A Geometric Approach to Personalized Recommendation with Set-Theoretic Constraints Using Box Embeddings
Authors: Shib Sankar Dasgupta
Michael Boratko
Andrew McCallum
Venue: ICML 2025 poster (2025)
Keywords: Box Embeddings
Personalized Query
Set-based embeddings
Recommendation
Primary area: general_machine_learning->representation_learning
TL;DR: Set-theoretic embeddings offer the appropriate inductive bias needed to effectively answer queries with set constraints.

Abstract: Personalized item recommendation typically suffers from data sparsity, which is most often addressed by learning vector representations of users and items via low-rank matrix factorization. While this effectively densifies the matrix by assuming users and movies can be represented by linearly dependent latent features, it does not capture more complicated interactions. For example, vector representations struggle with set-theoretic relationships, such as negation and intersection, e.g. recommending a movie that is “comedy and action, but not romance”. In this work, we formulate the problem of personalized item recommendation as matrix completion where rows are set-theoretically dependent. To capture this set-theoretic dependence we represent each user and attribute by a hyperrectangle or box (i.e. a Cartesian product of intervals). Box embeddings can intuitively be understood as trainable Venn diagrams, and thus not only inherently represent similarity (via the Jaccard index), but also naturally and faithfully support arbitrary set-theoretic relationships. Queries involving set-theoretic constraints can be efficiently computed directly on the embedding space by performing geometric operations on the representations. We empirically demonstrate the superiority of box embeddings over vector-based neural methods on both simple and complex item recommendation queries by up to 30% overall.

A Geometric Approach to Personalized Recommendation with Set-Theoretic Constraints Using Box Embeddings
Shib Dasgupta Michael Boratko | Andrew McCallum
Abstract
products, movies, articles, and more. In addition to generic recommendation, these platforms often present the option for the user to search for items, either via natural language or structured queries. While collaborative filtering methods like matrix factorization have proven successful in address- ing data sparsity for unconditional generation, they often fall short when attempting to combine them with more com- plicated queries. This is not unexpected, as vector embed- dings, while effectively capturing linear relationships, are ill-equipped to handle the complex set-theoretic relation- ships. Even advanced neural network-basse approaches, which are designed to capture intricate relationships, have been shown struggle with the-theoettic compositionally that underlie many real-world preferences.
Personalized item recommendation typically suf- fers from data sparsity, which is most often ad- dressed by learning vector representations of users and items via low-rank matrix factorization. While this effectively densifies the matrix by as- suming users and movies be represented by linearly dependent latent features, it does not cap- ture more complicated interactions. For example, vector representations struggle with set-theoretic relationships, such as negation and intersection, e.g. recommending a movie that "comedy and action, but not romance". In this work, we for- mulate the problem of personalized item recom- mendation as matrix completion where rows are set-theoretically dependent. To capture this set- theoretic dependence we represent each user and attribute by a hyper-rectangle or box (i.e. a Carte- sian product of intervals). Box embeddings can intuitively be understood as trainable Venn di- agrams, and thus not only inherently represent similarity (via the Jaccard index), but also natu- rally and faithfully support arbitrary set-theoretic relationships. Queries involving set-theoretic con- straints can be efficiently computed directly on the embedding space by performing geometric operations on the representations. We empirically demonstrate the superiority of box embeddings over bestor-based neural methods on both simple and complex item recommendation queries by up to 30% overall.
Let consider an example where a user named Bob wants to watch a comedy which is not a romantic comedy. Assum- ing we have prior watch history for users, standard collab- orative filtering techniques (e.g. low-rank matrix factoriza- tion) would yield a learned score function score(m, Bob) for each movie m. If we also have movie-attribute annota- tions, we could form the set of comedies C and set of ro- mance movies R and simply filter to those movies in C' \ R, however this assumes that the movie-attribute annotations are complete, which is rarely the case in practice. In fact, Dasgupta et al. (2023) show that item- att-ibtri matrices, even manually curated, are often incomplete, and remain sparse and noisy due to limited coverage and annotation effort.
A standard approach in setting with sparse data is to learn low-rank approximation for the attribute x movie ma- trix A, yielding a dense matrix Å. We can then form sets of movies based on this dense matrix using an (attribute- specific) threshold, e.g. C = {m Acomedy.m > Toomedy } and R = {m Aromance.m > Tromense). and then rank movies m E Õ') It according to score(m, Bob). While this approach does allow for performing the sort of queries we are after, it suffers from three fundamental issues:
1. Introduction
Recommendation systems are standard component of most online platforms, providing personalized suggestions for
Manning College of Information & Computer Sciences, UMass Amherst. Correspondence to: Shib Dasgupta <ssdas- gupta@ es edus..edu>.
1. Limited user-attribute interaction: Since the attribute classification is done independently from the user, any latent relationships between the user and attribute can- not be taken into account.
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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A Geometric Approach to Personalized Recommendation with Set-Theoretic Constraints Using Box Embeddings
Error compounding: Errors in the completion of at- tribute sets accumulate the number of sets involved in the query increase.
77
77
0
1
1
0
3. Mismatched inductive-bias. Our queries can be viewed as set-theoretic combinations of the rows, not linear combinations. As such, using a low-rank approxima- tion of the matrix may be misaligned with the eventual use.
:
0
0
...
0
1
0
In this paper, we formulate the problem of attribute-specifie recommendation as matrix completion where rows are not necessarily linear combinations each other but, rather, are ser-theoretic combinations of each other. More pre- cisely, given some attribute movie matrix A, the queries are considering are set-theoretic combinations of these rows (see Figure 1), For example, the ground -trnthh data for comedies which are not romance movies which Bob likes would be the vector æ E (0.1)IM) where I' = 1 if and only Uoo. = 1 and A Aromance.m
...
0
0
1
A
1
...
1
Q
0
movie interaction matrix U and
1
0
UN
A
1 0 0
Note that this not linear combination of the previous rows, and while the inductive bias of low-rank factorization has proven im- mensely effective for collaborative filtering we should not expect it to be directly applicable in this setting.
and comedy..m
Figure Standard matrix completion assumes you are given partial information about the user movie matrix U, and potentially incomplete information about the attribute X movie matrix A.
Instead, we propose to learn representations for the users and attributes that are consistent with specific set-theoretic axioms. These representations must also be compactly pa- rameterizable in lower-dimensional space, differentiable with respect some appropriate score function, and allow for efficient computation of various set operations. Box Em- beddings (Vilnis et al., 2018; Dasgupta et al., 2020), which are axis- saislllrllll n-dimensional hyperrectangles, meet these criteria (see Figure 2). The volume of a box is easily calcu- lated the product of its side-lengths. Furthermore, box embeddings closed under intersection (i.e. the intersec- tion of two boxes is another box). Inclusion-exclusine thus allows to calculate the volume of arbitrary set-theoretic combinations of boxes.
Vol
score(
Vol(
The contributions of our paper are as follows
1. We model the problem of attribute-specifi query rec- ommendation as "set-theoretic matrix completion". where attributes and users are treated as sets of items. We discuss the challenges faced by existing machine- learning approaches for this problem setup.
score B
Vol( A
Vol
2.
We demonstrate the inconsistency of existing vector embedding models for this task. Additionally, we estab- lish box embeddings as suitable embedding method for addressing such set-theorede problems.
Voll
B
Figure 2: Box embeddings represent the movies, users, and attributes as "boxes" (Cartesian products of intervals) in
We conduct an extensive empirical study comparing various vector and box embedding models for the task of ret-theoretic query recommendation.
2

