# An Interpretable Model for Multi-Target Predictions with Ordinal Outputs

Arthur Batel, C. Robardet, Marc Plantevit, Idir Benouaret

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Arthur Batel, C. Robardet, Marc Plantevit, Idir Benouaret. An Interpretable Model for Multi-Target Predictions with Ordinal Outputs. Machine Learning, In press. hal-05525069

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# An Interpretable Model for Multi-Target Predictions with Ordinal Outputs

Arthur Batel¹, Marc Plantevit², Idir Benouaret², Céline Robardet¹*

¹INSA Lyon, LIRIS UMR 5205, F-69621 Villeurbanne, France.

²EPITA Lyon, EPITA Research Laboratory (LRE), F-94276 Le Kremlin-Bicêtre, France.

*Corresponding author(s). E-mail(s): celine.robardet@insa-lyon.fr; Contributing authors: arthur.batel@insa-lyon.fr; marc.plantevit@epita.fr; idir.benouaret@epita.fr;

## Abstract

Multi-Target Prediction (MTP) aims to predict a class for multiple targets from a single input instance. In this paper, we focus on the general ordinal outputs setting, where targets may have different numbers of ordered classes. This scenario is under-explored yet critical for human-centered applications such as educational assessment or psychological profiling. In these domains, targets span multiple aspects of an evaluation, and capturing latent relationships between seemingly independent targets is essential for deriving meaningful user profiles. Beyond prediction accuracy, it is crucial that the resulting profiles are interpretable, as these applications directly impact humans. We introduce IMPACT, a novel method that extends existing binary MTP frameworks (such as CD-BPR) to the multi-class domain through a newly designed loss function. Rooted in a Bayesian modeling framework, IMPACT jointly embeds user profiles and targets within a shared vector space, providing theoretical rigor while explicitly optimizing for both predictive accuracy and interpretability. Furthermore, IMPACT offers a geometric interpretation of the embedding learning dynamics, giving insight into how the model captures relationships between users and targets and providing an intuitive understanding of profile formation in the latent space. Experimental results show that IMPACT outperforms state-of-the-art approaches in terms of profile interpretability while maintaining competitive prediction accuracy. An ablation study highlights the contribution of each component, demonstrating the benefits of extending the framework to multiple ordered classes, the Bayesian formulation, and the geometric interpretability in enhancing both performance and transparency.
# 1 Introduction

Multi-target machine learning models (Waegeman, Dembczynski, & Hüllermeier, 2019) are designed to predict multiple variables simultaneously within a single framework. By capturing relationships and dependencies among targets, these models leverage shared information across them to enhance predictive performance. This approach is particularly advantageous in complex domains where outputs are inherently interdependent, such as multi-label classification, recommender systems, and biomedical applications. Beyond improving predictive accuracy, multi-target models offer the potential for a more integrated understanding of the underlying data patterns, enabling more informed decision-making across diverse fields.

In the following, we focus on human-centered applications such as educational assessment (Feng, Heffernan, & Koedinger, 2009) and psychological profiling (Cella et al., 2010), where the goal is not only to accurately predict user responses, but also to uncover latent attributes that explain performance patterns across multiple tasks. This problem has traditionally been addressed by Cognitive Diagnosis Models (CDMs) (Wang et al., 2024), which aim to infer individuals' abilities from observable responses to diagnostic questions targeting user-specific skills or cognitive traits. While closely related to the task addressed by multi-target machine learning models, CDMs place a stronger emphasis on interpretability, ensuring that predictions provide actionable insights. These models infer latent variables by treating question responses as proxies for users' underlying traits. Although accurate response prediction indicates that the model captures relevant patterns, it does not guarantee that the inferred psychological traits are meaningful or consistent with expert standards. This gives rise to two critical challenges: (i) the user profile must reliably predict responses, and (ii) the learned traits must vary in ways that align with expert knowledge, ensuring interpretability and trust in the model's outcomes.

An additional challenge arises when targets span multiple output types that differ from one target to another, or exhibit highly imbalanced distributions. For instance, in educational assessments, targets may include skill mastery indicators with varying numbers of levels, and continuous performance scores. This heterogeneity complicates the modeling process, requiring methods that can integrate predictions across diverse target spaces while preserving interpretability.

Previous methods have struggled to overcome these difficulties. Statistical models of cognitive processes, such as MIRT (Reckase, 2009) or DINA (De La Torre, 2009), often produce clear respondent profiles but fall short in predictive performance compared to advanced machine learning techniques such as CD-BPR (Batel, Benouaret, Fruitet, Plantevit, & Robardet, 2024). Conversely, recent algorithms, including neural CDMs (NCDM) (Wang et al., 2020) and state-of-the-art regression techniques like SVD, excel in response prediction but fail to offer interpretable profiles. These methods face a recurring trade-off: they either rely on rigid, expert-defined dimensions that
constrain predictive performance, or they combine respondent-profile dimensions in complex, non-interpretable ways, thereby reducing their utility for domain experts. Our previous work, CD-BPR (Batel et al., 2024), successfully balanced predictive accuracy with the interpretability of user profiles, but was limited to binary prediction tasks.

In this paper, we introduce a novel CDM, named IMPACT (Interpretable Model for Predictions and Analysis of Concepts and Targets using Bayesian Personalized Ranking), designed to overcome these limitations. Building on CD-BPR, our approach effectively handles heterogeneous target outcomes while preserving the interpretability of each parameter and maintaining high predictive accuracy. By explicitly modeling the joint effects of expert-defined dimensions, the proposed framework achieves a principled balance between predictive performance and interpretability, making it well-suited for diverse and complex application domains.

The remainder of the paper is organized as follows. Section 2 presents the related work. In Section 3, we formalize the problem setting. Section 4 revisits the key principles behind CD-BPR, highlighting its strengths and limitations. In Section 5, we introduce IMPACT, a significant advancement over CD-BPR, designed to handle ordered multi-class outputs while maintaining both high prediction accuracy and interpretability. We present our quantitative experiments in Section 6, where IMPACT demonstrates superior regression performance and produces more interpretable user profiles compared to state-of-the-art baselines. An ablation study further dissects the contributions of each component, while a hyperparameter analysis explores how varying the number of output classes impacts the balance between accuracy and memory efficiency. Additionally, we introduce a memory optimization strategy to enhance scalability. Finally, in our qualitative evaluation, we illustrate how the structured and interpretable user profiles produced by IMPACT enable novel applications beyond traditional multi-target learning. Section 7 summarizes our contributions and outlines future research directions.

## 2 Related Work

Our review focuses on interpretable multi-target models, emphasizing the need for methods that not only predict outcomes but also provide transparent, actionable insights for identifying user profiles.

Cognitive Diagnosis Models (CDMs) (Wang et al., 2024) were originally developed for educational assessment, where the goal is to infer a student's mastery of specific skills from their responses to diagnostic questions. These models predict binary outcomes and produce interpretable profiles aligned with domain-specific categories (e.g., labeling a math item as “algebra” or “calculus”). Early CDMs, such as DINA (De La Torre, 2009) and MIRT (Reckase, 2009) explicitly modeled cognitive processes, while more recent approaches have incorporated machine learning. For instance, Neural CDM (NCDM) (Wang et al., 2020) leverages neural networks to capture complex interactions while preserving interpretability, and Item Response Ranking (IRR) (Tong et al., 2021) enforces interpretability through resampling. CD-BPR (Batel et al., 2024) further unified interpretability and predictive accuracy by embedding users and
items in a Bayesian Personalized Ranking (BPR) framework. However, CDMs remain largely limited to binary or categorical responses and often rely on rigid, expert-defined dimensions, which restrict their applicability in broader multi-target settings.

Multi-Target Prediction (MTP) generalizes this problem by predicting multiple target variables of potentially diverse types (binary, ordinal, or continuous) (Waege- man et al., 2019). DeepMTP (Iliadis, De Baets, & Waegeman, 2023), for example, introduced a flexible two-branch neural architecture that learns joint embeddings of instances and targets, achieving strong predictive performance. While such methods offer scalability and adaptability, they often sacrifice interpretability, making them less suitable for domains where human understanding of latent traits is essential.

A closely related line of work comes from collaborative filtering, where the task is to predict user preferences from past interactions. Matrix factorization (MF) techniques such as SVD (Koren, 2008), NMF (Lee & Seung, 2000), and PMF (Salakhutdinov & Mnih, 2007) decompose the user–item matrix into latent representations that capture key behavioral traits, with varying degrees of interpretability. Extensions based on deep learning, such as BiVAECF (Truong, Salah, & Lauw, 2021) and Graph Convo- lutional Matrix Completion (GCMC) (van den Berg, Kipf, & Welling, 2017), further improve performance by integrating, respectively, probabilistic modeling and graph representations.

Among these approaches, the Bayesian Personalized Ranking (BPR) framework (Rendle, Freudenthaler, Gantner, & Schmidt-Thieme, 2009) has been particularly influential because it directly optimizes ranking objectives in large-scale user–item interactions. This ranking-based perspective bridges recommendation and cognitive diagnosis, and it inspired the development of CD-BPR, which adapts BPR to unify accurate binary prediction with interpretable student profiling. Building on this methodological connection, our work extends CD-BPR to the ordinal-output set- ting, combining the interpretability of CDMs with the flexibility of MTP and the ranking-based rigor of BPR.

3 Formalization

Let $U$ denote the set of users and $Q$ the set of questions. Each question $q \in Q$ is associated with one or more categories $c \in C$ that represent the domain dimensions covered by that question. For example, the mathematical question “Solve $x^2 = y$” is associated with the “analysis” category, the rating for the movie “Titanic” is associated with the “romance” and “action” categories in cinema, while “stress” and “memory” are examples of categories in psychological questionnaires. We write $q \leadsto c$ to indicate that question $q$ is linked to category $c$.

For each user-question pair $(u, q)$, we observe a response $y_{uq}$ that belongs to a finite, ordered set of modalities $M_q = \langle 0, \dots, m, m+1, \dots, |M_q| - 1 \rangle$. For example, if a question asks “How often do you feel anxious?”, with response options {Never, Sometimes, Often, Always}, then $M_q = \langle 0, 1, 2, 3 \rangle$ after numerical encoding, where the order reflects increasing anxiety. Other questions may have different numbers of modalities, such as $M_q = \langle 0, 1 \rangle$ for binary responses.
The dataset is structured as response logs *L*, consisting of quadruplets (*u*, *q*, *y*, *c*), with *u* ∈ *U*, *q* ∈ *Q*, *c* ∈ *C* and *y* ∈ *M*<sub>q</sub>; we omit the subscripts from *y*<sub>uq</sub> when clear from context. For questions associated with multiple categories, the logs are expanded to include an entry for each category. A summary of notations is provided in Table 1.

<table><caption>**Table 1** Notation table.</caption><tr><td>U, u</td><td>The set of users, a single user.</td></tr><tr><td>Q, q</td><td>The set of questions, a single question.</td></tr><tr><td>M<sub>q</sub> and [[0, 1]]<sub>q</sub></td><td>The set of modalities for question q and its encoding in [0, 1].</td></tr><tr><td>C, c</td><td>The set of categories, a single category.</td></tr><tr><td>L</td><td>Log of users responses.</td></tr><tr><td>q &rarr; c</td><td>Question q is related to category c.</td></tr><tr><td>(u, q, y, c)</td><td>a log entry.</td></tr><tr><td>&pi;<sub>u,q,y,c</sub></td><td>The estimated preference of user u for responding y to question q related to category c.</td></tr></table><table><tr><td>E &in; R<sup>|U|&times;|C|</sup></td><td>The matrix of interpretable user embeddings.</td></tr><tr><td>E<sub>u</sub>[c] &in; R</td><td>The c-th component of the embedding of user u.</td></tr><tr><td>I &in; R<sup>|Q|*|M|&times;|C|</sup></td><td>The matrix of the embedding of question-response pairs.</td></tr><tr><td>I<sub>qy</sub>[c] &in; R</td><td>The c-th component of question-response pair (q, y) embedding.</td></tr><tr><td>d<sub>r</sub></td><td>The dimension of the reduced question-response subspaces.</td></tr><tr><td>I<sup>r</sup> &in; R<sup>|Q|*|M|&times;d<sub>r</sub></sup></td><td>The matrix of reduced embeddings for question-response pairs.</td></tr><tr><td>W<sup>c</sup> &in; R<sup>|C|&times;d<sub>r</sub></sup></td><td>Parameter matrix for reconstructing embeddings in the users' embeddings space from the reduced question-response embeddings of category c.</td></tr></table>

Our task consists of jointly predicting the responses to all questions that a given user has not answered yet. Therefore, we aim to develop a predictive Cognitive Diagnosis Model $f_{E,I}$ that shares information across targets by learning: user embeddings **E** and question-response embeddings **I**. This model serves two purposes: (i) predicting users' responses and (ii) constructing interpretable profile vectors. Specifically, each user *u* is associated with a profile $E_u \in R^{|C|}$, and each question-response pair (q, y) is represented by a vector $I_{qy} \in R^{|C|}$. In both cases, each component should correspond to a specific domain category *c* ∈ *C*, ensuring that profiles are grounded in expert-defined categories and that both users and questions live in the same interpretable latent space of dimension *|C|*. Our model is centered on two primary objectives:

* Prediction Accuracy (O1): The model learns user parameters $E_u$ and question-response parameters $I_{qy}$ to accurately predict unobserved responses, ensuring robust performance across categories.

