Adversarial Encoding Perturbation and Synthesis for Set Representation Auxiliary Learning
Authors: Yankai Chen
Xinni Zhang
Henry Peng Zou
Bowei He
Yangning Li
Philip S. Yu
Irwin King
Xue Liu
Venue: ICLR 2026 Poster (2026)
Keywords: Set Representation Learning
Auxiliary Learning
Adversarial Encoding Perturbation
Primary area: other topics in machine learning (i.e., none of the above)

Abstract: Sets are a fundamental data structure, and learning their vectorized representations is crucial for many computational problems. Existing methods typically focus on intra-set properties such as permutation invariance and cardinality independence. While effective at preserving basic intra-set semantics, these approaches may be insufficient in explicitly modeling inter-set correlations, which are critical for tasks requiring fine-grained comparisons between sets. In this work, we propose SRAL, a Set Representation Auxiliary Learning framework for capturing inter-set correlations that is compatible with various downstream tasks. SRAL conceptualizes sets
as high-dimensional distributions and leverages the 2-Sliced-Wasserstein distance to derive their distributional discrepancies into set representation encoding. More
importantly, we introduce a novel adversarial auxiliary learning scheme. Instead of
manipulating the input data, our method perturbs the set encoding process itself and
compels the model to be robust against worst-case perturbations through a min-max
optimization. Our theoretical analysis shows that this objective, in expectation,
directly optimizes for the set-wise Wasserstein distances, forcing the model to
learn highly discriminative representations. Comprehensive evaluations across
four downstream tasks examine SRAL’s performance relative to baseline methods,
showing consistent effectiveness in both inter-set relation-sensitive retrieval and
intra-set information-oriented processing tasks.

Published as a conference paper at ICLR 2026
ADVERSARIAL ENCODING PERTURBATION AND SYNTHESIS FOR SET REPRESENTATION AUXILI-
ARY LEARNING
Yankai Chen Xinni Zhang", Henry Peng Zou', Bowei Hel.2, Yangning Li Philip S. Yu', Irwin King". Xue Liu
MMBZUA, MMcGill University, 'The Chinese University of Hong Kong, Unniversity of Illinois Chicago, TTiinghua University
ABSTRACT
Sets are fundamental data structures, and learning their vectorized representations is crucial for many computational problems. While effective at preserving ba- sic intra-ser semantics. e.g., permutation invariance, existing methods may be insufficient in explicitly modeling inter-set correlations, which are critical for tasks requiring fine-grain grainga comparisons between sets. We propose SRAL, a Set Representation Auxiliary Learning framework for capturing inter-set correlations that is compatible with various downstream tasks. SRAL conceptualizes sets as nigh-dimensional distributions and leverages the 2-Sliced Wasserstein distance derive their distributional discrepancies into set representation encoding. More importantly, we introduce a novel adversarial auxiliary learning scheme. Instead of manipulating the input data, e.g., element dropout/addiiton, our method introduces adversarial perturbations for encoding synthesis at the feature level. Through min- max optimization, we compel the model to achieve robustness against worst-case perturbations. Our theoretical analysis shows that this objective, in expectation, directly optimizes for the set-wise Wasserstein distances, forcing the model to learn highly discriminative representations. Comprehensive evaluations across four downstream tasks examine SRAL's performance relative to baseline methods, showing consistent effectiveness in both inter-set relation-sensitive retrieval and intra-set information-orientee processing tasks.
INTRODUCTION
Set-structured data are prevalent in practice, as they represent complex data objects composed of simpler units (Vargas-Calderón, 2025). With the rapid development of machine learning techniques, leamning vectorized representations for sets is crucial (NaderiAlizade) and Singh, 2025). It not only benefits a variety of emerging applications, c.g., similar group matching in social networks (Shen al., 2012; Tang and Liu, 2022; Chen al., 2023b) and object retrieval in vector databases (Lee et al., 2019; Wang et al., 2022), but also demonstrates learning- based potential in addressing data management challenges (Hadjieleftheriou
al., 2008; Wang and He, 2019; Dong and Rekatsinas, 2018; Ge et al., 2019; Li et al., 2021; Zeakis et al., 2022; Yu et al., 2025).
Traditional algorithms for retrieving set-structured data are typically rule-based. For example, Set Similarity Join (Arasu et al., 2006; Mann et al., 2016) computes pairs of sets from data partitions to filter candidates with a given score threshold. Methods in this area design iterative verifiiciii frameworks (Deng et al., 2017: Zeakis et al., 2022) or propose mndex-basee solutions (Li et al., 2021) to improve efficiency. However, these algorithm algorithms often lack predictive capability to generalize from data to make inferences. To address this limitation, Set Representation Learning has emerged, Unlike naive approaches that merely sum element features, its fundamental goal is to learn holistic and fixed-size embedding that captures the intrinsic semantics of the whole unordered collection. This capability is essential for facilitating complex downstream tasks, for example, in E-commerce, enabling the model to interpret a product bundle cohesive semantic unit, e.g., "camping kit", rather than a loose collection of items, thereby improving recommendation accuracy. While early machine learning approaches leveraged kernel methods (Jebara et al., 2004; Gretton et al.,

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Published as a conference paper at ICLR 2026
Friindderr (Recallgeh0)
Sci STACTeR
WoASFPRR SET Foom Fassserr
Advesanial Man Stx Opistzzaion oner Set Ensoding Pettaliiions
Tesk IDA-3k (AIX)
Todk 1: Livel
M
821-1
Leter Set Conciatless
Sanila
Failurcs
Auxtlary
Laming
Takk
(Recallg220)
Task
Disiinika
Escoding Pertuiaation
Optiiizatios
Model
(AUC)
Maxaligaa
Youahu
Set Retneval
Simior Ponneres
12
BSAB-boscd
Natione
Proces Tado
(ACE)
Task Paased (ACC)
(Accali),20)
Main Tank Loennng
Figure 1: SRAL captures inter-set correlations for adversarial optimization (left); normalized ratios of second-best methods over SRAL are reported due to varying metric scales (right).
2006; Boiman et al., 2008; Muandet et al., 2012), recent deep learning methods focus on capturing diverse set cardinalities and ensuring invariance to element permutations (Xu et al., 2025; Skianis et al., 2020; Zaheer et al., 2017; Lee et al., 2019; Kim, 2022; Wang et al., 2023). They design a composition of permutation-eqquiariin neural network backhones and aggregation mechanisms to preserve set semantics that outperform traditional feature pooling techniques (Mialon et al., 2021; Murphy et al., 2018; Zhang et al., 2020). Detailed discussions are reported in Appendix B.
Despite these advancements, existing methods predominantly focus on fulfilling intra-set properties. While permutation invariance and cardinality independence are essential, these models are often insufficient for explicitly capturing the rich and complex inter-set correlations. This however is crucial for comparing set-wise similarities and differences in certain scenarios. For instance, in set retrieval tasks, identifying the nearest neighbors for query set inherently relies on a nuanced understanding of set-to-set relationships (Naderializadeh et al., 2021; Zhang et al., 2020). Similarly, in E-commerce applications, product bundles with overlapping items appeal to similar customer segments, where capturing the subtle relationships between different product sets can potentially enhance recommendation accuracy (Ma et al., 2022). Such set-wise correlation knowledge may not be naturally inherited through the mere encoding of intra-set properties, thus creating a notable gap in representation capability.
To bridge this gap, we introduce SRAL, a Set Representation Auxiliary Learning framework designed to learn representations with focus to capture inter-set correlations. Our framework formulates model optimization as a flexible auxiliary objective built upon two synergistic components. 1 First, we introduce novel set encoder grounded in optimal transport theory. By conceptualizing sets as empirical distributions, this encoder measures the set-wise distributional discrepancy using the 2-Sliced- Wasserstein metric (Lahn et al., 2025; Rabin et al., 201 I; Bonneel et al., 2015; Villani, 2009) and derives such distance information into the set embeddings. @ Second, and more critically, we propose an effective adversarial auxiliary learning scheme to forge discriminative representations by training the model to resist worst-case encoding perturbations. Specifically, our approach departs from conventional data manipulation strategies, e.g., element dropout/addition or subset sampling, by introducing adversarial perturbations directly to the set features. Our thcoretical analysis demonstrates that the learning over such perturbation is, in expectation, equivalent to optimizing for the 2-Sliced- Wasserstein distances between the underlying perturbed distributions. Then by training the model to be robust against worst-case perturbations via a min-max optimization, we compel the encoder to learn high quality set representations. As illustrated in Figure (left), these components jointly work to consolidate the fine-grained learning of inter-set correlations within a unified auxiliary framework.
To validate SRAL, conduct extensive experiments across four diverse downstream tasks. These tasks cover both inter-set relation-sensiivve retrieval applications, i.e., Learning to Rank Set Similarity and Bundle Recommendation, and intra-set information-oriente processing ones, i.c., Point Cloud Classification and Topic Set Expansion. As visualized in Figure 1 (right), SRAL not only excels its primary goal of capturing inter- set correlations but also performs well in processing intra-set information, which underscores the effectiveness and versatility of our framework.
2 PRELIMINARIES
Problem Description. Let S {S1, sets are drawn from finite universe 3 = 62.,........ }, which contains unique elements. Each element e, E is associated with a d-dimensional embedding vector 2j E (ž) 22...,, 2, In practice, depending on the availability of raw features, these embeddings z, are either initialized
Sm1 be a corpus of m sets. The elements in these
2

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using pre- pre-trained feature extractors, e.g.. word embeddings, randomly initialized to be learned from scratch. The primary objective of Set Representation Learning is to learn an encoder function that maps any set S, to a fixed-size vector embedding U, which preserves inter-set correlation information to facilitate downstream set retrieval and processing tasks. All notations are explained in Appendix C.