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A Geometric Approach to Personalized Recommendation with Set-Theoretic Constraints Using Box Embeddings
Box embeddings, with their geometric set operations, sig- nificantly outperform all vector-based methods. We also evaluate score multiplication and threshold-basse prediction for both vector and box embedding models, and find that performing set operations directly on the box embeddings performs best, solidifying our claim that the inductive bias of box embeddings provides the necessary generalization capabilities to address set-theoretic queries.
user and attribute data. While we could, in theory, proceed directly with logistic matrix factorization on this matrix, there are both practical and theoretical reasons to search for an alternative. First, the number of rows of this matrix is very large relative to the original data in our case we have [Q] = O[ but in general |Q] = (331)1IA)). This poses practical issues, both at training time (as there are an exponential number of elements of X to traverse) and inference time (storing the low-rank approximations re- quires ((IQ) ) memory, which is much larger than [A)+ There are also theoretical issues with the underlying assump- tion, as it is no longer reasonable to assume the rows of o (X) are linear combinations of some latent factors.
2. Task Formulation
2.1. Background
Matrix completion is a fundamental problem in machine learning, and arises in a wide array of tasks, from rec- ommender systems to image reconstruction. Formally, this problem is typically modeled as follows: Given a matrix X E |m= where only a subset of the entries are observed, find a complete matrix X m which closely approximates X on the observed entries. For the task of recommendation, this involves predicting user interactions with items they have not previously interacted with, and a common assumption is that the preferences of users and characteristics of the items can be expressed by a small number of latent factors, with the alignment of these latent factors captured via dootprrduct. This justifies the search for low-rank approximation X = BC, where B E RT*D and C E Dxn In the case where the original matrix is binary, X E {0, 1|mxn, it is common to perform logistic matrix factorization, where an elementwise sigmoid is applied after the dot-product of latent factors, which we denote (with slight abuse of notation) as X = (BC)..
3. Method
Our proposed solution to address these issues starts by defin- ing the sets of movies which comprise the queries of in- terest. Let, P(M) be the power set of movies M. Specif- ically, for each user u we can define the set M,, = {m | (u, m) E }. and for each attribute a we can define the set M2 = {m | (a, m) E DA). If we let M C P(M) be the collection of all such sets, then the set of movies corresponding to given query q are direct set-theoretic combinations of elements in M. Hence, the reasonable un- derlying assumption, in this case, is to model the elements of .M as sets via a map ʃ : M + R where R is also a set of sets, and the map f respects set- operations. i.e. ((STT) = ((S) and f(S) = ((S) ((T), etc. Such a map is referred to as a homomorphism of Boolean algebras, and the problem of learning such function was explored in general in (Boratko et al., 2022). In our work, we propose box embeddings as the function f which can be trained to obey the homomorphism constraints. As a result, user- aner-attrbbute-iem representations based on box em- beddings could serve as an optimal inductive bias for the proposed set-theoretic matrix completion task.
2.2. Set-Theoretic Matrix Completion
We will describe the task of set-theoretic matrix completion on the setting of movies, users, and attributes, though the formulation and our proposed model can be generalized to arbitrary domains. We are given a set Du C U x M of user-movie interactions, and set D C A x M of attribute- movie pairs. We assume both of these sets are incomplete.
3.1. Set-theoretic Representation Box Embeddings
As introduced in Vilnis et al. (2018), box embeddings rep- resent entities by a hyperrectangle in R., i.e. Cartesian product of intervals. Let the box embedding for user u be:
Our goal is to eventually be able to recommend movies based on some query, for example "comedy and not ro- mance". Such query for particular user can be repre- sented as A 01 A -a2, where u is the user, d1 comedy and d2 = romance. We let Q be the set of all queries of interest, which depends on which queries we anticipate evaluating at inference time. In this work, we will take Q to be queries of the form u. 41, u A 01, u A a₁ A 02, and A 01 A -22, where u E U and a1 0g E A.
D [ua und = [ui, will x x (bbup) C CRD, d=1
Box(u) = [mu
where [uyz un] is the interval for d-th dimension, ut < dp for d E (1....,))
The volume of an interval is defined as the length of the interval Vol (u², u2)) max(u² - uz.
With this formulation, pletion for matrix X E {0, 1) where the rows are derived by applying bitwise operators on the rows of
view task as matrix com-
Let, Box(m) = I mal be the box embeddings for a movie m. At dimension d, the volume of intersection
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A Geometric Approach to Personalized Recommendation with Set-Theoretic Constraints Using Box Embeddings
between user u and movie is defined
the ratio of hard volume calculations with the approximation to the expected volume,
Vollut((u) ua). (m2, mz))
Fau ((uz, u2), (m² ma): (T:)::(T))) ())
= max [inn(u) ma) - max(ua, my).0) 0).
LSE, (LSE (ug, ma) - LSE, (uz, ma)) 0)
LSE, (m mm';. 0)
When the movie interval [my. mi] is completely contained by user interval [u'g. ui]. then Vollin (( (mm...m))) Vol((mng. (mmmmmm))))
VollntGe (12z, u2), (mz, ma)s (T,
(3)
Volcn ((m) m)):
This objective creates a set-theoretic interpretation with box embeddings, where user Box(u) contains all the movie boxes related to u (Figure 2). The score for containment for a single dimension d is formulated as:
3.2. Training
We model each user, attribute, and movie as box in RD. and denote the map from these entities to their associated box parameters as 0, i.c., the trainable box embedding for user u is (()) == Box(u). Our goal is train these box representations to represent certain sets of movies which allow us to perform the sort of queries we are interested in. As motivated above, for a given user u, we train Box(u) to approximate the set M, via objective. Namely, for each (u, m) E DD, we have a loss term
TBox((ub) Ma); (mly m)))
VolInt((uz, u2), (mz, m2))
Vol((m). mz))
max(min(2 y m)) - max(u 2 m²).0) max(ma- my. 0)
(1)
cooise-cntrrativee estimation
The overall containment score is the multiplication of Foox for each dimension. The log of this score is referred to as the energy function as given:
(u.m)) (0) == Eos (u, m; 0)
BR log (1 -1- 11-pepp--0
exp(-EEB EGB (u, m: 0))]
EBo (u. = - log II d=1
FBox ((1ig unds (my, ma))- (2)
The first term is minimized when Box(u) contains Box(m). We approximate the second term via sampling, which en- courages Box(u) to be disjoint from Box(m) for a uni- formly randomly sampled movie m. We define analo- gous loss function ((.m) (0) for attribute- movie interactions, which trains Box(a) to contain the box Box(m) for each m such that (2, m) E Du.
This energy function is minimized when the user Box(u) contains the movie Box(m). Previous works have high- lighted the difficulty of optimizing an objective including these hard min and max functions (Li et al., 2019; Das- gupta et al., 2020). In our work, we use the latter solution, termed GUMBELBOX, which treats the endpoints x² and as mean of GumbelMax and GumbelMin random vari- ables, respectively. Given 1-dimensional box parameters =;] -= we define the associated GumbelMax ran- dom variables X with mean =t, and scale B, as well as the GumbelMin random variables X,, with mean v, and scale B. Dasgupta et al. (2020) calculates that the expected volume of intersection of intervals {[x,, X,Il can be ap- proximated by
The overall loss function is a convex combination of these loss terms:
[(0: Du,DA) = w
Σ
f(u,m) (0)
(u,m)EDU
+ (1 - w) *
Σ
laam) (0).
(a.m)EDA
for a hyperparameter w E [0. 1]. This optimization ensures that the movie boxes are contained within the corresponding user and attribute boxes, thereby establishing set-theoretic inductive bias. Both numbers of negative samples and are hyperparameters for training (Please Refer to Section 4, Appendix A.2) for further details. Training box embeddings is generally efficient, as the computation of box intersection volumes can be parallelized across dimensions. We provide training time details for the box embedding model and other rector-based baselines in Table 11 in Appendix C
E max min X,, max X., maxX,,, 0)]
~ LS (LSE_s(T), ..iih....,,..... ,x2)) - LSE (2i.................... (2))...,,,)))))
essentially replacing the hard min and max operators with smooth approximation, LSE, (x) = t log(E; e///). Ex- pected intersection volume in higher dimensions is just product of the preceding equation, as the random vari- ables are independent. We use this GUMBELBox (ab- brev GB) formulation in our work changing the notations Fooz. Vol, VolInt to FGB, VolGE, Vollnteg. We modify the per-dimension score function Foo in (2) by replacing
3.3. Inference
During inference, given the trained embedding model & and a user u we determine the user's preference for the movie
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A Geometric Approach to Personalized Recommendation with Set-Theoretic Constraints Using Box Embeddings
m by negating and exponentiating the energy function, score(m, u; 0) exp(
conjunction clauses. Each clause is handled by computing the volume of the intersection of the involved box embed- dings. Importantly, the model is trained only on pairwise user-item and attribute- ciribute-item interactions, yet naturally ex- tends its mechanism to arbitrary, unseen structured logical queries at inference time.
( EGB (u, m,;))
D
II d=1
Fan (0(u),, U)) E R20,
Time complexity Each DNF clause requires computing the intersection of multiple boxes. For box embeddings, this is implemented via log-sum-exp (LSE) over coordinate- wise minima and maxima. For clause involving k variables (user or attributes), the intersection cost is (kD)), where D is the embedding dimension. To score a full Boolean query with T DNF clauses, the total complexity is O(TD). In the worst case, where all combinations of n variables appear in disjunction, T 2 However, real-world queries are usually structured as conjunctions and simple negations, leading to far fewer terms. Additionally, we parallelize the LSE computations across dimensions and query terms, enabling efficient batched evaluation of logical queries. Our codebase includes these optimizations.
where (()) = (#g. 22)- Since the calculation is simply a product over dimensions, for notational clarity we will restrict our discussion for more complex queries to the one- dimensional case, and omit the explicit dependence on tem- perature hyperparameters, so
scoro(m, u; u; = VollutGe (0(m), (u))
Volen (f(m))
which is the proportion of (m) which is contained within (u)) (see Figure 2). It achieves it's maximum at 1 if ((u) contains {(m), and is minimized at 0 when they disjoint, corresponding to the motivation that O(u) represents the set of movies that user u has interacted with.
Given a query with conjunction between attributes (e.g- "comedy and action")) we denote the attributes involved d1 and 02. Similarly to the score for a single user query, we define the score for these attributes as the proportion of the movie box (m) which is contained inside of the (soft) intersection of boxes (uu). ((11), and ((2)), i.e.
4. Experiments
In our experiments, we evaluate all the models on item rec- ommendation across three domains: movies, songs, and restaurants. (4.1). We systematically generate queries of varying complexity from these datasets evaluate perfor- mance on set-theoretic tasks (4.2.1, 4.2.2). We train and select models based on the performance of the traditional personalized item prediction (4.3). Finally, we demonstrate that our set-based representation method is better suited for handling set-theoretic constraints recommendation tasks (5.1, 5.2).
score(m, urai/az; 0) VolInten (0(m), (u)), (a)). O(a2)) Volca (0(m))
Again, this score is maximized if (()) is contained inside ((u), O(a1), and ((22), and minimized when it is disjoint. In order to address queries with set differences, recall that, compute the
given two measurable sets S and T, volume of S T as Vol(S' \ T') Vol(S) Vol(S n T). Thus, if the query involves negated attribute (e.g. ""coedyy and not action"),, we define
4.1. Dataset
The datasets used in our study must contain two pri- mary components: Item-User interactions Du and Item- Attribute interactions DA We select datasets that offer rich ground truth annotations for both components. We uti- lize the MovieLens 1M and 20M datasets for personalized movie recommendations (Harper & Konstan, 2015). For the song domain, we employ subset of the Last-FM dataset, which is the official song tag dataset of the Million Song Dataset (Bertin Mahieux et al., 2011). In the restaurant do- main, we use the NYC-R dataset introduced by (Wang et al., 2018).
score(m, u a1 -a2;8)= = VolInten (0(m), (()), (a1))
Voles (f(m)
VolIntga (0(m), f()), (()) (a1), (a2))
Volaa (0(m)
This score maximized when ((m) is contained inside (u)
and (az) while being disjoint from ((22), and decreases when these conditions are not met.
We utilize the data curated by Dasgupta et al. (2023) to construct D, for the Movielens data. This dataset employs Wikidata (Vrandežić & Krötzsch, 2014) to generate ground truth attribute labels for movies!. For the Last-FM dataset, the authors use the Last.fm API (' ('gerTopTags'' to create
Our containment-basee scoring framework naturally gen- eralizes to more complex logical queries involving arbi- trary Boolean combinations of attributes. By leveraging the inclusion-xxclusin principle, any Boolean query can be converted into Disjunctive Normal Form (DNF). For exam- ple, the score for complex query such as u A 01 V d2 A 0g 04 can be rewritten as a sum of scores over several
http://iiiicoooooooorrrrrsoaaaee hhtps://www.lss.fmm
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A Geometric Approach to Personalized Recommendation with Set-Theoretic Constraints Using Box Embeddings
attribute tags. Likewise, the authors in (Wang et al., 2018) crawl restaurant review data from TripAdvisor to curate
query types- u d1 aaz and n di n aa2 e.g, Bob want to watch an Action Comedy movie, Alice want to watch Children but not Monster movie. Creating meaningful at- tribute compositions requires careful consideration, as not all combinations make sense. For instance, 'Sci-Fi' & 'Doc- umentary' might not be a meaningful combination, whereas 'Sci-Fi' & TTim--Travel is. Similarly, 'Sci-Fi' -' Fiction' doesn't 't make sense, but 'Fiction' - 'Sci-Fi' does. Some- times, even if the intersection is valid, it could be trivial and non-interesting, e.g., 'Fiction' & 'Sci-Fi'.
tags and ratings for restaurants in NYC. The sparsity of DA and Dy is comparable in the Movielens datasets. In contrast, the La and NYC-R datasets, designed with tag annotations in mind, exhibit much denser attribute-movie interaction. Thus, the selection of these three datasets not only encompasses diverse domains but also offers varying ground-truth distributions for our experiments.
We use the binarized implicit feedback data (Hu et al., 2008), indicating whether the user or the attribute has been associ- ated with the specific item. To ensure the quality of the data, we retain users/items with 5 or more interactions and at- tributes with frequency 20 or more all the datasets. Refer to Table I for a detailed description of the dataset statistics.
Intuitively, for two attributes a1 & az their intersection is interesting if |a n a2l is greater than combining any two random items set. Also, for their intersection to be non- trivial the size of the intersection (a) Nazl must be less than the individual sizes of the attributes i.e. a plag] and a (a2 I. Here.]. denotes the size of the item set corresponding to the attributes. E [0, 1] is design parameter, dedicated after manual inspection of the quality of the item sets for the com- binations In case of difference queries such as n -a2, we consider a22 to be the second attribute and carry out the same filtering strategy as done for the intersection queries. We denote the set of non-trivial and viable attribute pairs for the intersection to be An = {(a1, |ai N 02l > e, |a N a2) < aalal,, (0g n Nag < <alaz }. and for the differ- ence to be A, = {(1), (2)]|||] O -aal > C, laj n- < a al0z). |a1 n -021 < <al-az|> |. Using the above formulation, we generate the test set for the personalized complex queries QUnA,NA, and QUnA, using algorithm 2. Please refer to Table 2 for the detailed statistics. The link for the dataset is available at https: /tttp::///ithbb com/ ssdasqupta/ set-based
4.2. Dataset Splits& & Query Generation
To select models for each method, we train on a dataset split Disin & Dirain while evaluating on a held-out set Disal & Deval However, use these eval set pairs to construct compositional queries. Simple random sampling or leave- one-out data splits do not ensure substantial number of these queries. Therefore, we devise a data splitting tech- nique closely linked to query generation, which we discuss next.
4.2.1. 1. PERSONALIZED SIMPLE QUERY
This type of query corresponds to single attribute for particular user, e.g. Bob wants to watch comedy movie, More formally, given user u and an attribute a, the query type would be- u O a. Note that, these simple queries are set-theoretic combinations between the item sets corre- sponding to the users and the attributes. Let us denote the data corresponding to these queries as QUnA-
4.3. Training Details & Evaluation Criteria
We train all the methods on users and attributes jointly using Dirain Diginn U Dirain We use dimensions d 128 for vector-based models, and d = 64 for box models so that the number of parameters per user, attribute, and movie is equal.* 4 We perform extensive hyperparameter tuning for the learning rate, batch size, volume and intersection temperature of boxes, loss combination constant, etc. Please refer to the Appendix A.2 for details. We follow the standard sampled evaluation procedure described in Rendle al. (2020), only for model selection purpose. For each user- item tuple (u, m) in Do the model ranks m amongst a set of items consisting of the m together with 100 other true negative items w.r.t the user. Then we report on two different evaluation metries namely Hit Ratio@ (HR k) and NDCG. (a) HitRatio @ k: If the rank of m is less than or equals to l then the value of HR @k is I or 0 otherwise. (2) NDCG: if r is the rank of m, then 1/ log(r + 1) is the NDCG.
While constructing the Quna pairs need to ensure that- if an item is held out for evaluation for sim- ple query. the individual user-item and attribute-item pair should belong to the evaluation set as well. More formally, (u, a, i) E QunA + (u, i) E Di A (a, E D To ensure this train/test isolation, we use the sampling al- gorithm 1 that takes in Dv and DA and outputs QUnA. Dirin, DDimin Deyal Dexal (Refer to Appendix A.1 for more details). The detailed statistics for the splits are provided in Table 1. Also, the statistics for the Quna are present in Table 2
4.2.2. PERSONALIZED COMPLEX QUERY
The set- theoretic compositions that we consider here are the intersection and negation of attributes for a particular user. Given a user u and attributes d₁ and az, we consider the
*Recall that box embeddings are parameterized with two vec- tors, one for each min and max coordinate.
hhtpss/wwww.rrrpaaviio.ccmmm
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A Geometric Approach to Personalized Recommendation with Set-Theoretic Constraints Using Box Embeddings
Table I: Datasct Statistics, the Item-User interaction Dy & the Item-Attribute interaction DA-
The Train/Test split is created using algorithm to test set-theoretic generalization.
#Train Du
#Eval Dv
#Train
#Eval
Dataset
#Users
#Items
#Attributes
DA
DA
Last-FM
1,872
2417
490
60,497
8,857
34,374
4,240
NYC-R
9,597
3764
579
82.734
8,502
34,908
4,376
MovieLens 1M
6,040
3,705
57
963,554
36,655
10,273
1,545
MoVIeLens-20M
138,493
26,744
95
19,722.646
277.617
80,178
1,734
The model is selected based on the best- best-performing model on NDCG for the item prediction over the user-item valida- tion set Diyal, with the best-performing checkpoint saved for further evaluation on compositional queries. We follow the same evaluation protocol for the compositional queries as well, except, we rank amongst all items in the vocabulary rather than a sampled subset.
Table 2: Compositional Query Statistics
Personalized Simple Query u a
Personalized Complex Query
Dataset
Maj 00²
u aMa m02
Last-FM NYC-R
9,867
45.142
10,814
9,482
7,460
2,369
MI-IM
21,392
51,299
37,769
4.4. Baselines
MI-20M
35,368
42,355
47,374
The recommendation systems literature offers wide range of methods that represent users, and items in Rd, These methods then propose a compatibility score function be- tween the user and item, Φ Idd x Rd + R. common and effective choice for @ is the dot product, which under- pins matrix factorization (Rendle et al., 2020; Koren & Bell, 2015). To capture more complex interactions among users, items, and attributes, (He et al., 2017) extend matrix factor- ization by replacing the dot product with neural network- based similarity function. This method, called Neural Ma- trix Factorization (NEUMF), combines the dot product with an MLP. Similarly, (He et al., 2020) propose LightGCN (LGCN) to captures the user, items, and attribute interac- tion using Graph Convolution Network (Kipf & Welling, 2017) over joint graph of verritem-attribure We use MF, and, NEUMF L.GN as our baselines.
each movie related using the sigmoid function. For box embeddings, the en- ergy function is normalized by conditioning on the movie box volume (see Section 3.3). The score for negation is calculated by subtracting the normalized score from 1. The three methods using this technique are referred to as Box- MOX-PRODUCT, MF-PRODUCT, NEUMF-PRODUCH and LGCN-PRODUCE
attribute are normalized
3. GEOMETRIC: This approach leverages the geometry of the embedding space. For vector- based embeddings, learned through Matrix Factorization, addition, and sub- traction are often used for query composition (Mikolov et al., 2013). Box embeddings, on the other hand, nat- urally represent intersection operations, allowing to compute scores for any set-theoretic combination using box intersection and inclusion-exclusic principles. We refer to these methods as BOX-GEOMETRIC and MF- GEOMETRIC.
For personalized query, be it simple or complex, we need to devise a method to combine the individual scores of the user and the attributes involved the query. In this work, we compare three approaches to obtain an aggregated score:
this approach, we retrieve a list of items corresponding to the attributes based on the scores pro- vided by the embedding models. The list is generated by thresholding the scores, where the threshold is optimized by minimizing the Fl score between the training data and predicted scores. We refer to the methods using this ag- gregation technique as BOX-FILTER for box embeddings and MF-FILTER, NEUMF-FILTER, LGCN- -ILN-FIIt vector- based methods.
FILTER:
5. Results
After conducting extensive Deyal, we select the oop-eerfrrmnn model for each method based on NDCG scores (see Table 6 in the Appendix for the model selection details). This ensures that the chosen model is optimal for set-thcoretic query inference, with the following performance results.
search on
5.1. Set- Set-Theoretic Generalization
2. PRODUCT: In this method, the compositional score is computed by multiplying the scores for the individual queries. For rector-based embeddings, the scores for
We test the selected models for each method with the curated set-theoretic personalized queries (Detailed stats for the