* Interpretability (O2): The user profile $E_u$ should provide an understandable summary of the user's characteristics in the domain. Each component $E_u[c]$ reflects the user's standing in category *c*. A component is considered meaningful if (a) its consistency with observed responses can be quantified with sufficient confidence, and (b) it allows distinguishing different user behaviors within the domain.
The response logs are divided into two disjoint sets, $L_{train}$ and $L_{test}$, which are used to train and evaluate the model $f_{EI}$, respectively. The first objective (O1) is evaluated using standard machine learning metrics that measure prediction accuracy, such as Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), and Accuracy (ACC).

For objective (O2), interpretability is not an all-or-nothing property but rather a matter of degree: a user profile is considered more interpretable when its components behave consistently with the responses observed in the corresponding categories. Specifically, the value of a profile component $E_u[c]$ should tend to increase (or at least not decrease) as a user demonstrates stronger performance in category $c$. The closer this monotonic alignment is maintained across users, the more interpretable the resulting profiles become. Formally, this requirement can be expressed as a monotonicity condition between profile components and responses: for any two users $u$ and $v$, if user $u$ provides a higher response $y$ than user $v$ to a question $q$ in category $c$, then the profile component of $u$ in that category ($E_u[c]$) should also be greater than that of $v$ ($E_v[c]$).

Because monotonicity focuses solely on preserving the relative ordering of users, it does not depend on the absolute scale or variance of the responses. This makes it particularly robust in settings with heterogeneous questionnaires or unnormalized response distributions, where raw values may vary widely but the rank order of users still provides a consistent signal.

This dual-objective evaluation framework aims to ensure that the model achieves both accurate predictions and interpretable user profiles.

# 4 Cognitive Diagnosis Based on Bayesian Personalized Ranking for binary questions

In (Batel et al., 2024), we introduce CD-BPR, a Cognitive Diagnosis Model (CDM) built upon the Bayesian Personalized Ranking (BPR) framework (Rendle et al., 2009), originally designed for pairwise ranking tasks. In such tasks, the goal is to establish a relative ranking among user responses rather than predicting absolute scores. BPR, rooted in matrix factorization, achieves this by decomposing the user-item interaction matrix into low-rank representations, capturing latent factors in a shared embedding space.

CD-BPR extends BPR's ranking principle by integrating an additional learning objective explicitly designed to enhance interpretability, coupled with a simple yet effective parameter initialization strategy. The model employs two types of embeddings, **E** and **I**, both of dimension $|C|$, enabling a unified representation of users and question-response pairs within the same latent space. By leveraging Euclidean distance, CD-BPR ensures compliance with the triangle inequality, preserving meaningful relationships between users and responses. This geometric property enhances interpretability by maintaining consistency in user-question interactions, where proximity in the embedding space reliably reflects user performance and response similarity (S. Chen, Moore, Turnbull, & Joachims, 2012). In the following we present the principles of CD-BPR.
## 4.1 Maximizing prediction accuracy (objective O1)

Considering binary questions, the user's inclination towards one of the two responses is quantified by the negative squared Euclidean distance between the embeddings of the user and the corresponding response. Therefore, the closer the user's embedding is to the response's embedding, the more likely the user is to select that response:

$$ \widehat{\pi}_{u,q,y,c} \propto -\|\mathbf{E}_u - \mathbf{I}_{qy}\|^2 \quad (1) $$

The ranking-based learning objective ensures that the probability of predicting the response observed in a log ($u, q, y, c$) exceeds that of the alternative response. In the binary case, this alternative response is denoted by $\bar{y} = 1 - y$. To achieve this, the model is trained to prioritize the user's inclination toward the actual response $y$ for question $y$ rather than $\bar{y}$. Using the Maximum A Posteriori (MAP) principle (Bishop, 2006), this is formulated as maximizing the posterior probability of the embedding vectors $\mathbf{E}$ and $\mathbf{I}$, given the observed ranking on the question/response pairs:

$$ \max \mathbb{P}(\mathbf{E}, \mathbf{I} | \widehat{\pi}_{u,q,y,c} > \widehat{\pi}_{u,q,\bar{y},c}) \propto \max \mathbb{P}(\widehat{\pi}_{u,q,y,c} > \widehat{\pi}_{u,q,\bar{y},c} | \mathbf{E}, \mathbf{I}) \mathbb{P}(\mathbf{E}, \mathbf{I}). $$

The model seeks to identify the parameters $\mathbf{E}$ and $\mathbf{I}$ that maximize the likelihood of correctly predicting the user's response, ensuring that the probability of the chosen response $y$ is higher than that of $\bar{y}$. By making the simplifying assumption of independence between questions, responses, and users, Bayesian Personalized Ranking (BPR) estimates the model parameters using the Maximum A Posteriori (MAP) principle:

$$ \arg\max_{\mathbf{E},\mathbf{I}} \prod_{(u,q,y,c) \in L_{\text{train}}} \prod_{\bar{y} \neq y} \mathbb{P}(\widehat{\pi}_{u,q,y,c} > \widehat{\pi}_{u,q,\bar{y},c} | \mathbf{E}, \mathbf{I}) \mathbb{P}(\mathbf{E}, \mathbf{I}) \quad (2) $$

The probability $\mathbb{P}(\widehat{\pi}_{u,q,y,c} > \widehat{\pi}_{u,q,\bar{y},c} | \mathbf{E}, \mathbf{I})$ is approximated by $\sigma(\widehat{\pi}_{u,q,y,c} - \widehat{\pi}_{u,q,\bar{y},c})$, where $\sigma(z) = \frac{1}{1+e^{-z}}$ is the logistic sigmoid function, which approaches 1 when $\widehat{\pi}_{u,q,y,c}$ is greater than $\widehat{\pi}_{u,q,\bar{y},c}$. The prior distribution of the parameters is set to be a centered normal distribution $(\mathbf{E}, \mathbf{I}) \sim \mathcal{N}(0, \mathbf{I}/(2\lambda))$ where $\lambda$ is a regularization hyper-parameter. The comprehensive learning objective involves minimizing the following loss:

$$ \mathcal{L}_{\text{O1}}^{\text{bin}}(\mathbf{E}, \mathbf{I}) = \sum_{(u,q,y,c) \in L_{\text{train}}} \sum_{\bar{y} \neq y} -\log(\sigma(\widehat{\pi}_{u,q,y,c} - \widehat{\pi}_{u,q,\bar{y},c})) + \lambda \|\mathbf{E}, \mathbf{I}\|^2 \quad (3) $$

## 4.2 Maximizing interpretability (objective O2)

To encourage the monotonicity assumption during the learning process, we introduce an additional loss function that penalizes deviations from the desired order of user proficiency across different dimensions associated with a question. Considering a quadriplet $(u, q, y, c)$ and another quadriplet $(\bar{u}, q, y', c)$ where $y' \le y$, indicating that user $\bar{u}$ exhibited less proficiency for the same question, our objective is to ensure that, for the category $c$ associated with question $q$, the value of $\mathbf{E}_u[c]$ is greater than $\mathbf{E}_{\bar{u}}[c]$.
This regularization term promotes the desired monotonic relationship, steering the model toward learning embeddings that align with the desired interpretability:

$$ \mathcal{L}_{O2}^{bin}(\mathbf{E}, \mathbf{I}) = - \sum_{(u,q,y,c)} \sum_{\substack{\bar{u},q',y',c \\ y' \le y}} \log (\sigma(\mathbf{E}_u[c] - \mathbf{E}_{\bar{u}}[c])) \quad (4) $$

where $\sigma$ is the logistic sigmoid function. This function ensures $\mathbf{E}_u[c] > \mathbf{E}_{\bar{u}}[c]$, acting as a gatekeeper to maintain the prescribed hierarchy among the dimension-wise values. By regularizing the learning process, it ensures that the embeddings adhere to the desired monotonicity relationship, fostering interpretability by capturing nuanced distinctions in user performance across shared dimensions.

## 4.3 Cognitive Diagnosis based on Bayesian Personalized Ranking (CD-BPR)

The embeddings $\mathbf{E}$ and $\mathbf{I}$ are optimized using a variant of Stochastic Gradient Descent (SGD) which is the Adam optimizer. Each batch consists of a user $u$, their response $y$ to a question $q$, the opposite response $\bar{y}$ for the same question, and another user $\bar{u}$ whose response to $q$ is $(q, \bar{y})$. The corresponding embeddings are updated via backpropagation using the gradient of the following loss function:

$$ \mathcal{L}^{bin}(\mathbf{E}, \mathbf{I}) = \mathcal{L}_{O1}^{bin}(\mathbf{E}, \mathbf{I}) + \mathcal{L}_{O2}^{bin}(\mathbf{E}, \mathbf{I}) $$

By combining these two strategies, CD-BPR not only achieves strong predictive accuracy but also fosters embeddings that are inherently interpretable and aligned with real-world expectations of users and stakeholders. This dual emphasis on performance and interpretability is crucial for developing robust cognitive diagnostic systems. However, the main limitation of CD-BPR lies in its restriction to binary outcomes, where responses are modeled as either correct or incorrect. To address this shortcoming, we now introduce our extension, IMPACT, which generalizes the framework to handle ordinal outputs.

# 5 IMPACT: An Interpretable Bayesian Personalized Ranking Model for Ordinal Multi-Target Prediction

Designing a multi-target model capable of handling ordinal data requires constraining the learning process to preserve the natural order between modalities. Without such constraints, the model may treat ordered categories as independent, leading to inconsistent or implausible predictions. To address this, we introduce ordinal consistency constraints on the embeddings of question-response pairs. These constraints guide the model to respect the inherent ranking among response levels while capturing richer, more nuanced relationships between users and their answers. By structuring the embedding space in this way, the model learns representations that are both predictive and interpretable, ensuring that variations across modalities correspond to meaningful differences in user behavior.
Formally, we consider a response $y$ to a question $q$, encoded for the sake of notational homogeneity as a modality $m^*$ belonging to the discrete set of ordered modalities $M_q = \langle 0, \dots, m, m+1, \dots, |M_q|-1 \rangle$, $m^* \in M_q$. Incorporating this order into the BPR framework introduces two main challenges:

* **Preserving ordinal order:** The relative ordering between response modalities must be maintained to avoid inconsistent or erratic predictions. To achieve this, the loss function is adapted to explicitly encode these ordinal relationships.

* **Scalability:** As the number of modalities increases, the parameter space grows, which may lead to poor generalization. Regularization terms are introduced to control model complexity and stabilize training.

* **Interpretability:** User embeddings are designed to remain interpretable, capturing meaningful and explainable latent traits related to users' response behaviors.

## 5.1 Maximizing prediction accuracy in ordinal outputs setting

IMPACT does not aim to directly learn the exact preference scores $\pi_{u,q,m,c}, \forall m \in M_q$. Instead, it focuses on correctly ranking preferences for the modalities, and therefore, correctly ordering modalities in the multidimensional space. The key intuition is that user inclinations can be captured by comparing squared Euclidean distances between embeddings: if a user embedding $\mathbf{E}_u$ is closer to a question-modality embedding $\mathbf{I}_{qm}$, the model should favor modality $m$ over others. This ranking-based objective is particularly well suited to ordinal responses, as it provides flexibility in parameterization while prioritizing the prediction of the correct response.

We introduce the operator $>_{u,q}$, which ranks a user's predicted preferences $\pi_{u,q,m,c}$ over all possible modalities $m \in M_q$ for a given question $q$. Extending Eq. 2, the embeddings are optimized to ensure that the predicted preference for $m^*$, the ground truth, is consistently ranked higher than for any other modality:

$$ \arg\max_{\mathbf{E},\mathbf{I}} \prod_{(u,q,m^*,c)} \prod_{m \in M_q \setminus m^*} \mathbb{P}(\widehat{\pi_{u,q,m^*,c}} >_{u,q} \widehat{\pi_{u,q,m,c}} | \mathbf{E}, \mathbf{I}) \mathbb{P}(\mathbf{E}, \mathbf{I}) \quad (5) $$

under the assumption that logs are independent.

### 5.1.1 Joint optimization of user and item embeddings via ordinal-aware loss and regularization

To accurately model user profiles, the true modality for each user should consistently receive the highest ranking. Moreover, the model must distinguish between modalities that are close to the true one (e.g., $m^* \pm 1$) and those that are further apart (e.g., $m^* \pm 2$). Failing to account for this proximity can impair generalization, as the model may overfit to specific response values instead of learning a robust interaction function that captures smooth and meaningful transitions between modalities.