Take bundle recommendation concrete example. A set St represents product bundle, e.g., "camping kit", containing items such as a tent, a sleeping bag, a kettle, etc. The learning of bundle representations enables downstream systems to interpret the bundle captures the collective information of the items, rather than treating them as a loose collection of isolated products. Beyond the specific e-commerce scenario illustrated above. Set Representation Learning has found broad applications across diverse domains. These include computer vision, for aggregating multi-view images or video frames (Wang et al., 2022), bioinformatics, where proteins are modeled as sets of residues for property prediction INaderiAlizadee and Singh, 2025), and computational pathology, where whole-slide images are treated as sets of patches for cancer diagnosis (IIse et al., 2018; Carbonneau et al., 2018).
coherent semantic entity that
Distributional Distance Mcasurement. Wasserstein distance, derived from optimal transport (OT), provides a good measure for quantifying distributional distance (Tran et al., 2025; Rabin et al., 2011; Bonneel et al., 2015; Lv et al., 2024). It measures the minimum "cost" required to transform one probability distribution into another. Formally, given a probability distribution P. let the random variable follow the distribution P. i.e., X ~ P, X E Rd For the projection function 0 Rd R. pd represents the push-forward of P with 0 in one-dimensional space, defined as p (Y) P(æ 0(x) E Y)= P(O-1())) PY)- The a- o-Wasserstein distance between P and Q is defined inf D. (P,QQ) = (EPPans(
using La transport cost (Villani, 2009):
/ - ||--)())] (P()))))))(()) : a 11,
where the infimum is taken over all transport plans between P and Q. If a minimizer exists, denoted by g+, it is the solution to the OT problem. For one-dimensional distributions, closed-form transport solution for g from Q to P exists: (x() := Fp (Fo(z)), where Fq and Fp denote the cumulative distribution function (CDF) and the quantile function of P. respectively. To prevent numerical intractability for high-dimensional cases (Kolouri et al., 2019), the alternative metric of a-Sliced- Wasserstein distance has been recently studied (Rabin et al., 2011; Bonneel et al., 2015; Deshpande al., 2019):
1
= (..(P.))((A - (Da Q9)) (..(P))))
SD.
d0 a
(2)
Sd-1 denotes the unit d-dimensional hypersphere. The projection is (x)) = w To where w E Sd-1 is a unit vector in Rd. pd is the push-forward of P using ((x). This metric satisfies positive-definitensss symmetry, and triangle inequality (Kolouri et al., 2016; 2019), qualifying it for similarity/distance measurement (Yang et al., 2024; Kantorovich, 1960).
SRAL FRAMEWORK
3.1 OVERVIEW
Our SRAL framework is designed to capture inter-set correlations through the auxiliary learning objective, which seamlessly integrates with various downstream set-based tasks. The overall learning objective combines scenario-specifie main task loss LMain with our auxiliary loss L weighted by a hyper-parameter Ay. The complete objective function is:
L = LMain + LAux + 1 2-
where || is an Izzreeularizeer on all trainable parameters E to prevent over-fitting. The core of SRAL consists of two synergistic components. First, we introduce a novel set encoder based on the 2-Sliced- Wassersteisrnnnin distance, which represents sets as empirical distributions and embeds them based on their distance to a learned reference. Second, we propose a potent adversarial auxiliary learning scheme that perturbs the set features and encoding. By training the model to be robust against worst-case perturbations, this scheme forces the encoder to learn highly discriminative representations that capture fine-grained inter-set relationships.
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Macro view of set encoding
Micro view implementing 9
Fieture List Sef
Distribation Pt
Set Embedding
Reference
Reference O
SFE Modure
2
Feature List Vo
Distrbution Slicing
Optimal Transport Mapping from P." O
Figure 2: SFE module illustrations: macro and micro views of set representation encoding.
3.2 WSLLICE-WASSEEREE SET REPRESENTATION ENCODING
Our approaches begins by treating each set S; as an empirical distribution. Specifically, a set S; is represented by list of feature vectors of its elements, V, [=t, k=1 where 21, is the embedding of element G E S; We assume these feature vectors are sampled from an underlying true data distribution P., and the observed features V, define the empirical distribution P. In this work, consider P. to be valid approximation of P. and unify them as P; to simplify notation.
To learn stable and discriminative set representations, our encoder leverages the distributional distance between input set and a learnable reference distribution O. This reference design, characterized by И trainable embeddings Vo = [2h E R as a strategy shared by several previous methods (Naderializadch et al., 2021; Guo et al., 2021a; Mialon et al., 2021), serves as learnalle "origin" in the set embedding space.
3.2. SET FEATURE ENCODER (SFE)
Feature Mapping via Optimal Transport. Directly computing the Wasserstein distance between high-dimensional distributions P. and O is computational intractable. Therefore, we employ the iasserreensn distance, which circumvents this issue by slicing the high-dimensional distribu- tions into multiple one-dimensional ones. Each slice is defined by linear projection ((x) w x, where the unit vector w is uniformly sampled from the hypersphere Sd-1, This projection reduces P. and O to their one mee-imensional counterpart P. and O Recall the early introduction in $2, these sliced mne-dimensional distributions are compatible with the closed-ford solution. As stated in (Peyré et al., 2019; Kolouri et al., 2019; Naderializadeh et al., 2021; Deshpande et al., 2019), the optimal transport map g from the reference slice 0 to the input slice Py can be defined as:
9 g-(20 (z V; Fyod (F))(z)) where x E VO.
(4)
where 1 - T2. 2t,xkkm and Vg = [w" hm denote the sliced features of P and Oº, respectively, Here the CDF of 0 is: For (x: H Σ h= 5(x 2 w TT2,)) 2n., where ó returns 1 for zero input and 0 otherwise. We include the formal statement of Eq. (4) for readability in Appendix D.
[..1
For empirical distributions defined by samples, this theoretical solution can be intuitively interpreted as a rank-matching procedure. The term Fo (2²) essentially computes the rank percentile of z within the sorted values of V.g. and F Fp (-) then finds the value with the corresponding rank percentile in vi" This leads to the practical implementation g+ detailed as follows: Vz e Vg. let T(x V) be the rank of value in the ordered set V. The mapping procedure for g" is executed as:
|Sp|
gt (x0 Vi9) = arg min x'EV.
H T(x [:8)
(5)
We provide its derivation in Appendix D. The indicator function 7(-) can be pre pre-processed using the argsort function for v and the sort function for V88. Note that to accommodate cardinality differences of set features (i.e., when H + (S:), we employ linear interpolation, which effectively preserves the data continuity.
Constructing SFE Module. To avoid the infinite projections as required by theory in Eq. (2), employ a Monte Carlo approximation (Kingma et al., 2013; Metropolis et al., 1953) with R random projections. Let θ [ur denote all R sampled projection vectors. Consequently, our Set Feature Encoder (SFE) module constructs the set embedding v; by aggregating the results from all projections. For each projection w,, we implement the map git (w, 2n V,"-) for every point 2₁ in the reference set. Following these implementations with illustration in Figure 2, SFE concatenates
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along the innermost dimension to output the set embedding e, follows:
SFE(V, Vo (+) = Concal,=1..R.mt... k-1. (g (w, En V.
(6)
By capturing the inter-set distributional correlations, this SFE module provides the critical encoding process that leverage in our adversarial auxiliary learning framework.
3.3 ADVERSARIAL. SET ENCODING PERTURBATION AND OPTIMIZATION
To provide adaptability across downstream tasks, incorporate the auxiliary learning objective Aux within a self-supervised paradigm. The central idea is to produce stable representations when the constituent distribution slightly perturbed. Rather than using simple random noise, we employ an adversarial framework that compels the model to be robust against worst-case perturbations, forcing learn more informative inter-set information.
3.3.1 SELF-PERTURBATION ON SET FEATURE ENCODING
We begin by generating perturbed samples for each set. For the imput S; with features V; [S. [=,,]k=], the perturbation is constructed by adding small random noise to the element embeddings, where the norm of the noise is bounded by a hyper-parameter ..:
z = 2i,k + é/, where é is drawn from ||e||2 S n.
(7)
Based on the modification of zi,k, we generate series of perturbed distribution features, i.e., km By feeding this perturbed of elements into SFE, we obtain a perturbed set embedding oy SFE(V', Vo (). This would allow for more fine-grained simulation of feature variations, thereby providing high-quality perturbed samples for the subsequent learning process.
V = | E |d
3.3.2 ADVERSARIAL MIN-MAX OPTIMIZATION
With the perturbed embeddings, employ self- learning paradigm for optimization. To achieve this, we firstly construct two perturbed views for each set S,, yielding a pair of positive set embeddings t and v." We then implement with the InfoNCE loss (van den Oord et al., 2018) as follows:
exp( |1 - 12/12//1)
Lwd = Σ -log
(8)
Es ES exp(-llee 110
2/ 12//)))
S.ES
where db is Hlyer-paramette Typically, this loss term promotes consistency between the perturbed representations of the same set S,, while maximizing the Euclidean distance between embeddings of different sets, e-g., S; and Sy. This however raises a question: does this learning objective, which operates on perturbed embeddings, disrupt the SFE module's fundamental capability derived from the Sriced-Wassesseenn metric? Therefore, we introduce Remark I, which demonstrates that optimizing this objective is, in expectation, equivalent to optimizing the objective directly on the underlying distributional distances, thus consolidating the learning capability of our set embedding approach.
Remark 1. Let P` and P." denote perturbed distributions corresponding the input S., yielding perturbed set embeddings of and v"" S, v. SD2 denote the set database, a hyper-parameter, and the 2-Sliced- Wasserstein distance. For vS, E S, have:
-006--v00 exp(-1lo'-o"111//6 12/v)
exp(-ISSDgPP, PP) P/)
E
Σ Estes exceslxp(-1lv -2//l///)))
Es,es exp(-I|SD22P) P")112/1)
With proofs in Appendix D. Remark demonstrates that the Euclidean distance between the perturbed set embeddings is positively correlated in expectation with 2-Sliced-Wassersteed Wasserstein distance between their underlying distributions. This suggests that: by minimizing Towa the embedding space, we are implicitly optimizing for the alignment of distributional distances between sets, thereby enabling the model to capture fine-grained, distribution based inter-set correlations.
To further enhance representation robustness, go beyond merely resisting noise but actively seek "worst-case" perturbations that maximally disrupts the representation consistency. To this end, we elevate the self-supervised objective to adversarial min-max problem. Specifically, we seek an adversarial perturbation increment σ;; a is shared and applied to the perturbed features V and V?" that are generated earlier from Eq. (7). Our goal is to find "worst-case" perturbation that maximizes
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the loss in Eq. (8). Consequently, let =\t denote all trainable parameters, our final auxiliary learning objective, LAxx = max| (w, o). is to be minimized under such worst-case perturbation.