---

A Geometric Approach to Personalized Recommendation with Set-Theoretic Constraints Using Box Embeddings
Table 3: Hit Rate(5) f on Set-theoretic queries for datascts Last-FM MovieLens IM, NYC-R.
Table 4: Generalization Spectrum Gap for PERSONALIZED COMPLEX QUERY U MAAAA
Medhols
004
U0A,
Methodk
Hi Raic #50
Spectture Gap,
10
b0:0
h850
her0
ber00
ha:50
berto
bet50
Wiskest (W)
Waok--lser (W-U)
Werk-Amthuii:
Sr--hhomati (5)
(W-A)
(W Si/
LAST FM
MI FILTER
55.2
41,9
30.5
27.5
50.2%
MP PILTER
14.8
25.1
37.4
26.8
46.8
62.8
15.2
24.8
35.5
PRODUCT
67.4
38.5
39.3
26.1
MF-PRCOECT
9.0
21.7
48.0
14.3
56.8
73.2
4.8
14.8
43.4
GEOMETRIC
13.5
1.8
0%
95.615
MF DI--GEMMTTRIC
6.1
12.2
29.7
344
7.6
27.5
1.7
4.8
15.9
NEU CALMMF-PLLTE PILTER
48.4
33.1
40.4
35.9
58.5%
NeUME- FILTER
13.5
21.9
32.5
20.0
19.6
55.7
11.3
18.8
28.2
NEUME PRODUCT
.6.8
48.7
40.5
43.5
35.916
NEUMP PRODUCT
13.6
47.6
19.5
63.3
40.5
Box-Fu vea
52.7
44.5
30.3
18.5
45.9%
20.5
80.7
21.5
13
ROX-PREDDCC
BON- CON-GGOMMTTRICC
64.6
62.6
53.3
50.1
39.0
46.1
34.2
47.1%
26.4%
BOx. FILTER
22.9
31.5
39.0
52.7
46.5
55,9
22.0
32.1
40.3
BIN-PRODUCT
27.9
44.5
68.0
38.2
57.7
82.7
17.8
32.4
68.3
BON- GBOMETRIC
25,3
44.8
68.3
38,8
58.3
93.1
17.5
32.5
60.0
MOvIeLaNS-18 IM
as l increases, its performance declines across all model types (Box, MF, NeuMF) and datasets. This observation highlights the limitation of fixed threshold filter and advo- cates smoother aggregation techniques like PRODUCT and GEOMETRIC.
MF-FILTER
5.0
10.2
22.5
11.4
17.9
27.5
4.7
9.8
22.5
ME-PRONICT
4.3
8.5
20.4
5.1
10.6
26.1
3.4
7:3
19.3
MP GEOMETNIC
0.4
09
3.0
0.2
01.9
00.5
.27_
MEUMFFFITTRR
35.9
26.7
MOUMF-PaGOOCT
10.3
16.8
31.4
24.5
9.7
20.2
LGCN PILTER
822
12.5
20,9
11.4
15.6
24.0
9.9
13.8
21.9
PRCN-PRIDUCT
5.9
9.0
14.9
7.6
11.7
20.1
3.5
8.6
14.1
Box- FILTER
11.7
19.1
32.5
14.5
20.5
28.6
11.4
19.5
$4.0
BOx- PRODUCT
9.95
16.7
M1.
10.6
17.8
34.2
8.9
15.1
29.4
5.2. Spectrum of Generalization
11.0
34.2
46.1
31.0
NYC.E. R
The query generation process (refer Section 4.2.1) en- sures that for the target item m corresponding to query involving user u and attribute a, the pair (u, m) and (a,m) m) must not be in the training set (u,m) m) Diain and (a, m) & Digin The set-theoretic evaluation weak- e
ME PILTER
1.4
2.4
4.6
2.7
4.8
8.0
2.1
3.5
6.3
MI-PRODUCT
2.9
23.3
13.1
17.6
MF GHOMETRIC
1.5
012
3.5
12
3.7
NoUME- MILTER
388
5.6
9.2
2.5
3.2
4.5
4.2
63
10.8
NEUME PRODUCT
4.6
7.3
13:7
5.6
11.2
20.8
2.7
52
11.2
LGON- FILTER
488
788
17.2
12.2
16.9
21.8
5.4
8.6
16.4
5.0
87
18.1
12.1
17.6
35,1
4.9
800
132
ens when such pairs are added back to the training set. There are three different weakening settings applicable here, which we refer to as a spectrum - WEAKEST GEN- ERALIZATION ((,, m) E Dijin and (a, m) E DiA WEAK GENERALIZZAIINN-UE (a, & Dimin), WEAK
BOX-FITTER
4.9
788
13.4
9.9
13.5
20.4
4.4
7.1
12.5
BON-PEDDUCT
188
GEOMETXIC
17.6
(u.mm) Diyal and ((,,m) & Diginn and (a,m) E D We report Hi- tRate@50 50 performance on query type U n A n Ag for the Movielens- I IM dataset in Table 4 (More query types in Appendix- Table 9, 8).
queries in Table 2). We report the ranking performance in terms of Hit Rates at 10, 20, and 50. Please refer to for the results.
The Box Embedding- based method outperforms vector- based methods by a significant margin, showing av- erage 30% improvement when comparing the aggregated HR @50 performance of the best vector model (MF- to the box model (Box- GEOMERRC))) across all the three different domains.
The weaker the generalization setting the easier is for the models to achieve higher performance on the test set. In- deed, we observe that this is true across all the methods w.r. each of the aggregation settings, validating the correctness of the trained models.
However, we are interested in observing the performance gap when we go from the weakest to the strongest set- theoretic generalization. refer to the percentage gap Gen- eralization Spectrum Gap (hr(Weakest)
The UNA,NA, query is the most challenging, it requires accuracy in all three individual queries. For this difficult query, BOX-GEOMETRE shows the largest performance gap compared to other methods. Additionally, using vector addition and subtraction as geometric proxies for intersec- tion and difference performs significantly worse than all other rector-based methods, while geometric operations in the box embedding space outperform even other box embed- ding methods. This validates the set-theoretic inductive bias of box embeddings and confirms that geometric operations in this space provide valid set-theoretic operations, unlike vectors.
hr(Set-theoretic) hr hr(Weaeest) %). From Table 4 we observe that the best- performing box model Box- GEOMETRIC achieves the best Generalization Spectrum Gap for HR HR@50.
6. Related Work
6.1. Box Embeddings
Some of the recent works have tried to incorporate box embeddings in a recommendation systems setup.Xu et al. (2024); Wu et al. (2024); Zhang et al. (2021) use the side- length of the box embeddings as a preference range to obtain
The FILTER aggregation technique performs similarly to or better than other methods only for Hits @ 10. However,
8

