Mean Field Theory in Deep Metric Learning
Authors: Takuya Furusawa
Venue: ICLR 2024 poster (2024)
Keywords: deep metric learning
image retrieval
mean field theory
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
TL;DR: Application of statistical physics to deep metric learning

Abstract: In this paper, we explore the application of mean field theory, a technique from statistical physics, to deep metric learning and address the high training complexity commonly associated with conventional metric learning loss functions.
By adapting mean field theory for deep metric learning, we develop an approach to design classification-based loss functions from pair-based ones, which can be considered complementary to the proxy-based approach.
Applying the mean field theory to two pair-based loss functions, we derive two new loss functions, MeanFieldContrastive and MeanFieldClassWiseMultiSimilarity losses, with reduced training complexity.
We extensively evaluate these derived loss functions on three image-retrieval datasets and demonstrate that our loss functions outperform baseline methods in two out of the three datasets.

Published as a conference paper ICLR 2024
MEAN FIELD THEORY IN DEEP METRIC LEARNING
Takuya Furusawa ZOZO Research
1-3-22 Kioicho, Chiyoda-ku, Tokyo, Japan t kuya. furusawal zozo. com
ABSTRACT
In this paper, we explore the application of mean field theory, a technique from statistical physics, to deep metric learning and address the high training complexity commonly associated with conventional metric learning loss functions. By adapting mean field theory for deep metric learning, we develop approach to design classification-base loss functions from pair-based ones, which can be considered complementary to the proxy-based approach. Applying the mean field theory to two pair-based loss functions, derive two new loss functions, MeeaFFildCoonrrsiie with reduced training complexity. We extensively evaluate these derived loss functions three image-retrieval datasets and demonstrate that our loss functions outperform baseline methods in two of the three datasets.
and MeanFieldClass Muliimmttiiillarityy losses,
INTRODUCTION
Deep metric learning has emerged as a powerful technique for learning meaningful data representations in a variety of machine learning applications, such as image retrieval (Wang et al. 2014). face recognition (Schroff et al. 2015). and person re-idenfication (Hermans et al. (017). The primary goal deep metric learning is to provide order to the embedding space by bringing similar instances closer and pushing dissimilar ones further apart. Typically, this is achieved by optimizing a loss function that utilizes appropriate interactions based on the distance between data points. However, conventional metric learning loss functions often suffer from high training complexity scaling polynomially with the size of the training data (Schroff et al.) (2015). This challenge makes optimization of these loss functions difficult in large-scale applications and necessitates the development of sampling and mining strategies to find informative pairs.
The concept of order also plays crucial role in statistical physics, which studies the emergent behaviors of interacting many-body systems. These many-body systems exhibit various ordered phases of matter, such as solid state, magnetism, superfluidity, and supereonductivity, which cannot be predicted from their individual constituents (Anderson] 1972). While the interactions between the constituents are essential for hosting such nontrivial behaviors, they also make analyzing the systems challenging, analogous to the issue in deep metric learning.
Mean field theory (Weiss 1907) is a powerful approach for handling the challenge associated with interacting many-body systems and provides insightful framework for understanding their emergent behaviors. This theory introduces a mean field that represents the average behavior of constituent particles. The mean field is also known as the order parameter, its value helps distinguish the ordered phases. The mean field theory approximates their interactions as interactions with the mean field and significantly reduces the complexity of many-body systems.
this paper, we leverage mean field theory from statistical physics to tackle the complexity associated with deep metric learning. We develop the mean field theory for applications to loss functions in deep metric

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learning and find that mean field theory is better suited for losses without anchor concepts, as opposed to the proxy-based method introduced in Ahovshovitz-Attias et al. (2017). In this sense, it can serve as a complementary approach to the proxy-based method for designing classification-baas -ased loss functions from pair-based ones. Furthermore, we apply the mean field theory to two pair-based loss functions and propose two new loss functions with reduced training complexity. We evaluate the proposed mean field loss functions using benchmark protocol proposed in Musgravv et al. (2020a), which allows us a fair comparison with other baseline loss functions, and also using the traditional protocol utilized in Movshovitz- Attiia et al. 2017); Kim et al. (2020). The evaluation results indicate that our mean field losses surpass other methods in two out three image-retrieval datasets in the former protocol. Moreover, the latter evaluation protocol demonstrates that our losses not only exhibit performance improvements in three out of four image-retrieval datasets but also converge more rapidly compared to Archyr loss (Kim et al. 2020
The main contributions of this paper are three-fold: (1) the introduction of the mean field theory from statistical physics as tool to reduce the training complexity of pair-based loss functions based on an analogy between magnetism and deep metric learning; (2) the derivation of MeamFieldCCnttrstiiee and McanFieldClass MemnFoolClassSiiMMlllSSiimi WissMultiSimilariy losses by application of the mean field theory to two pair-based loss functions; (3) the demonstration that the derived loss functions are competitive with existing baseline losses in several datasets.
2 RELATED WORK
2.1 PAIR-BASED LOSS FUNCTIONS
Pair-based loss functions, a representative category of deep metric learning losses, leverage pairwise or triplet relationships among data points. Contrastive loss (Hadsell et al. 2006) is an early example this type, which utilizes positive and negative pairs of data and learns embeddings to place the positive pairs close and negative ones apart. Triplet loss (Weinberger & Saul, 2009) is an extension of Contrastive loss, which exploits triplets of positive, negative, and anchor data, and places an anchor embedding closer to a positive than a negative. These losses are further extended to incorporate interactions among pairs in mini-batcher to improve performance and convergence speed (Sohn 2016 Oh Song et al., 2016 Wang al. 2019a b). For instance, MultiSimilarity loss (Wang et al. (019b) takes into account multiple inter-pair relationships within a mini-batch, enabling more efficient learning of the embedding space.
However, a major drawback of the pair-based losses is their sensitivity to the choice of positive and negative pairs, which caused by the polynomial growth of the number of pairs and triplets with respect the number of training data (Schrolf et al. (015). This usually requires sophisticated sampling and mining strategies for informative pairs to improve performance and mitigate slow convergence (Schroff et al.) 2015 Shi et al. 2016, Hermans et al. 2017: Wu et al.] [017:: Yuan et al.) 2017 Harwood et al.] [017] Wang et al. 2019b). this paper, we pursue an alternative approach to reduce training complexity rather than investigate these strategies. We accomplish this by leveraging the mean field theory, concept from statistical physics.
2.2 CLASSTICATTION- LOSS FUNCTIONS
Dassification--asse loss functions utilize weight matrices and learn embeddings by optimizing a classification objective. Unlike pair- based losses, these losses do not face the complexity issue because they are computed in the same manner typical classification task.
A representative example is NormalizedSoftmax loss (Wang et al. 2017 Zhai & Wu 2018), which is obtained by the cross-entropy loss function with L2-normalized weight matrix. Its extensions include SphereFace (Liu et al. 2017). ArcFace (Deng et al. (2019), and CosFace (Wang et al. 2018a|bb b) losses obtained by modifying distance metrics and introducing margins. The losses with proxies such ProxyTriplee and
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ProxyNCA losses also belong to this category MMvssovvitz-Atias et al., 2017). Such proxy losses can be derived from corresponding pair-based losses by substituting positive and negative data points with learnable embeddings called proxies while retaining anchors. ProxyAnchor loss (Kim al. 2020) further considers interactions among samples in mini- and shows promising performance in popular public datasets, surpassing other classification-basse
and pair-based loss functions. These losses have been extended to incorporate refined structures among data, such graphs (Zhu et al. 2020), hierarchies (Yang et al. 2022).
and others (Qian et al. 2019 Teh et al., 2020; Li et al. 2022).