To address this challenge, we propose to learn the ranking of user inclinations between pairs of adjacent modalities (e.g., $m$ and $m+1$). This approach requires
only $|M_q| - 1$ pairwise comparisons per question but leverages the transitivity of ordinal relationships to infer that modalities further from the optimal one are less likely to be selected. Importantly, each comparison must account for whether a given modality is greater or smaller than the optimal one to preserve consistent ordering. This formulation ensures that ordinal structure is respected across all modalities. Consequently, Eq. 5 becomes:

$$ \arg\max_{\mathbf{E},\mathbf{I}} \prod_{(u,q,m^*,c)} \left[ \prod_{m<m^*} \mathbb{P}(\widehat{\pi_{u,q,m,c}} <_{u,q} \pi_{u,q,m+1,c} | \mathbf{E}, \mathbf{I}) \times \prod_{m \ge m^*} \mathbb{P}(\widehat{\pi_{u,q,m,c}} \ge_{u,q} \pi_{u,q,m+1,c} | \mathbf{E}, \mathbf{I}) \right] \mathbb{P}(\mathbf{E}, \mathbf{I}) \quad (6) $$

Boundary modalities ($m = 0$ and $m = |M_q| - 1$) participate in only one comparison, whereas all intermediate modalities appear in two. To ensure balanced learning, we introduce two sentinel modalities, $m = -1$ and $m = |M_q|$, excluded from prediction via a masking mechanism. These auxiliary embeddings stabilize gradient updates and improve convergence without influencing the model's outputs.

### Loss formulation.

To model the probability of correctly ranking two modalities, we employ the softplus activation function instead of the log-sigmoid used in Eq. 3, defined as softplus($x$) = $log(1 + e^x)$ (Dugas, Bengio, Bélisle, Nadeau, & Garcia, 2000). This smooth and differentiable approximation of the positive-part function ensures numerical stability and prevents vanishing gradient. Note that its derivative corresponds to the sigmoid function. The pairwise probability $\mathbb{P}(\widehat{\pi_{u,q,m,c}} > \pi_{u,q,m',c} | \mathbf{E}, \mathbf{I})$, with $m \ne m'$, is thus modeled as softplus($\widehat{\pi_{u,q,m,c}} - \pi_{u,q,m',c}$). Substituting this formulation into Eq. 6 yields the loss function:

$$ \begin{aligned} \mathcal{L}_{O1}^{ord}(\mathbf{E}, \mathbf{I}) &= - \log \mathbb{P}(>_{u,q} | \mathbf{E}, \mathbf{I}) \\ &= \frac{1}{|L|} \sum_{(u,q,m^*,c)} \frac{1}{|M_q| + 2} \left[ \sum_{m<m^*} \text{softplus}(\|\mathbf{E}_u - \mathbf{I}_{q(m+1)}\|_2^2 - \|\mathbf{E}_u - \mathbf{I}_{qm}\|_2^2) \right. \\ & \qquad \left. + \sum_{m \ge m^*} \text{softplus}(\|\mathbf{E}_u - \mathbf{I}_{qm}\|_2^2 - \|\mathbf{E}_u - \mathbf{I}_{q(m+1)}\|_2^2) \right] \end{aligned} \quad (7) $$

where normalization factors $\frac{1}{|L|}$ and $\frac{1}{|M_q|+2}$ facilitate numerical stability and enable fair comparison with alternative loss formulations.

### Regularization.

To prevent overfitting and ensure smooth embedding spaces, we place a centered Gaussian prior on all model parameters: $\mathbf{E}, \mathbf{I} \sim \mathcal{N}(0, \mathbf{I}/(2\lambda))$, where $\lambda$ controls the
strength of regularization. The corresponding penalty term, derived from the log-prior, is given by:

$$ \mathcal{R}_{\text{O1}}(\mathbf{E}, \mathbf{I}) = -\log \mathbb{P}(\mathbf{E}, \mathbf{I}) = \lambda [\|\mathbf{E}\|_2^2 + \|\mathbf{I}\|_2^2]. \quad (8) $$

This term encourages compact embeddings and prevents parameter blow-up, thereby improving generalization.

### 5.1.2 Model optimization

Let us analyze the behavior of the model during standard Stochastic Gradient Descent (SGD) optimization. Consider a training quadruplet $(u, q, m^*, c)$, where $m^*$ denotes the true modality for user $u$. We first analyze the updates of user embeddings and then those of modality embeddings.

#### User embedding updates

Let us define $\Delta_m = \|\mathbf{E}_u - \mathbf{I}_{q(m+1)}\|_2^2 - \|\mathbf{E}_u - \mathbf{I}_{qm}\|_2^2$, where $m$ represents an alternative modality. Let $s_m$ be a function that takes the value 1 if $m < m^*$ and -1 otherwise (i.e., if $m \ge m^*$). The step $\delta$ taken by the optimization algorithm during the user embedding update depends on the derivative of the loss function evaluated for the given training quadruplet:

$$ \delta(\mathbf{E}_u) = -\mu \frac{\partial \mathcal{L}_{\text{O1}}^{\text{ord}}}{\partial \mathbf{E}_u}(\mathbf{E}, \mathbf{I}) = \frac{2\mu}{|M_q| + 2} \sum_{m \in M_q} s_m \sigma(s_m \Delta_m)(\mathbf{I}_{q(m+1)} - \mathbf{I}_{qm}) \quad (9) $$

with $\mu$ the learning rate.

Equation 9 shows that the user embedding is updated by a sum of vectors whose directions are entirely determined by the set of modality embeddings (see Figure 1). Taking the sign terms into account, we observe that the user embedding tends, on average, to move from incorrect modalities toward the true one. The weight associated with each successive pair $(m, m+1)$ is given by $\sigma(\Delta_m)$, where $\sigma(t)$ denotes the logistic sigmoid function, which is also the derivative of the softplus function. Consequently, modality embeddings that are consistent with the user's response, that is whose distances to $\mathbf{E}_u$ follow the correct ordering with respect to the ground truth, receive higher weights and contribute more significantly to the update of $\mathbf{E}_u$. Conversely, inversely ordered (mismatched) modality embeddings yield weights close to zero, resulting in a negligible impact on the user embedding update. In this sense, the sigmoid function acts as a gating mechanism that suppresses updates driven by incorrectly ordered modality embeddings.

Since the factor $\sigma(\Delta_m)$ is precisely the modeled relative probability

$$ \sigma(\Delta_m) := \mathbb{P}(\widehat{\pi_{u,q,m,c}} <_{u,q} \widehat{\pi_{u,q,m+1,c}} | \mathbf{E}, \mathbf{I}), $$
<figure><img src="image_3.png" /><figcaption>**Fig. 1** Visualization of a single SGD update step for the user embedding, based on the gradient of $\mathcal{L}_{O1}^{ord}$, for one training log. The user embedding is updated by the sum of vectors between modalities (red arrow).</figcaption></figure>

the update propagates information primarily from responses that are comparatively likely under the data, thereby steering the user profile toward the true modality.

Furthermore, the loss function is globally convex with respect to $\mathbf{E}$, since it can be expressed as a sum of softplus terms, each being the composition of a convex and non-decreasing function with an affine transformation of $\mathbf{E}$ (via $\Delta_m$). Consequently, the optimization landscape admits no spurious local minima: every stationary point corresponds to the global minimum. In practice, convergence to this optimum can be effectively achieved using adaptive gradient methods such as ADAM.

## Modalities’ embedding updates

We apply the same methodology to the correct and incorrect modalities’ embedding to analyze their optimization:

* Let us examine how an incorrect modality embedding is updated during training. If $m < m^*$ (and thus $s_m = s_{m-1} = 1$), then we have:

$$ \delta(\mathbf{I}_{qm}) = -\mu \frac{\partial \mathcal{L}_{O1}^{ord}(\mathbf{E}, \mathbf{I})}{\partial \mathbf{I}_{qm}} = \frac{2\mu}{|M_q| + 2} [\sigma(\Delta_m) - \sigma(\Delta_{m-1})] (\mathbf{I}_{qm} - \mathbf{E}) \quad (10) $$
<figure><img src="image_4.png" /><figcaption>Fig. 2 Visualisation of a single SGD update of modality embeddings, illustrating incorrect vs. correct updates, based on the gradient of $\mathcal{L}_{O1}^{ord}$ for one training log.</figcaption></figure>

If $m > m^*$ (and hence $s_m = s_{m-1} = -1$), we obtain:

$$ \delta(\mathbf{I}_{qm}) = -\mu \frac{\partial \mathcal{L}_{O1}^{ord}}{\partial \mathbf{I}_{qm}}(\mathbf{E}, \mathbf{I}) = \frac{2\mu}{|M_q| + 2} [\sigma(-\Delta_{m-1}) - \sigma(-\Delta_m)] (\mathbf{I}_{qm} - \mathbf{E}) \quad (11) $$

Equation 10 and Equation 11 operate in the same manner, differing only by a sign: both updates are given by a scalar coefficient multiplied by the radial vector from $\mathbf{E}$ to $\mathbf{I}_{qm}$. Consequently, each modality embedding is either pushed away from or pulled toward $\mathbf{E}$. The corresponding update steps are illustrated in two dimensions in Figure 2. These equations show that both the direction and the magnitude of the update are determined by the interaction of the m-th modality with its two neighbors, due to the chained comparison between modalities ($\sigma(\Delta_m)$ and $\sigma(\Delta_{m-1})$). A comparison of these interactions reveals that $\mathbf{I}_{qm}$ moves in the direction that favors a consistent ordering of modalities $m-1, m, m+1$ with respect to the user $\mathbf{E}$ (distances decrease as $m$ approaches $m^*$). The magnitude of this movement depends on the level of disagreement between the expected ordering of modalities and their actual embedding ordering, until convergence to a stationary point where $\Delta_{m-1} \approx \Delta_m$. Interestingly, the comparison with the
right-neighbor modality $\Delta_m$ makes the loss function non-convex with respect to the incorrect modality embedding, thereby requiring good initialization and appropriate regularization for stable optimization.

From the previous optimization analysis, we can infer that the radius between modalities on either side of $m^*$ tends to shrink as the user embedding is pushed away. Since the optimization process preserves relative distances between modalities, this contraction may cause opposite modalities to collapse onto each other, harming the model's generalization ability. The following section introduces a dedicated regularization term to prevent this undesirable local minimum.

* Let us consider the equation governing how the correct modality embedding is updated during training

$$ \delta(\mathbf{I}_{qm^*}) = -\mu \frac{\partial \mathcal{L}_{O1}^{ord}}{\partial \mathbf{I}_{qm^*}}(\mathbf{E}, \mathbf{I}) = \frac{2\mu}{|M_q| + 2} [\sigma(\Delta_{m^*-1}) + \sigma(-\Delta_{m^*})] (\mathbf{E} - \mathbf{I}_{qm^*}) \quad (12) $$

Equation 12 shows that the modality embedding $\mathbf{I}_{qm^*}$ is systematically pushed toward the user embedding $\mathbf{E}$, since the scaling coefficient $[\sigma(\Delta_{m^*-1}) + \sigma(-\Delta_{m^*})]$ is always positive. The magnitude of this update is proportional to the consistency of the ordering of modalities $m^*-1, m^*$ and $m^*+1$. As with incorrect modalities, the loss is non-convex with respect to the true modality embedding.

The intuitive IMPACT optimization, which consists of linearly moving parameters in the joint embedding space, provides confidence in the model's ability to generalize. By further introducing a method to make this space interpretable in Section 5.2, we offer a simple understanding of IMPACT's learning principles that can be communicated to non-expert users. The ability to intuitively explain an efficient learning dynamics, through linear movements toward or away from a user vector, is rare in machine learning. These linear movements are driven by the preference score derivative: $\frac{\partial \pi_{u,q,m^*,c}}{\partial \mathbf{E}_u} = -2(\mathbf{E}_u - \mathbf{I}_{qm^*})$. Their effectiveness demonstrates that modeling preferences using squared distances is a viable approach.

### 5.1.3 Refinement of item embeddings through regularized ordinal consistency

From a broader perspective, IMPACT models modality choice based on Euclidean distances. To encourage the preservation of the modality order, it is desirable that successive modalities remain closer to each other than to more distant ones. Rather than enforcing this as a strict constraint, we implement it as a soft preference in the loss function, allowing the model to learn embeddings that approximately satisfy the ordinal relationships while maintaining flexibility to better generalize. Formally, for all modalities $m, n \in M_q$ with $m < n$, we encourage

$$ \|\mathbf{I}_{qm} - \mathbf{I}_{qn}\|_2^2 < \|\mathbf{I}_{qm} - \mathbf{I}_{q(n+1)}\|_2^2 \quad (13) $$

Violations of this ordering are penalized smoothly rather than prohibited, providing a differentiable mechanism for the model to respect ordinal structure.
Considering such a constraint would incur a computational cost of $\mathcal{O}(|M_q|(|M_q| - 2))$ per log in the training dataset, making it computationally expensive. To mitigate this, we propose constraining the order of distances exclusively with respect to the sentinel embeddings (see modalities $-1$ and $M_q$ defined in Section 5.1.1) at the extremities of the modality ordering. This strategy ensures an equal gradient distribution for the embeddings of the predicted modalities while efficiently preserving the modality order in the vector space through the double-chain constraint. Consequently, the computational cost per log in the training set is reduced to $\mathcal{O}(2(|M_q| - 2))$.