This derives the min-mat optimization problem as follows:
min max (E, a) min max ||a||
L (fog sses), where of SFE(V; + o, Vo ()). (10)
E |
However, in practice, it could be computational infeasible to exactly solve the min-max problem in Eq. (10). Therefore, employ first-order approximation (Goodfellow et al., 2015) efficiently
estimate the optimal perturbation. We linearize the loss function Lod by taking its first-order Taylor
expansion around σ 0:
(11) Maximizing this linear approximation under the norm constraint has closed-form solution where the perturbation aligned with the gradient direction. This motivates us to approximate the worst-case perturbation with a single step of gradient ascent and thus decompose the min-max problem with the following two alternating steps:
Lvd (E, (E, + VeLwa((E)e)
1. Inner Maximization to Find o. With the model parameters =\t fixed, compute the gradient of the loss with respect to small perturbation e evaluated at O:
Jr V Towd (ww
(12)
We then update the perturbation along the gradient direction to obtain initial adversarial perturbation â as Ô = Jo. where 7) is the ascent step size. To satisfy the constraint, we project this perturbation back onto the &g ball of radius
o - min
1, 111112
(13)
2. Outer Minimization to Update E. After identifying the adversarial perturbation increment o, we apply it to generate the final perturbed embeddings, e.g., 2. SFE(V' + a, Vo I)). Then, we compute the adversarial loss based on these examples, which is further combined with the main task loss Tmman- With 3 as the learning rate, the entire model's parameters E are updated via gradient descent:
ニ 3 V= V= + La + Ao E1
(14)
Through this procedure, method learns to acquire stable set representations from deliberate feature perturbation, with the following remark to formalize this intuition (proofs are in Appendix D).
Remark 2. Our min min-max optimization objective in Eq. (10) is approximately equivalent to implicit regularization of the SFE's s local Lipschitz continuity for representation stability.
Generally, method differentiates from conventional methods by adversarial perturbing the encod- ing process, and thus showcases its effectiveness and convergence efficiency in $4.3.2.
EXPERIMENTS
4.1
SETUPS
Tasks and Datasets. We conduct comprehensive experiments across four diverse downstream tasks. These tasks are selected to span two primary categories: inter-set relation sensitive tasks, i.c., Set Similarity Ranking, Bundle Recommendation, which require deep understanding of correlations between different sets, and intra-set information-oricnte tasks, i.c., Point Cloud Classification, Topic Set Expansion, which focus on processing the internal contents of single set. All tasks and datasets conducted in our experiments are in supervised setting. Due to the page limit, we supplement the detailed introduction of tasks and datasets in Appendix E.
relation-sensitive
Task 1: Learning to Rank Set Similarity. This task evaluates the model's ability to learn from known similar set pairs and predict new associations by ranking sets based on the Euclidean distance of their learned embeddings. We use two large-scale, real-world social network datasets: Friendster (Yang and Leskovee, 2015) and LIVEJ (Mislove al., 2007).
Task 2: Bundle Representation Learning for Recommendation. In this e-commerce scenario, the goal is to recommend bundles (sets of items) users. Effective bundle representation is crucial for prediction accuracy. Experiments are conducted on the Youshu (Chen et al., 2019) book bundle dataset and the NetFase (Cao et al., 2017) for music playlist recommendation.

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Published as a conference paper at ICLR 2026
Table 1: Performance comparison for Tasks I (left) and 2 (right), Best and second-best cases are highlighted. Statistically significant improvements (p < 0.05) are marked with
Task Set Similarity Learning
Friendster
LIVEJ
Model
R.6:00
N@
R 100
N80000
R:0:0
N:.20
R&
100
N8 100
Task Rundle Recommendation
SAP
72.41
68.13
85.32
12.15
79.86
77.35
86.94
85.34
Yousha
NetCase
DeepSet
SMP
70.78
68.94
81.61
72.22
69.89
78.95
75.45
76.70
74.55
86.83
79.76
K4.65
Model
R::20
100
N/620
R@
Repāet
80.63
76.56
86.49
74.92
82.15
79.63
88.41
83.12
MESPR
19.97
11.67
44.35
17.95
5.21
2.58
14.15
4.92
SAul
77:52
71,92
87.51
75:21
83.79
81.73
91.39
K5.07
DSBRee
20.46
12.01
45.34
18.12
5.51
3.04
14:76
5.14
83.18
81.25
86.15
20.53
45.58
15.38
3.11
5.12
Ser2Box
67.35
69.73
73.46
70.33
77.24
75.89
85.12
82.34
BundleNer
22:85
11.90
47.84
19.19
6.17
3.44
16.36
5.83
OTKE
79.53
73.68
86.64
79.59
81.45
79.82
87.95
85.10
BOCN
25.22
14.54
49.38
21.18
7.04
3.91
17:25
6.51
83.95
84.92
89.88
87.15
CroCURR
26.41
16.55
$1.00
23.30
7.21
4.08
18.32
6.77
PSWE
$3.05
84.26
88.59
85.77
83.52
84.61
89.45
86.67
SRAL+
26.92
52.18
23.64
4.21
7.01
FSPool
79:90
81.96
87.76
84.41
N5.36
87.17
93.07
91..9
Gui
1.9392
2.42%
0.54%
1,469
2.2258
3.19%6
1.86%
3.549)
FSW
67.23
SRAL
91.57
92.22
94.53
93.01
87.56
89.31
92.93
91.25
Gain
9.56%
9.28%
6.715%"
8.39%
2.5856
44
-0.15%
1.06%%
Task 3: Point Cloud Processing, A point cloud is a set of 3D data points representing an object's surface. This task aims to classify the object category based its point cloud representation. We utilize the standard ModelNet40 (Wu et al., 2015) benchmark dataset.
Task 4: Topic Set Expansion. Given a small seed set of keywords describing a topic, the objective is to expand this set with other semantically related keywords from a vocabulary. We use the LDA- Ik, LDA-3K, and LDA-5K datasets (Zaheer et al., 2017) with different sizes and scopes.
Implementation Configurations. For all tasks, we employ a consistent procedure for data preparation and training: partition the datasets into training and testing sets with an 8:2 ratio. The training set is then further subdivided into training and validation subsets using an 8:2 split to facilitate hyper- parameter tuning. All reported results are the average of five independent runs. Slyper-parameters and experiment configurations for reproducing are reported in Appendix E11-E.2.
Competing Methods. We compare SRAL against comprehensive list of baselines for four tasks. For the general set representation tasks, i.e., Set Similarity, Point Cloud Classification, and Topic Set Expansion, we include classic pooling methods SAP (Lin al., 2013), SMP (Lin al., 2013) and state-of-the-art deep learning models such as DeepSet (Zaheer et al., 2017), RepSet (Skianis et al., 2020), SAtt (Lee al., 2019). PoT (Guo et al., 20) Set2Box (Lee al., 2022), OTKE (Mialon et al.. 2021), DIEM (Kim, 2022), FSPool (Zhang et al., 2020), PSWE (Naderializadeh et al., 2021), and FSW (Amir and Dym. 2025). For the specialized Bundle Recommendation, we compare against established recommendation models: MFBPR (Rendle et al., 2012), DSBRec (Zaheer et al., 2017), DAM (Chen al., 2019), BundleNet (Deng et al., 2020), BGCN (Chang et al., 2020), and the state- of-the-art CrossCBR (Ma et al., 2022). For this task, we integrate SRAL into CrossCBR to enhance its bundle embedding module, denoting it as SRAL+. Detailed descriptions are in Appendix E.
4.2 EVALUATION RESULTS AND DISCUSSIONS
Task 1: Learning to Rank Set Similarity. We evaluate all models Task 1, where sets are ranked based on Euclidean distance in their embedding space. The results for Recall (R @R@ and NDCG (N@k) are summarized in Table (left) with threefold observations. O Deep set encoders, e.g., DeepSet, RepSet, SAtt, generally outperform conventional pooling methods like SAP and SMP. Moreover, general OT-based methods, e.g., PoT and OTKE, and particularly Bliced-Wassersteinn based methods such as PSWE and FSW, further elevate the overall performance. @ Our proposed SRAL consistently achieves ofate-of-the-art performance across nearly all metrics and datasets. On the Friendster dataset, SRAL improvements over the best baseline ranging from 6.71% % to 9.56%. On LIVEJ, it also demonstrates advantages, particularly capturing top-ranked items with gains of 2.58% in R @ 20 and 2.46% in N @ 20. @ The vast majority of SRAL's performance gains are statistically significant, confirmed by a Wilcoxon signed-rank test (Conover, 1999) at a 95% confidence level. For complete results, please refer to Appendix E.3.3.
Task 2: Bundle Representation Learning for Recommendation. The problem of this task is formulated with prediction function between bundle and user embeddings. Our analyses from Table I (right) yield three key findings. SRAL+ consistently outperforms baseline models on both Youshu and NetEase datasets, where the improvements are statistically significant across all metrics.
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Table 2: Performance comparison for Task 3: Point Cloud Processing.
Tisk 3: Point Cloud Processing
Backbone SAP SMP RepSet
SAll PoT Ser2Box OTKE DIEM PSWE #SPool FSW SRAL
Gain
MIP 57 37.65 86.35 83.45 85.89 85.20
82.15
85.92
85.58
86.41
85.76
86.38
86.53
+0.14%
ISAB
85,4 86.82 86.05 86.78 86.55
85.88
86.70
86.72
86.85
86.88
86,93
87.31
+0.44%*
Table 3: Performance comparison for Task 4: Topic Set Expansion.
4: Topic Set Expansion
Data
SAP
SMP DeepSet RepSet SAIl PoT Sec2Box OTKE DIEM PSWE FSPool FSW SRAL
Gain
LDA-1K 54.34 67.21
$4.98
57.32
58.55 58.94
50.59
62,95
63.58
58.36
75.67
64.56
80.94
+6.96%
LDD-3k 51.95
74.40
51,96
58.33
77.48 73.40
64,98
77.59
75.67
78.44
70.57
79.67
87,93
-0.337%
LDA.5k 51.34 80.65
52.05
61.39
74.59
75.11
65.67
72.57 76.96 78.81 71.16
80.94 86.20 +6.50%
@ We attribute this to the strength of CrossCBR backbone, which already excels at capturing user- bundle collaborative filtering signals (Rendle et al., 2012: He et al., 2017). Our SRAL+ enhances this by providing complementary signal, an explicit and semantically rich representation of the internal bundle structure. We recognize that this task ultimately aims to predict the user-bundle matching probability via learning the user-bundle interactions; therefore, this suggests that more integrated learning framework is a promising direction for future work, where both bundle semantics and collaborative patterns are co-optimized.
Task 3: Point Cloud Processing. We evaluate various set encoding methods on top of two distinct backbones: Multi-layer Perceptron (MLP) and the more advanced Induced Set Attention Block (ISAB) (Lee et al., 2019). The test accuracies are reported in Table 2. O We observe that SRAL achieves favorable results when paired with both backbones. With the standard MLP, SRAL yields a competitive accuracy of 86.53%. When the ISAB backbone is employed, the performance margin widens, and SRAL further reaches a statistically significant improvement of +0.44% over the second- best model FSW. @ While using ISAB backbone clearly improves performance for all methods, SRAL provides an additional performance lift in both settings. This indicates that SRAL effective with less impact from the backbone model selection and compatible with different feature extractors.
