A Geometric Approach to Personalized Recommendation with Set-Theoretic Constraints Using Box Embeddings

Thank you for recognizing the strengths of our work and for your constructive feedback. While we address your specific questions and concerns in the following rebuttal, we encourage you to review our general response, which highlights common themes regarding the scope of our study and ambiguity in notations.

The approach presented in this paper conflicts with the widely used matrix factorization model, which effectively leverages collaborative filtering signals between users and items. It is unclear how the proposed model addresses these signals.

All the models including our Box-based method train on Collaborative signal (user-vs-item matrix $U$) and Attribute signal (attribute-vs-item matrix $A$).

As you mentioned, the matrix factorization model effectively leverages collaborative filtering signals between users and items. They achieve that by training to increase the dot product between user and item representation when user and items interact and decrease the dot product similarity when they do not (on a negatively sampled version).

In the Box embedding-based method, we train on $U$ in a similar manner. Instead of dot product similarity, we use box containment as the similarity measure; i.e, if a user $u$ has watched an item $i$, then the collaborative signal encourages the item $\mathrm{Box}(i)$ to be inside user $\mathrm{Box}(u)$. We enforce this set containment by optimizing the following:

If the user $u$ has watched an item $i$, i.e., $U\_{u, i} == 1$, then, $\frac{ \operatorname{Vol}(\mathrm{Box}(u) \cap \mathrm{Box}(i))}{\operatorname{Vol}( \mathrm{Box}(i))} \rightarrow 1$

If a user $u$ has not watched an item $i$, i,e, $U\_{u, i} == 0$ (we sample the negatives), then, $\frac{ \operatorname{Vol}(\mathrm{Box}(u) \cap \mathrm{Box}(i))}{\operatorname{Vol}( \mathrm{Box}(i))} \rightarrow 0$

Here, $\rightarrow$ means "training to achieve the value". During training, we get values between $(0,1)$ for the above-mentioned expression. We train them to be as close to $1$ or $0$ according to the collaborative signal in $U$. We use binary cross-entropy to achieve this.

Note that, by doing this optimization we not only incorporate collaborative signals but also conceptualize the users as a set that contains all the relevant items.

The experimental baselines are not state-of-the-art; comparing the proposed method to more advanced recommendation models would better demonstrate its advantages.

In response to reviewer Vvma's suggestion, we have implemented LightGCN, a graph convolution-based solution for recommendation. Hyperparameter tuning is currently underway, and due to GPU constraints, it may require an additional two days to complete. We will update the draft with the new results and respond to this thread once the tuning is finalized.

Some equations are difficult to follow due to unclear notation explanations.

We have crafted a general response above titled General Response - Notations, where we have simplified the notations and provided a more detailed explanation for clarity. This response addresses similar questions raised by other reviewers regarding the notations. We kindly request that you review this section, and please let us know if any of the notations remain unclear. We are eager to receive your feedback and will revise our current draft accordingly.

Thank you for your constructive feedback! While we address your specific questions and concerns in the following rebuttal, we encourage you to review our general response, which highlights common themes regarding the scope of our study and ambiguity in notations.

The scope of this study is on a rather limited application, which the authors called "attribute-specific query recommendation".

While it is true that our study focuses on "attribute-specific query recommendation," this is a deliberate choice, as it addresses an unexplored and important problem within the broader landscape of recommendation systems. The ability to handle queries involving attributes and their logical combinations is critical for real-world applications, ranging from personalized e-commerce searches like “red winter coats under $100” to specific media preferences such as “dark comedies.” These scenarios highlight the need for systems that can handle both attribute-specific constraints and individual user preferences—an area where existing systems are limited. we see our focused contribution as the first step toward addressing an underexplored area with substantial real-world implications.

We also request the reviewer to refer to the “Scope of our Work” section in the “General Response.”