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A Geometric Approach to Personalized Recommendation with Set-Theoretic Constraints Using Box Embeddings
diverse recommendations for users, Mei et al. (2022a) utilizes the axis parallel nature of the box embeddings for faster retrieval. Sun et al. (2020a:b); Ren et al. (2020) are some of the recent works that focus on logical query over knowledge bases (KB). However, in this work, we frame collaborative filtering as a set-theoretic matrix completion problem, which helps us to achieve better generalization for the composition of personalized queries.
have compositional properties through vector arithmetic, for example in the empirical analysis of word2vee (Mikolov et al., 2013) and GLOVE (Pennington et al., 2014), and some theoretical analysis (Levy & Goldberg, 2014; Arora et al., 20. However, anecdotally, many have found that the compositional behavior of vectors is far from reliable (Rogers et al., 2017). Our paper provides comprehensive evaluation of vector embeddings on compositional queries and compares the results to region-based alternative.
6.2. Set-based queries in Search and group recommendation systems.
7. Conclusion & Future Work
While set-theoretic queries are commonplace in search, pop- ular question-answering (QA) benchmarks often do not in- clude them. We found QUEST (Malaviya et 2023) the most closely related study, introducing a benchmark for entity-seeking queries with implicit set-based semantics. However, QUEST does not focus on explicit constraints or personalization, which are central to our work.
In this work we presented the task of personalized recom- mendation with set-theoretic queries. We discussed how this problem can be viewed as set-theoretic matrix completion, and why the common approach of logistic matrix factoriza- tion is not aligned with the set-theoretic operations we wish to perform at inference time. We observed substantial im- provements over the vector/neural baselines when using box embeddings as the representation, validating our intuition regarding the necessary set- ottthhoreetic bias. Our empirical results confirm that box embeddings are ideally suited to the task of recommendation with set-theoretic queries.
6.3. Context Aware Recommendation
The concept of context-aware recommendation, as intro- duced in (Adomavicius et al., 2011), provides a general framework where "context" is broadly defined as any aux- iliary information. This framework emphasizes that user preferences for items can vary based on the context in which interactions occur, reflecting a user-centric view of contex- tual information.
In real-world recommendation systems- such as stream- ing platforms, e-commerce sites, or travel services - free- text queries (e.g., "funny action movies without clowns") are typically mapped to a curated set of item tags (e.g., genre, theme, metadata) via a natural language understand- ing (NLU) module. Our model operates downstream of this step, assuming structured query (e.g., Action A Comedy A Clowns) has already been derived. While our current focus is on the execution of structured set-theoretic queries, future work could explore tighter integration with front-end NLU systems. Large language models (e.g.. GPT-4o) have demonstrated strong performance in parsing natural lan- guage constraints, and we view such models as complemen- tary: performing query parsing and attribute identification, while our method serves as reliable and efficient back-end executor for the resulting set-theoretic logic. Bridging the gap between these stages offers a promising direction for building end-to-end systems that are both expressive and controllable.
Building on this foundation, recent works have explored specific instances of context-aware recommendation, such as "attribute-aware recommendation." " These approaches often leverage item or user attributes as contextual infor- mation to address various goals, including improving user profiling (Adomavicius et al., 2011 1), predicting missing item attributes (Wu et al., 2020; Chen et al., 2022), enhanc- ing recommendations for cold-start scenarios(Deldjo et al., 2019), or providing attribute- based explanations for recom- mendations (Xian et al., 2021).
Our work differs significantly in its focus and objectives. term "attribute-constrainee recommendation." which involves generating recommendations explicitly constrained by logical combinations of attributes. Unlike attribute-aware approaches, which aim to improve recommendation quality by incorporating attribute information as auxiliary data, our work directly targets the task of satisfying explicit attribute- based constraints posed by users.
As noted in Section 3.3, our model supports efficient eval- uation of complex queries. We construct benchmark of semantically plausible queries using statistical heuristics and manual filtering (Section 4.2.2), ensuring realistic and diverse combinations. While this allows us to test com- positional generalization, curating a benchmark of natural user-issued queries remains an important direction for future work.
6.4. Compositional Queries with Vector Embeddings
common in machine learning to represent discrete en- tities such as items or attributes by vectors (Bengio al., 2013) and to learn them by fitting the training data. Besides semantic similarity, some have claimed that learned vectors
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A Geometric Approach to Personalized Recommendation with Set-Theoretic Constraints Using Box Embeddings
Acknowledgments
References
The authors would like to thank the members of the In- formation and Extraction Synthesis Laboratory (IESL) at UMass Amherst, Steffen Rendle, and Zhang, for helpful discussions. This work was supported by IBM Research AI through the AI Horizons Network and National Science Foundation (NSF) under the Grant Numbers IIS-2106391. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of IBM NSF.
Adomavicius, G., Mobasher, B., Ricci, F,, and
Tuzhilin, A. Context-aware recommender systems.
AI Magazine, 32(3):67-80, Oct. 2011. doi: 10.1609/aimag..322..222
URL
Arora, S., Li, Y., Liang, Y., Ma, T. and Risteski, A. Linear algebraic structure of word senses, with applications to polysemy. Transactions of the Association for Computa- tional Linguistics, 6:483 6:483-495. 2018.
Impact Statement
Bengio, Y., Courville, A., and Vincent, P. Representation learning: A review and new perspectives. IEEE transac- tions on pattern analysis and machine intelligence, 35(8):
This paper aims to advance the field of Machine Learn- ing by introducing a geometric approach to personalized recommendation under set-theoretic constraints. Our pri- mary contribution methodological, focusing on improving representation learning for structured preference modeling- While recommendation systems have broad societal reach and their deployment may influence user behavior, faimess, or exposure to information, this work does not involve di- rect deployment or sensitive user data. As such, we do not societal risks
1798-1828, 2013.
Bertin-Mahieux, T., Ellis, D. P., Whitman, B., and Lamere, P. The million song dataset. In Proceedings of the 12th International Conference on Music Information Retrieval (ISMIR 2011), 2011.
Boratko, M., Patel, D., Dasgupta, S. S., and McCal- lum, A. Mcasure-theoreth set representation learn- ing. preprint from https://wet http::/wwwwmbbbatteeee www.atko.commooomomoooooooooo mtsrl. pdf, 2022.
identify any immediate or domain-spectie associated with this research. Nonetheless, we acknowledge the importance of responsible use and encourage future ap- plications of our method to consider faimess, transparency, and user control as core design considerations.
Chen, L., Cao, J., Wang, Y., Liang, W., and Zhu, Multi-view graph attention network for travel recommendation. Expert Systems with Appli- cations, 191:116234, 2022. ISSN 0957-4174. doi: http://o..or//0000b///ss.......0 URL hatemee//rrraceee//////997722222
Dasgupta, S., McCallum, A., Rendle, S., and Zhang, L. Answering compositional queries with set-theoretic em- beddings, 2023.
Dasgupta, S., Boratko, M., Zhang, D., Vilnis, L., Li, X. I.,, and McCallum, A. Improving local identifiability in probabilistic box embeddings. In Advances in Neural Information Processing Systems, 2020.
Deldjoo, Y., Ferrari Daerema, M., Constantin, M. G., Eghbal-Zadeh. H.. Cereda, S., Schedl, M., lonescu, B., and Cremonesi, P. Movie genome: alleviat- ing new item cold start in movie recommendation. User Modeling and User-Adapted Interaction, 29(2): 291-343, April 2019. ISSN 0924-1868, doi: 10.1007/ sl $11257-019-09221-- URL https: / /dwi. 100 1007 01t1///11157-11990999
Harper, F. M. and Konstan, J. A. The movielens datasets: History and context. ACM Trans. Interact. Intell. Syst., (4):19:1-19:19, December 2015. ISSN 21 2160-6455. doi:
10

---

A Geometric Approach to Personalized Recommendation with Set-Theoretic Constraints Using Box Embeddings
10.1145/2827872. URL hktp::p///doi. htt.:/iii.c..orr///// 1145/2827872.
doi: 10.11455346554473351200.. URL https://doo.ii 35112073.
org/10
He, X., Liao, L.,, Zhang. H., Nie, I.,, Hu, X., and Chua, T.S. Neural collaborative filtering. In Proceedings of the 26th International Conference on World Wide Web, WWW pp. 173-182, Republic and Canton of Geneva, CHE, 2017. International World Wide Web Conferences Steering Committee. ISBN 978 3781450349130 doi: 10. 1145/3003891 URI. https:/ /doi.org//////oooo/o////o/oooo 3052569.
Mei, L., Mao, J., Guo, G., and Wen, J.-R. Learning proba- bilistic box embeddings for effective and efficient ranking. In Proceedings of the ACM Web Conference 2022, WWW '22, Pp. 473-482, New York, NY, USA, 2022b. Associa- tion for Computing Machinery. ISBN 9781450390965. doi: 10.1 10.1145/34854443355220000 org/10. 3455533334444777777.... 3512073.
URL http::://///ece /doi.
10.
Mikolov, T., Chen, K., Corrado, G., and Dean, J. Efficient estimation of word representations in vector space. arXiv preprint 1rXiv:1301.3700 2013.
He, X., Deng, K., Wang. X., Li, Y., Zhang. Y., and Wang, M. Lightgnn: Simplifying and powering graph convolution network for recommendation. In Proceedings of the 43rd International ACM SIGIR Conference on Research and Development in Information Retrieval, SIGIR '20, pp. 639-648, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450380164. doi: 10. 0445/3397271.4400066 URL https: /tti:ddooooroooo///////..//oo/ot 10.1145/3397271..1 063.
Pennington, J., Socher, R., and Manning, C. D. Glove: Global vectors for word representation. In Proceedings of the 2014 conference on empirical methods in natural language processing (EMNLP), pp.
2014.
Ren, H., Hu, W., and Leskovec, J. Query2box: Reasoning over knowledge graphs in vector space using box beddings. In 8th International Conference on Learning Representations. OpenReview.net, 2020.
Hu, Y., Koren, Y., and Volinsky, C. Collaborative filtering for implicit feedback datasets. In Proceedings of the 2008 Eighth IEEE International Conference on Data Mining, ICDM '08, pp. 263-272, 2008
Rendle, S., Krichene, W., Zhang, L., and Anderson, J. Neural collaborative filtering vs. matrix factorization re- visited. In Proceedings of the 14th ACM Conference on Recommender Systems, RecSys '20. pp. 240-248, New York, NY, USA, 2020. Association for Comput- ing Machinery. ISBN 9781450375832. doi: 10.1145/ 3383313..412288.. URL https://doi. orr//00. 1145/3383313 3412488.
Kipf, T. N. and Welling, M. Semi-supervised classi- fication with graph convolutional networks. In In- ternational Conference on Learning Representations, 2017. URL
id=sJU4ayYgl.
Koren, Y. and Bell, R. Advances in Collaborative Filtering, pp. 77-118. Springer US Boston, MA, 2015. ISBN 978-1-4899-7637-6.
Rogers, A., Drozd, A., and Li, B. The (too many) problems of analogical reasoning with word vectors. In Proceed- ings of the 6th Joint Conference on Lexical and Com- putational Semantics (*SEM 2017), pp. 135-148, Van- couver, Canada, August 2017. Association for Compu- tational Linguistics. doi: 10.186531y11111--007,, URL wwtp://org/////1101011
doi: 10.1007/ 18-41--976377--6 3. URL uttpt::s: //tpi.oog/o......////oo/// 1007/978-1-44 74379-763776_..
Levy, O. and Goldberg, Y. Neural word embedding implicit matrix factorization. Advances in neural infor- mation processing systems, 27, 2014.
Sun, H., Arnold, A. O., Bedrax-Weiss, T., Pereira, F,, and Cohen, W. W. Faithful embeddings for knowledge base queries. In Proceedings of the 34th International Conference Neural Information Processing Systems, N0, Red Hook, NY, USA, 2020a. Curran Associates Inc. ISBN 97817 13829556..
X., Vilnis, L., Zhang, D., Boratko, M., and McCallum, A. Smoothing the geometry of probabilistic box embeddings. ICL.R, 2019.
Malaviya, C., Shaw, P., Chang, M.-W., Lee, K., and Toutanova, K. Quest: A retrieval dataset of entity- seeking queries with implicit set operations, 2023. URL org/ abs/ / 2005 11694.
Sun, H., Amold, A. O., Bedrax- T., Pereira, F., and Cohen, W. W. Guessing what's plausible but remember- ing what's true: Accurate neural reasoning for question- answering. 2020b.
https: /tatii:///rxvv....// arxiv.
Mei, I.., Mao, J., Guo, G., and Wen, J.-R. Learning proba- bilistic box embeddings for effective and efficient ranking. In Proceedings of the ACM Web Conference 2022, WWW '22. pp. 473-482, New York, NY, USA, 2022a. Associa- tion for Computing Machinery. ISBN 9781 9781450390965.
Vilnis, L., Li, X., Murty, S., and McCallum, A. Proba- bilistic embedding of knowledge graphs with box lattice measures. In Association for Computational Linguistics, 2018.