Task 4: Topic Set Expansion. With vocabularies encoded by word2vec (Mikolov, 2013), models classify elements based their semantic similarity to given query set. As shown in Table 3, SRAL consistently outperforms baselines in distinguishing intra-set semantics for topic expansion with AUC improvements ranging from 6.50% to 10.37%, demonstrating its effectiveness in capturing latent set semantics for classification. More importantly. in addition to its competitive performance on inter-set relation-sensitive applications (Task 1 & 2), SRAL also achieves superior results in intra-set information-orientel ones (Task 3 & 4), demonstrating its versatility as a good auxiliary representation learner.
4.3
EMPIRICAL ANALYSES OF SRAL
In this section, we delve into the design choices of SRAL through a series of empirical analyses. We use the Friendster dataset from Task as the primary testbed for these studies.
4.3.1 STUDY OF SRAL SET ENCODING DESIGN
Comparison with Other Designs. The design of our Set Feature Encoding (SFE) module is fundamentally based on the Wasserstein distance, which benefits from its implicit regularization to the embedding distance. To validate this design choice, we compare its performance against other common distributional distance metrics, namely the Kullback-Leibler (KL) Divergence, Jensen- Shannon (JS) Divergence, and the Sinkhorn Distance. Implementation details are reported in Appendix E.7. The results in Figure 3 (A) lead to two primary conclusions. ! SRAL and the Sinkhorn- based method emerge as the most competitive approaches. This highlights the overall superiority of the Wasserstein distance for this task. Furthermore, SRAL achieves the best performance, which we attribute to our novel SFE module that can capture complex set similarities more effectively than simpler methods. @ We observe that our method incurs relatively higher computational cost, i... time cost per training epoch. However, considering the performance improvement it delivers, we view this as acceptable trade-off between model effectiveness and computational efficiency.
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SML
Best Val. before carly stop
Perfmmane Metic (8gga
30
60
36
30-1
D1A
SPM
SPALL
2535 5000
7500
10000
12505 15000 17500 20100
SefoRAAN Mestown
Siee
(A) Results of varying perturbation approaches (B) Results of varying SSL losses
(C) Loss curves
Figure 4: Results of varying (A) perturbation approaches, (B) SSL losses, and (C) loss curves.
Setting R and H. We investigate the impact of Monte Carlo trials and the reference feature length H. with the results presented in Figure 3 (B). As observed, the model's performance improves as both R and increase. The performance is particularly sensitive to R. For instance, when fixing at 32, increasing R from 4 to 32 leads to significant rise in Recall @ 20 from 41 41.23% to 91.57%, which is attributed to accurate approximation distribution. But the performance improve- ment shows diminishing marginal returns larger parameter values, especially as the performance curve begins to plateau for R > 32. We also notice that varying based on fixed R provide less significant impact. Therefore, we select H = 32 and R 128 as our final configuration to strike an ideal balance between model performance and resource consumption.
RBI
the cumulative
(A) Varying distance metrics
(B) varying H
Figure 3: Results of varying (A) distance metrics and (B) R and H values.
4.3.2 STUDY OF ADVERSARIAL ENCODING PERTURBATION AND OPTIMIZATION
Perturbation Approaches. We introduce: Element-level Perturbation as random dropout or addition of set elements (DRA), and element replacement (EP). @ Set-level Perturbation as subset sampling (SS) (Yun et al., 2019), and set feature mixing (SFM) (Zhang et al., 2017). 2 Noise Injection as direct injection of noise (DIJ) into the encoded set embeddings. As shown Figure (A), both SRAL and SFM achieve competitive performance, outperforming simple data manipulation methods like EP and SS. This suggests that perturbing the set features yields more fine-grained and effective augmentations. Furthermore, compared to DJJ that directly perturbs final set embeddings, our strategy of perturbing the intermediate encoding process demonstrates superior effectiveness. Finally, while SRAL's performance is par with SFM, it is slightly more efficient (5.7min vs. 5.9min). This is because our approach perturbs individual sets, whereas SFM requires mixing multiple sets, allowing SRAL to achieve a better balance between performance and efficiency.
Implementation of Self-supervised Learning Loss. While the specific self-supervised learning (SSL) objective in Eq. (8) is not the primary focus of this paper, we investigate the flexibility of our framework with other loss functions. We tested several alternatives, including Set Triplet Loss, Soft-Nearest Neighbors Loss (Frosst al., 2019), and Barlow Twins Loss (Zbontar et al., 2021). The detailed formulations of these objectives can be found in Appendix E.7. As presented in Figure 4 (B), the results demonstrate that our framework compatible with various SSL objective functions, as these variants achieve competitive performance.
Convergence Analysis of Adversarial Encoding Optimization. Figure (C) presents the loss curves to illustrate model convergence. Although SRAL may have a higher complexity, compared to its ablated version without adversarial learning ("w/o AL"), we observe that SRAL converges faster and deeper in practice, reaching its best validation performance early in training. The baseline method requires a longer training duration and exhibits greater volatility. This experiment demon- strates SRAL's practical efficiency, effectively mitigating potential concerns about its computational overhead due to its rapid convergence. A more detailed analysis model scalability is provided in Appendix E.8.

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Table 4: Ablation study.
Variant
Task 1
Task 2
Task 3
Task
R 20
N@P0
R@
N@20
M-ACC
I-ACC
AUC
w/o SFE
67.02 (-26.81%) 69.38 (-24.77%) 25.31 (-5.98%)
15.14 (-10.68%) 66.45 (-23.20%)
72.48 (-16.99%) 73.39 (-16.54%)
w/o LI
75.45 (-17.60%) 74.29 (-19.44%) 25.96 (-3,57%)
15.83 (-6.61%) 74.77 (-13.59%)
79.75 (-8.66%)
72.44 (-17.62%)
w/o AEPO
77.13 (-15.77%) 79.42 (-13.88%) 26.22 (-2.60%)
16.38 (-3.36%) 86.56 (+0.03%)
86.46 (-0.97%) 66.21 (-24.70%)
w/o AI.
87.38 (-4.58%) 88.86 (-3.64%) 26,94
(+0.07%%)
16.97 (+0.115)
86.37 (-0.18%)
87.27 (-0.045)
83.53 (-5.005)
SRAL
91.57
92.22
26.92
16.95
86.53
87.31
87.93
4.3.3
ABLATION STUDY
We evaluate several ablation variants across four tasks. Due space constraints, report results for Friendster, Youshu, LDA-3k on Tasks 3, @R@21 20 and N @ 20) and Task 4 (AUC), while Task reports accuracy with MLP and ISAB backbones (M-ACC and I-ACC). From Table that: ! replacing SFE module with mean-pooling ("w/o SFE") results in substantial performance degradation despite the auxiliary learning with our encoding perturbation. This clearly validates the effectiveness and necessity of our SFE for capturing complex set features. @ Substituting linear interpolation in Eq. (3) with a two-layer MLP ("w/o L!"") yields less reliable dimension completion compared to our implementation. O Variant "w/o AEPO" retains the auxiliary learning objective but disables the adversarial optimization step. Specifically, we utilize the inner InfoNCE loss but remove the min-max strategy that generates worst-case perturbations via gradient ascent. The results show that it negatively impacts performance, particularly on Task 4 where the AUC drops by 24.70%. In contrast, removing the entire auxiliary learning ("w/o AL") by training the model solely with the main task supervision, results in a marginal performance improvement for Task 2.
observe
We attribute this to the specific nature of SRAL+, where its backbone model CrossCBR's graph structure is already highly optimized for capturing user- user-bundie bundle collaborative signals. Our adversarial learning, which focuses on the bundle-side representation by modeling inter-bundle relationships, may provide limited complementary information for interaction prediction goal. This suggests a more integrated framework for this interaction-centric scenario.
CONCLUSION AND FUTURE WORK
We introduced SRAL, an effective auxiliary learning framework for Set Representation Learning tha is compatible with set-based retrieval and processing problems. We first introduce a set encoder based distance, which effectively captures distributional discrepancies between sets. We then propose an adversarial learning paradigm that strengthens representations by generating and optimizing worst-case perturbations to set features and encoding. Experiments demonstrate the performance superiority of SRAL over competing methods and the efficacy of its constituent design components. promising future direction to investigate the integration of SRAL with approximate nearest neighbor search algorithms, such as set-vector index construction algorithm and index-based search approaches (Johnson et al., 2019), for efficient online retrieval settings. Another direction is to explore the synergy with Large Language Models (LLMs), such as utilizing context-aware LLM representations as element features, or employing SRAL as a structural adapter to generate compact soft prompts for LLM-basee set reasoning.
on the
ACKNOWLEDGMENTS
This work was supported in part by the Research Grants Council of the Hong Kong Special Admin- istrative Region, China (CUHK 2410072, RGC R1015-23), (CUHK 2300246, RGC C1043-24G), (CUHK 7010870), and by NSF under grants III-2106758 and POSE-2346158.
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A THE USE OF LARGE LANGUAGE MODELS (LLMS)
In preparing this manuscript, we employed Large Language Model (LLM) as an assistive tool mainly for refining prose for grammar, clarity, and conciseness. The LLM did not contribute to research conception, methodological development, result analysis, or scientific conclusions. All content, including final text and code, was thoroughly reviewed, edited, and validated by the authors, who retain full responsibility for the work's accuracy and integrity.
B EXTENDED RELATED WORK
Set Representation Learning. Traditional set-based problems, c.g., set similarity join and search, have long relied on rule-based algorithms to process set data efficiently (Arasu et al., 2006; Mann et al., 2016; Li et al., 2021; Deng et al., 2017; Zeakis et al., 2022). To enable predictive modeling, early learning-based approaches often adopt kernel methods for sets or distributions (Jebara et al., 2004; Gretton al., 2006; Boiman et al.. 2008; Muandet al., 2012). More recently, a large body of work learns permutation-invarian mappings via pooling or attention mechanisms (Zaheer et al., 2017; Murphy al., 2018; Skianis et al., 2020; Lee et al., 2019; Zhang et al., 2019; 2020). While some models learn an optimal permutation for prediction (Zhang et al., 2019; Rezatofighi et al., 2018), others define a canonical ordering through porting-basedd operators (Zhang et al., 2020). Set Transformer leverages Transformer-styl attention for permutation-invariant set modeling (Lee et al., 2019; Vaswani et al., 2017), and Perceiver further designs attention- pooling where cross- attention acts the permutation-invariiat aggregator (Jaegle et al., 2021). Beyond pooling/attention. trainable optimal-trans
embeddings have also been explored for feature aggregation (Mialon et al., 2021). Other representation families include similarity-preserving preserving box embeddings for sets (Lee et al., 2022) and measure- theurrett compact fuzzy set representations (Xu et al., 2025).