We derive a loss term from Eq. 13 in a manner similar to $\mathcal{L}_{O1}^{ord}$. However, it should be viewed as a regularization term, as it does not directly improve the prediction. Instead, it guides the model toward better solutions during training. Furthermore, we observed that this term often tends toward zero, reinforcing the notion that it acts as a structural constraint, which partially depends on the prediction error.

$$ \mathcal{R}_I(\mathbf{I}) = \sum_{(u,q,m^*,c)} \left[ \sum_{m \in M_q \setminus 0} \text{softplus}(\|\mathbf{I}_{q0} - \mathbf{I}_{qm}\|_2^2 - \|\mathbf{I}_{q0} - \mathbf{I}_{q(m+1)}\|_2^2) + \sum_{m \in M_q \setminus (|M_q|)} \text{softplus}(\|\mathbf{I}_{q(|M_q|)} - \mathbf{I}_{q(m+1)}\|_2^2 - \|\mathbf{I}_{q(|M_q|)} - \mathbf{I}_{qm}\|_2^2) \right] \quad (14) $$

Similar to $\mathcal{L}_{O1}^{ord}$, the regularization term $\mathcal{R}_I$ admits a geometric interpretation. Unlike $\mathcal{L}_{O1}^{ord}$, however, the relevant distances are measured from each question-modality embedding to a fixed sentinel point. As a result, $\mathcal{R}_I$ organizes the question-modality embeddings into an ordered chain within the embedding space, independently of the user embeddings.

## 5.2 Self-interpretable user profiles

The objective ($\mathcal{O}2$) aims to ensure that user profiles are interpretable by constraining their components to be monotonically increasing with respect to the users' responses in the corresponding dimension. To achieve this, we address the following optimization problem:

$$ \arg\max_{\mathbf{E}} \sum_{(u,q,y_u,c)} \sum_{(v,q,y_v,c)} \delta[(y_u - y_v)(\mathbf{E}_u[c] - \mathbf{E}_v[c])] \quad (15) $$

with

$$ \delta(x) = \begin{cases} 1 & x > 0 \\ 0 & x \le 0 \end{cases} $$

This optimization problem is not only non-convex but must also be addressed without significantly compromising predictive performance. Fortunately, the problem involves pairwise ordering, which is a fundamental principle underlying the design of IMPACT. To solve Eq. 15, we propose to relax the constraint to derive a continuous
learning objective that aligns well with our used differential optimization algorithm, Adam:

$$ \sum_{(u,q,y_u,c)} \sum_{(v,q,y_v,c)} \max(\mathrm{sign}(y_u - y_v)(\mathbf{E}_u[c] - \mathbf{E}_v[c]) - 1, 0) < \sum_{(u,q,y_u,c)} \sum_{(v,q,y_v,c)} \delta[(y_u - y_v)(\mathbf{E}_u[c] - \mathbf{E}_v[c])] \quad (16) $$

Using $-1$, the margin enforces the ordering of embeddings near 0. We transform the relaxed constraint into a loss function to minimize by applying $\max(-x, 0)$. Consequently, the proposed objective function is the Hinge loss, a canonical and well-suited loss for learning strict ranking (W. Chen, Liu, Lan, Ma, & Li, 2009).

$$ \mathcal{L}_{O2}^{ord}(\mathbf{E}) = \frac{\sum_{(u,q,y_u,c)} \sum_{(v,q,y_v,c)}^{1} \delta(y_u - y_v)}{\sum_{(u,q,y_u,c)} \sum_{(v,q,y_v,c)} \max(1 - \mathrm{sign}(y_u - y_v)(\mathbf{E}_u[c] - \mathbf{E}_v[c]), 0)} \quad (17) $$

Similarly to $\mathcal{L}_{O1}^{ord}$, a normalization factor is added. We note that, even if not explicit in the formula, this loss also leverage the transitivity of the comparison operator so that the sum of pairwise comparison allow to order each users' embeddings relatively to each other users' embeddings without needing to compare all possible pairs.

Similarly to the previous losses, we can give a geometrical interpretation to $\mathcal{L}_{O2}^{ord}$. Indeed, being minimized, the loss forces the users' embeddings to be correctly ordered along the axis of each question category (see Figure 3).

## 5.3 Losses combination

Our model incorporates four loss and regularization terms. Two are data-independent regularizations: $\mathcal{R}_{O1}(\mathbf{E}, \mathbf{I})$ constrains parameter magnitudes, while $\mathcal{R}_{I}(\mathbf{I})$ enforces sequential ordering of $\mathbf{I}_{qm}$ vectors in the embedding space. The other two are data-driven: $\mathcal{L}_{O1}^{ord}$ optimizes predictive performance, and $\mathcal{L}_{O2}^{ord}$ enhances interpretability.

Using a logarithmic formulation, the weighted sum of terms combines naturally. We propose the following strategy to address the problem of balancing the four weights: we treat $\lambda$, the weight of $\mathcal{R}_{O1}$, as a hyperparameter controlling parameter dispersion. Since embeddings are normalized, $\lambda$ is typically set to a very small value ($< 10^{-5}$). The weights of the remaining terms, $\mathcal{L}_{O1}^{ord}$, $\mathcal{L}_{O2}^{ord}$, and $\mathcal{R}_{I}$, are constrained to sum to one in order to stabilize their contribution with respect to $\mathcal{R}_{O1}$. We further distinguish regularization terms from loss terms by defining $\beta$ as the relative weight of $\mathcal{R}_{I}$, while the factor $1 - \beta$ is used to weight the two loss terms $\mathcal{L}_{O1}^{ord}$ and $\mathcal{L}_{O2}^{ord}$. A study of the $\beta$ hyperparameter (see Section 6.4) shows only a limited impact for values in the range [0.2, 0.5]. We selected $\beta = \frac{1}{3}$, which maximizes the MAE. Finally, to balance differences in magnitude and convergence rates between the remaining loss terms, we apply CoV-Weighting (Groenendijk, Karaoglu, Gevers, & Mensink, 2021) to $\mathcal{L}_{O1}^{ord}$ and $\mathcal{L}_{O2}^{ord}$, dynamically adjusted according to their relative variation such that they sum
<figure><img src="image_5.png" /><figcaption>Fig. 3 $\mathcal{L}_{O_2}^{ord}$: Visualization of the margin constraint to enforce interpretability. It can be observed that the higher $\mathbf{E}_v[c_0]$ compared to $\mathbf{E}_u[c_0]$, the closer they will be to there optimal question-modality.</figcaption></figure>

to $1 - \beta$:

$$w_{\mathcal{L}}(t) = \frac{\sigma_{\mathcal{L}}(t)}{\mu_{\mathcal{L}}(t)}, \quad w_{\mathcal{L}_{O_1}^{ord}}(t) + w_{\mathcal{L}_{O_2}^{ord}}(t) = 1 - \beta$$

with $\mu_{\mathcal{L}}(t)$ and $\sigma_{\mathcal{L}}(t)$ the mean and standard deviation of the loss at epoch $t$. This ensures losses with higher variation contribute more, improving training stability.

The total loss is thus:

$$\mathcal{L}(t) = w_{\mathcal{L}_{O_1}^{ord}}(t)\mathcal{L}_{O_1}^{ord} + w_{\mathcal{L}_{O_2}^{ord}}(t)\mathcal{L}_{O_2}^{ord} + \beta\mathcal{R}_I(\mathbf{I}) + \lambda\mathcal{R}_{O_1}(\mathbf{E}, \mathbf{I}), \quad (18)$$

## 5.4 Parameter Initialization

Proper initialization is crucial to ensure stable convergence and reasonable performance in both prediction and interpretability from the outset. To jointly satisfy these objectives, we initialize each component of the user embedding $\mathbf{E}_u$ as the average of the user's observed responses within the corresponding category. Similarly, question-modality embeddings $\mathbf{I}_{qm}$ are initialized by distributing them uniformly within their subspace, with components ordered according to the increasing modality indices. This initialization strategy provides a structured starting point that reflects the ordinal nature of the data and facilitates faster, more interpretable learning.
# 6 Empirical Validation

This section evaluates IMPACT's performance on both regression and interpretability metrics, comparing it to established methods. The assessment utilizes standard benchmark datasets as well as a new dataset introduced in this work.

<table><caption>**Table 2** Main characteristics of the MOVIELENS, POSTCOVID, PROMIS and PORTRAIT datasets, with $|C|/q$ the number of categories per question; $|U|/q$ the number of users per question; $|U|/c$ the number of users per category; $|Q|/c$ the number of questions per category; $|C|/u$ the number of categories per user; $|Q|/u$ the number of questions per user.</caption><thead><tr><th rowspan="2">Dataset</th><th rowspan="2">|L|</th><th rowspan="2">|U|</th><th rowspan="2">|Q|</th><th rowspan="2">|C|</th><th rowspan="2">max |M<sub>q</sub>|</th><th colspan="3">|C|/q</th><th colspan="3">|U|/q</th><th colspan="3">|U|/c</th></tr><tr><th>min</th><th>mean</th><th>max</th><th>min</th><th>mean</th><th>max</th><th>min</th><th>mean</th><th>max</th></tr></thead><tbody><tr><td>MOVIELENS</td><td>1562240</td><td>4231</td><td>2421</td><td>11</td><td>5</td><td>1</td><td>2</td><td>5</td><td>50</td><td>374</td><td>2731</td><td>3776</td><td>4043</td><td>4231</td></tr><tr><td>POSTCOVID</td><td>48583</td><td>746</td><td>180</td><td>17</td><td>22</td><td>1</td><td>1</td><td>3</td><td>38</td><td>192</td><td>460</td><td>38</td><td>206</td><td>460</td></tr><tr><td>PROMIS</td><td>1805842</td><td>16879</td><td>790</td><td>15</td><td>9</td><td>1</td><td>1</td><td>1</td><td>1839</td><td>2286</td><td>16558</td><td>12007</td><td>12348</td><td>16559</td></tr><tr><td>PORTRAIT</td><td>133632</td><td>1152</td><td>116</td><td>9</td><td>108</td><td>1</td><td>1</td><td>1</td><td>1152</td><td>1152</td><td>1152</td><td>1152</td><td>1152</td><td>1152</td></tr><tr><td colspan="17">Minimal thresholds</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td colspan="3">|Q|/c</td><td colspan="3">|C|/u</td><td colspan="3">|Q|/u</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><th>min</th><th>mean</th><th>max</th><th>min</th><th>mean</th><th>max</th><th>min</th><th>mean</th><th>max</th></tr><tr><td>MOVIELENS</td><td>50</td><td>50</td><td>50</td><td></td><td></td><td>62</td><td>340</td><td>1670</td><td>5</td><td>11</td><td>11</td><td>50</td><td>214</td><td>1832</td></tr><tr><td>POSTCOVID</td><td>20</td><td>20</td><td>5</td><td></td><td></td><td>7</td><td>14</td><td>22</td><td>1</td><td>5</td><td>10</td><td>20</td><td>46</td><td>92</td></tr><tr><td>PROMIS</td><td>1835</td><td>102</td><td>1</td><td></td><td></td><td>6</td><td>52</td><td>56</td><td>2</td><td>11</td><td>15</td><td>102</td><td>107</td><td>118</td></tr><tr><td>PORTRAIT</td><td>50</td><td>50</td><td>10</td><td></td><td></td><td>10</td><td>13</td><td>18</td><td>9</td><td>9</td><td>9</td><td>116</td><td>116</td><td>116</td></tr></tbody></table>

## 6.1 Experimental Setup

### 6.1.1 Datasets

We consider four real-world datasets comprising user responses to questions, along with associated dimensions linked to these questions. MOVIELENS (Harper & Konstan, 2015) captures movie ratings, reflecting individuals' expressed preferences for movies. Each preference is represented as a tuple, indicating a user's 0-5 star rating for a particular movie. Additionally, movies in this dataset are categorized into 11 distinct genres<fnref n="1" />. The POSTCOVID dataset contains ordinal responses to online neuropsychiatric health questionnaires focused on COVID-19 infection symptoms. Each question offers a different number of possible responses. The categories cover a range of COVID-related symptoms, such as stress, memory, and concentration issues<fnref n="2" />. The data were collected from the French population between 2020 and 2023 by Humans Matter, a company with which we collaborate. The PROMIS (Cella et al., 2010) dataset consists of ordinal responses to general physical, mental, and social health questionnaires gathered during a study by the American National Institutes of Health. Each question also presents a varying number of possible responses. The data were collected from a controlled panel of the American population between July 2006 and March 2007.

