. All p-values reported below, and while we don't 't want to make statements for single attributes, we can still notice that non-significant autribute-modalln pairs are often the same for different models.
Bell rmpariie
P-c.001
I-C
Blary reganive
One, rwothh
Yearigg IMC.
Fx 901
901
7- 361 000000
Hartche partwe
20.000
0..00 401
P+ 331
Fx 30
7x 36
Fx 831
Fx
371
901
000000
8:31
0+ 031
F-
531
Fx
300
74 00
CHANLpaPh
P9.000
631
30
F-C 301
P-: 001
S. CRick Wander partiee
F-:20 20:
7-:
331
F
301
27.000
9.6
6499
Fx XX
P4.03)
811
x- am
Dond Hereiieteeevvee
24 <81
831
Fx
531
30
221
1%
0.
Blae Syen Fmporike
1900
291
801
Str
70.001
94 331
K4:
P0000:
Px.01
100
<> 801
F- Xg
Mauth Me Co Pettr
Waug Far pocti
A4 0EL
531
1:11
6.00
C:11
Vany Hor pepdie
1.90
591
Mart' caepe 7o contart
2311
3:1
port
7
P0.00
Hot cwweteeee
040001
6- 201 FC: 30
Pk) W0:
16.00 06.00
P=.01
The Pagphe
T1 431
<- 001
Fr X0
partie
Figure 10. p-valucs associated with the KS test statistic for each (attribute, modality) pair, for areface, facenet, sphereface and adaface. We see how the the test result is significant for the majority of pairs.
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Attributes Shape the Embedding Space of Face Recognition Models
D. Image generation
3
2
Figure 11. Example of variations produced with Gan-control, row-wise: orientation, age, hair color, illumination, expression. We use subsequences of the rows projected embedding space to compute a vector field.
D.1. Gan control
In theory, all that is needed for GAN-control is an attribute classifier to train the pipeline, but we use the pretrained model and hence have access to the following attributes: pose, age, hair color, illumination and expression. These attributes are associated with specific, non overlapping subspaces of the latent space of the GAN. In addition, there are two other subspaces in the latent space: identity and extra. The extra bandwidth stores everything neither related to the identity nor the attributes, like the background. The identity subspace parametrizes the identity. Each attribute can be modified by the user and can be multidimensional (e.g. expression has 64 dimensions) not (e.g. age is single number). Figure 11 presents generation samples.
D.2. Low level augmentations
In addition the complex variations generated by GAN-control, we compute simpler augmentations: brightness, hue and image quality. A summary of the 8 types of augmentations is presented in Table 5
Attribute
Method
Start Value
End Value
Orientation (degrees)
GAN-control
-45
Age (years)
GAN-control
Hair Color (RGB)
GAN-contro
[52, 31, 10]
[215, 214, 124]
Illumination
GAN-control
Intense light from right Neutral
Intense light from left Wide Smile
Expression
GAN-control
Brightness
low level
-50
+50
Hue
low level
-4
+4
Image quality
low level
Blurring
Contrast enhancement
Table 5. Summary of augmentations of attributes and their ranges.
D.3. Generation details
We want to efficiently generate discrete curves of length L 3 in E along which a single attribute is varied, like in Figure 2. To do that we select range variation for each attribute and discretize in on K 7 values. Endpoints of this range
19

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Attributes Shape the Embedding Space of Face Recognition Models
indicated Table 5. For computational efficiency generate lattice composed of central hypercube spanning the range , K 1 and extended in dimension by _ 1 point each side. We checked that augmented images don't oon't move the embeddings far enough to change identity by analyzing id retention plots such as the ones reported on Figure 12. The distance should be compared to typical threshold value depending on the model (see Figure 5 for distances computed on LFW).
5.607
9J8
3.2
9.31
2.21
021
X21
632
5.
127
636
9.8
9:46
A00P
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9441
641
048
222
0.47
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318
11427
6:13
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5.11
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6200
0238
66-17
2311
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3.1
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Figure 12. Id retention heatmap for ArcFace: each cell represents distances measured between different variations of orientation (left) and hair color (right). Identity is fixed.
20

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Attributes Shape the Embedding Space of Face Recognition Models
E. Energy experiments
E.1. Details about the toy model
To obtain the toy model results of Figure 3, we trained 30 times a three layer MLP with hidden layer sizes 16, 16, respectively and 3-dimensional classification output. The training was performed using the Adam optimizer (Kingma Ba, 2015) with learning rate 10-4, with cross-entropy loss, dataset of 750 6-dimensional points divided in three classes shown in the main text.