---

A Geometric Approach to Personalized Recommendation with Set-Theoretic Constraints Using Box Embeddings
Vrandečić, D. and Krötzsch, M. Wikidata: free collabo- rative knowledgehase. Communications of the ACM, 57 (10):78-85, 2014.
21, pp. 716-724, New York, NY, USA, 2021. Associa- tion for Computing Machinery. ISBN 450349999
doi: 10.1 10.11457344796333441688 URL https: /doi.
Wang, X., He, X., Feng, F., Nie, L., and Chua, T.- S. Tem: Tree-enhanced embedding model for ex- plainable recommendation. In Proceedings of the 2018 World Wide Web Conference, WWW '18, pp. 1543-1552, Republic and Canton of Geneva, CHE, 2018. International World Wide Web Conferences Steering Committee. ISBN 9781450356398. doi: 10.1145/ 3178876.3186066. URL https: /tt////d...t// or0/00. 1145/3178876. 3186066..
Wu, C., Shi, S., Wang, C., Liu, Z., Peng, W., Wu, W., Kong, D., Li, H., and Gai, K. Enhancing recom- mendation accuracy and diversity with box embedding: A universal framework. In Proceedings of the ACM on Web Conference 2024, WWW '24, pp. 3756-3766, New York, NY, USA, 2024. Association for Comput- ing Machinery. ISBN 9798400701719. doi: 10. 145/
3589334.3645577. URL https: //dei. 10.
Wu, L., Yang, Y., Zhang, K., Hong, R., Fu, Y., and Wang, M. Joint item recommendation and attribute inference: An adaptive graph convolutional network approach. In Proceedings of the 43rd International ACM SIGIR Conference on Research and Develop- ment in Information Retrieval, SIGIR '20, pp. 679-688, New York. NY, USA, 2020. Association for Comput- ing Machinery. ISBN 9781 9781450380164, doi: 10.1
3397271.34011 3401144.
URL
1145/3397271.34011
Xian, Y., Zhao, T., Li, Chan, J., Kan, A., Ma, J., Dong, X. L., Faloutsos, C., Karypis, G., Muthukrishnan, S., and Zhang, Y. Ex3: Explainable attribute-aware item- set recommendations. In Proceedings of the 15th ACM Conference on Recommender Systems, RecSys '21, pp- 484-494, New York, NY, USA, 2021. Association for Computing Machinery. ISBN 9781 7781445384582. doi: 10. 114573460233..4442400. URL https: /tw./cc..oooooooooooo orgg 10. 10.1145/346022 3474244
Xu, Z., Qu. Y., Zhang, W., Liang, L., and zeng Chen, H. Inbox: Recommendation with knowledge graph us- ing interest box embedding. ArXiv, abs/2403.12649, 2024. URL https: httpp:/:/app..smaaiiieeseeeeeeee semanticscholar. org/Corpus ID: : 258532286.
Zhang, S., Liu, H., Zhang, A., Hu, Y., Zhang, C., Li, Y., Zhu, T., He, S., and Ou, W. Learning user rep- resentations with hypercuboids for recommender sys- tems. Proceedings of the 14th ACM International Conference on Web Search and Data Mining, WSDM
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A Geometric Approach to Personalized Recommendation with Set-Theoretic Constraints Using Box Embeddings
Table 6: Test NDCG on Di for selected models.
Algorithm 2 PERSONALISED COMPLEX QUERY Genera-
tion Algorithm
Dataset
MF
NEUMF
LGCN
Box
1: Compositional Query sets QUNAinAS QUnAin-Az
Last-FM
0.51
0.52
0.56
0.65
2: Non-Trivial attribute combination set A.
NYC-R
0.31
0.33
0.37
0.39
3: for each user-movie tuple in Eval set, i.e. (u, m) E UUEeat
ML-IM
0.51
0.53
0.55
0.58
{(u, ( = 1} do for each pair of attributes (a1, a2)
ML-20M
0.71
0.70
0.72
0.73
{(a1, A
and A = =1 do
5: if the pair is viable and non-trivial, i.e. (a1, (2)
A. Experiment Details
An then PEnAiAA,NNERRTT 01 Q2, m))
6:
A.1. Data Splits & Query Generation
7:
end if end for
8:
Algorithm 1 PERSONALISED SIMPLE QUERY (u a) gen- eration algorithm aa
9:
for each pair of attributes (a), a2) E
{(a a2) and Aaz.m = =0} do
A then 10:
if the pair is viable and non-trivial, i.e. (a1, (2)
1: Let the of users, attributes, and movies be A,
2: Marginal probability of an attribute in A, P(a)
11:
12: end if INSERTT au, 02,
Σm Aa.m/ Eo Em-Aa Apr.m
3: Marginal probability of an user u in U. P(u) =
13: end for
Em /a E.e Em Un'm
14:
end for
4: Marginal probability of an movie mn in U, P(m)
Σn / En, Em Un.'
5: Let U be the User Item matrix and A be the Attribute
x Item matrix.
A.3. Model Selection
6: Uraan &- U, ATaan < A
7: EEeal e 0, A Eval + 0
A.4. Set-Theoretic Generalization
8: Set of simple personalized queries, Q + &
9: while [Qural= < MAX SAMPLE SIZE do
Table 7: Hit Rate(%) for Set-theoretic queries for dataset ML-20M.
10: Sample an attribute a from A according P(a).
11: Sample a movie m from for the attribute a, i.e., Sam-
ple from I' 1}, according to P(m) Sample user from who has rated movie m, i.e.,
Methods
Und
UnA,
12:
ME-FILTER
8.1
16.1
0.4
2.9
13.7
Sample from Ju'[Um, = 1}. according to P(u)
MF- PRODUCT
4.1
7.5
15.6
3.3
6.6
16.4
2.7
5.1
11.4
MF-CROMETEII
0.1
03
0.6
0.0
0.0
0.0
0.3
0.6
1.4
13:
UTrain = 0, ATrain = 0, U Fual QUNA.NSERTT(U,,. m))
a.m. = 1, A Eval =
a,m
NEUME-
4.6
82
16.1
1.1
5.6
6.4
4.9
7,3
13,9
14:
NEUME PRODUCT
82
16.1
4.1
6.9
13.0
16.1
21.8
15: end while
BOX PRODUCT
82
16.1
21.8
7.1
BOX- GRONSTRIC 4.5
81
162
11.0
21.8
42.4
6.4
128
25.9
A.2. Training Details
A.S. Spectrum of Weak Generalization
Table 5: Hyper Parameter range for all the dataset. We run 100 runs for both models and select the best model on Aser-Movie validation set NDCG metric
Table 8: The spectrum of generalization for SIMPLE PERSONALIZED QUERY U n A. W: WEAKEST
GENERALIZATION, W-U: WEAK
Hypepannneess
Range Hox
Beet Valuo
Box
Range Watter
Bie Value Vezter
GENERALIZATION-USEE W-A: WEAK
Eateatlnn dim
125
GENERALIZATION- ATTRIBUTE, S: SET THEORETIC GENERALIZATION
Leaming
Io-i, Ie-2, Io-1, Ie-4, 1o-5
6001
Io-1 Io-2, 1e-3,
0.001
Butch Size
64, 128, 256, 1826
64, 128,
1024
Neuatives
L.
1.
Intesestion Teme
10,2.11
1e Je-3
2.0
Andiute coast
Volune Tomp
10.
0.3,
0.000
0.01
0.1.3....
Methods
WTWU
Hit Rate 1
WTW WAASS
Hi Rate OF 20
WT WATTS
HiERaee OF 30
MI-FILTER
3471671130130
363T13.3 2077333
542T30TT TEE3 122.3
MF-PRONSCT
233157188144
S47124 1501108121 20.4 15.1141774130
MF- GROOLIIMI
40109118 U.A
7911.1333199
Hyperparameters are reported in Table 5. Best parameter values are reported for Box Embeddings and MF method.
BoX-FILTER
30.3 MIST33.6 13333"
Box: PRODUCT
2521113.61
(2.2121
18. 11.5
25.4114.77 14.0
35.6 15.6123.11 2
52.21 41181 40.5 34.1
13

---

A Geometric Approach to Personalized Recommendation with Set- The-recccc Constraints Using Box Embeddings
utribute_joss_c
banch
embedsing.dm
internctior.
volume,temp
fandun
1,000
0.60
0.7
032
702
0.5
0.2
432
0.3
0.1
0.10
Figure Parallel Co-ordinate plot for different hyperparameters vs model performance. Lighter the color, better the model's performance.
Table 9: The spectrum of generalization for COMPLEX PERSONALIZED QUERY U n A1 n A2- W: WEAKEST
Kisakeal Chee Makionn
Waaak Ceneralicl on User Generalina
GENERALIZATION, W-U: WEAK
GENERALIZATION-USER W-A: WEAK
GENERALIZATION-ATTTIBBE S: SET THEORETIC
GENERALIZATION
WIWETWAAIS
WINTTWWA
MI FILTER
25311501124443
METTAT 193T93
S1.12931 1223"
ME-PRODECTT
Box- SOX-FIETU
24.1115:1110111
25312272211
Box- PRODUCT
21.1113711201199
30512177 19.31152
47.118.011 35.1 29.4
BOX-GUMNNITREE
Figure 4: Weak Generalization Illustration
Table 10: The spectrum of generalization for COMPLEX
PERSONALIZED QUERY U n A1 n Ag. W: WEAKEST
Geometric Success
Product Sucess
GENERALIZATION, W-U: WEAK
GENERALIZATION
W-A: WEAK
8975
GENERALIZATION- ATTRIBUTE S: SET THEORETIC GENERALIZATION
Hi
18500
WIWUTWAIS
WIWUTWAIS
WIWUTWAIS
ME-PLIER
153113661191112 11.
45.127.3
25.31
55.144.91 SUS 22.5
ME-PRODECT
MF--GEOMITTIC
Box- FILTER
30.1215511731146 14.5
41.11312
23.12205
52.714455
2135
PRODUCT
35.4123813441 10.6
47.014.5
21.11178
61.6152.1 39.0144.
140.0
5041
Filter Success
The Box- NOX-GEOMETRIC achieves the best Generalization Spectrum Gap for all types of queries.
Figure 5: Relationships of correct answers by the three box models on a1 A a2 queries.
B. Error Compounding Analysis
We further perform more granular analysis amongst the Box based methods with complex query type U n A Ag.
As claimed in our initial hypothesis, the FILTER method suffers from error compounding. the target movie m
14