RepSet is among the earliest approaches that explicitly incorporates optimal transport / matching structure into set representation learning by solving a bipartite matching problem between input elements and learnable hidden sets (Skianis et al., 2020). However, unlike this nigh-dimensional exact matching formulation, our SRAL employs -SSlicd-Wassersee bottleneck via efficient approximation (Bonneel et al., 2015; Kolouri al., 2019; Chen et al., 2024). Recent models also consider meta-learming settings (Guo et al., 2021a; Lee et al., 2023), and pooling by sliced-Wasserste Wasserstein embedding has shown strong empirical performance (Naderializadeh et al., 2021). Amir and Dym propose the Fourier Wasserstenn (FSW) embedding to provide injective or bi-L.isschttz representations for multisets and measures (Amir and Dym, 2025). In addition, quantum deep set modeling has been investigated Pargas-Calderón 2025), and aggregation has been applied to residue-level protein language model embeddings (NaderiAlizadeh and Singh, 2025).
metric to bypass the computational
Self-supervised Learning. Self-supervised learning generates augmented samples and derives supervision signals guide representation learning. Contrastive frameworks encourage similar views to cluster while pushing dissimilar ones apart (Chen et al., 2020c; He et al., 2020; Chen et al., b020a;b; Dwibedi et al., 2021; He et al., 2025; Zhang et al., 2024). Non-contrastive methods meaningful representations without feature collapse; for example, BYL minimizes positive-pair distances with momentum encoder (Grill et al., 2020), and Barlow Twins reduces redundancy by optimizing cross-correlation statistics between views (Zbontar et al., 2021). While most approaches manipulate input data, recent work modifies latent representations directly (Yüksel et al., 2021; Yu et al., 2022). Our work investigates perturbing the set encoding process, providing theoretical guarantees and outperforming existing augmentation designs.
also learn
Distributional Distance Measurement. Common distributional discrepancies include Kullback- Leibler divergence (Kullback and Leibler, 1951), Jensen-Shannon divergence (Endres and Schindelin, 2003), and Hellinger distance (Hellinger, 1909). The Wasserstein metric (Kantorovich, 1960; Villani, 2009), due to its strong theoretical properties, has gained increasing attention in modern machine learning (Arjovsky et al., 2017: Tolstikhin et al., 2018). However, exact Wasserstein computation is often intractable in high dimensions. Optimization-ban based approximations exist, such as entropic regularization (Sinkhorn) (Cuturi, 2013) and convolutional Wasserstein distances geometric domains (Solomon et al., 2015). To further improve efficiency, sliced- Wasserstein distances project high-dimensional distributions onto multiple one-dimensional subspaces and exploit foosdd-frrm
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solutions (Bonneel et al., 2015; Kolouri al., 2019; Chen et al., 2023a). This line has been applied across domains, including sliced- Wasserstein kernels and flows (Kolouri et al., 2016; Liutkus et al., 2019), and also underpins method for modeling inter-set distributional similarities (Naderializadch et al., 2021; ; Peyré et al., 2019).
C NOTATION EXPLANATIONS
We explain the key notations in Table 1.
D THEORETICAL PROOFS
Complete Formulation for Eq. (4). Let V." [w |wT2,,t 21, and Vi [w BuTel>K> denote the sliced features of P.P and 09, respectively. For any projected input x. E Vg. the OT solver is formulated as:
g (x
V ppp
(F((Fo(('))
Proof. The transport distance, one candidate plan for the t-Wasserstein distance (Eq,(1) in Section 2), can be computed as
Distance(g*)) = (/ 11 (x0 |1 (09(2))) 1/a
/ 11000 (Fo d0))))))))))))))))))))))))))))) 1/a
(IFdd - Fudl = -D P!",,
(2)
which equals Da(PP 08) by symmetry. This proves to be the optimal distance for the two slices.
Mapping Procedure for g+. Let T(= | V) be the rank of value x! the ordered set V. The
mapping procedure for gt is executed as follows:
arg min (r(Z' Vi = T(=" (()),
= [S,) arg min (r(a' Vi?) N [S.1 7(7" V8)), if H + |S]]. d'EV!
g* (x0) [1:0
ø'eV."
(3)
Proof. The empirical distributions of Vi" and Vgg e.g.,
1
For (x=")
H Σ 8(x w= 21),
(4)
are monotonically increasing. If H = |Sa), we can firstly modify the original form of the optimal
transport map Fpo Fox (x.)) to:
g+ (x
Vi9) arg min (Frę (x²)=)) where y Foe (x²). z'EV.
=
(5)
7(x" vig) is the ranking of each input z0 in the ascending sorting of v and then we can quantitatively replace the term Fpp (+):
gt (x [1,0 = arg min (7(x' k = 7 [V8) if H = |Sp.
(6)
eVVV'
If H * J.,). in this work applies the linear interpolation to the sorted positions. Therefore, the interpolation point complete the proof.
is set
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Table I: Notation Table.
Notation
Description
Basic Sets and Elements
& {er, en}
All elements. A corpus sets.
1S1,-. Sm
{ei, E E 19
Set St containing |S.| elements. Features for the input set S..
The final vector embedding for the set S..
Distributions and Optimal Transport
P.
The underlying distribution of elements for set S,.
0
A trainable reference distribution.
Vo = |2n. E B
Features for the reference distribution O.
P.
The push-forward distribution of P by function g. The Wasserstein distance between distributions P and Q. The a-Sliced- Wasserstein Distance between /' and Q.
()(P,Q) SD.(P.Q)
g*
The optimal transport plan.
ga-1
The unit d-dimensional hypersphere.
Set Feature Encoder (SFE)
((x)
w
A linear projection function parameterized by gd-1 ID distributions obtained by projecting Pt and O using 6.
p
0
V,
ww
ki,
15
Sliced features of P.P
V8
w 2b
Sliced features of O
Fr(-)
The Cumulative Distribution Function (CDF) of distribution P.
Fp
The quantile function (inverse CDF) of distribution P. The rank of projected value _ within the sorted values of V The optimal transport solver from reference slice O to input slice P.
7(
V8)
g **(x" V!")
SFE(V, Vo ())
The Set Feature Encoder function.
Adversarial Perturhation and Optimization
Pl P
Two perturbed distributions generated from the input St. Perturbed feature matrix of set St.
V
o of'
A pair of perturbed set emboddings for set St.
Random noise vector added to an element embedding.
The adversarial perturbation increment vector.
6
The initial adversarial perturbation before projection.
g-
The gradient the loss respect to perturbation.
of
The final adversarially perturbed set embedding.
Losses Hyperparameters
IMain
The main loss for the downstream task.
LAux
The auxiliary loss from our framework.
Lea
The InfoNCE- loss for self supersupe learning.
E
The of all trainable model parameters.
R, 1
Number of distribution slices and feature size of reference distribution.
The norm constraint (radius) for perturbations.
y) B
Step size inner maximization The temperature hyperparameter for the InfoNCE loss.
learning rate for outer minimization.
\1, Aa
Hyperparameters to halance the main loss, auxiliary loss, and regularizer.
Remark 1. Let Pl P" denote two perturbed distributions for input set S., yielding perturbed embeddings v/ v.f Let S be the set database, vb a hyperparameter, and SD2 the 2-Sliced Wasserstein distance. For all S, S.
exp( 110/ - -2(1//10)
Oxp(-I1SD((P) SyES (PP
E
SyES expl -111/ -1/1/11
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Proof of Remark 1. Let of denote the intermediate embedding derived by the function ,, || denotes the concatenation operation. Firstly, after setting a = 2, we have:
11 u2ll2 ll2 x
(w (i - g (w, 2n vighn))
R
H
k=1
F
(Foer (1x, 2n)) F Fpphor (Foe (2, 2/h plu
(8)
R
The inner summation is substituted by recognizing the relationship between the sample mean and the integral for an empirical distribution. Given an empirical distribution P(z) 1 Σ 5(r 2 Xx). the sample mean equals the integral: # Σ f(X,) = f(x) dP(x). Then we apply this to the inner summation, with t = w. 2n being a sample from O"::
1
plo (Fou. -F (Foe, (1))
F piio,
(9)
H
= =/
(F (Foe. (t)) -Fpjee (t)) d0%
(10)
For the outer summation, R co, we apply the Law of Large Numbers for integration over the hypersphere: T/ Σ f(0,) = foJ ((0 d0. This indicates that the continuous form in expectation of above equation can be derived
= (oo / (FEb(For(()) -Ppyl Foo (P d00 (1)
1/2
( 1'' (FE.lo)) - -Fiptu 2
1/2
dy d0
- -(1 D2(P, pmoo2 = =D2((P)
1/2
-
(11)
And the computation for P/, Pl", and P" are independent which thus complete the proofs.
Our min-max optimization objective in Eq. (10) is approximately equivalent to an implicit regularization of the SFE's local Lipschitz continuity for representation stability.
Remark 2.
Proof of Remark 2. To find the perturbation a that maximizes the linear approximation under the constraint |a]|2 ग, we align the perturbation with the gradient direction. This yields the optimal perturbation as:
Jo
V V.
σ
(12)
19
Lod (E:c)<-ol1
Substituting this back into the Taylor expansion of Eq. (11) gives the approximate value of the maximized loss as follows:
max 2
Laa(E, No (=, + - [V:
(13)
Thus, the original min-max objective can be approximated by the following minimization problem: min (I, क [
(14)
This formulation explicitly shows that the adversarial objective encourages the minimization of not only the standard loss Ludd but also penalizes the norm of the loss't 's gradient || (E: 0) This second term encourages the model to be less sensitive to input perturbations.
As for Lndd term, the loss Lan is essentially a function of the set embeddings. e.g., 8; which depend the perturbation via the SFE module: (e)c) SFE(V' e, Vo). We can apply the
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Table 2: Hyper-parameter settings.