In this paper, develop the mean field theory
technique to derive classification--ass loss from pair-based one, addressing the challenges of the latter. Although our approach is similar to the proxy-based method, it naturally adapts pair- wair-based losses without anchors, which have remained unexplored by the proxy-based method.
3 PROPOSED APPROACH
this section, we investigate the mean field theory and its application to deep metric learning. We first review the mean field theory for ferromagnet in statistical mechanics by following standard statistical mechanics textbooks (e.g., see Nishimori & Ortiz (2010).. Next, based on the analogy between the ferromagnet and deep metric learning, we apply the mean field approximation to the Contrastive loss and variant of the MultiSimilarity loss and derive classification-type loss functions with reduced training complexity.
3.1 MEAN FIELD THEORY FOR MAGNETS
Magnetism is of the representative phenomena in statistical physics that show a phase transition between ordered and disordered phases. A ferromagnet is composed of a large number of microscopic magnetic spins and shows macroscopic magnetization when macroscopic number of the magnetic spins aligned in the same direction.
To explain the mean field theory for ferromagnet, shall consider an infinite rangee mode whose Hamiltonian (or energy) takes the following form (Nishimori & Ortiz 2010):
N
J 2N
H
Σ sJ=1
S.
S.
(1)
with the total number of spins N N and the exchange interaction > Here, S, represents the i-th constituent magnetic spin, which is regarded vector living sphere. [ indicates that state where spins point in the same direction is preferred energetically"
According to statistical mechanies, a probabilistic distribution of the spin configuration at temperature T follows the Gibbs distribution: P((S.):) Z
-//T
7 7= z= /- II -.- H/T
(2)
and macroscopic properties of this system can be computed from the normalization factor Z. However, since the spins are interacting with each other, it does not look casy compute Z both analytically and numerically.
Note that readers might worry that this model appears too simple (e.g.. it lacks notion of lattice structure). However, it is sufficient for explaining the phase transition of ferromagnets and analogous to loss functions in deep metric lcarning. "Since the Hamiltonian (or energy) of magnetic moment m in an applied magnetic field B is typically given by H = m B (c.g., Zeeman effect (Sakurai & Commins, 1995)), Eq. ① is thought of as the interaction between spin S, and the magnetic field produced by the other spins. As result, the interaction between spins described by the cosine similarity, and this allows establish an analogy between discussions in statistical physics and deep metric learning.
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To address this difficulty, we introduce the mean field theory. The central idea of the theory is to approximate the Hamiltonian ①I) such that each spin interacts with an average field generated by the rest of the spins, rather than with other spins directly, thereby ignoring their fluctuations. More concretely, it means that we
expand H with respect to fluctuations {(S; M)): using the identity, S: = M + (S; M) and ignore the second-order terms of the expansion. This operation results
JN 2
H ~ HMFT
M M JM
S.
Since HMFT does not include interaction terms between spins, one can readily compute any information from the Gibbs distribution for this Hamiltonian as follows:
e -MMF//1 /T
III S Hurr/T
PMFT ((S:))
ZMFT
(4)
ZMFT
The value of the mean field M must be determined to minimize log MFT. This condition is justified because we can show that it is equivalent to the so-called self-consistent equation
1 N
M-
Σ E[S.]
by differentiating - log ZMFT with respect to the mean field. Here, we take the expectation value over the Gibbs distribution PMrT in Eq. (5). Since the mean field approximation is based on the expansion with respect the fluctuations around the mean field, Eq. ensures the consistency of expansion.
Overall, the mean field theory is a powerful tool that allows us to describe and analyze complex systems by approximating the interactions between individual constituents with an average field generated by the rest of the system. To draw parallel between the above discussion and deep metric learning, let us consider the - 0 limit. In this limit, the original problem of computing Z becomes one of finding a spin configuration that minimizes H. This is analogous to machine learning problem that seeks optimal parameters to minimize a loss function. Then, the mean field approximation reduces the problem one of minimizing HMFT with respect to both the spins and mean field. Therefore, this observation indicates that applications of mean field theory to deep metric learning problems introduce mean fields as parameters learned to minimize their loss functions.
3.2 MEAN FIELD CONTRASTIVE LOSS
To study how the mean field theory works for loss functions deep metric learning, let us begin by applying the mean field theory to Contrastive loss for the sake of simplicity and then proceed to discuss the mean field theory for a more complicated loss function.
In the following sections, we denote training data by D = {x,, y } composed of input data æ; and its class label yi E C [1,.... [C]]. We also denote a of data in class c C as De. We extract features from the input data using a machine learning model Fo. whose learnable parameters are represented by f. This model embeds the input into a manifold M, such as I2d or 5 with d E N. We also define the distance between two embeddings, F, F' E M, as d( FF F') 2 0. For instance, the distance can be given by the cosine distance for M 5d, taking the form d(F, F') = F. F' [[]]F]/ or by the Euclidean distance for M = Rd.
Contrastive loss is one of the primitive examples in deep metric learning, which is defined as
LCont
Σ
Σ
((F((z)). Fo(z;)) -mp]
2/C|
D
LEC
IJED.
(6)
MN d(Fo((.),
2C
De
De
iEDD.JEDy
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with [x]+ max(x, 0). Here, mp (my) hyperparameter that controls distances between positive (negative) instances. Note that Eq. 6 reduces to the Hamiltonian (I when |C] 1, mp < 0, and M = 5? and requires the (((D(2) training complexity as it is parallel to the situation in Sec. 3.1
This analogy encourages us to apply the mean field theory in order to obtain simpler loss function. Since we have multiple classes here, we shall introduce mean fields (Mej and expand Connt. with respect to fluctuations around them. Note that, in contrast to the single-class case, we must impose the following conditions to constrain relative distance among the mean fields:
mx d(M,, Mc) = (cxc).
(7)
This condition means that we should explore configurations of the mean fields which minimize Loont. at the zeroth order of expansions around the mean fields. practice, we take into account these constraints softly,
In the expansion around the mean fields, we ignore all cross-product terms of the fluctuations keeping any others so that we reduce the complexity while taking into account the higher-order terms of self-interacctions. By summing over the remaining terms, obtain McanFieldContrastin (MFCont.) loss, which takes the following form:
CMFCont.
([d(Fo(z)).M.)) mp)
+
Σ
mN d d(Fo(z.).M.)..)
C
D
aEC
iED
o' (((p)c)
+ AMF
mN ddM M
|CI
where impose the constraints 7 softly by Aur > O. Note that resummation here naively produces unstable terms. {-[mx- - (d(Me, M I+ Jo,e', but they vanish, thanks to the constraints (7) We emphasize that we must minimize CMFCout. by optimizing both M. and 0, and we can readily show that the optimal mean fields satisfy M. = ΣiED, Fo(x:)/(D. at the first order of fluctuations. Note that this equation is inherently satisfied by the optimal solution. This point should be contrasted to the center loss (Wen et al. 2016), which necessitates the updating of class centers in every batch. Furthermore, in contrast to the proxy-based method, which can applied only to pair-based loss with an anchor, mean field theory applicable to wider types of pair-based loss functions as it is based on the Taylor expansions.
3.3 MEAN FIELD CLASS- WISE MULTISIMILARIT LOSS
Lastly, we consider the mean field approximation of loss function which incorporates interactions within a mini-batch similar to MultiSimilarity (Wang et al.) (019b) and ProxyAnchor (Kim al.) (0200 losses. However, the mean field approximation relies on expansions around mean fields, and thus, a loss function symmetric with respect to x; and æy (i.e., without anchors) would be more desirable for our purpose. (See the supplement for the application to a loss with an anchor.) Since most loss functions do not exhibit such a symmetric property, we propose the following loss function that satisfies these requirements:
+ EnJED (a))
LowMs
Σ cEC
log
a/C
1+ DiED..iED (L
log
C
D Del
cife
Here, 'resummation' refers to the process of transforming infinite series back into function using method such Taylor expansion. An 'unstable term' term that could potentially violate the positivity of loss function.