The current manuscript lacks related work on "attribute-specific query recommendation".
(C1) To my understanding, "attribute-specific query recommendation" is the task where (1) item attribute values are partially observed and often missing, and (2) in the prediction phase, the positive items are conditioned not only on a user but also on a boolean query. The problem of (1) has been addressed in existing studies (e.g., [b,c]). I am not familiar with (2) in the context of item recommendation, but there might be existing research on it. A discussion/comparison of existing studies on these points would make it easier to understand the novelty of this work.

We appreciate your feedback highlighting the lack of discussion around "context-aware" and "attribute-aware" recommendations. Including this discussion would indeed help situate our work more clearly within the broader context of related research in this domain.

Context-Aware Recommendation: The concept of context-aware recommendation, as introduced in Adomavicius et al. (2011), provides a general framework where “context” is broadly defined as any auxiliary 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 contextual information.

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 information to address various goals, including improving user profiling (Adomavicius et al., 2011), predicting missing item attributes (Wu et al., 2020; Chen et al., 2022), enhancing recommendations for cold-start scenarios(Deldjoo et al., 2019), or providing attribute-based explanations for recommendations (Xian et al., 2021).

Our work differs significantly in its focus and objectives. We term “attribute-specific -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. We believe the explicit reasoning over attribute-constrained queries is underexplored and represents a meaningful contribution to the field.

We have ensured this distinction clearly in the revised draft of the paper.

Additional Baselines: We observe that many recent "attribute-aware" recommendation approaches leverage Graph Convolutional Networks to better model attribute-item-user interactions. Also, in response to reviewer Vvma's suggestion, we have implemented LightGCN, a graph convolution-based solution for recommendation. Hyperparameter tuning is currently underway, and due to GPU constraints, it may require an additional two days to complete. We will update the draft with the new results and respond to this thread once the tuning is finalized.

• Adomavicius et al., (2011) Context-Aware Recommender Systems
• Wu et al., (2020) Joint Item Recommendation and Attribute Inference: An Adaptive Graph Convolutional Network Approach
• Chen et al., (2022) Multi-view Graph Attention Network for Travel Recommendation
• Deldjoo et al., (2019) Movie genome: alleviating new item cold start in movie recommendation
• Xian et al., (2021) EX3: Explainable Attribute-aware Item-set Recommendations

Thank you for recognizing the strengths of our work and for your constructive feedback. While we address your specific questions and concerns in the following rebuttal, we encourage you to review our general response, which highlights common themes regarding the scope of our study and ambiguity in notations.

Despite showing promising recommendation results, the proposed method seems to be a direct application of box embeddings for personalized item recommendation with set-theoretic queries, which is somehow a limited contribution.

We acknowledge that this work indeed is an application of box embeddings, but it does so in a novel way—box embeddings have never before been tested empirically for their ability to generalize to arbitrary set-theoretic queries. As such, it was necessary to both (a) create a rigorously-justified set-theoretic task in multiple different domains, and (b) propose methods for leveraging the geometry to calculate scores from the boxes for these set-theoretic queries. Furthermore, understanding why vector-based embeddings fall short in faithfully representing set-theoretic relationships is an essential part of our contribution. Vector methods often struggle with maintaining set-theoretic consistency, a limitation that our approach with box embeddings directly addresses. By spotlighting these challenges and presenting an effective alternative, our work provides actionable insights that future researchers can build upon. We will reframe our contributions in the final version to emphasize these unique advancements and insights.

The paper relies heavily on the theory of box embeddings but some key concepts were not described sufficiently. For example, in line 191, how to guarantee x⌞<x⌝ for all dimensions d; what is the definition of VolIntGB in Equation 3?; what are the parameters of the model to optimize?

The following clarifications will also be incorporated into the revised version of the manuscript.

Definition of $\operatorname{VolInt_{GB}}$?

$\operatorname{VolInt}$ is the volume of the intersection between two boxes. When the box embedding parameters are defined using Gumbel random variable (abbrev $GB$), we replace $\operatorname{VolInt}$ with $\operatorname{VolInt}_{GB}$.