The attribute vector field for each dimension is computed by finite differences by perturbing the input dimension by h 0.05 i... e f(r1......... T6).
0 (c) = - 24 56) - ( 2i ,....,,6)) ..
then normalized to obtain unit-norm the vector fields v.
E.2. Details about the FR models energy computation
We computed the energy for each FR model by sampling 10* point from the 40 synthetic identity point clouds generated by GAN-control as it is described in Appendix D) For computational efficiency, fixing every point e, we compute the inner products between its vector and set of 100 randomly sampled points at distance less than the chosen scale e. The average distance for each point cloud estimated by randomly sampling 10000 pairs of points.
The complete results are shown in Figure 13, integrated with the additonal SphereFace model (Liu et al., 2022) for comparison.
foomd
Sprcndoon
Energh
wP
Dentaer
Narbuts
Figure Energies for the pretrained models considered the main text, with additional Sphereface architecture (Liu al., 2022).
21

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Attributes Shape the Embedding Space of Face Recognition Models
F. Fine-tuning details
F.I Datasets generation: train, validation and test
We generate similar train, validation and test sets of 10K identities each. Each identity is generated first from a fixed image, which is a unconditioned generated image with GAN-Control. For each attribute, we then generate variations of the fixed image with Gan-Control and low-level augmentations. For example, fixed image that has age value of 35 will yield two other images with age values 45 and 25 and so for other attributes for total of 1 + 8 x 2 = 17 images by identity. When finetuning on an attribute a, we include in the train set only fixed images and their variations of a. ArcFace and AdaFace have parametric losses: for each train identity i, a parameter e E R' is learned during training to represent centroid of P. which allows to compute the loss in a pointwise manner. We place ourselves in the open-set scenario, therefore our validation set contains (exclusively) identities not seen during training and hence we cannot directly measure the loss. To adapt this idea on hold-out data, we choose fixed images as parameters, making the loss parameter-free. Let /, and I2 be validation identities with fixed images z and 22 and augmented images 2f and z1 Then this parameter-fre loss decreases with d(f (xf), )xt and d(f((()) f(z )) It increases when d(f (x))) decreases.
F.2 Learning curves
We give details on our fine-tumi ane-tuning strategy and monitoring.
F.2.1. LOSS CURVES
We present sample of learning curves during fine-tuning on Figure 14 and Figure 15. In each plot, an attribute is chosen and the model is fine-tuned on it. Note the agreement between the attribute fine-tuned and the loss measured datasets containing different types of augmentations.
Pot sorner
janse
5
Figure 14. Validation losses for orientation, age and brightness finetunings of ArcFace
Figure 15. Validation losses for orientation, age and brightness finetunings of AdaFace
F.2.2. ROC CURVES
We also track ROC curves along fine-tuning, both on generated data and on CelebA
22

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Attributes Shape the Embedding Space of Face Recognition Models
Orientation ROC AUC validation
ROC_AUC, Neiebe
0.980
0.2
0.552
0.5
0.59
0.7
0.961
Figure 16. Left ROC AUC curve for ArcFace resnet 18 measured on a monovariant dataset of orientation variations. We see that the model fine-tuned on orientation performs the better. Right Along finetuning performance on CelebA decreases, but stays at decent level.
F.2.3. DISTANCE CURVES
Finally, we follow distances in embedding space along fine-tuning tineituning
44
Figure 17. Evolution of distances for specific variations during finetuning of AdalFace resnet 18. We see that the model fine-tuned on the corresponding attribute has smallest distances.
F.2.4. TEST METRICS
Evaluation of the best model checkpoint on the test set yields the following sanity check results:
1. A model fine-tuned on attribute a has a ROC AUC test equal or extremely close to 1 when evaluated on images subject to variations of a.
2. A model fine-tuned on attribute has the lowest test parameter-free loss of fine- tune-tuned models when evaluated on images subject to variations of a.