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pull
push
image sample
mean field
ClassWiseMutSSimilarryy
MeanFirddCCassWiisetttttiimmi
Schematic illustration of interactions in CWMS (left) and MFCWMS (right) losses. Each color indicates a class to which an embedding and a mean field belong.
Figure I:
with hyperparameters 0, > and 6 [-1, Since Eq. (9) takes similar form MultiSimilarity loss (Wang et al., 2019b) but incorporates interactions among negative samples in a class-wise manner, we refer Class WiseMuutiSimilarity (CWMS) loss.
Next, we derive the mean field counterpart of this loss function. Here, the logits in the first and second terms take forms similar to the positive and negative interactions found in Contrastive loss (6) Repeating the discussion in Sec. 3.2 we derive MeanFieldClass WisselassssMMMtiiiiiiiaagggg (MFCWMS) loss:
1+ ΣiED. ea(d(Fo(x.).M.))
MPOWWS
log
alC
Del
DEC
14 ΣiED. . +
EjED, Fo(s))
log
(10)
2
DDcl
DDel
cfce
AMP C
llog
+ e
where we also introduce the soft constraint for the mean fields to ignore unstable terms produced in the resummation | Compared to McanFieldContrastie loss, this loss function incorporates interactions of positive samples as well as those negative ones in a class-wise manner like ProxyAnchhor loss.
4 EXPERIMENTS
Let the effectiveness of the proposed mean field losses by evaluating their image-retrieval performance in several public datasets. We employ the recently proposed benchmarking scheme (Musgrave el al.) (020a) well as the traditional used in Movshovvitz-Arrn field losses and existing loss functions, such as MultiSimilarity and Proxy losses. We also explore the effect of hyperparameters on their evaluation metrics. our experiment, we use precision at 1 (P@ 1), R- R-precision (RP), and mean average precision at R (MAP@) R) evaluation metrics. In particular, we focus and RP) because it reflects ranking for retrievals and is suitable metric to evaluate the quality of the embedding space (Musgrave et al. 2020a). Note that we implement our experiments in PyTorc (Paszke et al. 2019) and utilize the PyTorch Metric Learning library (Musgrave et al. 2020bb to implement baseline models.
al. (2017): Kim al. (2020). We compare our mean
on MAP@R R in the main paper (see supplement for
correctness of the
-6) Je. simultancously. However, focusing on the region with B > 1, can casily find mean field configurations satisfying M.))) & 1 at the same time. This is enough ignore unstable terms in practice.
Rigaroussy speaking, we cannot minimize fe M..) (.)))
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4.1 DATASETS
In our experiments, we utilize four publicly available image-retrieval datasets, CUB-200-2011 (CUB) Wah et al. (011). Cars-196 (Cars) (Krause et al. 2013), Stanford Online Products (SOP) (Oh Song et al. 2016), and InShop (Liu et al. 2016). CUB comprises 11,785 images of birds categorized into 200 classes, while the Cars dataset consists of 16, 185 images of 196 car classes. In CUB, the first 100 classes (5,864 images) are used for the training dataset, and the remaining 100 classes (5.924 images) are allocated for the test dataset. Similarly, Cars split into 8,054 training images (98 classes) and 8,131 test images (98 classes). The SOP dataset contains 22,634 classes with 1 120,053 product images. The initial 11,318 classes with 59,551 images are used for training. and the remaining 11 11,316 classes with 60,502 images are allocated for testing. Lastly, the Inohop dataset features 52,712 images of 7,982 fashion products, with 25,882 images from 3,997 classes used for training and 26,830 images from 3,985 classes allocated for testing, which are further divided into query (14,218 images) and gallery (12,612 images) subsets.
4.2 IMPLEMENTATION DETAILS
As backbone embedding model Fø(x), we employ the inception network with batch normalization (BN- Inception) (loffe & Szegedy 2015), which is pretrained for the classification task on the ImageNet dataset (Rus) sakovsky et al., 2015). We reduce the embedding dimensions by inserting a fully-connecte activation functions in the first scheme and replacing its last linear layer with that of desired dimensions in the second scheme. In both cases, we apply random resized cropping and random horizontal flipping all inputs during training and only center cropping during evaluation.
layer with ReLU
In the modern benchmarking protocol, we perform 50 iterations of Bayesian optimization for hyperparameters in loss functions including the learning rate for proxies and mean fields for fair comparison. We split a dataset into train-valid (the first half classes) and test datasets (the remaining). The train-valid set was further divided into four partitions in a class disjoint manner, and we performed four-fold cross-vale validation based on the leave-one-out method in each iteration. In each cross- voos-vaalootion step, we train a model with embedding dimensions set to 1 128 and batch size set to 32 until MAP@R for the validation data converges. The Bayesian optimization aims to maximize the average of the four validation metrics. Note that we sample images so that each batih is composed of 32 classes (8 classes) and image (4 images) per class for classification-bast
spair-basel losses, and we utilize the RMSprop optimizer with learning rate 10- for the embedding model. In the test stage, we perform valos-valio again with the best hyperparameters, resulting in four embedding models. Using these models, we evaluate performance on the test dataset in the following two different ways: mean of the metrics computed from the 128-dimensional (128D) embeddings (separated) and those from 512D embeddings made of the four 128D ones (concatenated). We repeat this observation 10 times and report their average values with 95% intervals. We carry out the experiments on a single NVIDIA V100 GPU.
In contrast, in the traditional evaluation protocol, we use the predefined train-test splits described in Sec. 4.1 and train a model for up to 60 epochs with embedding dimensions 512 and batch size 128, setting the patience for early stopping to 5 to accelerate the experiments. In this case, we use AdamW optimizer (Loshchilov & Hutter (017) with the learning rate 10 for the embedding model, setting the learning rate for proxies 10 and that for mean fields to x 10-1 The hyperparameters for ProxyAnchor loss are fixed to (a, (32, 10-1), while we set (mp, m,, AMP) ) = (0.02, 0.3, 0) for MFCont. loss and (ax, B, 6, ^MF) (0.01, 80, 0.8, 0) for MFCWMS loss in default. We chose these default parameters according to the results of the Bayesian optimization and the discussion in Sec. Sec.[4.4]am and the supplement. We repeat the above procedure 10 times and report the averages of the metrics computed from the test embedding with the best MAP@R with 95% confidence intervals. The experiments on the CUB and Cars (SOP and InShop) datasets are carried out on a single NVIDIA V100 (A100) GPU. Note that we also present the results of the traditional evaluation protocol with VisionTransfoman Transformer (Dosovitskiy et al. 2020) in the supplement.
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Table I: MAP@R R obtained from the modern protocol in CUB, Cars, and SOP. We carry out test runs times and present the averaged metries along with their confidence intervals. The best result within each block is underlined, while the overall best results for all losses are highlighted in bold. ProxyAnchor loss failed to converge in SOP in our settings. See the supplement for complete results.
CUB
Cars
SOP
Loss
128D
512D
128D
512D
128D
512D
AreFace
21.5
2.2
18.3 - 0.1
27.6 0.1
4.2
47.4
CosFace
21.2
26.5 0.3
1.1
27.0 0.3
41.0
46.8
MS
2.2
2.2
0.3
27.2
4.2
46.7
MS+Miner
20.8 1 0.2
0.2
1.2
26.9
4.3
46.6 +.3
ProxyNCA
18.8 0.2
2.2
1.1
26.8
42.7
46.7
ProxyAnch.