Task
Task
Task
Task 4
Friendster
LIVEJ
Youshu
NetEase
MLP
ISAB
LDA-Ik
LDA-3K
LDA-5k
32
128
Н
128
128
32
1024
1024
128
128
128
R
32
32
64
128
256
256
32
32
32
m
5
10
1. 10
5 10
1.10
1 10
1:10
5 10
5 5:100
10
5.10
6
1.1001
1-10--
5 5-10-2
1-10-1
5 10
5-10-1
1.10
1-10-1
1. 1-10-1
7
1.10
1.10-2
1.10
1-10-1
1 1-1001
1.10
1.100
1-10
1. 1.10-1
5-10
1-10-2
1.10
1-10-3
1 10-
1.100
1-10-3
1-10-3
5-10-a
1.10
10
1.10
1-
1-10-3
1.10
5. 10
5- 10
50
Ag
1.10
1.
1.
10
10
1.10
1.
10
1.
multivariable chain rule to decompose the gradient. For simplicity, we consider the gradient's dependence on single embedding (e()). The total derivative of Lid with respect to e, evaluated = 0, can be expressed as:
VeLudd
E (V())(V()) V LedL iEBBua
(15)
Here V., Lund is the loss gradient with respect to the vector embedding vss and Jss=(V,) is the Jacobian matrix of the SFE function with respect to its input features. The regularizer in Eq. (13) penalizes the norm of this sum. For this norm be small, the model incentivized reduce the norms of its constituent components, most notably the spectral norm of the SFE Jacobian, ||JsFe||2.
This penalty the Jacobian norm is directly related the local Lipschitz continuity of the function. For a differentiable function, its local Lipschitz constant L over region V is bounded by the supremum of the spectral norm of its Jacobian within that region:
((SFE, E, V) = sup ||
(16)
VeV
By encouraging smaller Jacobian norm, our adversarial optimization implicitly regularizes the SFE to have a smaller local Lipschitz constant. smaller constant ensures that for any two nearby feature sets V4 and Vj. the distance between their embeddings is bounded:
|SFE(V)) - SFE(V))llz I L(SFE) I[V VV
(17)
This property means that small, non-semantic perturbations to the input features will only result in small, bounded changes to the output embedding, which defines the representation stability (Donchey and Farkhi, 1998).
E SUPPLEMENTARY DETAILS OF EXPERIMENTS
E. HYPER- PARAMETR SETTINGS
We report hyper- parameer settings in Table 2,
E.2 EXPERIMENT CONFIGURATIONS
We implement all models by Python 3.8 and PyTorch 1.12.0 with non-distributed training. We run on a Linux machine with 4 NVIDIA A A100-PCIF-40GB GPUs and 10 vCPU Intel Xeon Processor (Skylake, IBRS). We adhere the officially reported hyperparameter settings of all baselines and conduct grid search to models without prescribed configurations. The learning rate tuned in the range {10-4, 10-3 10 2). Optimization for all models is performed using the default Adam optimizer (Kingma and Ba, 2015). All results are averaged based on five-fold evaluations.
E.3 TASK 1: LEARNING TO RANK SET SIMILARITY
E.3.1 DATASET DESCRIPTIONS
We incorporate two large real-world datasets, i.c., Friendster (Yang and Leskovec, 2015) and LIVEJ (Mislove et al., 2007), for set similarity learning evaluation. We filter sets with fewer than three
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Table 3: Similarity metrics.
Jaccard
Cosine
NOverlap
Dice
[ai Ueg
[errwal
[srnaal max((e11])ea))
2es |ea|+|eal
Vaallea
Table 4: Dataset statistics of Task 1.
Dataset
#S
#E
# Avg. E/S
Friendster
889.839
5,501,401 ,01
11.29
LIVEJ 1,205,816
1,975,812
9.90
elements. Specifically, Friendster is the dataset from online social gaming site where each set corresponds to a group membership. LIVEJ is a dataset from a free online community LiveJournal..
With coefficients wi-wa randomly sampled from (0, 1), incorporate four widely-used similarity metrics, i.e., Jaccard, Cosine, Normalized Overlap (denoted as NOverlap), and Dice, as the score functions to construct similar sets. Their formulations are reported in Table 3. Given set, e.g., 8, its similarity value M to another set, c.g., $j. is computed as:
M)s MayMaaccard (st, 8z) + Wy.Mossimee (8t, 8y)+
(18) This construction process ensures that the evaluation is free from selection bias and captures wide spectrum of possible similar set configurations. With #S and #E denoting the numbers of sets and elements, the result data statistics are reported in Table 4.
Wy Mng.Mroverlappe (8,, (5) + (s;
E.3.2 METHOD DESCRIPTIONS
The methods for set similarity learning introduced as follows:
SAP (Lin et al., 2013) denotes the classic implementation with global average pooling methodology
for sets.
SMP (Lin et al., 2013) is the implementation with set max pooling.
DeepSet (Zaheer et al., 2017) is another classic set embedding method that learns permutation- invariant functions with deep neural networks. We implement global mean pooling the permutation-invariant function of DeepSet.
RepSet (Skianis et al., 2020) extracts set representations by computing the bipartite matching costs between the input set and a collection of learnable reference sets.
SAtt (Lee et al., 2019) is a state-of-the-ar method to embed set structures with self-attention mechanism and Transformer architecture.
PoT (Guo et al., 2021a) introduces a prototype-oriented optimal transport framework that learns set representations by minimizing the transport distance between the set's empirical distribution and learnable global prototypes.
Set2Box (Lee et al., 2022) maps sets into hyper-rectangular box embeddings to effectively capture logical relationships and set boundaries.
OTKE (Mialon et al., 2021) introduces trainable embedding scheme based kernelized optimal transport to aggregate set features.
DIEM (Kim, 2022) proposes differentiable framework learn informative set interactions and enhance representation distinctiveness.
FSPool (Zhang et al., 2020) is representative set embedding framework with carefuully-designed deep learning architecture.
PSWE (Naderializadeh et al., 2021) is of the state-of-the-art deep learning model for set representation learning.
httt:://ww..rieedstee
Phttp://ww.. hwwp::/wwwwwwwwaaoaaaaaaaa ivejournal
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Table 5: Detailed performance comparison for Task 1.
Task Set Similarity Learning
Friendster
LIVEJ
Model
R@200
N@:0
R 100
No 100
R
N@20
R@100
N@ 100
SAP
72.41
68.13
72.15
0.26
86.94 +:
0.12
SMP
70.78 1: 0.28
68.99 : 0.30
81.61 : 0.11
72.22
1 0.25
78.95
1 0.12
76.70 - 0.08
86.83 1: 0.10
84.65 0.09
DeepSet
63.20
60.75
76.60
1.64
69.89
74.55
79.76
RepSet
80,63 1- 0.1.19
76.56
0.21
86.49
0.14
74.92
1 0.23
82.15
0.18
79.63 - 0.22
88.41 1: 0.15
83.12 0.19
SArt
77.52
71,92 0.47
87.51
75.21
81.73
91.39
PoT
82.44 0.1.12
81.85 + 0.16
86.96
:: 0.15
81.47
0
83.18
0.14
84.25 + 0.17
89.33
1: 0.11
86.45 : 0.13
67.35
OTKE
79.53
0.1.14
0.21 73.68
86.64
0.11
79.59 + 0.17
81.45 4 0.16
79.82 + 0.19
87.95 4: 0.13
85.10 0.15
82,49
81.56
0.11
PSWE
83.05
1- 0.09
84.26
0.15
88.59
0.05
85.77
4 0.14
83.52
0.09
84.61 + 0.13
89.48 +: 0.02
86.67
0.09
FSPool
FSW
83.58
0.13
84.39
0.10
88.52
0.07
85.81
0.12
84.19
0.05
85.04
0.06
89.95 +: 0.09
87.23
0.10
SRAL
91.57 + 0.22
92.22
0.22
94.53
93.01 + 0.19
87.56 0.31
89.31 = 0.31
92.93 + 0.02
91.25 0.02
Gain
9.56%
9.28%
6.71%
8.39%
2.58%
2.46%
-0.15%
1.06%
Table 6: Data statistics of Task 2. S, U, and E denote sets (bundles), users, and elements.
Dataset
#S
# U
# E
# Avg. E/S
Youshu
4,771 22,864
8,039 18,528
32,770
37.03
NetFase
123,623
77.80
FSW (Amir and Dym, 2025) is another state-of-the-arr model that utilizes the Fourier transform in the frequency domain for "multisets".
E.3.3 DETAILED EXPERIMENTAL RESULTS
For completeness, we present the detailed experimental results for Task 1 in Table 5. These results including mean scores and standard deviations over multiple runs, substantiate the findings discussed in the main text, where our model SRAL consistently outperforms all baselines.
E.4
TASK 2: BUNDLE REPRESENTATION LEARNING FOR RECOMMENDATION
E.4.1 DATASET DESCRIPTIONS
Following recent works (Ma et al., 2022; Chang et al., 2020; Deng et al., 2020), we include two real-world datasets: Youshu (Chen et al., 2019) for book list recommendation and NetEase (Cao et al., 2017) for music playlist recommendation. Dataset statistics are reported in Table 6.
E.4.2 METHOО DESCRIPTIONS
We include the following bundle recommender models:
MFBPR (Rendle al., 2012) utilizes the Bayesian Personalized Ranking (BPR) loss within
Matrix Factorization framework to model collaborative filtering between users and bundles.
DSBRec is specialized implementation with DeepSet (Zaheer et al., 2017) and optimize with BPR loss.
DAM (Chen et al., 2019) attentively captures bundle representations from associated items and utilizes multi-task learning to optimize interactions between users and both items and bundles.
BundleNet (Deng et al., 2020) is traditional framework for bundle recommendation that constructs a tripartite graph of user-bundle-elemaa relationships and utilizes Graph Convolution Network (GCN) for representation learning alongside multi-task learning.
BGCN (Chang et al., 2020), bundle recommendation method, dissects user-bundle-elemen relationships into distinct perspectives: bundle-view and item-view graphs
CrossCBR (Ma et al., 2022) leverages contrastive learning to achieve cross-view alignment in the latent space.
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Table 7: Detailed performance comparison for Task 2.