F.3. Hyperparameters
For all finetunings, we always start from the same baseline, which is an ArcFace resnet 18 model trained on MSIMV3 (Deng et al., 2019a) for the first set of finetunings, and an AdaFace resnet 18 trained on VGGFace2(Cao et al., 2018) for the second one. We use ArcFace as a reference and when possible match end-of- of--ranninnggggggg configuration (the training done by the providers of the model). We train both models for a maximum of 15 epochs and we use callbacks on the validation loss get the best checkpoint, Both model have batch size of 42 identities, presenting always all images of an identity in the same batch. We let both models have the same optimizer parameters: a small learning rate 10 weight decay 5 x 10-4, momentum equal to 0.9. For ArcFace we use the default 0.5 rad for the margin while we set the learning rate of the loss to 10-3 For AdaFace, we let the default margin variability parameter, to 0.4, the concentration parameter, h to =0.333, the scale parameter, s to 64, the running batch mean coefficient to 0.01 and we set the learning rate of the loss to 10.
E44 Pipeline diagram
We show schematic representation of the microscale experiment on Figure 18.
23

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Attributes Shape the Embedding Space of Face Recognition Models
Leke
wemping
Preskred
Gavaran
Rowkuiig
Text
Figure 18. Complete pipeline of the microscale experiment.
F.5. Fine-tuned energies
We show the complete set of energies for all attributes, scales and fine-tunings in Figure 19.
Hair color
Expression
Age
Contract
94
Illmmiatioon
Brightness SeEcc
4
Huo
anglo
-5
200
Figure 19. Energy results for each attribute and each fine-tuning of ArcFace and AdaFace.
24


Reviews:
[{"review_id": "JPeMc9UsGS", "reviewer": "Reviewer_EQbt", "summary": "The paper describes a multi-scale geometric structure in embedding space created by Face Recognition (FR) models' feature embedding. The paper proposes a geometric-based approach to understand the influence of facial and image attributes to FR models. A physics-inspired alignment metric is also introduce. \nThe main findings help understand the models having some degrees of invariance across various attributes. This leading to deeper interpretability of the models' strengths and weaknesses.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "1. *\"There should be a section to conclude on the influence of attributes to both macroscale and microscale with specific examples and further analysis as suggested in the weaknesses section... Lacking of some further analysis on some attributes overlapping between macroscale and microscale analysis, e.g., hair color, age, expression.\"*\n\nThank you for the interesting comment. We don't have a detailed answer for the current work as the data type is different for the macroscale and microscale analysis (binary VS continuous); in addition, for the macroscale the attributes come from CelebA metadata, whereas for the microscale, the attributes are derived from GAN-Control; therefore, we think the mapping from both experiments is not straightforward. In general, however, the attributes in the micro and macro-scale are explored in different ways, even if they share the same name.  The microscale analysis describes the effect of *small* variations of images of the same identity, while the macroscale describes the effect of *large* variations for images belonging to different identities.\n\n2. *\"Missing reference to Fig. 1c.\"*\n\nThank you for the suggestion. We will include the missing reference in Sec. 3.1.\n\n3. *\"One potential strength of the invariance energy measure is that it could help guide the fine-tuning process to achieve a better FR model.\"*\n\nWe thank the reviewer for the comment. Indeed, one of the possibilities of making use of the proposed microscale invariance energy measure is to improve the recognition performance after measuring the invariance energy on meaningful attributes. In particular, as suggested, we can think of using the invariance measure to drive further fine-tuning on specific attributes.\n\n4. *\"Why only two FR models are chosen for the macroscale and microscale analysis in the paper? How about the other models shown in table 1?\"*\n\nFor the macroscale experiment, we filled the gap and conducted the experiment with all models mentioned, as it is computationally feasible.\nResults are reported in the updated Table 2 on this anonymized repository: https://shorturl.at/12ZFg.\nThe conclusions remain consistent: the embedding spaces' macroscale structures are most significantly influenced by attributes with the lowest intra-entropy, i.e., those most deterministically linked to an identity.\nFor the microscale experiment, we computed the energy for four models (AdaFace, FaceNet, SphereFaceR, ArcFace-ResNet18) as shown in Figure 12 of the Appendix. However, due to computational constraints, we performed fine-tuning validation only on two of the best-performing models.\nA comprehensive analysis of all combinations of losses and backbone architectures for the FR models is beyond the scope of this paper.\nWe believe the results discussed are sufficient to demonstrate that the proposed microscale and macroscale analyses are significant and valuable for exploring the embedding space through interpretable attributes.