<footnote n="1">¹MOVIELENS: Animation, Children, Drama/Romance/Comedy, Musical, Fantasy, Science-Fiction, Thriller/Action/Adventure, War, Crime, Mystery, Horror.</footnote>

<footnote n="2">²POSTCOVID: Anxiety, Sadness, Avoiding, Depression, Sleep disorders, Fatigue, Impact of psychological trauma (Mood), State of stress, Impact of psychological trauma (Hypervigilance), Perceived social support, Self-efficacy, Agency, Internal control, External control, Memory, concentration, Quality of life</footnote>
Example categories include emotional distress and pain<fnref n="3" />. The PORTRAIT dataset contains continuous responses to mental health questionnaires gathered in collaboration with Humans Matter. Answers were given on a scale from 0 to 10 using a numerical slider on an online platform. The categories cover a wide range of psychiatric symptoms, including sadness, anxiety, and inattention/hyperactivity, among others⁴. The data were collected from a controlled panel of the French population in 2024.

Each dataset comprises logs detailing a single numerical response to questions related to the respondent. All datasets underwent a thorough sanitization process: entries with missing user, questions, responses, or dimensions values were removed, along with duplicate records. Users, questions, or categories with insufficient entries were also excluded. Table 2 provides the basic statistics of the cleaned datasets, along with the thresholds applied during preprocessing. Responses for each question were rescaled to range from 0 to 1, ensuring equal intervals between consecutive modalities. Figure 4 shows the distribution of responses for each dataset.

<figure><img src="image_6.png" /><figcaption>Fig. 4 Responses distribution for each dataset.</figcaption></figure>

## 6.1.2 Baselines

To evaluate the performance of IMPACT, we compare it against three state-of-the-art matrix factorization techniques: NMF (Lee & Seung, 2000), SVD (Koren, 2008), PMF (Salakhutdinov & Mnih, 2007), as well as more established deep learning based approaches: BiVAE-CF (Truong et al., 2021) and GCMC (van den Berg et al., 2017). We also confront IMPACT to a multi-target prediction algorithm: DeepMTP (Iliadis et al., 2023).

All baseline hyperparameters were optimized using the Adam optimization algorithm. For each method, we set the number of latent space categories to match the number of categories $|C|$ in the dataset (see Table 2).

## 6.1.3 Training and hyperparameter setting

We employed a 5-fold cross-validation strategy, splitting the logs into training, validation, and test sets with proportions of 60%, 20%, and 20%, respectively. The optimal

<footnote n="3">³PROMIS: Pain-Interference, Pain-Quality, Pain-Behavior, Emotional Distress-Anxiety, Emotional-Distress-Depression, Emotional Distress-Alcohol Abuse, Emotional Distress-Anger, Physical function-Part I, Physical function-Part II, Physical function-Part III, Fatigue-Fatigue Experience, Fatigue-Fatigue Impact, Social Role-Performance, Social Role-Satisfaction

⁴PORTRAIT: Sadness/suicidal thoughts, Stress/fear/anxiety, Sleep disorders, Maniac, Impulsivity/anger, Hyperactivity/inattention, Mood Instability, Burn out, autistic traits</footnote>
number of epochs was determined using an early stopping mechanism based on the RMSE evaluated on the validation set. Hyperparameter tuning was performed using a Tree-structured Parzen Estimator (TPE) sampler, implemented in the Optuna library, with the validation RMSE as the optimization objective. Each method was evaluated over 100 trials per dataset. Model evaluation was then conducted on the test set to assess generalization performance. For the results in Table 3, the 5-fold cross-validation was repeated three times with different random seeds. The ablation study (see Table 4) was conducted using a single seed.

### 6.1.4 Evaluation measures

We evaluate the model's performance using standard regression metrics, including Root Mean Square Error (RMSE) and Mean Absolute Error (MAE). Since the number of ordinal values $|M_q|$ can be large and vary across questions, we normalize the question responses following (Iliadis, De Baets, & Waegeman, 2022; Salman & Kecman, 2012) by rescaling them to the unit interval:

$$ \llbracket 0, 1 \rrbracket_q = \left\{ \frac{m}{|M_q| - 1} \mid m \in M_q \right\}. $$

For instance, in a movie rating task where ratings range from 0 to 5 stars ($M_q = \langle 0, 1, \dots, 5 \rangle$), a rating of $m = 4$ corresponds to a normalized value of 0.8. This normalization ensures a consistent representation across heterogeneous response scales and facilitates fair comparison across questions. We also used the Accuracy (ACC) metric for datasets with discrete responses (MOVIELENS, POSTCOVID, and PROMIS). Because competitors predict real values, we discretize their predictions to the nearest modality in order to compute the accuracy.

To further assess the quality and interpretability of the user embeddings, we employ the Degree of Agreement (DOA) measure (Wang et al., 2020), as well as metrics based on the Pearson Correlation Coefficient (PC-ER). The Degree of Agreement (DOA) evaluates the alignment between users' embeddings and their responses in the response log. It measures the consistency of the model with observed user behavior, offering insights into the interpretability and reliability of the embeddings. Higher DOA values ($DOA \in [0, 1]$) indicate stronger alignment between the model predictions and the data. Formally, for a given domain category $c$, $DOA(c)$ evaluates pairs of users $(u, v)$ where the model predicts that $u$ performs better than $v$ ($\mathbf{E}_u[c] > \mathbf{E}_v[c]$). It calculates the proportion of questions related to domain category $c$ that both users have answered, for which $u$'s responses outperform $v$'s ones in the data.

$$ \begin{aligned} DOA(c) = & \frac{1}{\alpha} \sum_{u \in U} \sum_{v \in U} \delta(\mathbf{E}_u[c] > \mathbf{E}_v[c]) \\ & \times \frac{|\{q \in Q \mid (u, q, y_u, c), (v, q, y_v, c) \in L \text{ with } y_u > y_v\}|}{\beta(u, v)} \\ & \text{with } \alpha = \sum_{u \in U} \sum_{v \in U} \delta(\mathbf{E}_u[c] > \mathbf{E}_v[c]) \delta(\beta(u, v) > 0), \end{aligned} $$
$$ \beta(u, v) = |\{q \in Q | (u, q, y_u, c), (v, q, y_v, c) \in L \text{ with } y_u \neq y_v\}| $$

$$ \delta(x) = \begin{cases} 1 & \text{if } x \text{ is true} \\ 0 & \text{otherwise} \end{cases} $$

The PC-ER metric quantifies the correlation between two vectors, one that corresponds to the embedding value on category c for all users, and a second one that correspond to the average responses to questions of this category for all users. Specifically, we compute the Pearson correlation coefficient for category c defined as

$$ \rho_{\mathbf{F}_c, \mathbf{M}_c} = \frac{\mathrm{cov}(\mathbf{F}_c, \mathbf{M}_c)}{\sigma_{\mathbf{F}_c} \sigma_{\mathbf{M}_c}} $$

where $\mathbf{M}_c$ represents the average user response vector and is calculated as:

$$ \mathbf{M}_c[u] = \frac{\sum_{(u,q,y,c) \in L} y}{|\{q | (u, q, y, c) \in L\}|} \quad \text{and} \quad \mathbf{F}_c[u] = \mathbf{E}_u[c] $$

The PC-ER metric assesses the linear relationship between user embeddings and their average responses across categories, providing a quantitative evaluation of the alignment between the model's embeddings and the observed data. A higher PC-ER value indicates that the embeddings capture the underlying response patterns more reliably. The final PC-ER value is computed as the average correlation across all categories:

$$ \mathrm{PC-ER} = \frac{1}{|C|} \sum_{c \in C} \rho_{\mathbf{F}_c, \mathbf{M}_c} $$

### 6.1.5 Implementation

The code to reproduce the experiment is made available on a public repository⁵. It includes a comprehensive, ready-to-use PyTorch implementation of IMPACT. It also integrates the Cornac library for matrix factorization competitors (Salah, Truong, & Lauw, 2020) and the DeepMTP multi-target prediction library (Iliadis et al., 2023). IMPACT employs the Adam optimizer (Kingma & Ba, 2015), an adaptive stochastic gradient method that updates parameters along the negative loss gradient with per-parameter learning rates derived from first and second moment estimates. All experiments are run on a linux server with an INTEL x86_64 CPU and a NVIDIA RTX A6000 GPU.

## 6.2 Predictive performance and interpretability

Table 3 shows that IMPACT achieves competitive performance across all datasets on the RMSE, MAE, and ACC metrics compared to baseline models. Notably, the DeepMTP method failed to execute on the POSTCOVID dataset because certain modalities were missing in some cross-validation folds. Table 3 also shows that IMPACT achieves the best results on the interpretability measures.
<table><caption>**Table 3** Performance study: Regression (RMSE, MAE) and classification (ACC) performances, as well as interpretability metrics (DOA, PC-ER). Bold numbers indicate the best mean (lowest for RMSE/MAE, highest for the other metrics) and any result within this best method's standard deviation. ↑ and ↓ indicate metrics to be maximized and minimized, respectively.</caption><thead><tr><th>Methods</th><th>RMSE ↓</th><th>MAE ↓</th><th>ACC ↑</th><th>DOA ↑</th><th>PC-ER ↑</th></tr></thead><tbody><tr><td colspan="6" style="text-align: center; font-weight: bold;">MOVIELENS</td></tr><tr><td>IMPACT</td><td>0.2225 ± 0.0005</td><td><strong>0.1475 ± 0.0004</strong></td><td><strong>0.4993 ± 0.0009</strong></td><td><strong>0.6571 ± 0.0008</strong></td><td><strong>0.3728 ± 0.0056</strong></td></tr><tr><td>NMF</td><td><strong>0.2018 ± 0.0004</strong></td><td>0.1566 ± 0.0003</td><td>0.4880 ± 0.0010</td><td>0.4979 ± 0.0084</td><td>-0.0030 ± 0.0200</td></tr><tr><td>PMF</td><td>0.2035 ± 0.0004</td><td>0.1599 ± 0.0003</td><td>0.4932 ± 0.0010</td><td>0.4630 ± 0.0173</td><td>-0.0386 ± 0.0278</td></tr><tr><td>GCMC</td><td>0.2164 ± 0.0012</td><td>0.1685 ± 0.0010</td><td>0.3709 ± 0.0005</td><td>0.4891 ± 0.0163</td><td>-0.0043 ± 0.0343</td></tr><tr><td>SVD</td><td><strong>0.2018 ± 0.0004</strong></td><td>0.1566 ± 0.0003</td><td>0.4878 ± 0.0010</td><td>0.4979 ± 0.0086</td><td>-0.0030 ± 0.0200</td></tr><tr><td>BiVAECF</td><td>0.3855 ± 0.0628</td><td>0.3308 ± 0.0623</td><td>0.2773 ± 0.0054</td><td>0.5000 ± 0.0009</td><td>-0.0013 ± 0.0042</td></tr><tr><td>DeepMTP</td><td>0.2152 ± 0.0007</td><td>0.1693 ± 0.0007</td><td>0.3620 ± 0.0005</td><td>0.5010 ± 0.0238</td><td>-0.0158 ± 0.0607</td></tr><tr><td colspan="6" style="text-align: center; font-weight: bold;">POSTCOVID</td></tr><tr><td>IMPACT</td><td>0.2665 ± 0.0032</td><td><strong>0.1607 ± 0.0035</strong></td><td><strong>0.5616 ± 0.0101</strong></td><td><strong>0.7044 ± 0.0065</strong></td><td><strong>0.5052 ± 0.0185</strong></td></tr><tr><td>NMF</td><td><strong>0.2296 ± 0.0016</strong></td><td>0.1747 ± 0.0014</td><td>0.4835 ± 0.0052</td><td>0.5207 ± 0.0157</td><td>0.0557 ± 0.0433</td></tr><tr><td>PMF</td><td>0.2344 ± 0.0015</td><td>0.1779 ± 0.0016</td><td>0.4942 ± 0.0058</td><td>0.5105 ± 0.0308</td><td>0.0266 ± 0.0923</td></tr><tr><td>GCMC</td><td>0.3065 ± 0.0167</td><td>0.2516 ± 0.0143</td><td>0.2439 ± 0.0096</td><td>0.5035 ± 0.0169</td><td>0.0025 ± 0.0544</td></tr><tr><td>SVD</td><td><strong>0.2295 ± 0.0015</strong></td><td>0.1744 ± 0.0014</td><td>0.4845 ± 0.0046</td><td>0.5207 ± 0.0157</td><td>0.0555 ± 0.0433</td></tr><tr><td>BiVAECF</td><td>0.3645 ± 0.0369</td><td>0.3129 ± 0.0290</td><td>0.2229 ± 0.0309</td><td>0.5004 ± 0.0078</td><td>0.0007 ± 0.0184</td></tr><tr><td>DeepMTP</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td colspan="6" style="text-align: center; font-weight: bold;">PROMIS</td></tr><tr><td>IMPACT</td><td>0.1950 ± 0.0005</td><td><strong>0.0976 ± 0.0002</strong></td><td><strong>0.6846 ± 0.0008</strong></td><td><strong>0.8082 ± 0.0009</strong></td><td><strong>0.5703 ± 0.0008</strong></td></tr><tr><td>NMF</td><td>0.1788 ± 0.0003</td><td>0.1183 ± 0.0003</td><td>0.6411 ± 0.0013</td><td>0.4962 ± 0.0388</td><td>-0.0091 ± 0.0741</td></tr><tr><td>PMF</td><td><strong>0.1769 ± 0.0005</strong></td><td>0.1110 ± 0.0003</td><td>0.6696 ± 0.0011</td><td>0.4585 ± 0.0484</td><td>-0.0805 ± 0.0974</td></tr><tr><td>GCMC</td><td>0.1817 ± 0.0024</td><td>0.1227 ± 0.0028</td><td>0.1876 ± 0.0015</td><td>0.4817 ± 0.0196</td><td>-0.0198 ± 0.0659</td></tr><tr><td>SVD</td><td>0.1796 ± 0.0003</td><td>0.1187 ± 0.0003</td><td>0.6397 ± 0.0014</td><td>0.4962 ± 0.0393</td><td>-0.0092 ± 0.0751</td></tr><tr><td>BiVAECF</td><td>0.5483 ± 0.0546</td><td>0.4747 ± 0.0400</td><td>0.1211 ± 0.0157</td><td>0.5002 ± 0.0011</td><td>-0.0002 ± 0.0026</td></tr><tr><td>DeepMTP</td><td>0.1870 ± 0.0010</td><td>0.1262 ± 0.0023</td><td>0.1830 ± 0.0016</td><td>0.5022 ± 0.0173</td><td>-0.0017 ± 0.0443</td></tr><tr><td colspan="6" style="text-align: center; font-weight: bold;">PORTRAIT</td></tr><tr><td>IMPACT</td><td>0.2764 ± 0.0016</td><td><strong>0.1884 ± 0.0013</strong></td><td>-</td><td><strong>0.6459 ± 0.0028</strong></td><td><strong>0.4830 ± 0.0099</strong></td></tr><tr><td>NMF</td><td><strong>0.2598 ± 0.0010</strong></td><td>0.2044 ± 0.0011</td><td>-</td><td>0.4850 ± 0.0152</td><td>-0.0466 ± 0.0543</td></tr><tr><td>PMF</td><td><strong>0.2594 ± 0.0012</strong></td><td>0.2020 ± 0.0009</td><td>-</td><td>0.4776 ± 0.0429</td><td>-0.0768 ± 0.1392</td></tr><tr><td>GCMC</td><td>0.2643 ± 0.0025</td><td>0.2086 ± 0.0025</td><td>-</td><td>0.4972 ± 0.0313</td><td>0.0077 ± 0.1056</td></tr><tr><td>SVD</td><td><strong>0.2598 ± 0.0011</strong></td><td>0.2045 ± 0.0011</td><td>-</td><td>0.4849 ± 0.0155</td><td>-0.0467 ± 0.0553</td></tr><tr><td>BiVAECF</td><td>0.3609 ± 0.0356</td><td>0.3200 ± 0.0266</td><td>-</td><td>0.5004 ± 0.0025</td><td>0.0024 ± 0.0114</td></tr><tr><td>DeepMTP</td><td>0.2647 ± 0.0017</td><td>0.2099 ± 0.0017</td><td>-</td><td>0.5190 ± 0.0302</td><td>0.0531 ± 0.0974</td></tr></tbody></table>