21.7 - 0.2
0.2
19.4 0.2
26.8 0.3
Cont.
0.1
0.2
1.3
24.9 0.5
41.1 + 0.2
45.3 + 0.2 47.0 0.2
MFCont.
27.2
18.1 0.1
27.4 0.2
43.6 0.4
CWMS
21.5
26.9 + 0.3
19.3 + 0.3
27.8 + 0.3
41.5 + 0.2
45.1 + 0.2
MFCWMS
22.1 0.1
27.0 0.1
18.9 0.2
27.0 = 0.3
44.6 + 0.2
48.3 + 0.2
Table 2: MAP@R values and epochs with the best accuracies obtained using the traditional protocol on the CUB, Cars, SOP, and InShop datasets. The best result each column is underlined.
CUB
Cars
SOP
InShop
Loss
MAP@R R
Epoch
MAP@R
Epoch
MAP@R
Epoch
MAP@R
Epoch
ProxyAnch. 25.1 1 0.2 11.5 = 1.4
26.3 + 0.2 23.6 = 1.8 51.5
= 0.3 40.5 6.5
65.5 - 0.1 31.3 7.2
MFCont.
25.3 0.3
4.4
0.4
24.7 + 0.2
10.1 0.5
52.9
0.1
25.1 1.4
67.7
0.2
24.1 2.7
MFCWMS
25.3
4.7 +
0.6
24.0
0.2
8.5
0.8
52.7
+ 0.0 23.0
1.
67.5
0.4
20.6 2.1
4.3 BENCHMARK RESULTS
Based the first protocol, we study the performance of our loss functions three datasets; CUB, Cars, and SOP. We compare our loss functions with existing ones, such as Contrastive, MultiSimilarity. ArcFace, CosFace, ProXyNCA, and ProxyAnchor losses. Note that recently proposed losses with additional struc-
tures ZZheng et al.) 2021 Ko et al., 2021: Deng & Zhang 2022 Yang et al.) 2022 Li et al. 2022) are not included as they are out of our focus. The experimental results are summarized Table I (see the supplement for complete results). First, the mean field losses show better performance than their original pair-based losses in most cases, indicating that applying mean field theory not only reduces training complexity but also results in better embeddings. This is perhaps because the mean field losses can reduce the noise introduced in pairwise comparisons by the mean fields. Furthermore, the mean field losses consistently outperform other baseline methods in both separate and concatenated MAP@N R for the CUB and SOP datasets. However, in Cars, ProxyAnchor and CWMS losses show better performance than the mean field losses, which might imply the importance of interactions within batch samples in this dataset.
We also test the performance in the four datasets described Sec. 4.I following the traditional protocol. As shown in Table mean field losses outperform Anchor loss MAP@R except for the Cars dataset, which is consistent with the first experiment. The improvement in accuracy evident in the larger datasets. Besides, MFCont, and MFCWMS losses converge faster than ProxyAnchor loss in all the datasets.
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22
25
20
15
70
Wosgybeeee THE
0.7
Mesriii Eiweiiaaaaasiiiiiiii Controuted
0.8
0.9
100
25g E Embbodoing cin eiseersion
1024
Figure 2: MAP@R versus embedding dimensions in the CUB dataset.
Figure 3: MAP@ R against / and MFCWMS loss in the CUB dataset.
4.4 IMPACT OF HYPERPARAMETERS
Embedding dimensions. Since embedding dimensions are crucial hyperparameters controlling the perfor- mance of image retrieval, we investigate their effect on the accuracy (MAP@R). On the CUB dataset, we run the traditional experiments for Proxy Anchor,,,r, Cont., CWMS, MFCont., and MFCWMS losses by varying the embedding dimensions from 32 to 1024. The results monotonically increase their performance and show better performance than the baselines in most cases. Compared ProxyAnchor loss, the improvement is larger in relatively small dimensions (64, 128, and 256).
shown in Fig, 2 We find that mean field losses
3 and 8 of MFCWMS. We also explore the effect of and d of MFCWMS loss in the CUB dataset. We varied 3 from 50 to 90 and from 0.6 to 1, fixing (o, Axa) to (0.01, 0) and computed the MAP@R R for the test data. The results are summarized in Fig. 3 Fig. 3 ensures the competitive performance of the MFCWMS loss is stable against a choice of the hyperparameters. The preferred gradually decreases as d decreases.
5 CONCLUSION
this paper, we applied the mean field theory in statistical physics to Contrastive loss and ClassWiseuuliSmmilaara loss, a variant of MultiSimilarity loss (Wang et al., 2019b) without anchors, and derived MeanFieldContrastive and MennFretdClasssiiiMMuliiiiii WiseMuliiSmmilariy losses. We extensively evalu- ated the proposed loss functions and compared them with the existing baseline methods using both modern and traditional benchmark protocols. The evaluation results demonstrate that the proposed loss functions outperform the baselines in the CUB and SOP datasets in the former protocol, and in the CUB, SOP, and InShop datasets in the latter. These findings highlight the potential mean field theory as a powerful tool for simplifying and improving deep metric learning performance in various machine learning applications.
future work, it would be worthwhile to explore applications of the proposed approach to deep metric learning in the multi-label setting (Kobayashi 2023). Furthermore, since the mean field theory originally introduced to elucidate the phase transition and scaling laws in ferromagnets, it would be also interesing to apply the mean field theory explore the phase diagram of deep metric learning.
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ACKNOWLEDGMENTS
We thank Yuki Saito, Ryosuke Goto, Masanari Kimura and Yuki Hirakawa for their useful comments on our manuscript.
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A DETAILS OF MEAN FIELD THEORY
A.1 DERIVATION OF EQ.
Let us show the equivalence between minimizing - llg ZMFT and the self-consistent equation (5)
By differentiating
log ZMFT with respect to the mean field, we get
JN
N M.
Σ log / d
log ZMFT = dm
M.
jM
2T
J' S
JN
M
+
(11)
T
N
I d'
JN
M
+
Σ
ESS
N
and thus, log ZMPT/ /0M = 0 is equivalent to Eq.
(5)
A.2 SOLVING THE INFINITE-RANGE MODEL
For the infinite-range model 1 we can show that the mean field approximation becomes exact in the N - oo limit. To see this, we shall compute - log Z as follows:
-log& Z = - log log II d e See नेक Ex, = loo
/ II d S E,
-log log // II Se E,
/ d M.
12 (2TT/(JN))
- -og [f df
&See TNT Σ. E, Sie 2N M.M log / (2TTT((JN))
(12)
[f d / II & S NNT Σ, E: Sie - (M 24 ). (M- 22151)
-log|
+ +log v (2T1 (2TT/(JN)
log log [f PM
/ II d S # 8. + log V (2TT/(JN))3,
where we introduced a Gaussian integration in the second line and shifted the integration variable M by a vector Σ, S./N. Note that this trick is known the Hubbard- -traardodttt transformation in statistical physics (Hubbard 1959 Stratonovvch 1957). In the last line, the exponent of the integrand is equal to the mean field Hamiltonian (3), but the above equation differs from- log ZMFT by the integration over M.
However, the integration over M can be replaced by minimization with respect to M in the N + co limit. To see this, we first introduce
J M log / i
V
(13)
T
where the right-hand side is independent of the direction of M because the integration of S is performed over sphere. Then, we find
f dPM
/ As IN M-M+ /I M S.
f deM ff
M M (1)1))
(14)
Taking N + 0o, we see that the most contribution comes from M minimizing the integrand. Thus, Eq. log ZMFT in this limit.
reduces
A.3 DERIVATION OF MFCONT, AND MFCWMS LOSSES
understand the derivation of MFCont. and MFCWMS losses, us consider the expansion and resummation of (dd(Fo(z.) Fo (2x))) with an arbitrary function ø : R -> R and x; E De, E De. We expand this function by (ooF))

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Published conference paper at ICLR 2024
and (M. Fz(zz)), ignoring their cross terms, and obtain
d" Mc))
(d((() Mc))
Σ
OMck ... ..MMk (M. - -Fo(s;)) "..Me Fo(xi))
An 0" ("O(d(Me. Me))
(M - Fo((x))) ......... Fo(xz))k..