\n\n5. *\"What is the architecture of the ArcFace used in the analysis? Is it ResNet18 or ResNet50?\"*\n\nFor the macroscale, table 2 of the paper originally reported results for ArcFace with a ResNet50 backbone. The updated table 2 in the anonymized repository (same link as previous answer) now reports results for both backbones and other models. For the microscale analysis, AdaFace and ArcFace have the ResNet18 backbone.", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "Yes", "theoretical_claims": "There is no theoretical claims and proofs given in the paper. The paper defines metrics and formulas to obtain numbers for experimental analysis.", "experimental_designs_or_analyses": "Yes. The Macroscale and microscope analysis make sense as they're looking at different scale with regards of identity, global (across multiple identities) vs. local (within one identity).", "relation_to_broader_scientific_literature": "The key contributions of the paper provide some basic and metrics to be used to understand any issue of the pre-trained FR models and how fine-tuning may help to improve them.", "essential_references_not_discussed": "None", "other_strengths_and_weaknesses": "One potential strength is the invariance energy measure is that it could help guide the fine-tuning process to achieve a better FR model.\nHowever, the paper has some weaknesses/limitation as follows.\n1. Lacking of some further analysis on some attributes overlapping between macroscale and microscale analysis, e.g. hair color, age, expression. \n2. Missing a reference to a sub-figure in the paper, i.e. Fig. 1c.", "other_comments_or_suggestions": "There should be a section to conclude on the influence of attributes to both macroscale and microscale with specific examples and further analysis as suggested in the weaknesses section.", "questions_for_authors": "1. Why only two FR models are chosen for the macroscale and microscale analysis in the paper? How about the other models shown in table 1?\n2. What is the architecture of the ArcFace used in the analysis? Is it ResNet18 or ResNet50?", "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": {"ethical_review_concerns": "None"}}, {"review_id": "yxPKXLVVEh", "reviewer": "Reviewer_zixr", "summary": "This paper investigates the geometric structure of the embedding space in Face Recognition (FR) models, focusing on how human-interpretable facial and image attributes influence the learned representations. FR models, which use deep learning and contrastive losses, aim to map images of the same identity closer together in a high-dimensional space. However, the learned embeddings also encode other attributes such as hair color or image contrast, which can impact model performance and fairness. The study introduces a new physics-inspired alignment metric to analyze the dependence or invariance of FR models to these attributes.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "1. *\"The invariance energy measure is primarily evaluated on the CelebA dataset, which, while popular, may not encompass the full range of variability seen in real-world face recognition tasks.\"*\n\nWe want to clarify that we use CelebA for the macroscale experiment to compute the KS statistic on a real-world dataset, but GAN-Control is our main source of images for the invariance energy experiments.\nGAN-Control allows us to create a large number of variations of an image in a systematic and scalable way, which would not be possible with real images. \nIn particular, we obtained more than 120K images by identity, by varying over 8 continuous latent variables corresponding to the 8 attributes. We put the details in Appendix C.2 of the submitted paper.  \nStill, we acknowledge that the generated variability for each attribute is limited by the GAN-Control quality of generation.\n   \n2. *\"the study ... could further enhance the interpretability of the results by visualizing the learned embeddings in a more user-friendly way.\"*\n\nThe high-dimensional nature of embedding spaces in state-of-the-art models inherently limits the fidelity of visualizations. However, for the microscale analysis, we believe that visualizing the vector field over the embedding space, rather than just the embeddings themselves, can enhance interpretability.\nIn Figure 2c, we illustrate this concept using a toy model in a low-dimensional embedding space. Although this simplification doesn't fully capture the complexity of high-dimensional face recognition models, it provides a conceptual framework. For instance, an attribute like *contrast* might appear more \"disordered\" compared to \"head angle,\" aiding readers in understanding the underlying measure.\n\n3. *\"The study compares a few models.\"*\n\nWe kindly refer to the answer 4 given to reviewer EQbt, who raised a similar point.\n\n4. *\"multi-attribute data augmentation on the embedding space.\"*\n\nWhen creating synthetic identity point clouds, we augment the starting image with all combinations of attributes (see Appendix C.2 for details). However, when computing the invariance measure, we sample curves with \"infinitesimal\" variations along a single attribute to approximate the natural definition of the vector field.