---

A Geometric Approach to Personalized Recommendation with Set-Theoretic Constraints Using Box Embeddings
C. Time Efficiency analysis
nox-Geometric
Box-Product
Table 11: Training time (mmnss)) for single epoch are measured for different batch sizes with 5 negative samples on Movielens- 1M dataset. Experiments are conducted on Nvidia GTX 1080Ti gpus
3556
20408
Batch Size
MF
NEUMF
LIGHTGCN
Box
64
08:37
17:00
70:30
19:32
7067
128
04:32
09:46
38:40
11:40
2964
256
02:29
04:40
20:55
05:28
512
01:18
02:23
10:47
02:54
1024
00:40
01:20
05:24
01:12
18760
In Table 11, we observe that the MF, being the simplest ap- proach with minimal computational requirements, is consis- tently the fastest across all batch sizes. At the largest batch size (1024), it achieves the shortest training time of just 00:40. The Box-based method exhibits training times com- parable NEUMF. However, is significantly faster than LIGHTGCN, which relies graph convolutional compu- tations. The iterative message-passing operations required by LIGHTGCN result in considerably higher training times, particularly at smaller batch sizes (e.g., 70:30 at batch size of 64). As the batch size increases, training time for Box embeddings becomes almost as efficient as MF. For in- stance, a batch size of 1024, Box achieves a training time of 01:12, compared 00:40 for MF. This demonstrates that the computational complexity of box embeddings is of the same order as MF, making it a scalable and efficient choice.
Box-Filter Compounding Error
Figure 6: The Geometric method subsumes the benefit of the product in compounding error.
Box-Geometric
Box-Product
5587
994
26345
Box embeddings are generally quite fast because the compu- tation of box intersection volumes can parallelized over dimensions. Note that the training times above use Gumble- Box embeddings, which involve log-sum-exp calculations. However, this could be improved even further inference time by replacing these soft min and max approximations with hard operators. If such an optimized approach is de- sired, then training can accommodate this by regularizing temperature. For deployment in industrial set-up, we could take additional steps with Box Embeddings as outlined in (Mei et al., 2022b).
1130
933
9342
Box-Filter not Compounding Error
Figure 7: The effect is less for the non-compoudding error.
in the model's prediction list for A₁ but not for Ag or the other way round, we denote this error as compounding error. In figure 6, out the compounding errors, 34% solved by the Box- GEOMETRIC method and 26% by the Box- PRODUCT method. However, in figure for the error that is not due to compounding (where the model gets both A and A2 prediction wrong), only 18% are corrected by the Box- NOX-GEOMETRIC method and mere 10% of them are cor- rected by SOX-PRODUCT Refer to figure 6 7 for details. This demonstrates that the Box- GEOMETRIC significantly contributes to the correction of error compounding.
15