OM ... ..MMMe..
O" Mc.))
M(d(Me. Mc))
Σ
(M. Fo(xi))a, ..MM Fo(x,))
OMck ... -MMc..
8 (n M2))
(M - Fo((z))) ... (M - Fo(x,))
MM ... dMe. tnn
n=0 Rj
An
(odddd(MMM Mc)) o(d(Fo(a.) Mc)) ((d(MM
(15)
Here, the resummation induces the unstable term - Mcr)) ]) for e + c', which could result in non-convex loss function, but vanishes thanks the constraint (7). Note that for d' the unstable
becomes a constant since d(Mc, Mc) 0.
In the case of MFCont., o(d) is given by (d mp]+ [mn d)+, while it is given by ca(d-6) e in the case of MFCWMS. For instance, the first and second terms in MFCWMS loss TO are obtained as follows:
Σ log 1 EnJED (o(((Fo(a))) (((()
Σ LDD, ( Fo(s)))
ca(1-8)
log
+
2/Do|
2|Do|
2
(16)
Σ LiED, 8a (d.Fo
log
1
[Del
and
1+ DiEDWEED (dd 8)
Σ log
DD.
cxfe
(17)
ΣEDD e M(d( ((d.) 1 1
ΣjED, Fo(c().
Σ log
Da
DDer
cate
Note that the negative term in Eq. (10) can be ignored due to the mean field regularization.
A.4 APPLICATION TO PAIR-BASED LOSSES WITH ANCHORS AND COMPARISION WITH THE PROXY-BASED METHOD
In the main part of this paper, we focused on the application of mean field theory losses without anchors. We here apply it those with anchors, which typically take the following form:
L Σ Σ
Σ ((Fo(a)) (Fo(r))) (Fo(
RED P.D N.CD
(18)
ΣΣ
(e FLddB Folr)))
iED P.C N.CD
where Iú, P, and N; represent an anchor, a set of positive samples, and a set of negative samples, respectively. Note that we also introduced a normalized sum, faf(a) = Σ. f(a) (.,)E.. for simplicity. We then expand this loss function with respect to (M, (o())) Since Eq. (1) depends only on the distances between anchors and positive or negative samples, we must ignore cross terms of fluctuations of anchors and those of positive and negative samples. Taking the resummation over the remaining terms, we obtain
L ~~ LMPT =
Σ Σ OEC
[(Fozz,), Mc, (Mc ((t))) + ΣΣ Σ EEC PCD. NCD,D.
U(Me (Fo(z)) (Fo(z))
C
D.
LED.
(19)

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Published conference paper at ICLR 2024
Algorithm Mean field losses training procedure.
Initialize parameters 0 and mean fields M randomly.
2: for step = 1,... ,..... Nateps do
Sample a batch B = (r, yc from dataset
4: Compute LMF for batch B and classes C Update 9 and M using gradients of LMF
25-
Contrrative
Glsstiiieatt meartty
end for
10000
Tantimm
Figure 4: Accuracy (MAP@R) vs. runtime comparison on the SOP dataset.
with the constraint (1[Me, Mc, {Me [Me' (+()) = Ofeec While the proxy-based approach (Movshovitz-Atiaas et al. 2017) yields only the first term, our approach also gives us the second term, which represents interactions between mean helld and multiple positive and negative samples. This term is analogous to Proxy Anchor loss (Kim et al. 2020) and introduces interactions between samples. Furthermore, the second term comes from the expansion series with respect to the fluctuations of positive and negative data points, which are ignored in the proxy-based losses. Therefore, approximates the original pair-based loss better than the proxy-based one near the optimal solution.
expect that our mean field loss
For instance, let consider NCA loss (Goldberger et al. 2004). which takes the following form:
ΣjER e od(
CNCA - - E log
(20)
ΣjEP, e ad (oo((( + ΣkEN, e (xx)(FF(())..F((aa)))
RED PIC N.CD
Applying the above Eq. (88) leads to MeanFieldNCA loss
add(FF
LNCA LMENCA
log
C
Del
Edec (o.
CEC
iED
(21)
ΣjED, aaddM Cjdde-ooM..P Fo(oM) + ΣtEN Fod
log
PCD. NCD
Here, the first term is equal ProxyNCA loss and describes an interaction with the mean fields and each sample. On the other hand, the second term incorporates interactions among samples like ProxyAnchor and MFCWMS losses.
A.5 PSEUDOCODE AND RUNTIME
Algorithm delineates the procedure for updating the embedding model Fo(r)) and mean fields M. during training. Focusing on the T' 0) scenario as discussed in Section 3.1 we employ similar optimization strategy for our Iearnable parameters as utilized in other classification-base loss functions. This approach results in enhanced convergence efficiency, measured in runtime as well, as demonstrated in Figure 4
Lastly, let mention how optimize the partition function loog ZMFT for T >>> 0 in deep metric learning. In this case, the partition function can be written as
- log ZMFT = log / CaMP- (0ee-Grr(0..M.))11
(22)

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Published conference paper at ICLR 2024
which is similar to the objective function in the appetation-maximiiii (EM) algorithm. To facilitate optimization, we introduce distribution function Q(0) and minimize the following expression instead of log ZMrT:
M)). / d0Q(0) log
Q(0)
/ d0Q(0) log
(M.F((.M.D)))
/ZMPT (M)
log MMF (M)
(23)
Q(0)
LMPT (0,,.O.,,
log ZMFT (M) + KL Q(0) ZMF((M)
Thanks to KL(QIIP) 2 O, the lower bound for its minimum is the partition function itself. Minimizing F(Q,M) with respect to Q(0) (the E step) yields Q(0) e ( (0... MD/T /ZMFT(MMM Conversely, minimizing F(Q, M) with respect to M (the step) leads to the consisteney condition M. Eq CiED, / at the first order of the expansion. Iteratively performing the E and steps until convergence results in optimal M and ((0).
B ADDITIONAL EXPERIMENTAL RESULTS
In this section, we present the experimental results that cannot be shown in the main paper due to the page limit.
B.1 MLRC RESULTS
show the complete results of Table 1 in the main paper, which are obtained the modern benchmark protocol proposed in the "Metric Learning Reality Check" " (MLRC) paper (Musgrave et al. (020aa.. In the CUB-200-201 (CUB) dataset (Wah et al., 2011), MeanFieldContrastive (MFCont.) and MeanFieldClass MiseMutiSimillariy (MFCWMS) losses outperform the others in Mean Average Precision at R (MAP@) R) and R-Precision (RP), while ProxyAnchor loss (Kim et al. 2020) is better in Precision 1 (P@) the separated case. In contrast, in the Stanford Online Products (SOP) dataset (Oh Song et al. 2016), the MFCWMS loss shows the best performance in all the metrics.
Tables 3

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Published conference paper at ICLR 2024
Table 3: MLRC evaluation results in CUB. We carry out 10 test runs and show averaged metrics with their confidence intervals.