## 6.2.1 Analysis of regression and classification metrics

IMPACT consistently achieves superior MAE and ACC values across all datasets compared to baseline methods, while its RMSE values lag slightly behind. A closer look at the error distribution confirms that IMPACT makes fewer errors than its competitors, but that its errors are larger in magnitude. This is mainly due to the prediction loss $L_{O1}^{ord}$, given in equation Equation 7, which does not optimize RMSE as effectively as MAE because it focuses on local ranking consistency rather than penalizing large deviations quadratically.

## 6.2.2 Analysis of interpretability metrics

A key strength of IMPACT lies in its superior interpretability, as evidenced by high DOA and PC-ER scores, which reflect the quality of the learned embeddings. PC-ER values above 0.5 on the POSTCOVID and PROMIS datasets indicate a strong linear correlation between embeddings and user responses. On the PORTRAIT dataset, the

<footnote n="5">⁵https://github.com/IMPACT.git</footnote>
PC-ER value approaches 0.5, indicating a near-linear yet meaningful relationship. Likewise, DOA values exceeding 0.6 across all datasets confirm that the embeddings maintain a consistent monotonic ordering with respect to user responses, thereby enhancing interpretability.

Overall, IMPACT stands out by achieving both high predictive accuracy and high interpretability. This balance is particularly evident on the PROMIS dataset, where the model demonstrates that transparency and performance can be jointly optimized, unlike most competing methods which typically emphasize one at the expense of the other.

## 6.3 Ablation study of IMPACT

The ablation study was conducted to evaluate the contributions of the three main components of the algorithm. Specifically, we examined:

1. Parameter initialization: The initialization of parameters **E** and **I** plays a critical role in the algorithm's performance. To assess its impact, we replaced the custom initialization with Xavier normal initialization. Note that omitting initialization altogether results in a substantial degradation in performance, which can obscure the effects of other components being tested.

2. $\mathcal{R}_I$ loss term: The $\mathcal{R}_I$ loss term is designed to enforce specific regularization constraints, ensuring better structure and consistency in the embeddings. We evaluated the effect of excluding this term to understand its contribution to the overall performance.

3. The $\mathcal{L}_{O_2}^{ord}$ loss term: The $\mathcal{L}_{O_2}^{ord}$ loss term focuses on inter-domain relationships, facilitating better interpretability and alignment of the embeddings with domain-specific responses. We assessed the algorithm's behavior when this term is omitted to gauge its influence on the model's accuracy and interpretability.

Each ablation scenario was evaluated independently to isolate the impact of the respective component, providing a clear understanding of their individual contributions to the algorithm's overall performance. To easily refer to an evaluated configuration, we associate a code to each of them in the result Table 4.

### 6.3.1 General observations

From the results in Table 4, we observe that the impact of initialization varies across datasets. In particular, comparing trials without any additional component with trials using only initialization reveals a negligible gain on predictive performance but a significantly positive gain on interpretability (see configurations coded WIRL and WRL). Removing only initialization confirms this limited influence on prediction while showing that it improves interpretability for about half of the datasets (see WI). We further analyze how interactions with the $\mathcal{R}_I$ term reduce the role of initialization in enhancing interpretability.

The $\mathcal{L}_{O_2}^{ord}$ term has an essential impact on interpretability performance (see WL). Removing it leads to a substantial deterioration in interpretability metrics across
<table><caption>**Table 4** Ablation study: Regression (RMSE, MAE) and classification (ACC) performances, as well as interpretability metrics (DOA, PC-ER). Bold numbers indicate the best mean (lowest for RMSE/MAE, highest for other metrics) and any result within this best method's standard deviation. ↑ and ↓ indicate metrics to be maximized and minimized, respectively.</caption><thead><tr><th>Configuration</th><th>Code</th><th>RMSE ↓</th><th>MAE ↓</th><th>ACC ↑</th><th>DOA ↑</th><th>PC-ER ↑</th></tr></thead><tbody><tr><td colspan="7" style="text-align:center; font-weight:bold;">MOVIELENS</td></tr><tr><td>IMPACT</td><td>All</td><td><strong>0.2225 ± 0.0003</strong></td><td>0.1475 ± 0.0002</td><td>0.4991 ± 0.0006</td><td>0.6569 ± 0.0004</td><td>0.3728 ± 0.0052</td></tr><tr><td>Without init</td><td>WI</td><td>0.2276 ± 0.0005</td><td>0.1522 ± 0.0004</td><td>0.4880 ± 0.0012</td><td><strong>0.6740 ± 0.0012</strong></td><td><strong>0.4023 ± 0.0057</strong></td></tr><tr><td>Without <i>R</i><sub>I</sub></td><td>WR</td><td>0.2242 ± 0.0005</td><td>0.1492 ± 0.0002</td><td>0.4948 ± 0.0004</td><td>0.6631 ± 0.0008</td><td>0.3727 ± 0.0047</td></tr><tr><td>Without <i>L</i><sup>ord</sup><sub>O2</sub></td><td>WL</td><td><strong>0.2223 ± 0.0004</strong></td><td><strong>0.1468 ± 0.0003</strong></td><td><strong>0.5022 ± 0.0011</strong></td><td>0.5854 ± 0.0101</td><td>0.2501 ± 0.0242</td></tr><tr><td>Without init and <i>R</i><sub>I</sub></td><td>WIR</td><td>0.2244 ± 0.0007</td><td>0.1494 ± 0.0004</td><td>0.4944 ± 0.0009</td><td>0.6713 ± 0.0022</td><td><strong>0.3989 ± 0.0066</strong></td></tr><tr><td>Without init and <i>L</i><sup>ord</sup><sub>O2</sub></td><td>WIL</td><td>0.2254 ± 0.0005</td><td>0.1500 ± 0.0004</td><td>0.4938 ± 0.0009</td><td>0.5353 ± 0.0100</td><td>0.1030 ± 0.0247</td></tr><tr><td>Without <i>R</i><sub>I</sub> and <i>L</i><sup>ord</sup><sub>O2</sub></td><td>WRL</td><td><strong>0.2227 ± 0.0005</strong></td><td><strong>0.1471 ± 0.0004</strong></td><td><strong>0.5016 ± 0.0011</strong></td><td>0.5811 ± 0.0042</td><td>0.2354 ± 0.0117</td></tr><tr><td>Without init, <i>R</i><sub>I</sub> and <i>L</i><sup>ord</sup><sub>O2</sub></td><td>WIRL</td><td>0.2242 ± 0.0004</td><td>0.1485 ± 0.0002</td><td>0.4984 ± 0.0008</td><td>0.5076 ± 0.0206</td><td>0.0258 ± 0.0608</td></tr><tr><td colspan="7" style="text-align:center; font-weight:bold;">POSTCOVID</td></tr><tr><td>IMPACT</td><td>All</td><td><strong>0.2658 ± 0.0015</strong></td><td><strong>0.1602 ± 0.0031</strong></td><td><strong>0.5620 ± 0.0116</strong></td><td><strong>0.7044 ± 0.0043</strong></td><td><strong>0.5047 ± 0.0208</strong></td></tr><tr><td>Without init</td><td>WI</td><td>0.2694 ± 0.0035</td><td><strong>0.1630 ± 0.0035</strong></td><td><strong>0.5582 ± 0.0095</strong></td><td>0.6859 ± 0.0018</td><td>0.4542 ± 0.0180</td></tr><tr><td>Without <i>R</i><sub>I</sub></td><td>WR</td><td>0.2707 ± 0.0067</td><td>0.1653 ± 0.0079</td><td><strong>0.5510 ± 0.0208</strong></td><td>0.6940 ± 0.0093</td><td>0.4778 ± 0.0192</td></tr><tr><td>Without <i>L</i><sup>ord</sup><sub>O2</sub></td><td>WL</td><td>0.2756 ± 0.0047</td><td>0.1721 ± 0.0069</td><td>0.5306 ± 0.0192</td><td>0.6898 ± 0.0157</td><td><strong>0.5004 ± 0.0321</strong></td></tr><tr><td>Without init and <i>R</i><sub>I</sub></td><td>WIR</td><td>0.2681 ± 0.0026</td><td><strong>0.1613 ± 0.0040</strong></td><td><strong>0.5622 ± 0.0143</strong></td><td>0.6927 ± 0.0052</td><td><strong>0.4846 ± 0.0202</strong></td></tr><tr><td>Without init and <i>L</i><sup>ord</sup><sub>O2</sub></td><td>WIL</td><td>0.2708 ± 0.0023</td><td>0.1637 ± 0.0024</td><td><strong>0.5584 ± 0.0076</strong></td><td>0.5105 ± 0.0148</td><td>0.0303 ± 0.0373</td></tr><tr><td>Without <i>R</i><sub>I</sub> and <i>L</i><sup>ord</sup><sub>O2</sub></td><td>WRL</td><td>0.2732 ± 0.0013</td><td>0.1687 ± 0.0010</td><td>0.5408 ± 0.0035</td><td>0.6576 ± 0.0050</td><td>0.4205 ± 0.0155</td></tr><tr><td>Without init, <i>R</i><sub>I</sub> and <i>L</i><sup>ord</sup><sub>O2</sub></td><td>WIRL</td><td>0.2701 ± 0.0025</td><td><strong>0.1633 ± 0.0025</strong></td><td><strong>0.5582 ± 0.0070</strong></td><td>0.4957 ± 0.0230</td><td>-0.0172 ± 0.0589</td></tr><tr><td colspan="7" style="text-align:center; font-weight:bold;">PROMIS</td></tr><tr><td>IMPACT</td><td>All</td><td>0.1949 ± 0.0005</td><td><strong>0.0976 ± 0.0002</strong></td><td><strong>0.6846 ± 0.0006</strong></td><td>0.8079 ± 0.0010</td><td>0.5701 ± 0.0009</td></tr><tr><td>Without init</td><td>WI</td><td><strong>0.1938 ± 0.0006</strong></td><td><strong>0.0973 ± 0.0004</strong></td><td><strong>0.6842 ± 0.0012</strong></td><td><strong>0.8134 ± 0.0007</strong></td><td><strong>0.5759 ± 0.0018</strong></td></tr><tr><td>Without <i>R</i><sub>I</sub></td><td>WR</td><td>0.1952 ± 0.0005</td><td>0.0979 ± 0.0003</td><td><strong>0.6839 ± 0.0008</strong></td><td>0.8082 ± 0.0004</td><td>0.5705 ± 0.0022</td></tr><tr><td>Without <i>L</i><sup>ord</sup><sub>O2</sub></td><td>WL</td><td>0.1973 ± 0.0006</td><td>0.0999 ± 0.0007</td><td>0.6784 ± 0.0023</td><td>0.7586 ± 0.0021</td><td>0.4833 ± 0.0046</td></tr><tr><td>Without init and <i>R</i><sub>I</sub></td><td>WIR</td><td><strong>0.1940 ± 0.0005</strong></td><td><strong>0.0973 ± 0.0003</strong></td><td><strong>0.6844 ± 0.0009</strong></td><td><strong>0.8133 ± 0.0011</strong></td><td><strong>0.5735 ± 0.0013</strong></td></tr><tr><td>Without init and <i>L</i><sup>ord</sup><sub>O2</sub></td><td>WIL</td><td>0.1973 ± 0.0003</td><td>0.0999 ± 0.0002</td><td>0.6784 ± 0.0009</td><td>0.4664 ± 0.0074</td><td>-0.0805 ± 0.0176</td></tr><tr><td>Without <i>R</i><sub>I</sub> and <i>L</i><sup>ord</sup><sub>O2</sub></td><td>WRL</td><td>0.1975 ± 0.0004</td><td>0.0999 ± 0.0005</td><td>0.6790 ± 0.0019</td><td>0.7578 ± 0.0021</td><td>0.4851 ± 0.0048</td></tr><tr><td>Without init, <i>R</i><sub>I</sub> and <i>L</i><sup>ord</sup><sub>O2</sub></td><td>WIRL</td><td>0.1970 ± 0.0002</td><td>0.0997 ± 0.0002</td><td>0.6788 ± 0.0007</td><td>0.5010 ± 0.0311</td><td>-0.0006 ± 0.0625</td></tr><tr><td colspan="7" style="text-align:center; font-weight:bold;">PORTRAIT</td></tr><tr><td>IMPACT</td><td>All</td><td><strong>0.2765 ± 0.0017</strong></td><td>0.1882 ± 0.0014</td><td>-</td><td><strong>0.6463 ± 0.0027</strong></td><td><strong>0.4842 ± 0.0096</strong></td></tr><tr><td>Without init</td><td>WI</td><td>0.2808 ± 0.0014</td><td>0.1918 ± 0.0011</td><td>-</td><td>0.6342 ± 0.0028</td><td>0.4443 ± 0.0124</td></tr><tr><td>Without <i>R</i><sub>I</sub></td><td>WR</td><td><strong>0.2763 ± 0.0009</strong></td><td>0.1883 ± 0.0008</td><td>-</td><td>0.6367 ± 0.0032</td><td>0.4465 ± 0.0098</td></tr><tr><td>Without <i>L</i><sup>ord</sup><sub>O2</sub></td><td>WL</td><td>0.2820 ± 0.0012</td><td>0.1912 ± 0.0010</td><td>-</td><td>0.5687 ± 0.0034</td><td>0.2235 ± 0.0099</td></tr><tr><td>Without init and <i>R</i><sub>I</sub></td><td>WIR</td><td><strong>0.2766 ± 0.0007</strong></td><td><strong>0.1872 ± 0.0005</strong></td><td>-</td><td>0.6418 ± 0.0030</td><td><strong>0.4761 ± 0.0113</strong></td></tr><tr><td>Without init and <i>L</i><sup>ord</sup><sub>O2</sub></td><td>WIL</td><td>0.2879 ± 0.0018</td><td>0.1966 ± 0.0017</td><td>-</td><td>0.4866 ± 0.0096</td><td>-0.0226 ± 0.0317</td></tr><tr><td>Without <i>R</i><sub>I</sub> and <i>L</i><sup>ord</sup><sub>O2</sub></td><td>WRL</td><td>0.2804 ± 0.0013</td><td>0.1907 ± 0.0008</td><td>-</td><td>0.5495 ± 0.0014</td><td>0.1505 ± 0.0053</td></tr><tr><td>Without init, <i>R</i><sub>I</sub> and <i>L</i><sup>ord</sup><sub>O2</sub></td><td>WIRL</td><td>0.2821 ± 0.0016</td><td>0.1915 ± 0.0018</td><td>-</td><td>0.4659 ± 0.0240</td><td>-0.1126 ± 0.0845</td></tr></tbody></table>