Reviews:
[{"review_id": "b3j1gwopNF", "reviewer": "Reviewer_onQN", "summary": "This paper proposes a box embedding-based recommendation model that captures set-theoretic constraints in personalized recommendation tasks. Traditional vector-based embeddings struggle with expressing complex relationships like intersection and negation (e.g., \"comedy and action but not romance\"). The paper introduces an approach using box embeddings—n-dimensional hyperrectangles—to model user preferences and item attributes in a way that naturally supports set-theoretic operations.\n\nThe key contributions include:\n\nFormulating recommendation as set-theoretic matrix completion, where user-item interactions are represented via geometric set operations rather than linear factorization.\nBox embedding representations that allow efficient and faithful modeling of intersection, negation, and containment relationships.\nA novel training method that optimizes embeddings using a probabilistic volume function, ensuring semantic consistency with set-theoretic operations.\nExtensive empirical evaluation on MovieLens, Last.fm, and NYC restaurant datasets, showing that box embeddings outperform matrix factorization (MF), neural matrix factorization (NeuMF), and graph-based methods (LightGCN) by up to 30% in set-theoretic query performance.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewers for raising thoughtful and forward-looking questions regarding possible extensions of our approach. \n\nWe agree that a recommendation systems framework should be recognized not only for its current performance but also for its extensibility to address practical challenges such as deployability, handling cold start problem, content-aware recommendation, and evolving user behavior etc. While we focus our paper on a clearly defined core contribution of handling set-theoretic constraints, we touch upon these broader aspects in the rebuttal to highlight the extensibility and practical relevance of our framework. We will include these discussions in the final version of the paper.\n\n**Box operation in high dimensions and Deployment in a search system:**\n\nBox embeddings remain computationally efficient even in high dimensions, as box intersection volume calculation can be parallelized across dimensions. Our method uses GumbelBox embeddings during training, where soft intersections are computed via log-sum-exp operations, however inference speed can be further improved by substituting soft approximations with hard min/max operations, providing faster evaluation at deployment time.  This is achievable by annealing the Gumbel temperature to promote sharper boundaries during training (Capacity & Bias). Due to these advantages, box embeddings can actually provide even faster inference than approximate k-NN algorithms on vectors, as demonstrated in Mei et al. (2022b) which significant improvement over FAISS.\n\n**Content-aware recommendation / Handling Text image as side information / Cold start problem.**\n\nIn this work, we treat genres, tags, and metadata as item content, making our setup inherently content-aware with respect to structured attributes. We understand the reviewer’s interest in whether our box embedding framework can be extended to richer content modalities such as text or images.\n\nA practical approach is to encode text/image features using pretrained models like BERT or CLIP, and then project them into the box embedding space via a learnable transformation. Prior work supports this, to give a few examples; Onoe et al. (2021)  derived box embeddings for entity mention representation from BERT,  Rau et al. (2020) use ResNet50 features to construct box embeddings for visual scenes. Such multimodal integration would enable our model to handle unstructured content and is particularly promising for cold-start scenarios.\n\nWhile our current method assumes some user/item interactions during training (as in standard collaborative filtering), we are actually uniquely well-suited to handle cold-start settings. To do so, we could “pretrain” item representations such that they are contained within their attribute boxes. In typical cold-start fashion, a new user would generally provide a list of like/dislike annotations on attributes, which would directly correspond to set-theoretic operations on the attribute box representations.\n\n**Evolving user preferences over time:**\n\nOur current work does not explore incremental or online updates to user or item representations. However, our method can operate within the standard recommendation framework, where embeddings are typically retrained periodically using updated user-item and item-attribute interaction data. This batch retraining paradigm is widely adopted in practical systems, where new interactions are accumulated over time and incorporated during scheduled model updates. While we have not implemented online or streaming updates in this work, such extensions could be supported by fine-tuning user or item embeddings on recent interaction data.\n\n**Recommendation without constraint (implicit setting):**\n\nIn the implicit setting, the system defaults to using the user's historical preferences — captured via the user box learned from past interactions — as the query itself. This allows our model to function like standard collaborative filtering approaches, recommending items based on geometric containment between the user box and item points in the latent space.\n\n**Comparison to hybrid graph-based approaches:**\n\nWe acknowledge that many recent *attribute-aware* recommendation models use Graph Convolutional Networks (GCNs) to capture item-user-attribute interactions. In our work, we include **LightGCN** (He et al., 2020) as a baseline and adapt it to a *hybrid graph* where items are connected to both users and attribute tags as a form of graph. This ensures attribute information is incorporated, aligning with our study’s objectives. We agree that such hybrid graph-based methods are strong baselines and may offer partial inductive biases for compositional reasoning. Note, however, that our method outperforms LightGCN by >40% on average over all the query types.\n\nIf the reviewer has a specific hybrid graph-based method in mind that supports logical constraints or compositional queries, we would be happy to review and discuss it in the final version.", "claims_and_evidence": "Claim 1: Box embeddings better model set-theoretic constraints compared to vector-based methods.\nSupported by evidence: Empirical results in Table 3 show that Box-Geometric consistently outperforms vector-based models (MF, NeuMF, LightGCN) in Hit Rate@10/20/50.\n\nClaim 2: Box embeddings enable efficient computation of intersection and negation operations.\nSupported by evidence: The paper introduces a probabilistic volume-based approach (Equation 2) that ensures intersection and containment scores are computed efficiently.\n\nClaim 3: The model generalizes well to unseen user-item interactions.\n\nPartially supported: The Generalization Spectrum analysis (Table 4) shows that box embeddings exhibit a smaller performance gap when transitioning from weak to strong generalization settings.\n\nPotentially problematic claim:\nThe authors suggest that traditional vector embeddings inherently lack set-theoretic compositionality, but some recent models (e.g., graph-based hybrid models) can partially capture these relationships—a comparison with these approaches is missing.", "methods_and_evaluation_criteria": "Box embedding formulation:\n1. Represents users, items, and attributes as hyperrectangles in a latent space.\n2. Computes recommendation scores using geometric containment relationships.\n\nTraining objective:\n1. Optimizes embeddings with a noise-contrastive estimation loss, ensuring learned representations respect set-theoretic operations.\nEvaluation metrics:\n2. Hit Rate (HR@k) and NDCG@k for ranking-based evaluation.\n\nBenchmarking against:\n1. Matrix Factorization (MF), Neural Collaborative Filtering (NeuMF), and Graph Convolutional Networks (LightGCN).\n2. Experiments on three datasets (MovieLens, Last.fm, NYC-R) with varying sparsity levels.\n\nPotential improvement:\nIf the recommendation query is implicit, that is, the platform is only equipped with the user interactive history, how this method been applied rather than the current given query pattern.  Since the implicit recommendation is more common.", "theoretical_claims": "Seems solid.\n1. Set-theoretic matrix completion formulation ensures the inductive bias aligns with query constraints.\n2. Probabilistic volume estimation replaces hard min/max operations with smooth approximations to enable gradient-based training.\nContainment-based scoring function is proven to respect set relationships in Equations (1)–(3).", "experimental_designs_or_analyses": "Experiments systematically evaluate:\n1. Simple queries (e.g., “Bob wants a comedy movie”).\n2. Complex queries (e.g., “Bob wants a comedy & action movie but NOT romance”).\n\nGeneralization spectrum study:\n1. Tests how well the model handles unseen user-item interactions.\n2. Box embeddings retain higher accuracy than MF/NeuMF when tested on unseen combinations.\n\nBaselines include:\n1. Traditional MF, NeuMF, LightGCN.\n2. Three aggregation methods: filtering-based, product-based, and geometric.\n\nPotential improvement:\nSince the industry uses the user history sequence to build the recommendation system. Is this MF method still a solid baseline?", "relation_to_broader_scientific_literature": "Builds on prior work in box embeddings:\n1. Extends prior probabilistic box models used in knowledge graph reasoning to recommender systems.\n2. Connects to compositional reasoning models:\n\nChallenges traditional word2vec-style additive composition by demonstrating superior set-theoretic consistency.", "essential_references_not_discussed": "1. The paper does not cite recent work on hybrid graph embeddings for reasoning over structured constraints.\n2. LLM-based recommendation systems could be discussed as an alternative for compositional queries.", "other_strengths_and_weaknesses": "Strengths\n\n1. Conceptually novel: Reformulates recommendation as set-theoretic matrix completion.\n2. Box embeddings naturally handle complex queries.\n3. Empirical validation: Large-scale experiments show clear performance gains.\n4. Probabilistic smoothing improves training stability.\n\nWeaknesses\n\n1. No real-world user study: Effectiveness is validated only through automated ranking metrics.\n2. Limited generalization beyond movie/music domains: Would it work for news, e-commerce, or job recommendations?\n3. Cold-start problem remains: The approach still requires enough observed user-attribute interactions. How to solve the cold Start problem in this set embedding method", "other_comments_or_suggestions": "1. Would combining box embeddings with GNNs or LLMs improve performance?\n\n2. Investigate real-time feasibility: Can this be efficiently deployed in interactive recommendation scenarios?\n\n3. Consider user personalization beyond set-theoretic constraints: How can temporal user behavior be incorporated?", "questions_for_authors": "1. Box intersection operations could become expensive in high-dimensional spaces.\n2. How does it compare to hybrid graph-based approaches?\n3. Could it handle temporal or evolving user preferences?\n4. Can the box embeddings adapt over time to user behavior shifts?\n5. Would this approach work for content-aware recommendations?\n6. Could text/image embeddings be incorporated into the box representation?", "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": "rIjiv73Xu2", "reviewer": "Reviewer_CkQw", "summary": "1. This paper models the problem of attribute-specific query recommendation as “set-theoretic matrix completion”, where attributes and users are treated as sets of items.\n2. It demonstrates the inconsistency of existing vector embedding models for the above task, and establishes box embeddings as a suitable embedding method.\n3. It conducts an extensive empirical study comparing various vector and box embedding models for the task of set-theoretic query recommendation, and reports substantial improvements over the vector/neural baselines.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for going through the paper so meticulously — we truly appreciate the careful reading and thoughtful feedback. We're glad you found the framework novel and the paper well-written, and we appreciate your comments highlighting both the strengths and areas for improvement.\n\nWe apologize for the oversight in the last step of the equation — we’ve corrected it, and the fix will be reflected in the final draft.\n\n**Regarding supporting evidence for the limitations of vector-based embeddings in handling set operations:**\n\nWe found QUEST (Malaviya et al., 2023) to be a closely related study that introduces a benchmark for entity-seeking queries with implicit set-based semantics. Although QUEST does not focus on explicit constraints or personalization, which are central to our work, it empirically demonstrates that vector-based T5 embeddings struggle with set-theoretic operations — particularly with negation queries, and often treat conjunctions as disjunctions.", "claims_and_evidence": "Most claims in the paper are either backed by either explanations or experiments. However, it's better to provide evidence to support claims such as \"Even advanced neural network-based approaches, which are designed to capture intricate relationships, have\nbeen shown to struggle with set-theoretic compositionally that underlie many real-world preferences\".", "methods_and_evaluation_criteria": "1. The proposed method is based on box embeddings with GumbelMax and GumbelMin approximations.\n2. The training objective is based on a noise-contrastive estimation.\n3. The benchmark datasets are based on MovieLens 1M and 20M datasets, as well as a subset of the Last-FM dataset. In order to adapt these datasets to the attribute-specific query recommendation problem, the authors utilize various resources and preprocessing steps to construct user and attribute sets, along with personalized simple and complex queries.", "theoretical_claims": "The paper doesn't provide any theoretical claims or proofs.", "experimental_designs_or_analyses": "1. The proposed approach is compared with multiple baselines, including MF, NEUMF and LGCN.\n2. Three score aggregation methods are adopted: filter, product and geometric.\n3. The experiment results support the superiority of the box-embedding based approach over the baselines.", "relation_to_broader_scientific_literature": "The paper may be particularly interesting to communities addressing similar matching problem that can be set based.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "Strength:\n1. The paper is well written and easy to understand.\n2. It provides a novel framework based on set theory to address attribute-specific query recommendation problem.\n\nWeakness:\n1. There are typos in some equations.", "other_comments_or_suggestions": "1. Equation 1 last step is wrong.", "questions_for_authors": "N/A", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "UuahZ5Jdjp", "reviewer": "Reviewer_Qs1s", "summary": "This paper demonstrates the inconsistency of existing vector embedding models for personalized item recommendation task and discusses the challenges faced by existing machine learning approaches for this problem setup. To alleviate the issue, the authors formulated the problem of personalized item recommendation as matrix completion where rows are set-theoretically dependent. To capture this set-theoretic dependence the authors represented each user and attribute by a hyper-rectangle or box. Box embeddings can intuitively be understood as trainable Venn diagrams, and thus not only inherently represent similarity, but also naturally and faithfully support arbitrary set-theoretic relationships. Queries involving set-theoretic constraints can be efficiently computed directly on the embedding space by performing geometric operations on the representations. An extensive empirical study comparing various vector and box embedding models for the task of set-theoretic query recommendation is conducted and it demonstrates the superiority of box embeddings over vector-based neural methods on both simple and complex item recommendation queries by up to 30% overall.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the thoughtful and constructive review.\nWe appreciate your detailed engagement with our paper and the helpful suggestions on evaluation metrics.\n\n**Recommendation Task / NDCG evaluation / Hyperparameter**\n\nWe calculate the NDCG score for the set theoretic queries. Please refer to the tables below. \n\nMovielens-1M dataset\n| Method | $U \\cap A$ | $U \\cap A_1 \\cap A_2$ | $U \\cap A_1 \\setminus A_2$ |\n| --- | --- | --- | --- |\n| MF-Filter | 13.7 | 12.5 | 13.6 |\n| MF-Product | 15.4 | 16.4 | 15.3 |\n| MF-Geo | 11.0 | 10.3 | 11.2 |\n| NeuMF-Filter | 16.5 | 16.5 | 15.7 |\n| NeuMF - Product | 17.0 | 19.9 | 15.9 |\n| LGCN - Filter | 13.1 | 16.0 | 14.0 |\n| LGCN-Product | 12.0 | 11.8 | 13.4 |\n| Box- Filter | 16.7 | 16.5 | 16.9 |\n| Box - Product | 18.0 | 18.6 | 17.8 |\n| Box - Geometric | **18.6** | **21.5** | **17.9** |\n\nNYCR dataset. \n| Method | $U \\cap A$ | $U \\cap A_1 \\cap A_2$ | $U \\cap A_1 \\setminus A_2$ |\n| --- | --- | --- | --- |\n| MF-Filter | 9.6 | 10.4 | 10.3 |\n| MF-Product | 12.3 | 15.6 | 14.8 |\n| MF-Geo | 11.5 | 11.9 | 11.7 |\n| NeuMF-Filter | 11.1 | 10.1 | 11.8 |\n| NeuMF - Product | 13.9 | 15.7 | 12.9 |\n| LGCN - Filter | 10.8 | 10.8 | 9.1 |\n| LGCN-Product | 13.4 | 15.8 | 11.1 |\n| Box- Filter | 12.1 | 14.3 | 11.8 |\n| Box - Product | **14.4** | 18.6 | 14.6 |\n| Box - Geometric | **14.4** | **19.1** | **14.9** |\n\nLastfm dataset also shows similar trends (omitting it for character constraint).\n\nIn addition to the logical queries, we also perform a standard recommendation task (i.e. recommend an item for a user), reported in terms of NDCG score (Table 6. in Appendix B2)  on all four datasets where we observe that box embeddings outperform the baselines across all datasets. We posit this is because the box embedding space is more accurately able to capture the item-attribute information.\n\nWe provide the hyperparameter search details in Appendix B.2. We use wandb.ai for hyperparameter tuning (See Figure 3. for the hyperparams for Box).\n\n**Baseline consideration:**\n\nWe have attempted to compare against formidable baselines which seem most capable of capturing set-theoretic semantics. The baselines chosen—MF, NeuMF, and LightGCN—are standard, well-established methods in recommendation systems. They represent varying levels of neural complexity and address user-item and attribute-item interactions effectively. More recent baselines either do not directly provide any particular advantage for set-theoretic queries, or use significant external information (eg. pretrained LLMs). We feel the selection of baselines covers the existing space fairly well (matrix, neural, and graph-based methods) without diluting the central contribution of introducing a method tailored to set-theoretic compositionally or introducing exogenous information.\n\n**Query Generation Bias:**\n\nWe agree that not all attribute combinations reflect realistic user intents, and we explicitly address this concern in our **query generation process** through **statistical heuristics combined with manual inspection**, as described in Section 4.2.2.\n\nSpecifically, to ensure **semantic plausibility**, we define **viable and non-trivial attribute pairs** for intersection and difference queries based on two criteria:\n\n1. **Relevance**: We require that the overlap between the attributes is **greater than random chance**, i.e., $|a_1 \\cap a_2| > \\varepsilon$, ensuring the pair co-occurs frequently enough to represent realistic co-preference.\n2. **Non-triviality**: We further enforce that the overlap is **not overly large**, i.e., $|a_1 \\cap a_2| < \\alpha \\cdot \\min(|a_1|, |a_2|)$, to avoid trivial combinations where one attribute almost subsumes another (e.g., \"fiction\" ∧ \"sci-fi\").\n\nThis formulation reflects the **plausibility of attribute combinations**, avoiding unlikely or degenerate cases such as \"sci-fi and documentary\". In practice, the resulting combinations include interpretable and realistic pairs like (\"action\", \"comedy\") or (\"children\", \"not monster\"), which align with user behavior patterns. To further ensure the quality of queries, we have manually verified the plausibility of the selected attribute combinations to the best of our knowledge and effort. We also plan to release the full dataset upon paper acceptance to support reproducibility and facilitate future work in this space.\n\n**Changing User preferences:**\n\nOur current work does not explore incremental or online updates to user or item representations. However, it follows the standard recommendation framework, where embeddings are periodically retrained using accumulated user-item and item-attribute interactions — a paradigm widely adopted in practical systems. While we have not implemented streaming updates, such extensions could be supported via fine-tuning on recent data or by leveraging continual learning techniques common in embedding-based models.", "claims_and_evidence": "The claims made in the submission are supported by clear and convincing evidence.", "methods_and_evaluation_criteria": "The methods and evaluation criteria proposed in this paper are reasonable on the whole, but there are the following major problems: \n1. This paper mainly uses Hit Rate@k (HR@10, HR@20, HR@50) as an evaluation measurement. Although it can reflect the hit rate of the model in the recommendation list, it cannot measure the recommendation quality comprehensively. Please add more measurements such as NDCG for a comprehensively evaluation.\n2. This paper generates personalized query through algorithm 1 and algorithm 2\nIt is not sufficiently discussed whether the distribution of these queries truly reflects the actual needs of users. For example, some attribute combinations (such as \"science fiction\" and \"documentary\") may rarely appear in actual scenes, but the rationality of these combinations is not verified in the paper.", "theoretical_claims": "Yes, the theoretical claims are checked.", "experimental_designs_or_analyses": "The experimental design and analysis of this paper are reasonable on the whole, but there are some problems such as query generation bias, single evaluation measurement, lacking transparency of hyperparameter tuning.", "relation_to_broader_scientific_literature": "The key contributions of the paper are closely related to existing scientific literature, especially in the application of box embedding, the processing of set theory queries, and the study of context-aware recommendation systems. The work in this paper not only extends the application range of box embedding, but also fills the gap of the existing recommendation system in handling set theory queries, and provides a new solution for personalized recommendation.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "Strengths: \n1.An innovative embedding method (box embedding) is introduced to represent users and items as hyperrectangles (i.e. \"boxes\" in geometric space), breaking through the limitations of traditional vector representations such as matrix factorization.\n2.Set theory relations (such as intersection, negation) are directly integrated into the embedded space of the recommendation system, and the modeling problem of complex logical constraints (such as \"A and B but not C\") is solved. The efficient computation of complex queries is achieved through geometric operations such as box intersection and complement, which is intuitive and theoretically attractive..\n3.The authors designed an additional task to learn a state augmentation model that can generate informative embedding to assist the agent in obtaining prior knowledge of the environment and accelerate the convergence of agent networks.\n4.The method is designed with mathematical rigor: box embedding is defined by the Cartesian product interval and supports the closed geometry operations of set operations (such as Jaccard similarity calculation), and the theoretical derivation is clear.\n5.Experiments have verified the effectiveness of the method, and it can improve over the vector method on simple and complex queries. The experimental design is reasonable and the results are credible.\nWeakness:\n1.Verify that the experiment includes a personalized recommendation task for a real scenario (e.g., Top-K recommendation), not just a test of a synthetic or specific logical query.\n2.The baseline methods employed for comparison in the paper are relatively limited in number and somewhat outdated. This restricts the comprehensiveness of the performance evaluation and may not fully highlight the advantages of the proposed method over more recent advancements in the field.\n3.User interests may change over time (e.g., from a preference for \"action\" to \"drama\"), but the paper does not indicate how the box can be updated to accommodate such dynamic changes. In real recommendation systems, data usually arrive incrementally (such as real-time user interaction), and the paper does not verify whether the method supports online learning.", "other_comments_or_suggestions": "None.", "questions_for_authors": "Please see the above comments.", "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": "AAraJpuKrC", "reviewer": "Reviewer_BTfm", "summary": "This paper proposes a geometric approach to personalized item recommendation using box embeddings. Unlike traditional vector-based methods that struggle with set-theoretic operations, the authors introduce axis-aligned hyperrectangles (boxes) to model users, items, and attributes. This enables logical operations such as AND, OR, and NOT to be represented geometrically. The recommendation task is formulated as set-theoretic matrix completion, where items are recommended based on box containment and intersection.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the reviewer’s suggestion to clarify how natural language queries are mapped to structured forms, as well as their thoughtful concerns regarding generalization, scalability, and efficiency.\n\n**Mapping from natural language to attributes:** \n\n In real-world recommendation systems — such as streaming platforms, e-commerce sites, or travel services — free-text queries (e.g., “funny action movies without clowns”) are typically mapped to a curated set of item tags (e.g., genre, theme, metadata) via a natural language understanding (NLU) module. \n\nOur model operates **after** this step, assuming a structured query (e.g., `Action ∧ Comedy ∧ ¬Clowns`) has already been derived. While we do not include an NLP module, it is **modular and complementary** to our framework. Recent LLMs (e.g., GPT-4o) can perform this front-end task — identifying attribute constraints from text — while our model serves as the **back-end executor** for set-theoretic reasoning.\n\nWe thank the reviewer for this helpful suggestion; including such an example would indeed strengthen the narrative, and we will incorporate a clarifying illustration in the final version.\n\n**Extending the Framework for more complex queries:**\n\n**1. Support for Arbitrary Logical Queries.**\n\nUsing the **inclusion-exclusion principle**, any Boolean query can be transformed into disjunctive normal form (DNF). For example, a query like $(\\text{User} \\wedge A_1) \\vee (\\text{User} \\wedge A_2)$ can be evaluated as:\n\n$\\text{Score}(((U \\cap A_1) \\cup (U \\cap A_2)) \\cap i) = \\text{Score}(U \\cap A_1 \\cap i) + \\text{Score}(U \\cap A_2 \\cap i) - \\text{Score}(U \\cap A_1 \\cap A_2 \\cap i)$, $i$ is the item.\n\nOur **containment-based scoring function** remains applicable to any logical query formed over box embeddings, including negations and disjunctions.\n\n**2. Time complexity / Efficient Implementation via Gumbel Boxes.**\n\nEach term in the DNF expansion involves computing the **intersection of multiple boxes**, which we implement using a **log-sum-exp (LSE)** operation over the min and max coordinates. For a query involving $k$ attributes, the largest term in the DNF requires intersecting $k$ boxes, resulting in a complexity of $\\mathcal{O}(k \\cdot d)$, where $d$ is the box embedding dimension.\n\nNow to compute scores for the  full Boolean query with $t$ inclusion-exclusion terms: $\\mathcal{O}(t \\cdot k \\cdot d)$. In worst case — e.g., a disjunction over $k$ variables — $t$ can be $\\mathcal{O}(2^k)$. \n\n- However, typical queries involve conjunctions and negations, resulting in **far fewer DNF terms** than the worst case. E.g.,(User∧Children∧Comedy∧¬Clowns)∨(Monster∧¬Animation) results into 8 terms in its DNF form.\n- Additionally, **each term is independently computable**, and we parallelize the LSE computation **across dimensions and query terms for** intersection volumes calculation, enabling efficient batched evaluation. These optimizations are provided in our codebase.\n\nThere would be no difficulty in operating on more complex queries, such as $U \\cap A_1 \\cap A_2 \\cap \\neg A_3$. Note that our model does not need to be **trained on complex queries** — it is trained only on pairwise interactions but naturally generalizes to a broad space of **unseen logical compositions.** As noted above, evaluating on deeper queries is computationally efficient, and so we could certainly evaluate our model on such queries. The main challenge would be identifying deeper queries which would be representative of a real user query distribution, which is non-trivial.\n\n**Why Gumbel max instead of softmax-max:**\n\nWe initially explored using the softmax-based approximation for max/min operations, but encountered numerical instability and a lack of probabilistic consistency. Specifically, softmax-max computes a convex weighted average that always underestimates the true max:\n\n$\\operatorname{SM-Max}(\\mathbf{x}; \\tau) =\\sum_{i=1}^n\\underbrace{\n\\Bigl(\\frac{e^{x_i/\\tau}}{\\sum_j e^{x_j/\\tau}}\\Bigr)}_{w_i}\\,x_i.$\n\n$\\operatorname{SM-Max}(\\mathbf{x}; \\tau) < \\operatorname{max}(\\mathbf{x})$, \n\n$\\operatorname{SM-Min}(\\mathbf{x}; \\tau) = -\\operatorname{SMM}(-\\mathbf{x}; \\tau) > \\operatorname{min}(\\mathbf{x})$ \n\nThis becomes problematic when computing box intersections, which rely on min/max operations over interval boundaries. For example, if box $[c,d]$ is fully contained in $[a,b]$ we should have $\\operatorname{Vol}([a,b] \\cap [c,d]) = \\operatorname{Vol}([c,d])$. But the softmax-based approximation inflates the intersection to $[x, y]$ where $x < c$ and $y > d$, violating containment and producing invalid probabilities (e.g., $\\operatorname{P}([x,y] \\mid [c,d]) > 1$).\n\nIn contrast, GumbelBox embeddings have the opposite quality, they upper bound the max operation ensuring the $\\operatorname{P}([x,y] \\mid [c,d]) < 1$. \n\nWe thank the reviewer for highlighting this point — it indeed reflects an interesting theoretical contribution. We will provide the detailed proof in the final version of the paper.", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "Yes", "theoretical_claims": "The paper provides a clear mathematical framework for representing user preferences and item attributes as boxes. Set operations correspond to geometric operations (intersection for AND, union for OR, complement for NOT), aligning well with set theory. This grounding makes the approach interpretable.", "experimental_designs_or_analyses": "1. The authors compare their box embedding model against vector-based baselines like matrix factorization (MF), Neural Matrix Factorization (NeuMF), and LightGCN. The box model shows improved performance on queries with set constraints, highlighting its advantage in those scenarios.\n\n2. Ranking metrics such as Hit Rate@K and NDCG@K are used, which are appropriate for measuring recommendation quality. The box model’s gains on these metrics, particularly for composite queries, support the claim that it better captures user intent for complex filters.\n\n3. An ablation on the use of the Gumbel-softmax (comparing different temperature settings or using a hard version) demonstrates its effect on training stability and performance. This helps justify the inclusion of the smoothing technique.", "relation_to_broader_scientific_literature": "1. The paper builds on prior work in geometric embeddings for logic and set reasoning. For example, in knowledge graphs, Query2Box embeds complex logical queries as boxes to answer multi-hop queries with AND/OR operators​. Similarly, in NLP semantics, Word2Box shows that box embeddings can capture set relations among word meanings (e.g., \"red cars\" is a subset of \"cars\" in embedding space)​. This work extends such ideas to the recommendation domain.\n\n2. It contrasts with traditional collaborative filtering and neural embedding approaches. Standard latent factor models (e.g., matrix factorization or neural CF) represent users and items as point vectors, which struggle to naturally perform set operations. By using regions (boxes) instead of points, this model can directly encode logical constraints that would be hard to achieve with dot-product similarity alone.", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "Strengths:\n\n1. The conceptual novelty of casting recommendation as a set-theoretic problem and using boxes to represent sets is a significant strength. It offers a new perspective that is both intuitive (via geometry) and potentially powerful for complex preference queries.\n\n2. Empirical validation on multiple datasets strengthens the work. By showing improvements on MovieLens, Last.fm, and NYC-R, the authors demonstrate that the approach is not tailored to a single scenario.\n\n3. The mathematical formulation (energy function and training procedure) is sound and clearly derived. The use of an energy-based model with appropriate smoothing shows careful consideration of the optimization challenges.\n\nWeaknesses:\n1. Scalability to more complex queries is not fully explored. While the method works for the queries tested, it’s unclear how performance or accuracy holds up with queries that combine many conditions or involve deeper logical structure.\n\n2. Mapping from natural language to attributes is a practical concern not addressed. In a deployed system, users might issue queries in text (e.g., “find me action movies from the 90s not starring actor X”), which requires translating into the appropriate box intersections. The paper assumes structured attribute inputs, leaving this mapping as future work.", "other_comments_or_suggestions": "1. It would strengthen the paper to explain how a user’s query (in words) would be processed. Are attributes manually selected by the user, or could an NLP model tag the query with the corresponding attributes to form the box query? Providing even a brief discussion or an example would make the approach more concrete.\n\n2. The authors could consider extending their framework to support more complex logical expressions. For instance, how would the model handle a query like (User AND Attribute1) OR (User AND Attribute2), or multiple negations? If the method naturally extends (or if not, what the challenges are), mentioning this would be insightful for readers interested in general logical query support.\n\n3. Including a complexity analysis (even just Big-O) or some timing experiments for query evaluation would be helpful. For example, how does the recommendation runtime grow with the number of items or attributes? Understanding this would indicate the practicality of the approach for large-scale use.", "questions_for_authors": "1. How does the system handle free-text user queries or preferences? Do you assume a fixed set of attributes that users filter by, or is there a component to dynamically infer attribute constraints from text?\n\n2. Your experiments focus on queries with one or two attribute filters. How would the model scale to queries with multiple attributes and negations (e.g., User AND A1 AND A2 AND NOT A3). Would there be any challenges in training or evaluation with deeper logical compositions?\n\n3. ou chose Gumbel-softmax to approximate the min/max. Did you consider other continuous relaxations (e.g., sigmoid approximations of indicator functions, or other softmin/softmax approaches)? If so, what made Gumbel-softmax the most suitable for this problem?", "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:

Meta-review: This paper presents a geometric approach to recommender systems using box embeddings where users, items, and attributes are all represented as boxes in the latent space to facilitate set-theoretic operations, such as negation and intersection, e.g., recommending a comedy but not romance. 

Overall the paper received positive-leaning scores where all reviewers praise the innovative usage of the box embeddings in the context of recommender systems. There are some shared concerns on the baseline selections. Overall I believe the positive overweights the negative. I also skimmed the paper myself (albeit likely not as carefully as a reviewer would). I have one comment: the authors claim that the item-attribute matrix tends to be incomplete (just like user-item matrix), hence an imputation approach should be used here to create an "imputed" attribute for filtering. However, I think in practice, the item-attribute matrix can often be considered "complete" since they are often curated manually. Furthermore, I don't think the underlying assumption behind matrix completion for user-item matrix (user A and B interacted with similar items, hence recommending items that user A interacted with but not B to B is a good idea) necessarily translates to item-attribute matrix -- if two movies A, B have similar attributes, it doesn't mean B should also have the additional attributes that only applies to A. Because of this, it seems to me that the most natural baseline would be simply do hard filtering. 

Another thing that I want to point out is that due to the unique topic combinations involved in this paper, none of the reviewers are actually very familiar with the box embedding related literature (myself included). Therefore our assessment might be less confident than normal. I am still voting for accept but open to be bumped down.
Decision: Accept (poster)