\nWhile considering multiple attributes simultaneously is intriguing, it is unclear what geometric construct would best represent this scenario. One could explore the energy of a k-vector field or average the vector fields across different attribute augmentations. Although this presents an interesting research direction, we defer this investigation to future work.\n\n5. *\"Intuitive understanding of the change in embedding space.\"*\n\nVisualizing changes in high-dimensional spaces can be challenging, even for experts. Our goal in this work is to provide quantitative insights on how embeddings behave when slightly varying an interpretable attribute.\nIf modifying an attribute shifts the embeddings in a \"random\" way, the energy will be high. Conversely, an attribute modification yielding very predictable changes in embeddings would correspond to an ordered embedding space.", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "The paper presents a comprehensive methodology for analyzing the geometric structure of the embedding space in Face Recognition (FR) models, with a focus on attribute invariance. However, there are several areas where both the methods and experiments could be improved or expanded:\n\n1. The invariance energy measure is primarily evaluated on the CelebA dataset, which, while popular, may not encompass the full range of variability seen in real-world face recognition tasks. The dataset is limited in terms of facial attributes and may not capture all the complexities of face recognition across different demographics, environments, and conditions.\n\n2. While the study emphasizes the geometric structure of the embedding space, it could further enhance the interpretability of the results by visualizing the learned embeddings in a more user-friendly way.\n\n3. The study compares a few models (FaceNet, ArcFace, AdaFace), but it would benefit from a more thorough cross-model analysis.\n\n4. While the study focuses on fine-tuning with single attribute augmentation, it would be better to fully explore the effect of multi-attribute data augmentation on the embedding space.\n\n5. Although the paper provides some insights into the geometric structure of the embedding space, it could be difficult for non-experts to intuitively understand how the changes in embedding space occur due to different attributes.", "theoretical_claims": "Yes.", "experimental_designs_or_analyses": "Yes. Please see \"Methods And Evaluation Criteria\".", "relation_to_broader_scientific_literature": "/", "essential_references_not_discussed": "None", "other_strengths_and_weaknesses": "Strengths:\n1. The writing of this paper is good, and the structure is easy to follow.\n\n2. The proposed method sounds reasonable.\n\n3. The experimental results look good.\n\nWeaknesses:\n\n1. The invariance energy measure is primarily evaluated on the CelebA dataset, which, while popular, may not encompass the full range of variability seen in real-world face recognition tasks. The dataset is limited in terms of facial attributes and may not capture all the complexities of face recognition across different demographics, environments, and conditions.\n\n2. While the study emphasizes the geometric structure of the embedding space, it could further enhance the interpretability of the results by visualizing the learned embeddings in a more user-friendly way.\n\n3. The study compares a few models (FaceNet, ArcFace, AdaFace), but it would benefit from a more thorough cross-model analysis.\n\n4. While the study focuses on fine-tuning with single attribute augmentation, it would be better to fully explore the effect of multi-attribute data augmentation on the embedding space.\n\n5. Although the paper provides some insights into the geometric structure of the embedding space, it could be difficult for non-experts to intuitively understand how the changes in embedding space occur due to different attributes.", "other_comments_or_suggestions": "none", "questions_for_authors": "none", "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": "El4UUIHSBF", "reviewer": "Reviewer_MCoe", "summary": "This paper provides comprehensive analysis of the embedding space of face recognition models. Through a geometric structure perspective, this work analyses the macroscale and microscale structures of the embedding space, and quantifies how human interpretable attributes influence the structures. Experimental results show that the proposed \"invariance measure\" can well quantify the sensitivity of FR models to different attributes, and targeted finetuning strategy guided by the \"invariance measure\" effectively brings performance gains on targeted attributes.\n\n## update after rebuttal\nThe authors have provided detailed and reasonable responses. Though it would be better if constraints of the proposed method could be relaxed and more interesting analysis could be observed, this paper in current form is ok for ICML. So, I would like to keep \"Weak accept\" rating unchanged.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "1. *\"The calculation of 'invariance measure' relies on strictly controlable face images, so that it can only be calculated and analysed on GAN-generated images currently, which may limit its' real-world applications.