all datasets. This suggests that $L_{O2}^{ord}$ is integral to maintaining interpretability. The results also confirm that the $L_{O2}^{ord}$ learning objective is effectively derived from the monotonicity assumption. Surprisingly, $L_{O2}^{ord}$ improves predictive performance on POSTCOVID, PROMIS and PORTRAIT datasets (see WL). This implies that the monotonicity assumption holds better on those dataset. An observation reinforced by the significantly greater PC-ER values of IMPACT on those datasets compared to MOVIELENS.

Combined, initialization and $L_{O2}^{ord}$ account for most of the interpretability performance. Indeed, removing $R_I$ has little impact on interpretability, whereas omitting initialization and $L_{O2}^{ord}$ leads to a marked drop (see WR and WIL). Although the effects of initialization and $L_{O2}^{ord}$ are not strictly additive (see WI, WL, WIL). We can see on MOVIELENS and PROMIS that initialization improves interpretability along with $L_{O2}^{ord}$, even when it has a negative impact alone. It appears to prepare the embeddings effectively by leveraging a simple yet efficient approximation of monotonicity requirement, where average responses for each dimension initialize user embeddings. Both initialization and $L_{O2}^{ord}$ are essential for interpretability; without them, IMPACT becomes less interpretable than baseline models like NMF or SVD. Together, they also improve predictive performance, demonstrating the validity of hypothesis O2 and the model's
ability to leverage distinct learning objectives. These results validate once again the strategy of using ranking losses both for the regression and the interpretable objective.

The $\mathcal{R}_I$ term improves regression performance on the MOVIELENS, POSTCOVID and PROMIS datasets but has no significant effect on PORTRAIT (see WR). Its impact on interpretability and accuracy is negligible. This supports the notion that $\mathcal{R}_I$ primarily aids in structuring embeddings and stabilizing the learning process, while not being critical, similar to other regularization terms. Removing both $\mathcal{R}_I$ and initialization, we observe that their combined effect is negligible, except on MOVIELENS, where their combination improves predictive performance and DOA (see WIR). This indicates that the embedding distribution induced by initialization does not significantly prepare the embeddings for the $\mathcal{R}_I$ term. We also observe that there is no significant interaction between $\mathcal{R}_I$ and $\mathcal{L}_{O_2}^{ord}$ (see WL and WRL).

### 6.3.2 Dataset-specific observations

IMPACT does not reach on PORTRAIT dataset the prediction and interpretability performances observed on other datasets. However, the ablation study shows that the three tested components are efficiently improving both objectives, except for the $\mathcal{R}_I$ term. This may be due to the highly imbalanced nature of PORTRAIT which diminish all competitors performances. The MOVIELENS dataset appears particularly sensitive to initialization.

The observed variations in response prediction metrics, compared to interpretability metrics (especially between MAE and DOA) highlight the importance of the dynamic loss weighting used in IMPACT. This approach accounts for the convergence rate of metrics relative to their average value and range, ensuring balanced optimization. These results underscore the necessity of multi-objective evaluation to fully capture the trade-offs and strengths of IMPACT.

## 6.4 Hyperparameter study

IMPACT introduces four hyperparameters, as summarized in Table 5. Among these, the learning rate and $\lambda$ are standard machine learning hyperparameters selected using grid search strategy. Importantly, the learning rate scheduler implemented in IMPACT reduces the sensitivity of the learning process to the initial learning rate value. The third hyperparameter, $\beta$, weights the $\mathcal{R}_I$ term in the total loss. The fourth hyperparameter, the maximum number of modalities per question ($|M|$), controls the granularity of response discretization and applies exclusively to continuous datasets. For ordinal datasets, IMPACT dynamically adapts the number of modalities for each question, making $|M|$ irrelevant once it exceeds the actual maximum number of modalities observed in the dataset.

We first evaluate the impact of the $\beta$ hyperparameter on the PORTRAIT dataset. We can see from Fig 5 that below a threshold of 0.7, the hyperparameter has almost no impact on performance and interpretability. Beyond 0.7, the regularization term becomes too dominant in the total loss, resulting in underfitting.

We then evaluate the impact of the number of modalities ($|M|$) on the PORTRAIT dataset, the only continuous dataset suitable for this analysis. As shown in Figure 6
<table><caption>**Table 5** IMPACT hyperparameters.</caption><thead><tr><th>Hyperparameters</th><th></th><th>Ordinal datasets</th><th>Continuous datasets</th></tr></thead><tbody><tr><td>Learning rate</td><td>Determines the magnitude of each parameter update during optimization.</td><td>✓</td><td>✓</td></tr><tr><td>λ</td><td>Determines the magnitude of the regularization term <i>R</i>(<i>E</i>, <i>I</i>) in the loss.</td><td>✓</td><td>✓</td></tr><tr><td>β</td><td>Determines the magnitude of the regularization term <i>R</i><sub>I</sub> in the total loss.</td><td>✓</td><td>✓</td></tr><tr><td>|<i>M</i>|</td><td>Maximum number of modalities per question.</td><td>✗</td><td>✓</td></tr></tbody></table>

<figure><img src="image_7.png" /><figcaption>**Fig. 5** Average regression accuracy and interpretability performance (left axis) of IMPACT on the PORTRAIT dataset across 5-fold cross-validation, evaluated as a function of the β hyperparameter.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Fig. 6** Average regression accuracy and interpretability performance (left axis), along with the memory requirements (right axis), of IMPACT on the PORTRAIT dataset across 5-fold cross-validation, evaluated as a function of the |M| hyperparameter.</figcaption></figure>

(a), |M| substantially affects RMSE, which consistently improves at higher |M| values but eventually plateaus once |M| becomes large. This pattern stems from the discretization strategy: increasing |M| mitigates discretization artifacts, and since RMSE penalizes large errors, its reduction is more pronounced (see Section 6.2). The MAE metric (Figure 6 (b)) also improves up to |M| = 14 but becomes unstable beyond this threshold. Because PORTRAIT is a continuous dataset, MAE benefits less from an
exact match between the predicted modalities and the ground truth, as there are more
continuous values than discrete bins. Interestingly, RMSE and MAE exhibit oppo-
site peaks for |M| ∈ {12, 14, 18, 20}, indicating a trade-off between large and average
errors.

From a resource perspective, the number of parameters increases by nearly 30% between |M| = 5 and |M| = 12, suggesting a sizable memory overhead associated with improving RMSE and MAE. Meanwhile, the interpretability of the model’s embedding (Figure 6 (c)) remains largely unaffected by |M|. As expected, larger |M| extends both training and testing times, driven by the growing number of pairwise comparisons and embeddings (Figure 7). Indeed, the second axis in that figure shows that the total number of model parameters doubles from the smallest to the largest value of |M|. Notably, the high peak at |M| = 20 corresponds to a similar spike in MAE, implying convergence difficulties and reinforcing the observed erratic behavior for |M| > 14. We hypothesize that beyond this threshold, the additional resolution offered by large |M| captures mainly noise, rather than meaningful information.

<figure><img src="image_9.png" /><figcaption>**Fig. 7** Average training and testing time (left axis) and memory requirements (right axis) of IMPACT on the PORTRAIT dataset across 5-fold cross-validation as a function of the $|M|$ hyperparameter. The black dotted line highlights the chosen hyperparameter value.</figcaption></figure>

**6.5 Optimization of the model size**

One potential drawback of the approach is the size of the embeddings, and consequently, the overall model size, which may pose challenges for fitting into memory, especially for larger datasets. We compare the parameter complexity of our model with that of competing approaches across the four datasets considered in the experiments. The results are presented in Figure 8. We observe that IMPACT consistently lies on the Pareto frontier when considering the trade-off between MAE performance and model size, indicating competitive accuracy for its level of complexity. However, this comes at the cost of a comparatively high number of parameters. This drawback does not appear to be directly related to the computational complexity of IMPACT, as supported by the training and testing time experiments provided in the accompanying Git repository. To address this issue, we propose and evaluate a solution inspired by
(Rothe & Schütze, 2016), which involves reducing the size of the item embeddings and learning a linear layer that rescales them at inference time, mapping them to the same dimensional space as the user embeddings. In the following, we present this approach and evaluate its impact on model performance.