Separated (128D)
Concatenated (512D)
Loss
MAP@R R
P@]
RP
MAP@R
P@ 1
RP
ArcFace
21.46 + 0.13
59.98 = 0.22
32.31 + 0.14
26.39 + 0.16
67.11 + 0.23
37.23 = 0.17
CosFace
21.19 + 0.22
59.74 0.28
32.00 0.23
26.54 0.29
67.14 + 0.29
37.38 0.28
MS
20.98 + 0.16
59.38 + 0.27
31.84 1:84+0.11 0.15
+
67.34 + 0.35
36.99 0.16
MS+Miner
20.78 0.17
59.02 = 0.25
2.194
67.08 0.32
36.77 = 0.16
ProxyNCA
13.75:0.1.
57.06 = 0.27
29.64
2.28
65.60 0.28
34.82 0.25
ProxyAnch.
21.67 + 0.22
60.80 + 0.33
32.53 + 0.23
26.48 - 0.23
67.72 0.30
37.30 = 0.23
Cont.
21.02 + 0.14
59.35 = 0.33
31.80 + 0.15
26.37 + 0.18
67.67 + 0.25
37.10 0.19
MFCont.
22.01 + 0.10
60.29 0.23
32.85 0.10
27.16 0.07
67.64 + 0.27
37.95 0.07
CWMS
21.48 + 0.27
60.09 = 0.27
32.32 - 0.26
26.94 0.29
68.24 + 0.42
37.69 = 0.27
MFCWMS
22.11 + 0.08
60.28 + 0.10
32.96 + 0.08
27.03 + 0.12
67.63 + 0.21
37.83 0.12
Table 4: MLRC evaluation results in Cars. We carry out 10 test runs and show averaged metrics with their confidence intervals.
Separated (128D)
Concatenated (512D)
Loss
MAP@R
P@1 1
RP
MAP@R
P@] 1
RP
ArcFace
18.25 + 0.12
71.12 0.36
28.63 + 0.13
27.63 0.15
84.39 0.15
37.45
0.15
CosFace
18.49 + 0.13
74.66 = 0.21
28.75 0.12
26.96 + 0.25
85.29 + 0.26
36.80 0.24
MS
18.66 + 0.30
71.89 0.33
0.29
27.19
84.03 0.30
37.39 0.36
MS+Miner
18.49 + 0.23
71.99 + 0.28
0.22
26.89 0.38
$3.89 0.36
37.09 0.33
ProxyNCA
17.43 + 0.11
70.96 0.26
27.85 +.110
66.78#0.11
84.31 + 0.24
36.83 0.17
ProxyAnch.
19.44 + 0.17
76.15 + 0.25
29.89 + 0.18
26.81 + 0.27
85.53 + 0.30
36.76 0.26
Cont.
17.04 + 0.26
69.77 + 0.40
27.48 + 0.26
24.93 0.46
81.87 0.35
35.12 0.42
MFCont.
18.12 +: 0.13
71.77 + 0.28
28.54 + 0.14
27.37 + 0.18
84.56 + 0.21
37.19 0.18
CWMS
19.27 + 0.26
74.19 + 0.30
29.95 + 0.25
27.80 + 0.33
85.18 + 0.28
37.89 + 0.29
MFCWMS
18.85 + 0.16
73.02 = 0.20
29.55 + 0.15
26.98 0.31
84.00 0.22
37.11 0.27
Table 5: MLRC evaluation results in SOP. We carry out 10 test runs and show averaged metrics with their confidence intervals. We removed ProxyAnchor because it failed to converge in our settings.
Separated (128D)
Concatenated (512D)
Loss
MAP@R
P@1 1
RP
MAP@R R
P@1
RP
ArcFace
41.47 + 0.24
71.39 + 0.20
+4.35+0.22. + 0.23
47.37 + 0.23
76.13 + 0.16
50.22 + 0.22
CosFace
11.01+0.22
71.03 0.22
43.89
46.77 0.20
75.69
0.13
49.63 = 0.20
MS
41.87
71.10 0.18
45.00
46.70
75.21 0.15
49.70 + 0.17
MS+Miner
4
71.08 + 0.25
15.05+0.2.
46.57
75.09 0.19
49.57 = 0.28
ProxyNCA
0.17
71.77 - 0.08
4..72
0.133
75.24
49.61 = 0.13
Cont.
4.109
7
44.18
45.35 + 0.19
73.88 + +
48.28 + 0.19
MFCont.
43.62 0.36
72.74 = 0.29
46.55 0.35
+
75.57 - 0.16
49.85 = 0.20
CWMS
41.53 1 0.20
70.76 - 0.16
44.50 0.21
45.13 0.16
73.99 + 0.11
47.99 - 0.16
MFCWMS
44.57 + 0.16
73.32 + 0.11
47.53 + 0.16
48.33 + 0.18
76.38 + 0.14
51.17 + 0.18
Moreover, we performed the T-test for MAP@R of MFCont. and MFCWMS losses against ArcFace and Prchy losses at the 95% confidence level as shown in Tables. 6 and ) 7 MFCWMS loss shows greater performance in all scenarios in CUB and

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Published conference paper at ICLR 2024
Table 6: T-test results for MAP@R of the mean field losses compared to ArcFace loss are presented. v indicates statistical significance, while denotes non-significance. Additionally, 1 signifies that the mean field losses outperform ArcFace loss, whereas i indicates the opposite.
CUB
Cars
SOP
Loss
128D
512D
128D
512D
128D
512D
MFCont.
MFCWMS
/
Table 7: T-test results for MAP@R R of the mean field losses compared to ProxyAnchor loss are presented. v indicates statistical significance, while denotes son-significance. Additionally, t signifies that the mean field losses outperform ProxyAnchor loss, whereas J. indicates the opposite.
CUB
Cars
SOP
I Loss
128D
512D
128D
512D
128l)
512D
MFCont. MFCWMS
/ t
/
t
V
/
f
V
x f
Table 8: Impact of batch size to MAP@R for MFCWMS loss in CUB and Cars.
Table 9: Impact of batch size to MAP@R for MFCWMS loss SOP and InShop
Batch size
CUB
Cars
Batch size
SOP
InShop
30
23.3 + 0.3
22.0 + 0.3
30
51.1 + 0.1
67.4 0.2
60
24.4
0.1
0.3
60
52.0
0.2
90
23.6 0.1
90
5.14
67.6
120
0.2
2.2
120
5..
67.8
150
0.3
24.2
+ 0.3
150
52.8
0.1
67.0
180
25.4 + 0.2
24.2
0.2
300
53.0
0.3
67.0 0.4
SOP datasets. Additionally, our simpler loss function, MFCont loss, although slightly less competitive than MFCWMS loss, still holds up well against these benchmarks.
B.2 BATCH SIZE
We also investigate how batch size affects the accuracy of our models. We compare the test performance of MFCWMS loss by changing the batch size between 30 and 180 for CUB and Cars (Table and 30 and 300 for SOP and InShop (Liu et al. 2016) (Table The accuracy metric basically improves as the batch size increases. The improvement mostly saturates around 150 and 180 for the smaller datasets and around 150 and 300 for SOP. the InShop dataset, we observe that the accuracy drops around batch size 150 with relatively large variance. Note that the accuracy of ProxyAnchor loss in the InShop dataset also saturates between 60 and 150 and starts to decrease gradually for large batch sizes. Moreover, Table IT shows the MAP@R R in ProxyAncho and MFCWMS for the InShop dataset without the query-gallery split of test data. Accuracies of both losses start to decrease gradually around batch size 150, which consistent with Table 10
B.3 a OF MFCWMS
In addition 3 and of MFCWMS loss, we studied the effect of We varied a from 0.01 40, fixing (8, AMP) (80, 0.8.0 0) and computed the MAP@R and the best epoch in the CUB dataset. The result is summarized in Table. 12 and indicates the performance of MFCWMS loss glows slowly as a increases until a z 20 while the convergence speed becomes slower for larger

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24.8 + 0.2 Published conference paper at ICLR 2024
Table 10: Accuracies in InShop with the query-gallery
Table 11: Accuracies in InShop without the query-gallery
split.
split.