Task Bundle Recommendation
Youshu
NetEase
Model
R4220
N
R 100
N@100
R6
N@
R 100
N@ 100
MEBPR
19.97 1: 0.45
11.67
1: 0.52
44.33
0.33
17.95
0.41
0.115
2.98 :: 0.11
14.15
0.188
DSBRec
20.46 1: 0.41
12.03
1. 0.31
45.34
0.32
18.12
0.28
5.51
1.100
3.04
1 0.08
14.76
0.13
5.14
0.14
DAM
20.83 1.83+40.88
11.99
+ 0.22
45.58
0.26
18.38
+ 0.33
4.14
3.11
+ 0.09
14.98
+.0.6
5.12
0.10
BundleNet
22.85 1: 0.35
11.90
1. 0.25
47.84
0.22
19.19
0.31
6.17
0.11
3.44
1
16.26
0.111
5.83
1.33..0.00 0.08
BGCN
35.22-40.11
14.34
0.10
49.38
0.26
21.18
+ 0.29
7.04
4.110
3.1 + 0.08
17.23
6.51
+.00
CrossCBR
26.41
16.55
0.44
23.30
0.25
7.21
0.11
18.32
0.13
6.77
0.07
SRAL
26.92 + 0.09
16.95 + 0,08
$2.18
0.19
23.64
0,06
7.37 0.11
4.21 = 0,05
18,66 + 0.01
7.01 = 0,06
Gain
193%
2.42%*
0.54%
1.46%*
2.22%
3.19%
1.3%%
3.54%
Table 8: Test accuracy (%) of SRAL and competing methods for Task 3.
Thin 3 Petne Cloud Pricesiig
Settion
MIP
1.553
1.0.
86.6
8.6
E.4.3
DETAILED EXPERIMENTAL RESULTS
Table catalogs the precise outcomes both the Youshu and NetEase datasets, with standard deviations included to underscore the stability of our findings. The results show that SRAL* consistently outperforms other rival methods with low standard deviations across all
E.5
TASK 3: POINT CLOUD PROCESSING
E.5.1 DATASET DESCRIPTIONS
For this study, we utilize ModelNet40 dataset Wu et al. (2015), comprising 3D point clouds extracted from triangular meshes of 12,31 computer-aided design models across 40 distinct object categories. Each object is represented by set of 1024 points, following methodologies outlined in Guo et al. (2021b); Qi et al. (2017).
E.5.2 DETAILED EXPERIMENTAL. RESULTS
For Task 3, Table 8 summarizes the comparative test accuracies using both MLP and ISAB backbones. The inclusion of standard deviations further illustrates the stability of these outcomes.
E.6 TASK 4: TOPIC SET EXPANSION
E.6.1 DATASET DESCRIPTIONS
We leverage three datasets i.e., IDA-IK, LDA-3K, and LDA-5K, from previous work Zaheer et al. (2017) that are originally processed from latent Dirichlet allocation Blei et al. (2003). They respectively contain 2,000, 6,000, and 10,000 sets and 17,016, 37,718 18 and 61,127 vocabulary elements. Their average elements per set around 25.
E.6.2 DETAILED EXPERIMENTAL RESULTS
The complete results with standard deviations of Task 4 evaluation are reported Table
E.7
IMPLEMENTATION OF SELF-SUPERVISED LEARNING Loss
We implement the following self-supervised learning losses for comparison:
Set Triplet Loss:
thub.
http://twwtt..oo/ooa//aaaaa//a// data/1da
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Table 9: AUC results (%) of SRAL and competing methods for Task 4.
Touk Taeee Pxpamiee
SMP
Reydee
Repler
540
Pil
Seltten
IXMM
PSP
50.41
frem
50.90 7e1
00.9 0
1048
54394337
19.31
72.77 239
78.96 2333
71.5
9034
06.30
+0.001"
Table 10: Runtime comparison for Task and Task 2.
Task I: Set Shmilarity Learning Runtime
Task 2: Bundle Recommendation Runtime
Friendster
LIVEJ
Youshu
NetEase
Model
In Total
Per-epoch
In Total
Per-epochh
Model
In Total
Per-epoch
In Total
Per-epoch
PSWE
3.1h
1.6min
5.1h
2.7min
BGCN
12.6min
6.3%
23.4min
9.7%
FSPool
2.hh
1.5min
6.5h
2.
CrossCBR
10.4min
4.5s
19.5min
8.4s
SRAL
3.9h
5.6min
7.7h
7.9min
SRAL
24.5min
42.2s
44.8min
57.4s
C Σ max(d(ex, 2//) d(us, 12)+), + 0).
(19)
S,S,ES
where a the hyperparameter.
Soft-Nearest Neighbors Loss:
C = log exp(sim(v/, w/)/T)//))))))))
exp(sim( ( w///T) Σ j exp(sim(e)) (u, "///T)
log Ejj' exp(sim(u, uf}/T)
(20)
where we use inverse Euclidean distance to implement the sim function. For this experi- ment, it is important note that these metrics cannot be directly integrated into SFE architecture. Therefore, we made adaptations for them as follows: for sets of varying sizes, we first aggregate their element embeddings using mean pooling to obtain a single vector representation for each set. Subsequently, we employ each respective metric to calculate the distributional distance. The training objective is to align these distances with the ground-trun truth similarity rankings from Task 1.
Barlow Twins Loss:
Σ k=
CKN) +
>>
(Cn)
(21)
1
where
C'
BatchSize norm(V') norm(V").
(22)
Here V and 4" denote the perturbed batch set embeddings.
E.8 SCALABILITY STUDY SRAL
To assess the scalability of SRAL, derived nine proportionally-sizee sub-datasets from Friendster. Figure presents scalability analysis of SRAL on subsets of the Friendster dataset. We have two main observations. First, the training time per epoch (blue bars, left axis) exhibits a gradually growing trend as the data volume increases. This prevents prohibitive computational costs on larger datasets and confirms the model's acceptable efficiency. Second, the model's performance, measured by Recall @ 20 (orange line, right axis), remains generally consistent with slight fluctuation. These results demonstrate the effectiveness of our SRAL model, making it practical solution for large-scale settings.
E.9 COMPUTATION EFFICIENCY ANALYSIS
E.9.1 COMPUTATION CoST COMPARISON
We report the training time cost of SRAL and compare it with two most competitively performing models across all tasks. As shown in Tables 10 and although our SRAL model incurs higher
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Table 11: Runtime comparison for Task 3 and Task 4.
Point Processing Runtime
Task 4: Topic Set Expansion Runtime
MLP
ISAB
LDA-IK
LDA-3k
LDA-SK
Model
Total
Per-cpoch
In Total
Per-opoch
Model
In Total
Per-pooc
Tocal
Par-epoch
In Tocal
Per-poch
PSWE
1.6h
27.4s
188h
29.5s
PSWE
10.58
0.3s
15.68
0.3s
18.56
0.3s
FSPool
1.26
23.9s
1.6h
26.3s
14.4s
SRAL
2.3b
1.6min
2.7h
1.7min
SRAL
14,38
0.58
23.5s
0.5s
31.68
0.6s
Reollgg00
Time
Altering
dataset
Figure 1: Results with varying data sizes.
"Per-epoch" computational cost compared to the baselines, its "In Total" training time remains comparable. This is because our proposed Adversarial Encoding Perturbation and Optimization mechanism well promotes the model convergence, we analyzed earlier in Section 4.3.2.
E.9.2 MODULE COMPUTATION COST
To offer deeper insights into the computational overhead, we break down the per-epoch training time for individual modules using the Friendster dataset from Task as follows:
As shown Table 12, the Adversarial Encoding Perturbation and Optimization (AEPO) module incurs the highest computational cost among all components. This overhead primarily arises from the adversarial perturbation generation and the iterative min-max optimization process. Nevertheless, considering its substantial contributions to accelerating convergence and significantly improving training stability, we believe that this computational expenditure constitutes a worthwhile trade-off.
E. MODULE COMPUTATION COST
To offer deeper insights into the computational overhead, we break down the per-epoch training time individual modules using the Friendster dataset from Task as follows:
As shown in Table 12, the Adversarial Encoding Perturbation and Optimization (AEPO) module incurs the highest computational cost among all components. This overhead primarily arises from the adversarial perturbation generation and the iterative min-max optimization process. Nevertheless, considering its substantial contribution to accelerating convergence and significantly improving training stability, we believe that this computational expenditure constitutes worthwhile trade-off.
E.11 COMPARISON WITH VANILLA OT-BASED SOLUTIONS
While Optimal Transport (OT) has been explored set representation learning, e.g., RepSer Skianis et al. (2020), our approach distinguishes itself through the utilization of the Wasserstenn (SW) distance and a tailored adversarial enhancement mechanism. RepSet formulates the set distance as a bipartite matching problem, which is equivalent to exact OT, and offers approximation variant (ApproxRepSer) by relaxing constraints to a semi-relaxed OT problem. To provide the comparison, conducted experiments the Friendster dataset (Task 1) covering three aspects: (1) direct performance comparison against RepSet and ApproxRepSer; (2) evaluating RepSet variants as encoders within our SRAL framework. The results summarized in Table 13.
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Table 12: Training time cost of each major module.
SRAL (complete) 5.6min
SRAL (SFE Module)
SRAL (AEPO)
SRAL (Main Loss)
Youshu
1.8min
3,3min
0.1min
Table 13: Performance comparison with RepSet and its variants.
RepSer
SRAL (RepSet) ApproxRepSet
SRAL (ApproxRepSee)
SRAL 91.57
2c
80.63
82.25
77.81
78.93
Performance Analysis. We observe that our SRAL outperforms the vanilla RepSer baseline; and the approximation variant ApproxRepSet exhibits inferior performance. We attribute this gap to the nature of the "semi-relaxed" approximation employed in ApproxRepSet. As noted in Skianis et al. (2020), dropping constraints to achieve computational efficiency may sacrifice rigorous metric properties, such as the triangle inequality, and makes the optimization prone to local optima. In contrast, our SW-based approach preserves key geometric properties while remaining computationally efficient, resulting in superior representation stability.