<figure><img src="image_10.png" /><figcaption>**Fig. 8** Regression performance (MAE) as a function of the number of model parameters for IMPACT and competing methods across four datasets.</figcaption></figure>

### 6.5.1 Restricting the size of item embeddings

To encourage parsimony and reduce the number of parameters, we optimize the embedding size for question-modality pairs, particularly for item-modality embeddings where interpretability is not a primary requirement. Inspired by a work on ultra-dense subspaces for world embeddings (Rothe & Schütze, 2016), our approach embeds question-modality pairs into a lower-dimensional space of size $d_r < |C|$, followed by a linear transformation that rescales these embeddings into $\mathbb{R}^{|C|}$ for computing $\pi_{u,q,m,c}$. By leveraging similarities among questions tied to the same concept, we define a dedicated subspace for each concept. The linear transformation mapping question-modality embeddings to the user embedding space is parameterized by a weighted matrix $\mathbf{W}^c \in \mathbb{R}^{|C| \times d_r}$, where $c \in C$ represents the concept. This transformation is expressed as: $\mathbf{I}_{qm} = \mathbf{W}^c \mathbf{I}_{qm}^r$, with $\mathbf{I}_{qm}^r \in \mathbb{R}^{d_r}$.

To ensure that introducing question-modality subspaces leads to an overall reduction in the total number of parameters, the dimensionality of the subspace ($d_r$) must satisfy the following condition, accounting for the additional parameters introduced
by the transformation matrix:

$$ \underbrace{d_r |C|^2}_{\text{transformation matrices}} + \underbrace{(\sum_{q \in Q} |M_q| + 2) d_r}_{\text{reduced questions}} < \underbrace{(\sum_{q \in Q} |M_q| + 2) |C|}_{\text{original questions}} $$

$$ \iff d_r < \frac{(\sum_{q \in Q} |M_q| + 2) |C|}{|C|^2 + (\sum_{q \in Q} |M_q| + 2)} \quad (19) $$

$$ \implies d_r < |C| \quad (20) $$

In real cases, we observed that the squared number of categories is negligible compared to the number of question-modalities ($|C|^2 \ll \sum_{q \in Q} |M_q| + 2$), that is to say, the addition of transformation matrices is negligible compared to the gain of parameters due to the dimension reduction. We can simplify the exact Eq. 19 by the straight forward Eq. 20, which means that as soon as the subspace dimensionality is lower than the user embedding dimensionality, then the number of model parameters decreases.

Hence, **W** parameters is given the same prior as **E** and **I**: **W** ~ N(0, Id/(2λ)) and a new regulation term is introduced in Eq. 8:

$$ \mathcal{R}_{\text{O1}}(\mathbf{E}, \mathbf{I}, \mathbf{W}) = -\log \mathbb{P}(\mathbf{E}, \mathbf{I}, \mathbf{W}) = -\lambda [\|\mathbf{E}\|_2^2 + \|\mathbf{I}\|_2^2 + \|\mathbf{W}\|_2^2] \quad (21) $$

### 6.5.2 Evaluation of the optimization procedure

We evaluate the extent to which the embedding dimension $d_r$ minimizes the number of parameters in the MOVIELENS dataset, while also assessing its impact on computational efficiency, regression accuracy, and interpretability. Our experiments demonstrate that selecting an appropriate embedding dimension for question-modalities ($d_r$) effectively reduces the memory cost of IMPACT without degrading regression performance and, in some cases, even improving RMSE (Figures 9 (a) and (b)). Similarly, interpretability metrics benefit from a well-chosen $d_r$ (Figure 9 (c)).

Interestingly, execution time decreases as $d_r$ increases (Figure 10), potentially due to the greater ease of escaping local minima with embeddings in higher dimensions. Notably, the execution time for the selected $d_r$ remains comparable to its standard deviation and does not significantly exceed that of the default setting ($d_r = |C|$).

Since reducing the embedding size does not degrade performance, we could apply the elbow method to the RMSE curve (Figure 9 (a)) to automatically determine an optimal $d_r$ value, adapting it to the specific characteristics of each dataset.

## 6.6 Qualitative study

While quantitative experiments in Section 6.2 highlight the superior monotonicity of IMPACT's user profiles, questions remain about the interpretability of these embeddings. In particular, how closely does each embedding dimension relate to predicting
<figure><img src="image_11.png" /><figcaption>**Fig. 9** Average regression and interpretability performances (left axis), along with the memory requirements (right axis) of IMPACT on MOVIELENS dataset over 5-fold cross-validation, as a function of *d*<sub>r</sub>.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Fig. 10** Average training and testing time (left axis) and memory requirements (right axis) of IMPACT on the MOVIELENS dataset across 5-fold cross-validation as a function of the *d*<sub>r</sub> hyperparameter. The black dotted line highlights the chosen hyperparameter value.</figcaption></figure>

the responses in its associated category? And how can the extracted user profile offer
deeper insights into individual responses?

We first investigate whether the accuracy of response prediction strongly depends on the specific component of the user embedding that corresponds to a question’s category. If this assumption holds, gradually injecting noise into a user’s embedding component should degrade the RMSE and MAE for questions associated with that component’s category. Therefore, we conduct the following experiment: for the *k*<sup>th</sup> component of a user’s embedding, we add a random variable drawn from a normal distribution with a mean of 0 and a standard deviation expressed as a fraction of the original embedding standard deviation (shown on the *x*-axis). We then compute RMSE and MAE separately for each category. The top row of Figure 11 presents the results for the MAE metric<sup>6</sup>. We can see that IMPACT’s error varies significantly across categories. However, most of the curves increase, indicating that response prediction does rely on the specific embedding component associated with the category.

Next, we aim to assess how much faster the MAE grows in the perturbed cat-
egory compared to the others. For this purpose, we compute the ratio between the

<footnote n="6"><sup>6</sup>Similar trends are observed for RMSE.</footnote>
category-specific MAE and the MAE averaged over the remaining categories. If the noise primarily affects category-related questions, these ratios should increase with the noise level. To emphasize the relative evolution rather than absolute values, each ratio is normalized by its corresponding MAE at zero noise. The bottom row of Figure 11 shows these normalized MAE ratios. The results highlight several key insights. First, curves are strictly increasing in most cases, indicating a stronger dependence on category-specific dimensions for response prediction. Second, we observe that PROMIS, on which IMPACT performs indisputably best for MAE, Accuracy, DOA and PC-ER, also demonstrates the highest category-specific dependence for prediction. These findings confirm IMPACT’s internal consistency: its user profiles not only capture the cognitive processes underlying user responses through monotonic relationships but also influence the prediction of unobserved responses in a structured manner.

<figure><img src="image_13.png" /><figcaption>**Fig. 11** Influence of noise on IMPACT's user embeddings and its effect on MAE performance. Each curve illustrates the impact of progressively injecting noise into a single embedding component. In the top row, the MAE is computed exclusively for the questions associated with the category of the perturbed component. In the bottom row, the MAE is computed as the ratio between the error on the perturbed category and the error on the remaining categories, normalized by the initial value (results averaged over 5-fold cross-validation × 3 random seeds).</figcaption></figure>

To demonstrate the interpretability of IMPACT’s user embeddings and their utility as user profiles, we examine the embeddings generated for the MOVIELENS dataset. Figure 12 presents a heatmap in which the color intensity shows each user’s embedding magnitude across different categories. Users are ordered according to an index $i_u$ that
counts how often a user's embedding component exceeds those of other users:

$$i_u = \sum_{c \in C} \sum_{v \in U \setminus u} \delta(E_u[c] > E_v[c]), \quad \text{where } \delta(x) = \begin{cases} 1 & \text{if } x = \text{True}, \\ 0 & \text{otherwise.} \end{cases}$$

Categories (columns) are arranged in ascending order of their average embedding values $\mu_c$,

$$\mu_c = \sum_{u \in U} E_u[c],$$

and each column's scale is normalized to [0, 1] to facilitate comparison.

The heatmap reveals a gradient from more "critical" users, who systematically rate movies lower, to less "critical" users, who tend to give higher ratings overall. Although identifying a user's favorite categories requires individual inspection of their embedding profile, one striking pattern emerges: "Musical" movies appear to be widely appreciated among MOVIELENS users.

<figure><img src="image_14.png" /><figcaption>**Fig. 12** MOVIELENS users' embedding. Users are ordered by the number of dominated components, and categories are sorted by their average values.</figcaption></figure>

Figure 13 shows the embeddings of four randomly selected users alongside their three favorite movies. To account for each user's general tendency to rate movies
highly or poorly, we rescaled the embedding components. We then compared each user's profile with their chosen favorites (arbitrarily capped at three). The results indicate a strong alignment between the categories of the users' top films and their highest embedding components, underscoring the embeddings' interpretability despite the small sample size.

A key challenge in this dataset arises from the wide array of categories that can be assigned to a single movie, at times producing contradictory labels. For example, *Star Wars: Episode V – The Empire Strikes Back* (1980) is simultaneously classified under “War” and “Drama–Romance–Comedy,” which complicates fine-grained category-based analysis.

<figure><img src="image_15.png" /><figcaption>**Fig. 13** Rescaled embeddings of viewers alongside their three favorite movies based on ratings. The categories of the favorite movies generally align with the users' profiles, reflecting the interpretability of the embeddings.</figcaption></figure>
To conclude, the user profiles extracted by IMPACT consist of components that are directly involved in predicting responses to questions associated with their respective dimensions. Moreover, since the embeddings are learned through comparisons between users, they not only capture individual characteristics but also reveal broader tendencies within the overall user population, ensuring both personalized and generalizable insights.

## 6.7 Summary of results

Quantitative experiments in Section 6.2 demonstrate that IMPACT achieves competitive regression performance while offering superior interpretability. This is supported by embedding quality metrics, which indicate strong monotonicity between the user profiles constructed by the model and the corresponding response vectors. Additionally, IMPACT outperforms all competitors on the MAE metric across all datasets, while achieving slightly lower performance on RMSE. The ablation study in Section 6.3 confirms that each component of IMPACT effectively achieves its intended objective, with the exception of the PORTRAIT dataset. As shown in Table 3, all methods perform significantly worse on PORTRAIT compared to other datasets, primarily due to its highly imbalanced distribution (even after preprocessing) illustrated in Figure 4. This raises concerns about the nature of the information captured by different methods on this dataset and their ability to generalize beyond predicting a default value (e.g., 0). The hyperparameter study in Section 6.4 underscores the trade-off between memory and training costs versus regression accuracy when adjusting the number of modalities in originally continuous datasets. To mitigate concerns about parameter growth, we introduced techniques in Section 6.5 to reduce embedding size, effectively lowering the overall parameter count while maintaining performance. Finally, our qualitative analysis in Section 6.6 highlights the intuitive properties of the user profiles generated by IMPACT.

## 7 Conclusion

In this paper, we introduced IMPACT, a new Bayesian framework for Multi-Target Prediction (MTP) that extends existing approaches to the ordinal setting. Our method embeds users and targets in a shared vector space, enabling deeper insights. To address the complexity of ordered multi-class MTP, we developed a loss function inspired by Bayesian Personalized Ranking, complemented by a novel regularization term and a hinge-based loss that together enhance interpretability. We evaluated IMPACT against six state-of-the-art baselines on four benchmark datasets. Results show that it achieves competitive regression performance while consistently outperforming existing methods in terms of interpretability, demonstrating that accuracy and interpretability can be jointly optimized. An ablation study highlights the contribution of each component, and a hyperparameter analysis quantifies the trade-off between predictive accuracy, memory cost, and training efficiency. To address the practical challenge of model scalability, we further introduced a parameter-constraining strategy and measured its impact. Finally, qualitative analyses reveal that IMPACT's embeddings capture meaningful and interpretable structures, effectively synthesizing user characteristics and
preferences, as illustrated on the MOVIELENS dataset. These findings confirm that learning to rank within a unified vector space enables IMPACT to achieve the dual objectives of accuracy and interpretability. Looking ahead, several extensions could further enhance the proposed framework. First, while the current formulation relies on maximum a posteriori estimation, exploring variational inference may provide a richer characterization of parameter uncertainty and improve uncertainty calibration. In addition, future work will explore alternative formulations of the local comparison loss to further improve predictive performance across metrics beyond MAE. In parallel, integrating IMPACT into computerized adaptive testing frameworks represents a promising application direction, where dynamically selected questions could accelerate profile convergence and reduce user burden. These avenues constitute complementary methodological and practical extensions that we leave for future work.

# 8 Acknowledgment

This work was supported by the French National Research Agency (ANR) through the France 2030 program, under "PORTRAIT" (ANR-22-CE23-0006) project. It was granted access to the HPC resources of IDRIS under the allocation 2025-AD010616269 made by GENCI.

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