\"*\n\nOur work requires the ability to act meaningfully on the input space, which can be challenging. However, while GANs provide a powerful tool for generating controlled variations in complex attributes, they are not strictly necessary. Low-level data augmentation techniques, such as rotations or brightness adjustments, can also be used effectively to create variations in input data. The brightness, contrast, and hue in Fig. 4 belong to this class of attributes. Moreover, our proposed methodology is not limited to image data. It can be applied to interpretable inputs, such as tabular data (cf. our toy model validation experiment, where we don't use a GAN). In this case, sensitivity analysis methods are well-established, but our approach, specifically aimed at measuring invariances for a set of continuous attributes, could be complementary.\nFinally, transforming complex input data meaningfully remains challenging, but architectures like GAN-Control show that the ability to predict is enough to gain some controllability. Rapid advances in generative modeling may further enhance the applicability of our approach.\n\n2. *\"Unsurprising results for the invariance energy.\"*\n   \nIn Fig. 4, panels c and d are about the fine-tuning validation, so we assume your comment refers to panels a and b. The results shown in panels a and b align with expectations and, in fact, mirror findings from the macroscale experiment: they show that interpretable attributes have a greater influence on FaceNet's embedding space compared to ArcFace or AdaFace. \nHowever, the point we want to highlight is the methodology, which is different. The invariance measure allows us to analyze continuous data, for which scales are incomparable in a principled manner. For instance, it's challenging to directly compare the scale of age with that of hair color. By focusing solely on directional information, our approach provides coherent results without assuming a relation between data scales, but while, concretely, both macro- and microscale experiments use testing image sets, the tools to analyze them differ.\n\n\n3. Unrelatedly, here we provide a link to an anonymous repository https://shorturl.at/12ZFg to share updated figures and tables following the comments of other reviewers", "claims_and_evidence": "N/A", "methods_and_evaluation_criteria": "N/A", "theoretical_claims": "N/A", "experimental_designs_or_analyses": "N/A", "relation_to_broader_scientific_literature": "N/A", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "[Strengths]\n- This work provides deeper and interesting interpretability of FR embeddings from a geometric structure perspective.\n- The proposed \"invariance measure\" can well quantify the sensitivity of FR models to different attributes,  so that  can suggest new avenues for improving the robustness of FR systems. \n- This paper is well written and easy to follow.\n\n[Weaknesses]\n- The calculation of \"invariance measure\" relies on strictly controlable face images, so that it can only be calculated and analysed on GAN-generated images currently, which may limit its' real-world applications. \n- Though sounds reasonable, the observations in Figure 4 are not surprising. Analysis of sensitivity of face recognition models with respect to attributes can also be accomplished by constructing face recognition testing sets featured with specific face attributes. Some \"non-trivial\" observations through the proposed \"invariance measure\" may help highlight the contribution of this work.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "N/A", "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": "x1rUdN5LpF", "reviewer": "Reviewer_43PU", "summary": "The paper studies the organization of embeddings in face recognition task with respect to facial attributes. The study is mainly focussed on ArcFace and FaceNet models, while the study uses results on LFW and CelebA.\n\n## Update after rebuttal\nIn my initial review, most of the local issues raised were w.r.t clarity. They have been answered and while I cannot pinpoint to one as being wrong or as having not addressed completely the issues, the answers have raised the clarity only partially. The explanation are ample, with many mathematical formulation that need to be matched with the larger explanation from the paper. Thus the paper needs to be rebuild with all new information in its place to have a better understanding.\n\nOverall, I have raised my grade from 1 to 2, but am I not confident to raise it in the \"acceptance range\" because:\n - while authors have provided their arguments that FR theme is treated in NIPS and ICML, I maintain that this paper is better suited in a \"Face\" or \"Biometric\" conference. The mentioned papers are scarce and distributed over several years. In my view, a paper  would need to be very strong on a niche theme attract auditorium in ICML. \n - the clarity is improved. But now there are too many pieces on the table. I feel that I need to reread a complete paper to be certain that the paper is clear. I am still not convinced that the energy has value for a reader.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Here $E$=embedding space; $A$=attribute/s; $MS$=macroscale; $ms$=microscale \n\n1.*\"Figures too small, unreadable\"*\n\nThank you for the comment. \nAnonymous repo shows improved Figs and Tables: https://shorturl.at/12ZFg\n\n2.*\"Paper uses ArcFace and FaceNet while uses results on LFW and CelebA. Meaning of $MS$ and $ms$\"* \n\nWe point to the summaries provided by the 3 other reviewers.