Encoder Compatibility. When integrating RepSel as the encoder within our framework, we observe performance gain but still falls short of native SRAL model. This empirical finding validates our theoretical analysis in Remark I: our AEPO mechanism is specifically tailored for the Set Feature Encoder (SFE) based AEPO aim to maximize discrepancies in the embedding space, which are positively correlated in expectation with the Sliced- Wasserstein distance. In contrast, such a direct perturbation may not hold for the bipartite matching objective in RepSet, rendering the adversarial optimization less effective in capturing distributional semantics.
metric. The adversarial perturbations generated by
26


Reviews:
[{"review_id": "ZeA0dMBhx4", "reviewer": "Reviewer_du3f", "summary": "This paper proposes SRAL (Set Representation Auxiliary Learning), a novel framework for set representation learning that explicitly captures inter-set distributional relationships rather than focusing solely on preserving intra-set structures. SRAL introduces a learnable reference distribution O and encodes each set according to its 2-Sliced-Wasserstein distance to this common reference, thereby aligning all sets within a shared geometric space. Moreover, SRAL incorporates a self-supervised objective with adversarial perturbations applied in the aligned feature space to obtain more robust and discriminative set embeddings.", "questions": "- How were the hyperparameters chosen, and how computationally costly is the proposed method? The approach requires tuning the hyperparameters λ1 and λ2, and according to Table E.1, the optimal values of λ1 vary considerably across different tasks (from 5e-1 to 1e-3). For comparison, do the baseline methods such as FSPool and CrossCBR in the benchmark tables also require similar levels of hyperparameter tuning to achieve the reported performance? It would be interesting to see how the proposed method performs in terms of computational efficiency and cost.\n\n- In Eq. (5), the mapping g+ appears to be non-differentiable. Did you use any soft approximation of the sorting operation, or how do you propagate the gradients with respect to the parameters.\n\n- Typos and minor notes\n  - L.107: g: R^d -> R^d\n  - L.155: should be \"true data distribution P_i\" (without a hat)\n  - L.159: (perhaps clear from context, but to confirm) inside delta, it should be theta(x) >= theta(z), or x and z should be one-dimensional scalars.\n  - L.257: the correct reference should be https://arxiv.org/abs/1412.6572\n\n\nI will adjust the score based on replies from the authors.", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- Proposes a theoretically grounded approach to set representation learning based on the Wasserstein distance.\n- Achieves superior performance compared to existing methods across various benchmark datasets.", "weaknesses": "Motivation for introducing reference set O is a bit unclear to me. In the manuscript, it is only briefly explained: “To learn stable and discriminative set representations, our encoder leverages the distributional distance between an input set and a learnable reference distribution O.” For instance, if my understanding is correct, one could compute an L_wd-like loss without introducing a reference distribution, by directly calculating the sliced Wasserstein distance between each pair of P_i and P_j (with perturbations), and replacing the term -||v_i′ - v_j″||_2 with that sliced Wasserstein distance. Could the authors clarify this design choice or point out what I might be missing?", "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": "RZ7WHLFAN6", "reviewer": "Reviewer_9yXr", "summary": "This paper proposes an optimal transport-based approach for learning cardinality- and permutation-invariant representations for set-structured data. A set encoder based on sliced Wasserstein distance is used, and a contrastive-style adversarial loss is added to the objective to learn more informative set embeddings based on worst-case perturbations of the element-wise embeddings. Numerical results on four different tasks show the superiority of the proposed approach over several other set learning methods.", "questions": "- Is it possible to simplify Eq. (13) by setting $\\epsilon^+ = \\frac{\\pi}{\\\\|g_{\\epsilon^+}\\\\|} g_{\\epsilon^+}$?\n- Could you please compare your method's performance with some more recent approaches, such as FSW [B]?\n\n[B] Amir, Tal, and Nadav Dym. \"Fourier Sliced-Wasserstein Embedding for Multisets and Measures.\" In The Thirteenth International Conference on Learning Representations, 2025.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- Learning representations of sets is a critical problem in many application areas of machine learning, so the considered problem is timely and important.\n- The paper is written clearly and relatively straightforward to read and follow, even for an audience with minimal background in set representation learning.\n- The scope of the presented experiments is broad and acceptable in my view.", "weaknesses": "- It is unclear how the optimal adversarial perturbations are applied to the sets. On line 249, the authors mention that the perturbations are shared between all sets and both views of each set. However, the two views of each set will be identical if the perturbations are equal. Is one view unperturbed and the other view perturbed by $\\epsilon^+$? Is $\\epsilon^+$ the perturbation used for all sets in a given training batch?\n- In the abstract and introduction section, the authors mention that perturbations are not applied to the input features but occur during the encoding process. However, as presented, it does seem that the perturbations are indeed applied to the features *before* the encoding process. If my understanding is correct, the claims in the abstract and introduction need to be revised. Otherwise, if the encoding process itself is to be perturbed, one option is to perturb the projection parameters $\\Theta=[w_r]_{r=1}^R$; would that be feasible with your adversarial approach?\n- If I understand correctly, the SFE module introduced in Section 3.2.1 is now new and is the same as the one introduced by PSWE [A]. Could you please explain if there are any distinctions between SFE and PSWE?\n- The notation at the beginning of Section 2 is confusing. Is the universe $\\mathcal{E}$ finite and contains $n$ elements, or is $n$ the cardinality of the set $S_i$? What is the difference between $e_i$'s and $z_i$'s?\n- The contribution of Remark 1 is unclear to me. SFE and PSWE both imply that the Euclidean distance between embeddings approximates the Sliced Wasserstein distance before the encoding process, which is unrelated to the used InfoNCE loss function. Also, could you please explain more precisely how you go from Eq. (8) to Eq. (9) (from summation to integration)? Do you need assumptions on, e.g., an infinite number of projections $R$?\n\n[A] Navid Naderializadeh, Joseph F Comer, Reed Andrews, Heiko Hoffmann, and Soheil Kolouri. Pooling by sliced-wasserstein embedding. NeurIPS, 34:3389–3400, 2021.", "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": "ekKlmYWGvV", "reviewer": "Reviewer_jqC5", "summary": "This paper presents an set representation learning approach that uses optimal transport to compare input sets and learnable references. The optimal transport is approximated via sliced Wasserstein distance (SWD) where the number of slices R is a hyperparameter (that trades between accuracy and computation time). In addition to the sliced Wasserstein distance used to encode sets (thus achieving permutation invariance and cardinality-independence), the authors also use adversarial-robust learning (in a min-max formulation) to be robust against the perturbations of encoded set features. The authors show that in a variety of tasks, using SWD to encode sets together with the adversarial-robust learning exceeds state of the art results.", "questions": "- I found the intro to be hard to crack, given that I have a background in OT and deep learning but not set representation learning (SRL). I had to look into some of the author's references to understand the topic better. Perhaps the problem description in Preliminaries could be extended with a figure or a motivational example could be given to introduce SRL to a wider audience.\n- how do you define a CDF in d-dimensions (line 159) ? You can only do this after projecting into 1-d (unless you introduce 'copulas'!)\n- not clear from intro how the 'reference distribution' O is learned!\n- For Proposition 1 it is enough to cite a standard book on OT, e.g. Peyre and Cuturi's book. As it is a well known result, no need to cite it as a Proposition, because it's not the author's contribution.\n- Similarly for Proposition 2, sorting and matching the quantiles is well known in the literature as a solution, no need to state it as a proposition.\n- It is not clear how the features z_{i,k} of the sets are obtained from the intro.\n- Besides Skianis et al., the papers by Mialon et al. (2021) [OTKE] and Kim (2022) both discuss OT-approaches in a kernel-method context. These and other OT-based methods should be discussed in detail and compared against.\n- Are all datasets in the experiments supervised? Discussing the experiments a bit better in the main text would be better.\n- Figure 3: A and B captions should be swapped\n- very hard to read most of the figures, please revise\n- not clear what is the difference between AEPO and AL in the ablation study.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "This is a nice experimental paper, that extensively evaluates their proposed method and shows good results in all the tasks considered. The ablations also support the conclusion that both the OT-based encoding and adversarial-robust learning help to improve the performance in the tasks considered.", "weaknesses": "Unfortunately, OT in set-representation learning is not new, see Skianis et al. for what seems to be the first OT-approach. The authors do cite this paper in passing but not in detail. In that paper the OT approach is presented as a bipartite graph optimization, which is equivalent to the linear programming formulation for solving OT problems. The authors should discuss that paper in detail and compare their method against theirs (in a fair way, e.g. using the same encoders etc.) Interestingly, whereas the authors use the sliced Wasserstein distance to approximate the OT, the authors in that paper take another approach: by dropping one of the constraints, Skianis et al. reduces OT to the semi-relaxed case (unbalanced OT with one of the penalties going to infinity). The authors should also extend their comparisons to Skianis et al. by comparing the various approximations.", "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": "AIePGxM6DB", "reviewer": "Reviewer_kDXr", "summary": "The paper presents a new method, SRAL, for set representation learning. To encode a set, the authors first extract distributional features from it using sliced wasserstein distances rank-matching to a learned canonical set. These set representations are then inserted into a contrastive loss for learning the canonical set embeddings. To choose the positive pairs, the authors augment each set by adding noise to its elements and then extracting the new noisy set embeddings. Specifically, to increase the augmentations effect, they optimize an adversarial noise direction in an inner loop.", "questions": "- How does the authors view an integragtion of a powerful LLM into this problem? A discussion on that could be helpful. \n\n- Except the embeddings of the canonical set V_o, what other parameters are learned in SRAL?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 1, "contribution": 2, "strengths": "- While writing clarity could be improved, the proposed method appears simple and intuitive for set representation.\n\n- The analsys presented in remark 1, showing the connection between the proposed loss metric and a sliced wasserstein based contrastive loss, is interesting.", "weaknesses": "- **Writing clarity of method sections.** Sections 2-3 would benefit from simplified presentation. Although the proposed method appears simple, it is hard to follow. Specifically, it is notation-heavy (the notation table in the appendix is almost a page long) which further reduces clarity. Rewriting these sections in a simplified manner can greatly improve the paper's clarity.\n\n- **Limited baseline comparisons.** The experimental section is missing several key comparisons to more recent methods. Moreover, when SRAL is compared to a more recent method, CrossCBR, it shows only marginal gains. The paper would be strengthened by adding comparisons to [1,2,3] on tasks 1 and 4 to evaluate SRAL against more recent approaches.\n\n- **Marginal gains in tasks 2 & 3.**  The results of SRAL on tasks 2 and 3 show only marginal gains over the baselines. Additionally, on Task 2, SRAL is integrated into CrossCBR rather than evaluated standalone. Providing standalone results for Task 2 would help clarify SRAL's direct contribution.\n\n\nMinor remarks:\n\n- **Figures and tables sizing.** Most figures and tables are quite small and difficult to read, particularly in printed form. Enlarging key results would improve readability.\n\n\n[1] Lee, Dong Bok, et al. \"Self-supervised set representation learning for unsupervised meta-learning.\" The Eleventh International Conference on Learning Representations. 2023.\n\n[2] Guo, Dandan, et al. \"Learning prototype-oriented set representations for meta-learning.\" arXiv preprint arXiv:2110.09140 (2021).\n\n\n[3] Lee, Geon, Chanyoung Park, and Kijung Shin. \"Set2box: Similarity preserving representation learning for sets.\" 2022 IEEE International Conference on Data Mining (ICDM). IEEE, 2022.", "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)