Batch size
ProxyAnchor
MFCWMS
Batch size
ProxyAnchor
MFCWMS
30
63.6 + 1.4
0.2
30
61.7 = 0.6
64.7 0.1
60
65.7 + 0.2
6.26
60
62.8 - 0.4
6.21
90
65.5 = 0.3
67.6
90
62.9 - 0.3
6..
120
65.6 = 0.3
67.8
120
62.9 0.3
65.0 6.5
150
65.5 0.2
67.0 + 0.6
150
62.7 0.1
300
64.5 + 0.2
67.0 + 0.4
300
61.9 0.2
64.7 0.4
500
63.3 0.2
67.1
500
60.6 = 0.2
64.60..1
Table 12: MAP@R values and epochs with the best accuracies against the hyperparamter a of MFCWMS loss in the CUB dataset.
0.01
0.1
10
20
40
MAP@R 25.3 0.3
25.3 + 0.3
25.4 0.3
25.5 + 0.3
25.5
0.2
Epoch 5.0 + 0.9
4.8 + 1.1
4.7 + 0.8
8.7
+ 1.9
14.0 + 2.1
16.7 + 4.5
Table 13: MAP R against the regularization coefficient AMP in Eq. (T) in the CUB dataset.
ÀMF
0
0.01
0.1
1
10
100
MFCont.
25.2 0.3
25.2 = 0.3
25.3
0.3
25.2 + 0.2
25.2
25.0 + 0.3
MFCWMS 25.3 + 0.2 25.3 + 0.3
25.3 + 0.3
25.3 + 0.3
25.2 + 0.3
25.1 + 0.3
B. MEAN FIELD REGULARIZATION
We also studied the impact of the regularization term in Eq. TO).. We investigated the test performance of MFCont. and MFCWMS losses in the CUB dataset varying the scale of the regularization coefficient and summarized in Table [3] The results indicate that the regularization does not offer statistically significant difference in performance. This is because the constraint (7) should be satisfied naturally when the main parts of the losses are minimized. This point can be confirmed by Figs. Fags.5a
anddsaaandSS and [b] where we plot the values of the mean field regularization terms of MFCont. and MFCWMS losses without AMF during the training in the CUB dataset. These figures show that the values become zero after two epochs even for AMP = 0. Therefore, we conclude that the constraints in MFCont. and MFCWMS losses are significant theoretically, but less relevant in practice.
B.5 LEARNING CURVES
Figure 6 (shows learning curves obtained in the traditional evaluation protocol (Movshovitz-Atiias et al., 2017 Kim et al. 2020 in fixed seeds. Both MFCont. and MFCWMS losses show faster convergence than ProxyAnchor loss well as Contrastive and CWMS losses. In the smaller datasets (CUB and Cars), accuracies of our mean field losses seem to decrease faster while don't see such behaviors in the larger datasets (SOP and InShop). This phenomenon might be caused by strong repulsive interactions with negative mean fields. For larger datasets, the embedding spaces may be sufficiently populated to balance the repulsive force, while this may not be the case for smaller datasets. might not occur for ProxyAnchor loss since repulsive forces for Proxy Anchor loss are weighted depending on distances between proxy and negative samples.

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Published conference paper at ICLR 2024
0.010
0.0025
0.1
0.1
0.008
0.0020
10
100
0.004
0.0011
0.007
0.000
0.0000
30
12
6
10 12
Epoch
Epoch
(a) MFCont.
(b) MFCWMS
Figure 5: Plot of the Mean Field regularization term of (a) MFCont. and (b) MFCWMS (AMF is not included) in CUB-200-2011. Even if AMF the constraint imposed by the Mean Field regularization satisfied in few epochs.
Table 14: MAP@R values and epochs with the best accuracies obtained using the traditional protocol with Vision TrsionTrannfommr (ViTB16) in the CUB, Cars, SOP, and InShop datasets. The best result in each column is underlined.
CUB
Cars
SOP
InShop
Loss
MAP@R
Epoch
MAP@R
Epoch
MAP@R
Epoch
MAP@R
Epoch
ProxyAnch. 30.1 + 0.5
3.1 = 0.7
27.2 + 0.2
6.6 0.8
56.8 = 0.3
25.3 = 3.6
66.4 = 0.1
31.9 - 5.9 11.9
MFCont.
2.4 0.3
26.9 : 0.1
5.6 - 0.4
8.2 58.2
68.1 0.2
MFCWMS
0.3
4.6
27.4
0.2
8.7
57.9
12.2 2.2
69.4 + 0.1
26.2 23
B.6 TRADITIONAL EVALUATION WITH VISIONTRANSFORMER
We also performed the traditional evaluation protocol replacing the BN-Inception loffe & Szegedy 2015) by VisionTransformer (VTTB16) (Dosovitskiy et al. 2020) pretrained on ImageNet (Russakovsky al.) 2015) in the four datasets described in Sec. 1.1 As shown in Table 14 MFCWMS loss outperforms ProxyAnchor loss in MAP@R R in all the datasets, and the improvement accuracy is evident in the larger datasets observed in Sec. 4.3 On the other hand, MFCWMS loss converges relatively slower in the CUB and Cars datasets. Note that we set a = 20 for MFCWMS loss these experiments.
B.7 PROXYNCA VS. MEANFIELDNCA
Lastly, also evaluated ProxyNCA and MFNCA losses in the CUB dataset in the modern benchmark protocol compare losses from the proxy-based and mean field theory approaches. Table [5 summarises the results and shows MFNCA loss outperforms ProxyNCA loss consistently. In particular, the is improved better compared the other metrics. The experimental result implies the better approximation of original loss functions discussed in Sec. A.4 is likely a key to enhancing overall performance in deep metric learning.

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Published conference paper at ICLR 2024
26
25
24
22
20-
MAP3R
18
15
16.
ProxyAncher
10
ProkyAnchor
MeanFellCCnttaastie
McanReloContrstivee
MconFicid Www.iiiddeessieeeittiiiimanng
Contrastive
Contrastive
ClasWiiluubiSmmllrrryy
2
12
14
18
18
30
42
Epoch
Epoch
(a) CUB
(b) Cars
60
50
40
M6PBR
40
35
ProxyAncher
MeonFellCCotrrstive
MeanficldClassWWie
20-
MeanReldCi
Contrastive
Contrastive
25
ClessWuuerunSimllartyy
10
ClassWieMubbSimllattyy
16
24
32
40
40
56
5
10
15
20 Epoch
23
30
35
40
Epoch
(c) SOP
(d) InShop
Figure 6: The test accuracy (MAP@R) plotted against the number of epochs for the (a) CUB, (b) Cars, (c) SOP, and (d) InShop datasets, comparing ProxyAnchor, MFCont., MFCWMS, Cont., and CWMS.
Table 15: Comparison of MLRC evaluation results of ProxyNCA and MENCA losses in the CUB dataset. We carry out 10 test runs and show averaged metrics with their confidence intervals.