\nIn brief, we study how interpretable $A$s of images impact the geometry of the $E$ of FR models. \nWe clarify that we use LFW to compare the intra or inter-class distances ($d_b$, $\\overline{d}$) for various models (Table 1). \nThis motivates us to study $E$ through the lenses of $A$s at two scales (lines 137-143): $ms$ and $MS$, focusing on intra or inter-class geometry.\nThe $ms$ relates to single identity point clouds (images as points); $MS$ to the whole $E$, (identities as points).\nWe use CelebA to analyze $MS$ as it is endowed with interpretable $A$s. \nWe explore the $ms$ through mapping augmented data on $E$ according to $A$s given by the latent space of GAN-Control and other $A$s (brightness, contrast, hue).\n\n3.*\"Vague claim 1; unclear Fig 1d. The paper sends the reader to Appendix. Vague Table 2\"*\n\nWe agree to improve Sec. 3 discussing $MS$ (see anon. repo) and the Appendix.\nHowever, we clarify that the results are in the main paper: Table 2 and Fig. 1d.\nDue to space constraints, Sec. 3 shows:\n(a) How to compare the $E$ of ArcFace and FaceNet through KSdist from the lenses of binary interpretable $A$s.\n(b) How FaceNet differs from ArcFace since the KSdist are higher (see new Fig. 1d, scatterplot higher than the diagonal), showing that for FaceNet, the $E$ has higher structural dependency on the $A$s. \n(c) Significant negative correlation (Table 2) between the intra-entropy of an $A$ (= *how much the $A$ varies inside each identity point cloud*) and the KSstat (= *how much an $A$ shapes the global geometry of $E$*), confirmed by the \"bubble-size\" of the scatterplot of Fig. 1d: the more the $A$s vary within identities, the less they shape $E$ at $MS$.\n\nFig.6 analyzes only the CelebA $A$s, no mapping on $E$: the $A$s with greatest (bald, male) or lowest (mouth-slightly-open, smiling) intra-entropy from new Fig. 6b have high and low KSstat on Fig. 1d, respectively.\n\n4.*\"Claim 2: derivation and interpretation of the energy not appealing. From Sec. 4, unclear if \"lager is better\"*\n\nWe refer to Sec. 2, then Sec. 4.1 and Fig. 2c.\nSec. 2 observes that FR models achieving better identity recognition should be approximately invariant to $A$s. \nThus, in Sec. 4.1, we conjecture that an FR model has high sensitivity to an $A$ if the associated local vector-field derived from the data augmentation on the same $A$ has low energy (Fig. 2c \"Aligned\" $\\mathcal{E}=0.0$ compared to $\\mathcal{E}=0.75$ \"Unaligned\"). \nWe conclude that the model is more invariant to $A$s if the energy is higher (\"higher energy is better\").\n\n5.*\"Energy is not something that one can take away to explore their models.\"*\nOur methods rely on either (1) an attributed dataset or (2) controllable data augmentation. \nIf one wishes to diagnose how a particular $A$ is treated by a model, one needs to measure and/or control $A$. \nSince metadata labeling is expensive, controllable augmentation is the easiest way.\nAs noted by Reviewer EQbt, one of the energy usages is to guide the fine-tuning process to achieve a better FR model.\nThe energy measures the $A$s over which an FR model is less invariant. \nThus, it not only shows the model's limitations but is also a quantitative tool for investigating potential biases and providing directions for model improvement and resilience. \n\n6.*\"Paper might be better suited to FR conference. Results less relevant for the general ML community.\"*\n\nWe note that our references encompass previous works covering solely the FR domain and appearing in general ML conferences:\n(a) ICML2021 \"Larnet: Lie algebra residual network for FR\";\n(b) NeurIPS2024 \"TopoFR: A closer look at topology alignment on FR\".\nInspired by those, we suggest that FR is a significant domain for general ML development, specifically for representation learning. \nIndeed, it deals with a high-dimensional domain, where the low-dimensional manifold hypothesis has been verified (e.g., \"The intrinsic dimension of images and its impact on learning.\" ICLR2021), so that embedding projection and metric learning are meaningful approaches. \nIn addition, it motivates us to interpret $E$ because of the sensitive nature of identity recognition for face images.\nFurthermore, past works have driven a valid effort to collect images with metadata, thus enabling further analysis of $E$.\nMoreover, the hierarchical distinction of $MS$ and $ms$ can be extended to other open-set metric learning frameworks.\n\n7.*\"Distance for Table 1? Pretraining on which database?*\n\nSee Table 5 in anon. repo listing the used models.\nFaceNet with Euclidean, others with cosine.\n\n8.*\"Eq2 circle\"*\n\nComposition of functions.", "claims_and_evidence": "The paper makes relatively vague claims. They can be inferred from page 2 l88-l108, left column. They would be:\n 1. \"a procedure to check if attributes shaping the relations inter-identities are the ones most deterministically linked to an identity\"\n 2. \"an invariance energy measure to quantify the invariance of the embedding model to each attribute\"\n\nA problem with both is that steps are taken towards achieving them, but the results are not clear. \n\nFor claim 1, this is treated in section 3.2 but the results are in appendix B (outside the paper and therefore not mandatory to be read for  review). In Appendix B (where space is unlimited) we find  figure 6 (where writing and all graphical signs are too small) and without explanation. What is better? What is worse? In section 3.2 it is mentioned that there are two models yet in the appendix there is a single set of results. Some results are showed again in figure 1d, but that is again too small and it is not quite clear what is there (w.r.t what is expected).