To elaborate, Dasgupta et. al. 2020 propose GumbelBox, where the min and max corners are replaced with Gumbel random variable solving this gradient issue. This modification boils down to using use logsumexp which is a smooth approximation of max operations. logsumexp is defined as $LSE_t(\mathbf x) := t \operatorname{log}(\sum_i e^{x_i/t})$, $t$ is known as the temperature of the GumbelBox.

Under the GumbelBox (abbrev $GB$), we change the notations $F\_{Box}, \operatorname{Vol}, \operatorname{VolInt}$ to $F\_{GB}, \operatorname{Vol}\_{GB}, \operatorname{VolInt}\_{GB}$.

Do we need to enforce $x^{\llcorner} < x^{\urcorner}$?

We do not enforce $x^{\llcorner} < x^{\urcorner}$ when the box embedding is defined as the Gumbel box.

In Gumbel box, the $x^{\llcorner}, x^{\urcorner}$ is replaced with max-Gumbel and min-Gumbel random variables $X^{\llcorner}, X^{\urcorner}$ with mean $x^{\llcorner}, x^{\urcorner}$ respectively. Under these Gumbel distributions, any point in the real line has a positive probability to be inside the box, i.e., $\operatorname{Prob}(X^{\llcorner} <z< X^{\urcorner}) > 0, \forall z \in \mathbb{R}$.

Thus, when the end points flip, i.e., $x^{\llcorner} > x^{\urcorner}$, the value of $\operatorname{Prob}(X^{\llcorner} <z< X^{\urcorner})$ becomes negligible, although still remains positive. This is also observed in our $\operatorname{Vol}\_{GB}([x^{\llcorner}, x^{\urcorner}]) = LSE_t(x^{\urcorner} - x^{\llcorner}, 0)$ equation; $LSE\_{\tau}(x, 0) > 0$, even if $x^{\urcorner} - x^{\llcorner} < 0$, i.e, $x^{\llcorner} > x^{\urcorner}$. Thus we need not ensure the $x^{\llcorner} < x^{\urcorner}$, when we use $\operatorname{Vol}\_{GB}$ instead of $\operatorname{Vol}$.

Trainable parameters of box embeddings?

The upper and lower corners $x^{\llcorner}, x^{\urcorner}$ are both trainable parameters for box embeddings. The training signal is provided by the volume of intersection ($\operatorname{VolInt}\_{GB}$) between two boxes. To ensure a fair comparison, we compare the performance of $d$ dimensional box embeddings with $2*d$ dimensional vector-based embeddings. Note that, the temperature $t$ corresponding to the Gumbel Box is treated as hyperparameters.

Representative baselines such as LightGCN [1] and MultiVAE [2] were not considered. Including more recent and advanced baselines will further ascertain the strength of the proposed method.

We have already implemented the LightGCN and currently running the hyperparameter tuning for all the datasets for faithful final results. Due to GPU constraints this process would take two more days. We will update the results in our revised manuscript, and respond back in this thread.

Missing descriptions of some important experimental settings, e.g., how many negative sampled required to train Equations in lines 233 and 240. Moreover, ablative analysis of key hyper-parameters is also not presented. For instance, how the number of negative samples affect the model accuracy? The same questions for in line 240.

We perform extensive hyperparameter tuning for the learning rate, batch size, volume, and intersection temperature of GumbelBoxes ($\nu, \tau$), the number of negative samples for noise contrastive training (Equation in line 233), and the attribute loss constant $w$ (Equation in line 240). Specifically, we vary the number of negative samples in {1, 5, 10, 20} and the attribute loss constant $w$ in the range {0.1, 0.3, 0.5, 0.7, 0.9}. For each method, 100 hyperparameter runs are conducted, with each run drawing a random hyperparameter combination from the grid of all possible combinations.