128D
512D
Loss
MAP@R
P@
RP
MAP@R
P@
RP
ProxyNCA
18.8 + 0.2
57.1
+ 0.3
29.6 + 0.2
23.8 0.2
65.6 0.3
34.8 0.2
MENCA
19.0 - 0.2
57.8 = 0.3
29.9 - 0.2
24.2 0.3
66.2 0.4
35.1 + 0.3


Reviews:
[{"review_id": "TiLZ1cgHMH", "reviewer": "Reviewer_zMTp", "summary": "This paper proposes new DML losses that are inspired from the Mean-Field Theory (MFT), which is a concept from statistical physics. Specifically, the authors follow the constructive loss and the multi-class loss to implement two MFT losses. Their extensive experiments demonstrate the efficiency of the new losses on popular DML benchmarks.", "questions": "Is there any assumption or internal connection between the point distances and the spin configuration?\n\nPlease respond to the above weakness.", "limitations": "", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "The idea of introducing unique losses based on the theory of statistical physics looks interesting and novel. No prior research has taken on this particular task.\n\nThe proposed method is evaluated on several popular DML benchmarks. The authors evaluate their method on advanced MLRC metrics, making their results convincible.", "weaknesses": "My major concern is that the proposed theory does not seems to be solid when it is applied on DML task. There is not enough theoretical clue that the mean-field theory (MFT) would directly benefit the DML task compared with the proxy-based losses. The authors should provide more analysis to explain the intrinsic connection between the interaction between the magnetic spin and the similarity (distance) between the data points in DML task.\n\nThe relation and comparison between the proposed loss and proxy-based loss is still not clear. The intuition behind the MFT looks similar to the proxy-based loss, where they both compare a sample with an anchor instead of all class members. Thus, a systematic compare between it and other close related losses should be provided. \n\n\nIt seems the computation complexity to get the mean field in Eq.5 is higher than compared with the proxies. Thus, a comparison of performance and running time may be essential to be discussed. To further clarify its arithmetic progress and complexity, it is also suggested to list the pseudo-code of the loss.\n\n\nExperimental results show that the improvement in some datasets is not significant (as illustrated in table 2, figure 2). But this is not a big issue for me.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "dFCz5yFHUd", "reviewer": "Reviewer_aqYA", "summary": "This paper explores the mean field theory into metric learning  by designing two loss functions to train deep neural networks. The model's performance is evaluated on various benchmarks, including CUB, Cars, and SOP. While the paper is generally easy to follow, it lacks a sufficient level of novelty and performance improvement.", "questions": "The clarification of model novelty.\nThe performance improvement.", "limitations": "", "rating": 3, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "This paper explores the mean field theory into metric learning  by designing two loss functions to train deep neural networks. The model's performance is evaluated on various benchmarks, including CUB, Cars, and SOP, by comparing several other methods. The paper is generally easy to follow.", "weaknesses": "The major concern is that the paper lacks a sufficient level of novelty and performance improvement.\n\nFirst, they mainly explore mean filed theory into metric learning. Such metric is close to central loss [R1]. \n[R1]. Wen, Yandong, et al. \"A discriminative feature learning approach for deep face recognition.\" Computer Vision–ECCV 2016: 14th European Conference, Amsterdam, The Netherlands, October 11–14, 2016, Proceedings, Part VII 14. Springer International Publishing, 2016.\n\nSecond, the performance is not significant. In table I, compared with ArcFace and ProxyAnch, it is hard to justify the significant improvement. It is essential to do t-test.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "fnYKuslAGb", "reviewer": "Reviewer_uk7M", "summary": "The manuscript proposes two new metric learning algorithms inspired by mean-field analysis from physics. In pair-based algorithms, the loss function pushes pairs of the same class to be closer and pairs of different classes to be apart, and need to be calculated over many training pairs; in contrast, for the proposed mean-field approach, the loss pushes each sample to be close to the class mean and away from other classes' mean, and further pushing class means away from each other. The underlying technical derivation is quite general, through taking a derivating of an energy function. Thus, the same method can be applied to other cases, such as proposing a loss function using class means in a minibatch setting.", "questions": "* Is it always favourable to optimize the means `M` and the parameters `theta` together? Maybe alternating between optimisation steps would be superior in numerical terms.\n * The method seems similar to classic soft-clustering approaches, which were optimised by alternating between two optimisation steps, such as the EM algorithm. Can you make the connection more explicit?\n * Are the new methods faster or slower than other methods?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "* Regardless of any inspiration taken from physics, replacing pair-based methods with mean-based methods seems a scalable approach, well grounded in statistics.\n * The proposed class of methods is possibly prudent in the sense that they can be used to derive loss functions, taking into account mean-class information for other problems.\n * The simulations performed seem great, and the authors explain the optimisation carried out on both their method and other methods used for comparison. The results suggest the proposed class of methods achieve very competitive results. \n * The writing is clear, almost tutorial-like, and easy to follow.", "weaknesses": "* The authors hide the actual derivation in the appendix, so they do not detail enough their technical approach. On the face of it, the modified loss function could have been suggested just through statistical intuition, not derived from an energy approximation (Hubbard-Stratonovich, saddle-point approx, etc.), so it is a pity the authors don't sketch their methodology in the main text.\n * It is hinted that the method can be more efficient (e.g. due to the lack of pair sampling or anchor points choice), but it is not reported if the training time is superior to the other methods or if the optimisation over `M`-s causes a substantial overhead (which may be justifiable with the improved results).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "z4OVPAtfIT", "reviewer": "Reviewer_pubU", "summary": "The paper considers mean field theory approximations of pair-based loss functions for metric learning. Inspired by techniques in statistical physics, pairwise calculations are approximated by comparing to a mean approximation, ie, turning a summation over $i$ and $j$ to only that of $i$. Using such an approach, two different mean field contrastive loss functions are proposed. Empirically, the proposed loss functions are evaluated and are shown to even out perform their non-mean field counterparts.", "questions": "- What are the runtimes of the mean field variants compared to their regular counterparts? Does the additional complexity of requiring optimization of mean fields $\\mathbf{M}_c$ outweigh the reduction of computational complexity via the mean field approximation?\n- One of the major interesting components of the paper is that mean field approximation performs better than its non-mean field counterparts. Is there a good hypothesis for why this may be the case? Have you found a good characterization of when one out performs the other? Additional insight here would be great since the paper claims in the empirical section that the mean field losses provide better \"training complexity but also results in better embeddings\".\n- I am unsure exactly what the authors mean be \"resummation\" and \"unstable terms\". Clarification here would be great.\n- Furthermore, what do that authors mean by \"... the above discussion implies the mean field theory is independent of the concept of anchors ...\"?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The approximation technique for reducing pairwise summations to their mean field approximation intuitively makes sense. This concept is particularly well illustrated in Figure 1.\n- The proposed approach seems promising. Surprisingly, the mean field approximations perform better than their non-mean field counterparts in many circumstances.", "weaknesses": "- Part of the motivation for the approximation (not including its statistical physics analogy) was the reduction in runtime complexity pair-based loss function in metric learning. However, there is no runtime values reported in the paper.\n- Some terms in the paper are not explained. (See questions below).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Fgo9BKpaLr", "reviewer": "Reviewer_q1qw", "summary": "In this paper, the mean field theory is introduced into the domain of deep metric learning. By incorporating foundational components such as the Contrastive loss and Class Wise Multi-Similarity loss, the authors construct the Mean Field Contrastive loss and Mean Field Class Wise Multi-Similarity loss. The proposed method is evaluated through extensive experiments on benchmark datasets, covering two benchmark protocols. The results demonstrate the effectiveness of the proposed approach.", "questions": "How can the mean field theory be extended to encompass other commonly used loss functions in deep metric learning?", "limitations": "", "rating": 5, "confidence": 5, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. The authors have integrated the mean field theory into the realm of deep metric learning. \n\n2. They have introduced two pair-based loss functions, namely the Mean Field Contrastive loss and the Mean Field Class Wise Multi-Similarity loss. \n\n3. The experiments and ablation studies conducted in the paper are comprehensive.", "weaknesses": "Deep metric learning involves a range of loss functions, and it is unclear whether the mean field theory can be applied to other loss functions commonly used in this context. It would be valuable for the authors to specify under what conditions and contexts the mean field theory is applicable and offer practical guidance for its implementation in other scenarios.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (poster)