\n\nFor claim 2, the paper indeed proposes an energy, but the derivation results and interpretation are not very appealing. Following section 4, it is not clear to me if \"lager is better\" or viceversa.", "methods_and_evaluation_criteria": "The benchmarks make sense. The metrics make less sense. Alternatively, the paper might revisit the explanation and make it more clear.", "theoretical_claims": "The paper does not contribute significantly on theoretical side.", "experimental_designs_or_analyses": "The experimental design is arguable. While the benchmark and the problem make sense, the assumed methodology, results presentation and explanation provided are not fluent. The proposed energy is not something that reader can take away and used it to explore their models.", "relation_to_broader_scientific_literature": "The paper approaches a deconstruction of the model used to provide embeddings in the face recognition tasks. There are previous results within the same trend, but the precise direction of the paper is not, to my best knowledge, previously explored. \n\nOn other hand, the specific of the paper, the findings... I am not sure that ICML is the best place to present them; they seem to be more suitable in a Face dedicated conference (IEEE Face&Gesture, Biometrics, etc.)", "essential_references_not_discussed": "I believe the paper is fine on this criteria.", "other_strengths_and_weaknesses": "The paper approaches an interesting direction that definitely is worth of investigation. My concerns are the following:\n 1. instead of  ICML the paper might be better suited to \"Face\" or \"Biometrics\" dedicated conference. The significance of the results  is less relevant for the general Machine Learning community and more for the face recognition.\n 2. the paper is not clear. The method used to investigate the relevance of attributes and their impact over the mapping  providing embeddings lack clarity. On the formal aspect, the paper sends the reader too many times to inform in the appendices, but some details are omitted there, too. Figures are too small and it is not explained what should be seen. On the information, the paper chooses to use a mathematical formulation, but often fails to clarify things. For instance why do we need to define a space of a transform  of  a specific attribute, which is than simulated with a GAN?\n3. The findings are not clear. There are conclusions that try to summarize the findings, but they are vague and poorly correlated with the results (or with results presentation?)", "other_comments_or_suggestions": "I have not notice typos.", "questions_for_authors": "- What is meant by \"macroscale\" and respectively by \"microscale\"?\n\n- What distance (i.e. L2, Cosinus, etc) is used for Table 1? And the pretraining process was done on which database? This may be related to information presented in Appendix A (\"The models where the embedding space is equipped with cosine distance (AdaFace, ArcFace, SphereFaceR) show the distance in degrees\") but a plain explanation would be better.\n\n- Similar questions for table 2. The caption and text explanation make things very vague (\"we find significant negative Spearman correlations between the intra-entropies for each attribute and the previously obtained KSa (Table 2)\")\n\n- Eq (2) what is meant by \"\\circle\" between f and \\alpha?", "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": {"ethical_review_concerns": "The paper approaches the face recognition theme, which is a sensitive one, but in my view, the approach, the proposal and the finding do not raise any ethical flag."}}]

Author rebuttal:

Meta-review: This paper was reviewed by 4 experts in the field who provided detailed suggestions. The reviewers’ recommendations were somewhat divergent, with EQbt recommending Accept, reviewers zixr, and MCoe recommending Weak Accept, and reviewer 43PU recommending Weak Reject. 

The reviewers noted the paper's strengths in providing deeper interpretability of face recognition embeddings from a geometric perspective (MCoe) and proposing a novel metric (invariance measure/energy) to quantify model sensitivity to attributes, potentially guiding model improvement (EQbt, zixr, MCoe). 

The reviewers did have some concerns, however, which included issues with clarity, presentation, and the interpretation of the results, particularly the proposed energy measure (43PU, zixr). The significance or novelty of some findings and the paper's relevance to ICML were also questioned (MCoe, 43PU). The authors provided a detailed rebuttal, which helped clarify some of details, causing 43PU to raise their initial recommendation from Reject to Weak Reject.  

I read the reviews, rebuttal and feedback. I believe study of face embeddings is of value to the community, from the perspectives of both face rec. and interpretability, and accordingly side with the positive recommendation or Accept.
Decision: Accept (poster)