Due to the random sampling nature of the hyperparameter tuning, different runs with varying numbers of negative samples are likely to differ in other hyperparameter settings as well. This makes it challenging to isolate the impact of specific hyperparameters, such as the number of negatives or $w$, under the current scheme.

We have already provided this discussion on hyper-parameters in Section 4.3 and Appendix B.2. We will update the writing around the loss equations, to incorporate a discussion about the negative sampling and attribute constant hyper-parameters.

Edit (Sun Nov 24): Updated Appendix B.2 with a Parallel co-ordinate plot with space of hyperparameters vs Box model's performance on ML20M.

Training with the encoded similarity

• We train on binary cross entropy-based loss where the cross entropy is calculated between the observed data and the model’s similarity output (incorporating both collaborative and attribute data).
  • Parameters: For the MF method, the vector corresponding to the users, items, and attributes are trainable parameters. For the NeuMF method, the neural network is trained along with these parameters. For box embeddings, the vectors corresponding to the lower left ($\llcorner$) and upper right ($\urcorner$) corners are the trainable parameters.
  • Hyperparameters: We perform extensive hyperparameter tuning for the learning rate, batch size, volume and intersection temperature of Gumbelboxes($\nu, \tau$), number of negative samples for the noise constrastive training(. Please refer to Appendix B.2 for detailed hyperparameter space. For each method, we conduct 100 hyperparameter runs, varying the values of the aforementioned hyperparameters.

Inference:

• During inference, we calculate the volume of the box embedding region corresponding to the set-theoretic query.

  • Inferring on $U \cap A$: We calculate score as: $\textrm{score}(u \wedge a, i)_d = \frac{\operatorname{VolInt}\_{GB}(\textrm{Box(u)}\_d, \textrm{Box(a)}\_d,\textrm{Box(i)}\_d)}{\operatorname{Vol}\_{GB}(\textrm{Box(i)}\_d)}.$
  • Inferring on $U \cap A_1 \cap A_2$: We calculate score as: $\textrm{score}(u \wedge a_1 \wedge a_2, i)_d = \frac{\operatorname{VolInt}\_{GB}(\textrm{Box(u)}_d, \textrm{Box}(a_1)\_d, \textrm{Box}(a_2)\_d,\textrm{Box(i)}\_d)}{\operatorname{Vol}\_{GB}(\textrm{Box(i)}_d)}.$
  • Inferring on $U \cap A_1 \cap \neg A_2$: We use inclusion-exclusion to calculate scores as: $\textrm{score}(u \wedge a_1 \wedge \neg a_2, i)_d = \textrm{score}(u \wedge a_1, i)_d - \textrm{score}(u \wedge a_1 \wedge a_2, i)_d$

• For vectors, there is no prescribed way for calculation for these set operations, so we explore the following options. 1. Filter: Instead of representing the set operation in the embedding space we use post hoc aggregation. 2. Product: Normalized prediction scores can be multiplied to get the joint query score. 3. Geometric: Vector addition and subtraction for intersection and negation respectively.

• $F_{\textrm{Box}}$ involves min and max operations, which hinder with gradient, making it hard to learn the parameters of the box embeddings.
• Dasgupta et. al. 2020 propose GumbelBox, where the min and max corners are replaced with Gumbel random variable solving this gradient issue. This modification boils down to using use logsumexp which is a smooth approximation of max operations. logsumexp is defined as $LSE_t(\mathbf x) := t \operatorname{log}(\sum_i e^{x_i/t})$, $t$ is known as the temperature of the GumbelBox.
• Under the GumbelBox (abbrev $GB$), we change the notations $F\_{Box}$ to $F\_{GB}$, $\operatorname{VolInt}$ to $\operatorname{VolInt}\_{GB}$, $\operatorname{Vol}$ to $\operatorname{Vol}_{GB}$.
• With the new formulation, the per-dimensional similarity score becomes the following: $F_\{GB\}(\textrm{Box(u)}_d, \textrm{Box(i)}_d; (\tau, \nu)) := \frac{LSE\_\nu(LSE\_{-\tau}(u_d^{\urcorner}, i_d^{\urcorner}) - LSE\_\tau(u_d^{\llcorner}, i_d^{\llcorner}), 0)}{LSE\_\nu(i_d^{\urcorner} - i_d^{\llcorner}, 0)} =: \frac{\operatorname{VolInt}\_{GB}(\textrm{Box(u)}_d, \textrm{Box(i)}_d; (\tau, \nu))}{\operatorname{Vol}\_{GB}(\textrm{Box(i)}_d; \nu)}$
  • $\nu, \tau$ are the volume and intersection temperature of GumbelBox. As $\nu, \tau \rightarrow 0$, the GumbelBox becomes the originally proposed Box. We tune these as hyperparameters.

We appreciate the reviewers' observations regarding the clarity of our explanations of the notations. We acknowledge that some key concepts were not described as thoroughly as they could have been. In this general response, we provide an outline of the methodology of our work and clarify these concepts. These updates will be incorporated into the revised version of the manuscript. We kindly request the reviewers to review the general response before addressing our answers to the specific questions.

Problem Set up

• The user-item matrix is the collaborative signal $U$
• The attribute information is coded in $A$.
• We want personalized recommendations with attribute constraints.

  • User would like what comedy movies? This is the same as completing rows in $U \cap A$.

  • What romantic-comedy movie that the user would like? This is same as completing rows in $U \cap A_1 \cap A_2$

  • What comedy movie the user would like that is not romantic? This is same as completing rows in $U \cap A_1 \cap \neg A_2$

Similarity Function

• Similarity Functions for encoding the collaborative and attribute signals.

  • For vector, we represent the user with vector $\mathbf{u} \in \mathbb{R}^D$ and items with vector $\mathbf{i} \in \mathbb{R}^D$, the similarity is calculated as $\sigma(\mathbf{u}^T.\mathbf{i})$ of for any neural method $\sigma(f(\mathbf{u}, \mathbf{i}))$.

  • For box embeddings, we represent the user $u$, item $i$ and attribute $a$ as hyper-rectangles or cartesian product of intervals: $\textrm{Box}(u)=\prod_{d=1}^D[u_d^{\llcorner}, u_d^{\urcorner}], \textrm{Box}(i)=\prod_{d=1}^D[i_d^{\llcorner}, i_d^{\urcorner}], \textrm{Box}(a)=\prod_{d=1}^D[a_d^{\llcorner}, a_d^{\urcorner}]$. where, $u_d^{\llcorner} < u_d^{\urcorner},i_d^{\llcorner} < i_d^{\urcorner}, a_d^{\llcorner} < a_d^{\urcorner}$ .

  • Also the $d$ th dimension of $Box(u)$ is an interval denoted as $\textrm{Box(u)}_d := [u_d^{\llcorner}, u_d^{\urcorner}]$.

  • Note that, the total volume of a rectangle is a mere multiplication of the length of each dimension. So the rest of the discussion will be on a single dimension $d$, we can multiply the scores for each dimension to calculate the final score.

    • Volume of item box at dimension $d$ is given by the length of the interval. $\operatorname{Vol}( \textrm{Box(i)}_d) =\max(i_d^{\urcorner}-i_d^{\llcorner}, 0)$.
    • Volume of intersection between user and item box is given by, $\operatorname{VolInt}(\textrm{Box(u)}_d, \textrm{Box(i)}_d) = \max(\min(u_d^{\urcorner}, i_d^{\urcorner}) - \max(u_d^{\llcorner}, i_d^{\llcorner}), 0)$.
    • To understand the above equation better, the intersection of two intervals is again another interval, whose min co-ordinate $=$ max (min-coordinate of the two intervals), max co-ordinate $=$ min (max-coordinate of the two intervals).

  • The volume of Intersection between the user and the item box decides how much the item is related to the user. We normalize the similarity score by item box volume. The similarity value for the dimension $d$ is given as: