# Training-free Graph-based Imputation of Missing Modalities in Multimodal Recommendation

Daniele Malitesta, Emanuele Rossi, Claudio Pomo, Tommaso Di Noia, Fragkiskos D Malliaros

► To cite this version:

Daniele Malitesta, Emanuele Rossi, Claudio Pomo, Tommaso Di Noia, Fragkiskos D Malliaros. Training-free Graph-based Imputation of Missing Modalities in Multimodal Recommendation. IEEE Transactions on Knowledge and Data Engineering, 2026. hal-05520677

HAL Id: hal-05520677

https://hal.science/hal-05520677v1

Submitted on 20 Feb 2026

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
# Training-free Graph-based Imputation of Missing Modalities in Multimodal Recommendation

Daniele Malitesta*, Emanuele Rossi†, Claudio Pomo‡, Tommaso Di Noia‡, Fragkiskos D. Malliaros*

*Université Paris-Saclay, CentraleSupélec, Inria, Gif-sur-Yvette, France

†VantAI, Barcelona, Spain

‡Politecnico di Bari, Bari, Italy

Email: \*name.surname@centralesupelec.fr, †emanuele.rossi1909@gmail.com, ‡name.surname@poliba.it

**Abstract**—Multimodal recommender systems (RSs) represent items in the catalog through multimodal data (e.g., product images and descriptions) that, in some cases, might be noisy or (even worse) missing. In those scenarios, the common practice is to drop items with missing modalities and train the multimodal RSs on a subsample of the original dataset. To date, the problem of missing modalities in multimodal recommendation has still received limited attention in the literature, lacking a precise formalisation as done with missing information in traditional machine learning. In this work, we first provide a problem formalisation for missing modalities in multimodal recommendation. Second, by leveraging the user-item graph structure, we re-cast the problem of missing multimodal information as a problem of graph features interpolation on the item-item co-purchase graph. On this basis, we propose four training-free approaches that propagate the available multimodal features throughout the item-item graph to impute the missing features. Extensive experiments on popular multimodal recommendation datasets demonstrate that our solutions can be seamlessly plugged into any existing multimodal RS and benchmarking framework while still preserving (or even widen) the performance gap between multimodal and traditional RSs. Moreover, we show that our graph-based techniques can perform better than traditional imputations in machine learning under different missing modalities settings. Finally, we analyse (for the first time in multimodal RSs) how feature homophily calculated on the item-item graph can influence our graph-based imputations.

**Index Terms**—Multimodal recommendation, graph learning, missing modalities.

## I. INTRODUCTION AND MOTIVATION

Recommender systems (RSs) are powerful machine learning algorithms designed to derive useful preference patterns of customers over products or services on online platforms (e.g., Amazon, Netflix, Zalando). Despite their large success, mostly built around the collaborative filtering (CF) paradigm, they still fail in inferring users' preferences when the training data is highly sparse [1], a phenomenon that occurs quite often in the literature and real-world settings.

Among popular solutions to address the issue, one of the most effective ways is to leverage products' side information to empower the RSs of useful knowledge regarding users and items in the system. In this respect, multimodal recommender systems [2]–[7] have long settled as the state-of-the-art in personalized recommendation, largely outperforming traditional CF-based recommender systems in providing accurate recommendations. In domains such as fashion [8], music [9],

food [10], and micro-video [11] recommendation, multimodal RSs leverage multimodal data accompanying items (e.g., product images and descriptions) to extract high-level multimodal features and inject them into the recommendation framework.

Nevertheless, any multimodal RS seeks high-quality multimodal data to derive useful information for the downstream recommendation task, but this requirement cannot always be ensured. In fact, in real-world settings, multimodal data might be noisy or (even worse) missing [12]. For instance, let us consider a common scenario in e-commerce where, for each product in the catalog, the figure and the description are displayed on the product webpage. It is not rare to encounter situations where the image or also the description are not available. Indeed, the phenomenon is not only limited to real-world settings, as analysing the Amazon Reviews dataset [1] (Table I) suggests that this also happens in academic research.

In this respect, the standard practice in multimodal recommendation is to drop items with missing modalities and, after this pre-filtering phase, train and evaluate the recommendation model on a subset of the original recommendation data. We maintain that this procedure, despite being common practice in the related literature, becomes counteractive to the idea of enhancing the recommendation data through multimodal data to tackle dataset sparsity. To further support the statement, we also show some motivating experimental results (that will be better detailed in Section VI-A). As depicted in Figure 1, we report the performance improvement (Recall@20) of a multimodal recommender (NGCF-M [12]) over its non-multimodal counterpart (NGCF [13]) on Music and Beauty categories of the Amazon Reviews dataset, when we drop items with missing modalities and when we impute them (as a pre-processing step). Indeed, we observe that in the imputed setting, the performance improvement of NGCF-M over NGCF is much higher (and even reverted) than the one calculated between the same models in the dropped setting. This demonstrates that, also empirically, imputing missing modalities in multimodal recommendation can be beneficial to widen the performance gap between traditional and multimodal RSs, as extensively demonstrated in the related literature [1], [14]–[21].

On such a basis, it becomes fundamental to design a proper imputation method to recover missing modalities in multimodal recommendation. The problem of missing information in machine learning has been historically discussed and addressed over the past few decades [22]. When it comes
<figure><img src="image_2.png" /><figcaption>Fig. 1: Motivating example where we show the performance improvement (Recall@20) between NGCF (blue) and NGCF-M (brown) in the dropped and imputed settings, for the Music and Beauty datasets. As is evident, the performance improvement of NGCF-M over NGCF becomes wider (or even reverts on the Music dataset) with the imputed setting.</figcaption></figure>

to recommendation, several works have tackled the issue of missing information in the user-item interaction data [23]–[25] and, in a very few cases, users’ and items’ metadata [26], [27]. However, despite the extensive literature on missing modalities in other multimodal learning tasks [28]–[30], the problem has encountered limited (and only recent) attention in multimodal recommendation. The few existing approaches addressing the missingness of modalities in multimodal recommendation [31]–[35] work within the end-to-end training of the multimodal recommender model by either learning how to impute the missing features or making the recommender system robust to the absence of modalities. However, such solutions are bind to the specific multimodal recommender system they come with, and cannot be applied to any other existing multimodal recommender systems out there. We consider this as a critical limitation since the literature outlines a consistent number of state-of-the-art multimodal recommender systems demonstrating high recommendation performance in standard settings without missing modalities.

To address the outlined issues, in this work, we propose a family of novel approaches that can be plugged into any multimodal recommender system to impute the missing multimodal features as a pre-processing step before the training of the multimodal recommendation model. Then, unlike the existing solutions, our approaches leverage the graph structure of the user-item recommendation data. Concretely, by considering the item-item co-occurrence graph connecting items that have been interacted by the same users, we propose to re-sketch the problem of missing multimodal features as a graph features imputation problem [36]. That is, for each modality, we impute the missing features through the available ones by propagating them at multiple hops in the item-item graphs.

Our contributions can be summarized as follows:

1) We provide, for the first time in the literature, a problem formalisation for missing modalities in multimodal recommendation, which explicitly outlines differences with missing information in traditional machine learning.

2) We propose a group of novel imputation approaches that leverage the item-item co-occurrence graph, thus re-interpreting the problem as a task of graph features

interpolation. These solutions can be plugged into any existing multimodal RSs and act in a pre-processing manner (i.e., they are training-free methods).

3) Extensive experiments conducted on six recommendation datasets from the Amazon catalog with 2 modalities, one short-video recommendation dataset with 3 modalities, and 14 (multimodal) recommendation models empirically demonstrate that imputing missing modalities through our approaches can preserve (or even improve) the performance gap between traditional and multimodal RSs. Noticeably, we implement the (multimodal) recommenders either with Elliot [2], MMRec [37], or with their own codebase: this further demonstrates how easy and effective is to apply our imputation pipeline to any already-existing framework in the literature.

4) Further results prove that our graph-based imputations can outperform traditional imputation methods in machine learning for several settings of missing modalities.

5) By recognizing the importance that feature homophily [38] (on the item-item graph) can hold on our graph-based imputations, we provide, for the first time in multimodal recommendation with missing modalities, an empirical investigation on their interdependence.

Code is accessible at: https://github.com/danielemalitesta/Graph-Missing-Modalities-TKDE.

## II. RELATED WORK

In this section, we present the related literature on the two main topics for this work: (i) multimodal recommendation and (ii) missing information in recommendation.

### A. Multimodal recommendation

In diverse domains such as e-commerce [39], fashion [8], music [9], food [10], and micro-video [11] recommendation, the inclusion of multimodal content associated with items has proven to significantly enhance the representational capability of recommender systems. With the advancements in multimodal learning [40], multimodal recommender systems strive to address persistent challenges in personalized recommendation, such as data sparsity and cold-start [1], [9]. Additionally, exploiting multimodal content can help uncover user-item interactions and intentions through attention mechanisms, thereby enhancing the interpretability of recommendations [8]. In the following paragraphs, we provide a comprehensive overview of the multimodal recommendation domain, as highlighted in recent surveys on the topic [2]–[7].

Given the recent surge in graph neural networks for recommendation [41], several techniques have begun integrating multimodality into user-item bipartite graphs and knowledge graphs [42]. These techniques refine the multimodal representations of users and items through various approaches employing the message-passing schema. While initial efforts involved simply injecting multimodal item features into the graph-based pipeline [14], more advanced methods now learn separate graph representations for each modality and disentangle users' preferences at the modality level [43], considering both global and local user's interests [20]. Further approaches
also concentrate on unveiling multimodal structural distinc-
tions among items in the catalogue [16], [17], [44], [45],
sometimes leveraging self-supervised [18], [19], [21], [46],
[47] and contrastive [48], [49] learning. Finally, recent works
propose to introduce powerful generative approaches such as
variational autoencoders [50], [51], diffusion models [52],
and large language models [53], while others translate the
multimodal features to the frequency domain [54], [55], trans-
fer the multimodal knowledge from larger to smaller model
architectures [56], [57], or enhance the recommendation of
long-tail items from the catalog [58].

In this work, we propose a group of training-free graph-
based imputation methods to address missing modalities in
multimodal recommendation. Such approaches can be plugged
into any multimodal recommender systems presented above.
Thus, we select representative (both well-established and re-
cent) multimodal recommender systems and test our solutions
on top of them: VBPR [1], NGCF-M [12], LightGCN-M [19],
FREEDOM [17], BM3 [18], MGCN [46], MMSSL [19],
LGMRec [20], DiffMM [52], and MENTOR [21].

B. Missing information in recommendation

In a broader sense, the issue of missing information has al-
ways been debated in recommendation. The literature outlines
two main research directions regarding missing information
on (i) user-item feedback and (ii) content/metadata.

Indubitably, the missing feedback issue is widely recognised
as a crucial challenge. On the one hand [59], the underneath
assumption is that ratings in real-world recommendation data
are missing at random since the missing probability does
not depend on the ratings themselves; in such a context,
learning from observed (not-missing) data is sufficient to
provide optimal predictions. On the other hand [59], a more
complex and interesting scenario is the one of missing not
at random ratings, where multiple mechanisms determine the
missing feedback ratings; hence, the recommendation algo-
rithm should not ignore these mechanisms during the training.
Indeed, the latter setting has been largely investigated over the
last few decades [23]–[25], for instance in multicriteria [60],
social [61], and movies recommendation [62].

Conversely, the issue of missing content in recommendation
has been poorly explored in the related literature so far. Based
on a careful review, the two principal application scenarios
involve items' metadata [26], [27]. Such works [26], [27]
address the problem of missing attribute entries in the users'
and items' metadata following two different strategies, namely,
feature sampling towards model's robustness [26] and graph
convolutional networks propagating features on a tripartite
graph of users, items, and attributes nodes [27].

Finally, we consider the problem of missing multimodal
information in multimodal recommendation [31]–[35], that
may fall into the second categorization we made from above.
To the best of our knowledge, the literature regarding this
aspect is very limited and quite recent [12]. The solution
presented in [31] is one of the first approaches, and works
through a generative technique which is trained to reconstruct
specific missing modalities embeddings; the only modalities

involved are visual and textual. Second, the authors from [32]
design a train a novel multimodal model making it robust to
the absence of modalities through modality-specific variational
autoencoders. Then, the approach proposed in [33] leverages
inter- and intra-modality views to impute missing modalities
and combine the two outcomes into what pseudo-features
replacing the missing ones. More recently, the works in [34],
[35] exploit single-branch networks and invariant learning to
address missing modalities, but they to not perform any actual
multimodal imputation within their pipeline.

Unlike the outlined works, our proposed solutions show sev-
eral advantages: (i) they act during the pre-processing phase,
being completely model-agnostic and avoiding any end-to-end
training within the recommendation pipeline; (ii) they leverage
information coming from the iter-item co-purchase graph, thus
exploiting, for the first time in the related literature, graph-
based imputation techniques. Note that, since our approaches
are conceptually and technically different from the related
works in the literature (i.e., operate during the pre-processing
instead of being trained end-to-end), we purposely decide not
to compare them one another, as their comparison would be
unfeasible and unfair for the above reasons.

III. BACKGROUND NOTIONS

In this section, we provide useful background notions for
our paper and methodology. Specifically, we will formalise
the problem of missing information in machine learning and
present popular methods to address it. Then, we will formalise
the multimodal recommendation task.

A. Missing information in machine learning

The problem of missing data information in machine learn-
ing has been historically discussed for decades [22]. Without
loss of generality, let $\mathbf{X} \in \mathbb{R}^{S \times C}$ be a matrix of $S$ ob-
served samples, each characterised by $C$ features. Then, let
$\mathbf{Y} \in \mathbb{R}^{S \times K}$ be a vector mapping each sample to a one-hot
encoding class label, where we consider $K$ possible classes. In
a multi-class classification problem, we aim to learn a machine
learning model (e.g., a neural network) such that it can predict
(with a certain approximation error) the class of each label:

$$
\hat{\mathbf{Y}} = \mathrm{ML}(\mathbf{X}), \qquad (1)
$$

where $\hat{\mathbf{Y}} \in \mathbb{R}^{S \times K}$ is the predicted vector of labels for each
data sample by the learned ML(·) function.

In real-world settings, the input data may be missing some
of (or all) its features. While a possible workaround could
be to drop those samples having missing information, another
popular procedure involves **imputing** those missing values in
the input data. Let $\mathbf{B} \in \mathbb{R}^{S \times C}$ be a binary matrix such that
$\mathbf{B}_{sc} = 1$ if the $c$-th feature value is available for sample $s$, $0$
otherwise. Thus, if $\mathbf{B}_{sc} = 0$, we design an $\text{IMPUTE}(\cdot)$ function
that approximates the missing value of $c$ for the sample $s$:

$$
\hat{\mathbf{X}}_{sc} = \text{IMPUTE}(\mathbf{X}_{*c}[\mathbf{B}_{*c}]), \quad (2)
$$

where we select, from the input data, all values for feature
c and retrieve (with the binary mask and the view operator
[·]) only the available values and impute the missing one. In
general, to impute missing values on the feature c, we use the available values from the **same** feature. We indicate such an imputed value with $\hat{\mathbf{X}}_{sc}$. The IMPUTE(·) function may be designed in different manners. For instance, we could approximate the missing data with all zero values (i.e., ZEROS), with the mean of the available data (i.e., GLOBALMEAN), or random values (i.e., RANDOM).

Once we imputed all missing values from the input data, we train the machine learning model to provide output predictions based on the available and imputed data:

$$ \hat{\mathbf{Y}} = \text{ML}(\mathbf{X}[\mathbf{B}], \hat{\mathbf{X}}). \quad (3) $$

## B. Multimodal recommendation

Let $\mathcal{U}$ and $\mathcal{I}$ be the sets of users and items, respectively, where $|\mathcal{U}| = U$ and $|\mathcal{I}| = I$. Then, assuming an implicit feedback scenario, we indicate with $\mathbf{R} \in \mathbb{R}^{U \times I}$ the user-item interaction matrix, where $\mathbf{R}_{ui} = 1$ if there exists a recorded interaction between user $u \in \mathcal{U}$ and item $i \in \mathcal{I}$, 0 otherwise.

As in any latent factor-based approaches for recommendation, we map users' and items' IDs to embeddings in the latent space, where $\mathbf{E}_u \in \mathbb{R}^D$ and $\mathbf{E}_i \in \mathbb{R}^D$ represent the D-dimensional embeddings for user $u \in \mathcal{U}$ and item $i \in \mathcal{I}$, with $D \ll U, I$. On such a basis, the recommendation task is about reconstructing the user-item interaction matrix $\hat{\mathbf{R}}$ by learning meaningful representations for the user and item embeddings $\mathbf{E}_u$ and $\mathbf{E}_i$ through the existing user-item matrix $\mathbf{R}$:

$$ \hat{\mathbf{R}} = \text{RECSYS}(\mathbf{R}, \mathbf{E}_u, \mathbf{E}_i) \quad \forall u \in \mathcal{U}, i \in \mathcal{I}. \quad (4) $$

In any multimodal recommendation setting (e.g., fashion, song, micro-video, or food recommendation), items' representation may be suitably enriched by considering the multimodal content describing them, such as products' images, descriptions, or audio tracks. Thus, let $\mathcal{M}$ be the set of modalities (with $|\mathcal{M}| = M$ the number of admissible modalities), and $\mathbf{F} \in \mathbb{R}^{I \times M \times C}$ be the multimodal feature tensor of all items, where $\mathbf{F}_{i*} \in \mathbb{R}^{M \times C}$ is the tensor of all multimodal features of item $i \in \mathcal{I}$, $\mathbf{F}_{*m} \in \mathbb{R}^{I \times C}$ is the tensor of all item features accounting for the modality $m \in \mathcal{M}$, and $\mathbf{F}_{im} \in \mathbb{R}^C$ is the feature of item $i$ accounting for the $m$ modality (e.g., the visual feature extracted from the product image of item $i$ through a pre-trained deep convolutional network [63]). In both machine learning and multimodal recommendation settings, $C$ refers to data features, standing for data columns (e.g., age, gender, job) in the former, and latent embedding dimension of multimodal features (e.g., 2048 as in the case of ResNet50 to extract visual features from product images) in the latter.

On such preliminaries, the multimodal recommendation task is about reconstructing the user-item interaction matrix $\hat{\mathbf{R}}$ by learning meaningful user and item embeddings $\mathbf{E}_u$ and $\mathbf{E}_i$ through the existing user-item interaction matrix $\mathbf{R}$ and the multimodal items' features $\mathbf{F}$:

$$ \hat{\mathbf{R}} = \text{MMRECSYS}(\mathbf{R}, \mathbf{E}_u, \mathbf{E}_i, \mathbf{F}) \quad \forall u \in \mathcal{U}, i \in \mathcal{I}. \quad (5) $$

# IV. PROPOSED METHODOLOGY

Unlike other data types in machine learning, recommendation data is modeled by a bipartite and undirected graph

representing various relations: user-item, user-user, and item-item. Based on these foundations and motivated by recent advancements in graph learning [36], we propose to frame the problem of imputing missing modalities in recommendation as a problem of **graph feature interpolation** on the **item-item** graph. Specifically, we aim to impute missing multimodal features of an item through the existing (i.e., non-missing) multimodal features of the items in the item-item graph.

This section is structured as follows. To begin with, it provides a formal definition (for the first time in the literature) of missing modalities in multimodal recommendation. Then, it re-interprets it under the lens of graph feature interpolation. Finally, it presents our pipeline for missing modalities imputation in multimodal recommendation leveraging graph-aware imputation methods. In Figure 2, we visually represent the main stages of our graph-based imputation strategies.

## A. Missing modalities in multimodal recommendation

The problem of missing modalities in multimodal recommendation refers to the unavailability of some (or all) multimodal feature vectors for a specific subset of items (see again Table I). For instance, when considering a twofold modality setting (e.g., visual and textual), it could happen that item $i$ is missing either the visual or textual modality or both; in every such case, we say item $i$ has some missing modalities.

The purpose is to design an imputation function IMPUTE(·) that, similarly to the definition provided in Section III-A, can approximate the missing multimodal features through the available ones. Nevertheless, unlike traditional machine learning, we consider another version of the binary mask as $\mathbf{B} \in \mathbb{R}^{I \times M \times C}$ such that:

$$ \mathbf{B}_{im} = \begin{cases} \mathbf{1}_C & \text{if modality } m \text{ is available for } i \\ \mathbf{0}_C & \text{otherwise,} \end{cases} \quad (6) $$

where $\mathbf{1}_C = [1, 1, \dots, 1] \in \mathbb{R}^C$ and $\mathbf{0}_C = [0, 0, \dots, 0] \in \mathbb{R}^C$. The above outlines that, in the considered setting, the **whole** feature vector can be available (missing) for item $i$ and modality $m$. Thus, if $\mathbf{B}_{im} = \mathbf{0}_C$, we design an IMPUTE(·) function that approximates the modality $m$ for item $i$:

$$ \hat{\mathbf{F}}_{im} = \text{IMPUTE}(\mathbf{F}_{*m}[\mathbf{B}_{*m}]), \quad (7) $$

where we select, from the input multimodal features, those accounting for modality $m$, and retrieve (with a binary mask) only the items with this available modality. Similarly to the definition provided for machine learning, we assume again that we can impute missing multimodal features with the available features from the **same** modality (e.g., visual with visual, and textual with textual). Nevertheless, in the most general setting, we may want to impute missing multimodal features with **any** available multimodal features (e.g., visual with textual and vice-versa). We will devote future work to test strategies which apply **cross-modal** imputation, following similar solutions in the literature which propose the alignment across modalities [18], [19], [21], [47].
<figure><img src="image_3.png" /><figcaption>Fig. 2: Visual representation of our graph-based imputations. Starting from the user-item graph with incomplete multimodal features (a), we project and sparsify it to obtain the item-item co-purchase graph (b). Then, we propagate the available multimodal features in the obtained item-item graph (c), which allows us to impute the missing features (d).</figcaption></figure>

Once we imputed all missing modalities, we train the multimodal recommender system to provide output predictions based on the available and imputed multimodal data:

$$ \hat{\mathbf{R}} = \text{MMRECSYS}(\mathbf{R}, \mathbf{E}_u, \mathbf{E}_i, \mathbf{F}[\mathbf{B}], \hat{\mathbf{F}}) \quad \forall u \in \mathcal{U}, i \in \mathcal{I}. \quad (8) $$

## B. Graph-aware imputation

We propose to frame the problem of imputing missing modalities in multimodal recommendation as a problem of **graph feature interpolation** on the **item-item** co-interactions graph. As observed in the literature [36], graph feature interpolation requires some prior assumptions on the input graph. One of the common assumptions is the feature homophily, which states that the features of each node should be close to those of its neighbors. Translated into our scenario, we assume (with support from the literature [16], [17], [44]) that **co-interacted items are likely to present semantically-similar multimodal features**, aligning with the same principle behind collaborative filtering (users sharing similar preferences tend to interact with the same items). In Section VI-D, we provide a finer-grained evaluation of this aspect, assessing the performance of our approach to different levels of feature homophily [38].

Starting from the user-item bipartite graph, we build the item-item graph as the projection on the item partition: $\mathbf{R}^I = \mathbf{R}^\top \mathbf{R}$ where $\mathbf{R}^I \in \mathbb{R}^{I \times I}$ and $\mathbf{R}_{ij}^I$ is the number of users who interacted with both items i and j, namely, the co-interactions among items from the catalog. As the projected item-item graph may be too dense (i.e., with noisy item-item

interactions), we perform the sparsification of the item-item matrix through TOPN(·) sparsification (compare with [16]):

$$ \bar{\mathbf{R}}_{ij}^I = \text{SPARSE}(\mathbf{R}^I, N) = \begin{cases} 1 & \text{if } \mathbf{R}_{ij}^I \in \text{TOPN}(\mathbf{R}_i^I) \\ 0 & \text{otherwise,} \end{cases} \quad (9) $$

where $\bar{\mathbf{R}}_{ij}^I$ is the sparsified item-item matrix, while TOPN(·) is the function returning the N highest values of a matrix row-wise, with N setting the sparsification rate. Regarding this operation, we acknowledge that the proposed sparsification strategy may lead to possible popularity bias issues where popular items are largely represented in the projected graph; thus, more refined solutions using dynamic thresholding or taking into account interaction frequency might limit the issue [39]. However, our graph-based imputation strategies apply some ad-hoc matrix normalisations, which might effectively prevent this problem (see the following for further details).

Our goal is to design the graph-aware imputation function IMPUTE(·) able to approximate the missing multimodal features. Unlike the definition provided in Equation (7), the function takes the available multimodal features **and** the item-item co-interaction matrix. In other words, the IMPUTE(·) function approximates the missing multimodal features of an item node through the multimodal features of its neighbors. In general, these latter features may be missing themselves.

To address this aspect, we design the IMPUTE(·) function taking as inputs the **whole** set of multimodal features, and the item-item co-interactions matrix. For the missing multimodal features, and inspired by [36], we initialise their embeddings to the zero vector. Thus, to impute the missing features on a
given modality $m \in \mathcal{M}$, we do the following:

$$ (i): \mathbf{P} = \mathbf{F}_{*m}[\mathbf{B}], \quad \mathbf{F}_{*m}[\neg\mathbf{B}] = \text{ZEROS}(\mathbf{I}) $$

$$ (ii): \mathbf{F}_{*m} = \text{IMPUTE}(\overline{\mathbf{R}}^{\mathcal{I}}, \mathbf{F}_{*m}) \quad (10) $$

$$ (iii): \mathbf{F}_{*m}[\mathbf{B}] = \mathbf{P}, \quad \hat{\mathbf{F}}_{*m} = \mathbf{F}_{*m}[\neg\mathbf{B}], $$

where we (i) save the available features in a placeholder vector $\mathbf{P}$ and initialise the missing features with the zero vector<fnref n="1" />; (ii) impute the missing features with the item-item co-interaction matrix, and all features accounting for that modality; (iii) we extract the imputed missing features and restore the available ones to their original values. In the following, we list four alternatives to design the IMPUTE(·) function.

1) **NEIGHMEAN**: We extend the GLOBALMEAN imputation through the mean of multimodal features from the one-hop neighborhood (NEIGHMEAN):

$$ \mathbf{F}_{*m} = (\mathbf{D}^{\mathcal{I}})^{-1} \overline{\mathbf{R}}^{\mathcal{I}} \mathbf{F}_{*m}, \quad (11) $$

where $\mathbf{D}^{\mathcal{I}} \in \mathbb{R}^{I \times I}$ is the degree matrix for the item-item co-interaction matrix.

2) **MULTIHOP**: As we leverage the item-item graph, we can also consider **multi-hops** relationships between items. Thus, our second graph-aware method iteratively aggregates features from neighborhood nodes (MULTIHOP):

$$ \mathbf{F}_{*m}^{(t)} = (\mathbf{D}^{\mathcal{I}})^{-0.5} \overline{\mathbf{R}}^{\mathcal{I}} (\mathbf{D}^{\mathcal{I}})^{-0.5} \mathbf{F}_{*m}^{(t-1)}, \quad (12) $$

where $\mathbf{F}_{*m}^{(t-1)}[\mathbf{B}] = \mathbf{P}$ to ensure that, at each new iteration $t \in \{1, \dots, T\}$, we re-initialize the available features to their original values [36]. Note that we are normalising the aggregated features at each iteration to ensure low-degree nodes are not over-penalised over high-degree ones. This ensures that items initially quite popular in the user-item graph will not be overrepresented during the item-item imputation, thereby limiting the popularity bias problem.

3) **PERSPAGERANK**: While the normalisation operation from above only accounts for the 1-hop neighborhoods, the recent literature has demonstrated that **graph diffusion at multiple hops** [64] may improve graph learning; the idea is to leverage an enhanced representation of the interaction matrix, as personalized PageRank [65].

Hence, we propose a third graph-aware imputation that modifies the MULTIHOP strategy by incorporating **personalised PageRank** as a normalisation of the item-item co-interactions matrix (PERSPAGERANK):

$$ \mathbf{F}_{*m}^{(t)} = a \left( \mathbb{1} - (1-a)(\mathbf{D}^{\mathcal{I}})^{-0.5} \overline{\mathbf{R}}^{\mathcal{I}} (\mathbf{D}^{\mathcal{I}})^{-0.5} \right)^{-1} \mathbf{F}_{*m}^{(t-1)}, \quad (13) $$

where $a \in [0, 1]$ is the probability of following an edge from the current item or jumping back to a specified set of nodes.

4) **HEAT**: With similar rationales to the PERSPAGERANK approach from above, in the HEAT strategy we use **heat diffusion** [66] for the normalisation of the item-item co-interactions matrix:

$$ \mathbf{F}_{*m}^{(t)} = \exp\left(-b\left(\mathbb{1} - (\mathbf{D}^{\mathcal{I}})^{-0.5} \overline{\mathbf{R}}^{\mathcal{I}} (\mathbf{D}^{\mathcal{I}})^{-0.5}\right)\right) \mathbf{F}_{*m}^{(t-1)}, \quad (14) $$

where $\exp(\cdot)$ is the exponential function over matrices, and $b$ is the diffusion time.

### C. Differences with item-item recommendation strategies

Our graph-based imputation solutions present analogies with already-existing multimodal recommendation approaches leveraging item-item similarities, such as [16], [17], [44], [45]. Thus, it is important to distinguish those methodologies from the proposed ones, evidencing how our graph-based imputations settle as novel techniques in the current literature. We clarify it under two main perspectives.

* **The task is different.** Unlike other approaches from the literature, our techniques are specifically tailored to address the imputation of missing modalities in multimodal recommendation. To the best of our knowledge, this is the first attempt in the literature to solve this problem through the graph topology of the user-item dataset.

* **The item-item structure is different.** While the resulting item-item similarity graph may appear similar to those proposed in related works, the structure of this graph is technically and conceptually different. The item-item similarity graph implemented in the literature leverages only the multimodal signals, as it is built on the similairities (i.e., cosine similarity) among multimodal embeddings of items. Conversely, in our setting and scenario, some items are missing (part of) their multimodal embeddings, thus it would not be technically and conceptually possible to build an item-item similarity graph through them. That is why we decide to exploit an additional signal, the collaborative one encoded in the user-item dataset, and derive item-item similarities based upon users' interactions (i.e., two items are connected if at least of user has interacted with both of them).

### D. Further discussion on the proposed imputations

While effective, our graph-based imputation strategies may, in principle, suffer from inherent limitations stemming from their dependence on the quality of user-item feedback (used to construct the item-item co-interaction graph) and on the availability and reliability of multimodal embeddings, which are exploited to infer missing representations. In this regard, well-known challenges in recommender systems, such as user and item cold-start as well as noisy information, may negatively impact the effectiveness of the proposed imputations. Specifically, items with very limited interactions may be poorly represented in the item-item graph (cold-start), while noisy multimodal embeddings may lead to the unintended propagation of noise during the imputation process.

That said, we emphasize that the primary focus of this work is the problem of missing multimodal information in recommendation, an aspect that remains largely underexplored in both prior and recent literature. We acknowledge that none of the related studies simultaneously address the three challenges discussed here, namely missing multimodal information, cold-start, and noisy data, within a unified framework. Nevertheless, we plan to extend the proposed imputation methodologies to explicitly account for cold-start and noise-related issues, with

<footnote n="1">¹We use the “-” operator to indicate the logical negation operator for the binary matrix.</footnote>
<table><caption>TABLE I: Datasets statistics, where we show items with missing Visual and Textual information, before and after they have been dropped (without and with the apex), and the missing percentage over all the items in the catalog.</caption><thead><tr><th rowspan="2">Datasets</th><th rowspan="2">|U| / |U'|</th><th rowspan="2">|I| / |I'|</th><th rowspan="2">|R| / |R'|</th><th colspan="3">Missing</th></tr><tr><th>Visual</th><th>Textual</th><th>Tot. %</th></tr></thead><tbody><tr><td>Office</td><td>4,905 / 4,891</td><td>2,420 / 1,746</td><td>53,258 / 35,185</td><td>0</td><td>674</td><td>28%</td></tr><tr><td>Music</td><td>5,541 / 5,349</td><td>3,568 / 2,453</td><td>64,706 / 51,516</td><td>2</td><td>1,114</td><td>31%</td></tr><tr><td>Baby</td><td>19,445 / 19,440</td><td>7,050 / 6,382</td><td>160,792 / 138,149</td><td>13</td><td>667</td><td>9%</td></tr><tr><td>Toys</td><td>19,412 / 19,410</td><td>11,924 / 11,089</td><td>167,597 / 146,640</td><td>17</td><td>832</td><td>7%</td></tr><tr><td>Beauty</td><td>22,363 / 22,293</td><td>12,101 / 11,124</td><td>198,502 / 165,772</td><td>7</td><td>977</td><td>8%</td></tr><tr><td>Sports</td><td>35,598 / 35,526</td><td>18,357 / 15,633</td><td>296,337 / 251,201</td><td>70</td><td>2,701</td><td>15%</td></tr></tbody></table>

the goal of ensuring their robustness and effectiveness also in these particularly challenging scenarios.

## V. EXPERIMENTAL SETTINGS

In this section, we describe the experimental settings we adopted in terms of selected recommendation datasets, recommendation baselines and (graph-based) imputation methods, and the reproducibility details to fully replicate our results.

### A. Datasets

For the sake of this paper, we select six categories of the famous Amazon Reviews dataset [1]: Office Products (Products), Digital Music (Music), Baby, Toys & Games (Toys), Beauty, and Sports & Outdoors (Sports). For each dataset, we consider two versions: (i) **dropped**, where we remove items with at least one missing modality, along with interactions involving those items and users having interacted only with those items; (ii) **imputed**, where we impute the items' missing multimodal features. We report the complete statistics of the selected datasets in the two settings, where we distinguish the dropped version from the imputed one through an apex (Table I). From the available items' multimodal data (i.e., product images and textual descriptions), we extract their visual [63] and textual embeddings [67].

### B. Baselines

To test our methodologies, we select recommendation baselines accounting for *traditional models* (without the adoption of multimodal information) and *multimodal ones*. These are: BPRMF [68], NGCF [13], LightGCN [41], and SGL [69] for the traditional recommendation, and VBPR [1], NGCF-M [12], LightGCN-M [19], FREEDOM [17], BM3 [18], MGCN [46], MMSSL [19], LGMRec [20], DiffMM [52], and MENTOR [21] for the multimodal case. Specifically, BPRMF, NGCF, LightGCN, SGL, VBPR, NGCF-M, LightGCN-M, FREEDOM, BM3, and MGCF are implemented in the framework Elliot [2], while LGMRec and MENTOR with the framework MMRec [37], and finally MMSSL and DiffMM with their original codebase. This further demonstrates how our (graph)-based imputations can be seamlessly plugged into any existing multimodal recommendation framework.

As for the imputation approaches, we use *classical machine learning* methods, generative approaches based upon *autoencoders*, and the proposed four *graph-aware* techniques.

The classical machine learning methods are: (i) ZEROS imputes all missing multimodal embeddings to the all-zeros vector. (ii) RANDOM imputes all missing multimodal embeddings to the all-random values vector, by choosing a uniform random distribution. (iii) GLOBALMEAN imputes the missing embeddings from one modality to the average of the available embeddings from the same modality.

Then, for the autoencoder-based imputations, we select: (i) AUTOENC [28] trains an autoencoder to reconstruct missing features from one modality $m_1$ through the available features from another modality $m_2$. The model is trained on the portion of items having both modalities, and then used in inference on the actual missing features. (ii) GRAPHAUTOENC applies a similar approach to AUTOENC, with the difference that it uses a graph-based encoder (i.e., a graph convolutional network [70]) operating on the item-item co-purchase graph, obtained through the projection and TOPN sparsification from the user-item graph on the item partition. Then, the decoder is again a multi-layer perceptron as in the AUTOENC approach.

Finally, our proposed graph-based imputations are (see Section IV-B): (i) NEIGHMEAN, (ii) MULTIHOP, (iii) PERSPAGERANK, and (iv) HEAT.

### C. Reproducibility

For the training and test of the baselines models, we split the recommendation datasets following the hold-out strategy into train and test sets (80%/20%). As for the hyper-parameter search of the recommendation models, we select a shared grid search space with the learning rate in [0.0001, 0.0005, 0.001, 0.005, 0.01] and the regularization term in [1e-5, 1e-2], leaving the other parameters to the best values for each model, and fix the batch size at 1024 and the epochs at 200. Regarding the hyper-parameters for graph-based imputation approaches, we explore the TOPN sparsification in [10, 20, ..., 100] and the propagation hops in [1, 2, ..., 20]. We use the Recall@20 and the nDCG@20 as test metrics and retain 10% of the train set as validation, with Recall@20 as a validation metric.

## VI. RESULTS AND DISCUSSION

In this section, we aim to answer the following research questions (RQs): **RQ1)** Can the proposed graph-based imputation methods be beneficial in standard multimodal recommendation settings from the literature? **RQ2)** How do the traditional, autoencoder-, and graph-based imputation methods differently affect the final recommendation performance? **RQ3)** What is the impact of different sparsification rates
and propagation hops on the item-item co-purchase graph for graph-based imputation methods? **RQ4**) How does the level of feature homophily in the item-item co-purchase graph influence the imputation performance for graph-based impu- tation approaches? **RQ5**) What is the performance variation under various percentages of missing modalities for traditional, autoencoders-, and graph-based imputation methods? **RQ6**) Do the imputation strategies effectively extend to multimodal recommendation datasets with more than two modalities?

A. RQ1) Graph-based imputations in standard multimodal recommendation

We analyse the impact of our proposed graph-based impu-
tation methods in standard multimodal recommendation set-
tings. We conduct this investigation through two experiments,
described in the following.

In the first investigation, we seek to assess whether and
to what extent such imputation methodologies can widen
the performance gap between traditional and multimodal rec-
ommendation systems. To this end, we select three popular
recommendation models, namely, BPRMF, NGCF, and Light-
GCN, alongside their multimodal versions where we only
inject the multimodal items' features (i.e., VBPR, NGCF-
M, and LightGCN-M, respectively). We choose these three
models because, in their multimodal versions, we do not
perform any additional modifications to their original mod-
els' architectures apart from the injection of the multimodal
features; thus, we can precisely assess the impact of our
graph-based imputations of missing modalities and avoid the
performance influence from other strategies (e.g., denoising
procedures as in FREEDOM).

Table II shows the performance improvement on Recall@20 and nDCG@20 for the three multimodal recommender systems over their original versions, both in the **dropped** and in **imputed** settings, where the imputed setting uses the best graph-based imputation according to the Recall@20 calculated on the validation set. We highlight that our analysis focuses on the **performance improvements** between each non-multimodal recommender and its multimodal counterpart, rather than on the single performance because the dropped and imputed datasets should be regarded, in general, as different datasets (see again Table I), so we could not provide a fair and consistent comparison otherwise. As observable in the table, results largely confirm that the imputed setting not only can preserve the performance gap between traditional and multimodal recommender systems but, in several cases, can widen or even revert it (e.g., consider the case of BPRMF/VBPR on Music or LightGCN/LightGCN-M on Office and Beauty).

As a confirmation of the outlined outcomes, in our second
investigation (i.e., Table III), we provide an extensive bench-
marking of traditional and multimodal recommender systems
from the literature in the dropped setting. Once again, the
missing modalities were imputed through the best graph-
based imputation method according to the Recall@20 on the
validation set. Results generally highlight the performance
predominance of multimodal recommender systems over tra-
ditional approaches. Moreover, more recent recommendation

techniques (e.g., LightGCN and SGL on the traditional side,
MMSSL, LGMRec, and MENTOR on the multimodal side)
can provide higher recommendation performance.

All things considered, the reported results demonstrate that
the application of our proposed graph-based imputation meth-
ods for missing modalities can preserve the trends observed
in the literature, or even amplify/revert the performance gap
between traditional and multimodal approaches. Indeed, the
advantages of the imputed setting are to retain important
information from the datasets (without the need to perform any
filtering/dropping on them) and recover the missing modalities
through high-quality strategies.

B. RQ2) Impact of traditional, autoencoder-, and graph-based imputations

In the current analysis, we compare the recommenda-
tion performance of selected multimodal recommender sys-
tems in the dropped setting when equipped with traditional,
autoencoder-, and graph-based imputation strategies. Table IV
displays the results on all the recommendation datasets in
terms of Recall@20 and nDCG@20.

From a general overview, a consistent trend across all
datasets and recommendation models suggests that graph-
based imputation strategies (the ones proposed in this pa-
per) settle as the best or second-best imputation approaches
compared to all other techniques. However, it is important
to highlight that for some recommendation models, such as
VBPR, MGCN, and MMSSL, autoencoder-based imputations
can still provide sufficient performance on specific datasets,
maybe thanks to their ability to learn from the data how to
recover one missing modality from the other available.

Imputation-wise, we observe that, for traditional imputation,
there is no recognisable trend, where each of the three ap-
proaches can provide improved performance depending on the
considered recommendation model and dataset. Conversely,
in terms of autoencoder-based imputation, we notice that, in
most cases, GRAPHAUTOENC is the most successful solution
among the two tested ones; this further confirms our intuition
to adopt the item-item co-purchase graph structure to impute
missing modalities. Finally, when it comes to graph-based
imputation, approaches that propagate multimodal features at
multiple hops (i.e., MULTIHOP), with more refined normali-
sation solutions (i.e., PERSPAGERANK and HEAT) tend to be
the most powerful techniques.

In conclusion, our proposed training-free graph-based impu-
tation methods largely outperform other families of imputation
approaches on all the recommendation models and datasets.
Moreover, we highlight the importance of leveraging the
item-item co-purchase graph in the imputation, especially by
exploring its topology at multiple hops. Finally, even against
strong imputation strategies (such as the autoencoder-based
ones which are trained to perform their task) our solutions
can still provide high-quality imputations while being much
more computationally-efficient (as they are training-free).

C. RQ3) Impact of sparsification rates and hop propagations

This study serves to provide a hyper-parameter sensitivity
analysis for the sparsification rate (TopN) and number of
<table><caption>TABLE II: Performance improvement of VBPR, NGCF-M, and LightGCN-M over BPRMF, NGCF, and LightGCN (respectively) in the **dropped** and **imputed** settings. For multimodal recommender systems, the selected graph-based imputation method is the one performing the best on the validation set.</caption><thead><tr><th rowspan="3">Models</th><th colspan="6">Dropped</th><th colspan="6">Imputed (Ours)</th></tr><tr><th colspan="2">Office</th><th colspan="2">Music</th><th colspan="2">Beauty</th><th colspan="2">Office</th><th colspan="2">Music</th><th colspan="2">Beauty</th></tr><tr><th>Recall</th><th>nDCG</th><th>Recall</th><th>nDCG</th><th>Recall</th><th>nDCG</th><th>Recall</th><th>nDCG</th><th>Recall</th><th>nDCG</th><th>Recall</th><th>nDCG</th></tr></thead><tbody><tr><td>BPRMF</td><td>7.74</td><td>3.65</td><td>26.50</td><td>14.24</td><td>9.81</td><td>5.00</td><td>8.51</td><td>4.55</td><td>24.59</td><td>13.69</td><td>9.43</td><td>4.85</td></tr><tr><td>VBPR</td><td>9.99</td><td>4.82</td><td>25.22</td><td>13.71</td><td>11.12</td><td>5.86</td><td>9.47</td><td>5.23</td><td>26.25</td><td>14.58</td><td>10.92</td><td>5.71</td></tr><tr><td>Improvement (%)</td><td>+29.07%</td><td>+32.05%</td><td>-4.83%</td><td>-3.72%</td><td>+13.35%</td><td>+17.20%</td><td>+11.28%</td><td>+14.95%</td><td>+6.75%*</td><td>+6.50%*</td><td>+15.80%*</td><td>+17.73%*</td></tr><tr><td>NGCF</td><td>9.90</td><td>4.49</td><td>24.39</td><td>12.61</td><td>8.64</td><td>4.28</td><td>8.18</td><td>3.91</td><td>24.12</td><td>13.06</td><td>8.92</td><td>4.56</td></tr><tr><td>NGCF-M</td><td>13.10</td><td>6.36</td><td>24.12</td><td>12.73</td><td>9.68</td><td>4.94</td><td>11.07</td><td>5.59</td><td>25.04</td><td>13.53</td><td>11.37</td><td>5.97</td></tr><tr><td>Improvement (%)</td><td>+31.32%</td><td>+41.65%</td><td>-1.11%</td><td>+0.95%</td><td>+12.04%</td><td>+15.42%</td><td>+35.33%*</td><td>+42.97%*</td><td>+3.81%*</td><td>+3.60%*</td><td>+27.47%*</td><td>+30.92*</td></tr><tr><td>LightGCN</td><td>13.95</td><td>6.70</td><td>27.86</td><td>14.74</td><td>11.79</td><td>6.13</td><td>11.55</td><td>6.12</td><td>26.61</td><td>14.49</td><td>11.61</td><td>6.02</td></tr><tr><td>LightGCN-M</td><td>13.18</td><td>6.36</td><td>28.06</td><td>15.29</td><td>11.38</td><td>5.99</td><td>11.55</td><td>6.38</td><td>26.82</td><td>14.89</td><td>12.12</td><td>6.32</td></tr><tr><td>Improvement (%)</td><td>-5.52%</td><td>-5.07%</td><td>+0.72%</td><td>+3.73%</td><td>-3.48%</td><td>-2.28%</td><td>+0.00%*</td><td>+4.25%*</td><td>+0.79%*</td><td>+2.76%</td><td>+4.39%*</td><td>+4.98%*</td></tr></tbody></table>

<table><caption>TABLE III: Recommendation performance in the imputed setting. For multimodal recommender systems, the selected graph-based imputation method is the one performing the best on the validation set. **Boldface** and **underlined** are the best and second-best values, respectively.</caption><thead><tr><th rowspan="2">Models</th><th colspan="2">Office</th><th colspan="2">Music</th><th colspan="2">Baby</th><th colspan="2">Toys</th><th colspan="2">Beauty</th><th colspan="2">Sports</th></tr><tr><th>Recall</th><th>nDCG</th><th>Recall</th><th>nDCG</th><th>Recall</th><th>nDCG</th><th>Recall</th><th>nDCG</th><th>Recall</th><th>nDCG</th><th>Recall</th><th>nDCG</th></tr></thead><tbody><tr><td>BPRMF</td><td>8.51</td><td>4.55</td><td>24.59</td><td>13.69</td><td>4.56</td><td>2.23</td><td>8.55</td><td>4.55</td><td>9.43</td><td>4.85</td><td>5.32</td><td>2.73</td></tr><tr><td>NGCF</td><td>8.18</td><td>3.91</td><td>24.12</td><td>13.06</td><td>5.59</td><td>2.81</td><td>8.22</td><td>4.23</td><td>8.92</td><td>4.56</td><td>4.97</td><td>2.52</td></tr><tr><td>LightGCN</td><td>11.55</td><td>6.12</td><td>26.61</td><td>14.49</td><td>7.36</td><td>3.76</td><td>10.22</td><td>5.43</td><td>11.61</td><td>6.02</td><td>8.26</td><td>4.34</td></tr><tr><td>SGL</td><td>10.28</td><td>5.81</td><td>27.59</td><td>15.86</td><td>6.08</td><td>3.27</td><td>10.89</td><td>6.05</td><td>12.20</td><td>6.72</td><td>7.87</td><td>4.27</td></tr><tr><td>VBPR</td><td>9.47</td><td>5.23</td><td>26.25</td><td>14.58</td><td>5.90</td><td>3.03</td><td>10.36</td><td>5.63</td><td>10.92</td><td>5.71</td><td>6.65</td><td>3.39</td></tr><tr><td>NGCF-M</td><td>11.07</td><td>5.59</td><td>25.04</td><td>13.53</td><td>6.96</td><td>3.46</td><td>9.21</td><td>4.88</td><td>11.37</td><td>5.97</td><td>7.43</td><td>3.77</td></tr><tr><td>LightGCN-M</td><td>11.55</td><td>6.38</td><td>26.82</td><td>14.89</td><td>7.72</td><td>3.91</td><td>11.24</td><td>6.10</td><td>12.12</td><td>6.32</td><td>7.50</td><td>3.89</td></tr><tr><td>FREEDOM</td><td>12.01</td><td>6.56</td><td>26.53</td><td>14.48</td><td>8.34</td><td>4.36</td><td>13.01</td><td>6.77</td><td>13.59</td><td>7.14</td><td>9.26</td><td>4.75</td></tr><tr><td>BM3</td><td>11.20</td><td>5.94</td><td>23.99</td><td>13.10</td><td>7.89</td><td>4.01</td><td>9.87</td><td>5.17</td><td>11.51</td><td>6.00</td><td>8.55</td><td>4.44</td></tr><tr><td>MGCN</td><td>11.63</td><td>6.25</td><td>24.60</td><td>13.41</td><td>8.44</td><td>4.26</td><td>12.35</td><td>6.59</td><td>12.73</td><td>6.88</td><td>8.53</td><td>4.49</td></tr><tr><td>MMSSL</td><td><strong>13.50</strong></td><td><strong>11.38</strong></td><td><strong>29.28</strong></td><td><strong>21.92</strong></td><td>8.80</td><td><strong>6.55</strong></td><td>12.04</td><td><strong>8.69</strong></td><td>13.72</td><td><strong>10.06</strong></td><td>9.22</td><td>7.10</td></tr><tr><td>LGMRec</td><td><u>13.19</u></td><td><u>7.28</u></td><td><u>29.05</u></td><td><u>17.11</u></td><td><strong>8.91</strong></td><td><u>4.58</u></td><td>12.74</td><td>6.82</td><td>13.76</td><td>7.56</td><td>9.23</td><td><u>4.89</u></td></tr><tr><td>DiffMM</td><td>12.60</td><td>6.93</td><td>28.49</td><td>16.46</td><td><strong>8.81</strong></td><td>4.49</td><td>11.92</td><td>6.33</td><td>13.27</td><td>7.13</td><td>9.29</td><td><strong>4.90</strong></td></tr><tr><td>MENTOR</td><td>12.76</td><td>7.25</td><td>27.63</td><td>15.76</td><td>8.73</td><td>4.47</td><td><strong>13.23</strong></td><td><u>7.04</u></td><td><strong>14.04</strong></td><td><u>7.68</u></td><td><strong>9.33</strong></td><td>4.87</td></tr></tbody></table>

explored hops (Hops) in our graph-based imputation strategies. Specifically, we consider the performance variation on VBPR and MGCN equipped with the HEAT imputation on the Sports recommendation dataset (Figure 3).

Starting from VBPR, we see that the performance tends to slightly degrade when retaining bigger portions of neighbors for each node (i.e., higher TopN values). This aligns with the rationale that retaining only the most similar neighbors for each node should improve the informativeness level of the item-item co-purchase graph, thus improving the graph-based imputation and the recommendation performance. Regarding the Hops, we acknowledge that even with a few explorations, the graph-based imputation can reach a sufficiently high-quality reconstruction of missing features in VBPR, as we see improved recommendation performance for those settings and negligible performance improvements with increasing Hops.

Trends are more or less aligned on MGCN in terms of Hops. However, focusing on the TopN, we clearly notice that retaining more neighbors for each node in the item-item co-purchase can, this time, improve the imputation and, consequently, the final recommendation performance. We ascribe this difference to the different nature of VBPR and MGCN, where the latter is a graph-based multimodal recommender system working on

the user-item graph. In this respect, the careful reader would note that the item-item co-purchase graph used in our paper carries unexplicit information from the user-item graph (i.e., items that have been interacted by the same users). Thus, it seems reasonable that the HEAT imputation working on a graph-based multimodal recommender system such as MGCN can benefit from larger neighbors to efficiently support the graph-based multimodal recommender system.

All in all, the above investigation outlines that the proposed graph-based imputation require careful hyper-parameter selection depending on the strategies and methodologies adopted by the multimodal recommender system.

### D. RQ4) Impact of feature homophily

Inspired by [36], we study the impact of feature homophily of the item-item co-purchase graph on the proposed graph-based imputation methods. To the best of our knowledge, this is the first time such an investigation is conducted in multimodal recommendation with missing modalities.

As the traditional and more common definition of node homophily [71] is not applicable to our setting, where no class nodes are available, we leverage the definition of feature

<footnote>* The performance improvement is higher in our imputed setting than in the dropped one.</footnote>
<table><caption>TABLE IV: Impact of all imputation methods on the performance of selected multimodal recommender systems for all the tested datasets. **Boldface** and **underlined** stand for best and second-best, respectively.</caption><thead><tr><th rowspan="2">Models</th><th rowspan="2">Imputation</th><th colspan="2">Office</th><th colspan="2">Music</th><th colspan="2">Baby</th><th colspan="2">Toys</th><th colspan="2">Beauty</th><th colspan="2">Sports</th></tr><tr><th>Recall</th><th>nDCG</th><th>Recall</th><th>nDCG</th><th>Recall</th><th>nDCG</th><th>Recall</th><th>nDCG</th><th>Recall</th><th>nDCG</th><th>Recall</th><th>nDCG</th></tr></thead><tbody><tr><td rowspan="4">VBPR</td><td>+ZEROS</td><td>9.49</td><td>5.24</td><td>24.85</td><td>14.02</td><td>5.03</td><td>2.49</td><td>10.07</td><td>5.42</td><td>9.80</td><td>5.08</td><td>6.31</td><td>3.15</td></tr><tr><td>+RANDOM</td><td>8.57</td><td>4.75</td><td>25.65</td><td>14.24</td><td>5.24</td><td>2.64</td><td>9.80</td><td>5.37</td><td>10.09</td><td>5.22</td><td>6.52</td><td>3.30</td></tr><tr><td>+GLOBALMEAN</td><td>9.46</td><td>5.30</td><td>24.81</td><td>14.03</td><td>4.98</td><td>2.46</td><td>9.76</td><td>5.33</td><td>10.55</td><td>5.55</td><td>6.31</td><td>3.15</td></tr><tr><td>+AUTOEnc</td><td>9.34</td><td>5.02</td><td>24.76</td><td>13.95</td><td>5.01</td><td>2.48</td><td>9.45</td><td>5.09</td><td>10.13</td><td>5.33</td><td>6.30</td><td>3.13</td></tr><tr><td rowspan="4">FREEDOM</td><td>+GRAPHAUTOEnc</td><td><u>10.11</u></td><td><u>5.46</u></td><td>24.88</td><td>13.70</td><td>5.63</td><td>2.80</td><td><b>10.43</b></td><td><b>5.68</b></td><td>10.87</td><td><b>5.74</b></td><td>6.54</td><td>3.30</td></tr><tr><td>+NEIGHMEAN</td><td>9.43</td><td>5.20</td><td>24.77</td><td>13.69</td><td>4.99</td><td>2.48</td><td>10.10</td><td>5.45</td><td>9.80</td><td>5.08</td><td><b>6.71</b></td><td><b>3.42</b></td></tr><tr><td>+MULTIHOP</td><td>10.09</td><td>5.41</td><td><u>26.03</u></td><td><u>14.56</u></td><td>5.68</td><td>2.81</td><td>10.31</td><td>5.61</td><td>10.83</td><td><u>5.73</u></td><td>6.56</td><td>3.38</td></tr><tr><td>+PERSPAGE RANK</td><td>9.47</td><td>5.23</td><td><b>26.25</b></td><td><b>14.58</b></td><td><u>5.83</u></td><td><u>2.95</u></td><td><u>10.36</u></td><td><u>5.63</u></td><td><u>10.91</u></td><td><b>5.74</b></td><td><b>6.66</b></td><td><b>3.40</b></td></tr><tr><td></td><td>+HEAT</td><td><b>10.21</b></td><td><b>5.64</b></td><td>25.23</td><td>14.39</td><td><b>5.90</b></td><td><b>3.03</b></td><td>10.35</td><td><b>5.63</b></td><td><b>10.92</b></td><td>5.71</td><td>6.65</td><td>3.39</td></tr><tr><td rowspan="4">BM3</td><td>+ZEROS</td><td>11.10</td><td>5.87</td><td>25.41</td><td>13.57</td><td>7.30</td><td>3.73</td><td>9.66</td><td>5.18</td><td>10.65</td><td>5.54</td><td>7.48</td><td>3.94</td></tr><tr><td>+RANDOM</td><td>12.30</td><td>6.53</td><td>26.49</td><td>14.72</td><td>8.37</td><td>4.30</td><td>12.71</td><td>6.63</td><td>13.12</td><td>6.87</td><td>9.00</td><td>4.66</td></tr><tr><td>+GLOBALMEAN</td><td><u>12.36</u></td><td><u>6.63</u></td><td><b>26.84</b></td><td><b>14.92</b></td><td>8.25</td><td>4.23</td><td>12.52</td><td>6.53</td><td>13.08</td><td>6.80</td><td>8.89</td><td>4.63</td></tr><tr><td>+AUTOEnc</td><td><b>12.42</b></td><td>6.51</td><td>25.99</td><td>14.20</td><td>7.51</td><td>3.81</td><td>9.46</td><td>5.08</td><td>12.00</td><td>6.32</td><td>7.61</td><td>4.02</td></tr><tr><td></td><td>+GRAPHAUTOEnc</td><td>12.23</td><td>6.42</td><td><u>26.73</u></td><td>14.50</td><td>7.60</td><td>3.87</td><td>9.96</td><td>5.34</td><td>12.11</td><td>6.45</td><td>7.65</td><td>4.01</td></tr><tr><td rowspan="4">MGCN</td><td>+NEIGHMEAN</td><td>12.30</td><td>6.45</td><td>26.57</td><td>14.55</td><td><u>8.63</u></td><td><u>4.42</u></td><td>12.69</td><td>6.67</td><td>13.23</td><td>6.88</td><td>8.96</td><td>4.63</td></tr><tr><td>+MULTIHOP</td><td>12.31</td><td><b>6.68</b></td><td>26.53</td><td>14.48</td><td>8.54</td><td>4.39</td><td>12.86</td><td><u>6.78</u></td><td><b>13.59</b></td><td><b>7.14</b></td><td><b>9.11</b></td><td><b>4.71</b></td></tr><tr><td>+PERSPAGE RANK</td><td>12.01</td><td>6.56</td><td>26.69</td><td>14.72</td><td>8.34</td><td>4.36</td><td><u>12.94</u></td><td><b>6.79</b></td><td>13.28</td><td>6.98</td><td><b>9.26</b></td><td><b>4.75</b></td></tr><tr><td>+HEAT</td><td>12.25</td><td>6.48</td><td>26.52</td><td>14.43</td><td><b>8.68</b></td><td><b>4.44</b></td><td><b>13.01</b></td><td>6.77</td><td><u>13.40</u></td><td><b>7.00</b></td><td>8.96</td><td>4.67</td></tr><tr><td rowspan="4">MMSSL</td><td>+ZEROS</td><td>11.31</td><td>5.93</td><td>24.64</td><td>12.99</td><td>7.64</td><td>3.87</td><td>9.75</td><td>5.02</td><td>11.31</td><td>5.92</td><td>8.43</td><td>4.43</td></tr><tr><td>+RANDOM</td><td>10.76</td><td>5.80</td><td>24.29</td><td>12.93</td><td>7.53</td><td>3.84</td><td>9.64</td><td>5.07</td><td>11.43</td><td>5.94</td><td>8.49</td><td><b>4.45</b></td></tr><tr><td>+GLOBALMEAN</td><td>10.93</td><td>5.79</td><td>24.27</td><td>12.88</td><td>7.70</td><td>3.94</td><td>9.84</td><td>5.13</td><td>11.37</td><td>5.87</td><td><u>8.56</u></td><td><b>4.45</b></td></tr><tr><td>+AUTOEnc</td><td>10.92</td><td>5.87</td><td><b>25.04</b></td><td><b>13.48</b></td><td>7.86</td><td>3.99</td><td>9.98</td><td>5.24</td><td>11.30</td><td>5.92</td><td>8.25</td><td>4.40</td></tr><tr><td></td><td>+GRAPHAUTOEnc</td><td>10.86</td><td>5.92</td><td>24.53</td><td>13.00</td><td>7.85</td><td><u>4.00</u></td><td><b>10.17</b></td><td><u>5.26</u></td><td>11.41</td><td>5.95</td><td>8.52</td><td>4.45</td></tr><tr><td rowspan="4">DiffMM</td><td>+NEIGHMEAN</td><td>10.91</td><td>5.71</td><td>24.46</td><td>12.95</td><td>7.62</td><td>3.91</td><td>9.77</td><td>5.09</td><td>11.43</td><td>5.94</td><td>8.36</td><td>4.35</td></tr><tr><td>+MULTIHOP</td><td><b>11.40</b></td><td>5.60</td><td>24.67</td><td>13.14</td><td>7.87</td><td><b>4.01</b></td><td>9.87</td><td>5.17</td><td>11.49</td><td><b>6.04</b></td><td>8.46</td><td>4.42</td></tr><tr><td>+PERSPAGE RANK</td><td>11.20</td><td><u>5.94</u></td><td>23.99</td><td>13.10</td><td><b>7.89</b></td><td><b>4.01</b></td><td>9.79</td><td>5.12</td><td>11.31</td><td>5.92</td><td>8.55</td><td>4.44</td></tr><tr><td>+HEAT</td><td><u>11.35</u></td><td><b>6.04</b></td><td>24.41</td><td>12.97</td><td><b>7.89</b></td><td><b>4.01</b></td><td><u>10.12</u></td><td><b>5.27</b></td><td><b>11.51</b></td><td><b>6.00</b></td><td><b>8.63</b></td><td><b>4.49</b></td></tr><tr><td rowspan="4">DiffMM</td><td>+ZEROS</td><td>11.17</td><td>5.80</td><td>20.90</td><td>10.97</td><td>5.64</td><td>2.79</td><td>5.51</td><td>2.88</td><td>10.77</td><td>5.92</td><td>5.13</td><td>2.75</td></tr><tr><td>+RANDOM</td><td>11.64</td><td>6.06</td><td>21.16</td><td>11.75</td><td>8.28</td><td>4.24</td><td>11.93</td><td>6.35</td><td>12.69</td><td>6.83</td><td>8.31</td><td>4.34</td></tr><tr><td>+GLOBALMEAN</td><td>11.45</td><td>6.12</td><td>22.00</td><td>11.89</td><td>8.20</td><td>4.20</td><td>11.91</td><td>6.30</td><td>12.62</td><td>6.72</td><td>8.29</td><td>4.36</td></tr><tr><td>+AUTOEnc</td><td>11.76</td><td>6.21</td><td>23.27</td><td>12.91</td><td>5.93</td><td>2.98</td><td>7.47</td><td>4.01</td><td>12.21</td><td>6.67</td><td>5.70</td><td>2.99</td></tr><tr><td></td><td>+GRAPHAUTOEnc</td><td>11.38</td><td>6.05</td><td>23.68</td><td>13.09</td><td>6.15</td><td>3.12</td><td>7.19</td><td>3.88</td><td>12.22</td><td>6.71</td><td>5.65</td><td>3.01</td></tr><tr><td rowspan="4">DiffMM</td><td>+NEIGHMEAN</td><td><b>11.81</b></td><td>6.20</td><td>22.43</td><td>12.07</td><td>8.21</td><td>4.21</td><td><b>12.40</b></td><td><b>6.62</b></td><td>12.54</td><td>6.74</td><td>8.54</td><td>4.45</td></tr><tr><td>+MULTIHOP</td><td>11.79</td><td>6.13</td><td><u>24.15</u></td><td><u>13.28</u></td><td>8.40</td><td><b>4.28</b></td><td><u>12.35</u></td><td>6.59</td><td>12.63</td><td>6.80</td><td>8.53</td><td>4.49</td></tr><tr><td>+PERSPAGE RANK</td><td>11.41</td><td><u>6.24</u></td><td>23.46</td><td>12.93</td><td><b>8.44</b></td><td><b>4.26</b></td><td>12.33</td><td><u>6.60</u></td><td><b>12.76</b></td><td><b>6.89</b></td><td><b>8.55</b></td><td><b>4.50</b></td></tr><tr><td>+HEAT</td><td>11.63</td><td><b>6.25</b></td><td><b>24.60</b></td><td><b>13.41</b></td><td>8.26</td><td>4.21</td><td>12.17</td><td>6.48</td><td><u>12.73</u></td><td><b>6.88</b></td><td><b>8.59</b></td><td><b>4.51</b></td></tr><tr><td rowspan="4">DiffMM</td><td>+ZEROS</td><td>13.40</td><td><u>11.36</u></td><td>28.83</td><td>21.42</td><td>8.84</td><td>6.55</td><td><u>12.06</u></td><td>8.65</td><td><u>13.76</u></td><td>10.05</td><td>9.11</td><td>7.00</td></tr><tr><td>+RANDOM</td><td>12.37</td><td>10.83</td><td>28.03</td><td>20.70</td><td>8.52</td><td>6.35</td><td>11.54</td><td>8.47</td><td>13.36</td><td>9.78</td><td>8.93</td><td>6.93</td></tr><tr><td>+GLOBALMEAN</td><td>13.07</td><td>11.00</td><td>28.89</td><td>21.55</td><td>8.86</td><td>6.58</td><td>12.00</td><td>8.68</td><td>13.58</td><td>9.87</td><td>9.21</td><td>7.09</td></tr><tr><td>+AUTOEnc</td><td><b>13.50</b></td><td><b>11.38</b></td><td>28.85</td><td>21.47</td><td>8.83</td><td>6.58</td><td>11.73</td><td>8.55</td><td>13.61</td><td>9.98</td><td>9.05</td><td>7.01</td></tr><tr><td></td><td>+GRAPHAUTOEnc</td><td>13.22</td><td>10.93</td><td>28.85</td><td>21.41</td><td><b>8.95</b></td><td><b>6.65</b></td><td>11.97</td><td>8.64</td><td>13.72</td><td>10.06</td><td>9.17</td><td><u>7.12</u></td></tr><tr><td rowspan="4">DiffMM</td><td>+NEIGHMEAN</td><td>13.29</td><td>11.31</td><td>28.94</td><td>21.48</td><td>8.91</td><td>6.62</td><td><u>12.06</u></td><td>8.73</td><td>13.69</td><td>10.07</td><td>9.17</td><td>7.07</td></tr><tr><td>+MULTIHOP</td><td>13.16</td><td>11.11</td><td>29.28</td><td><b>21.92</b></td><td>8.90</td><td><u>6.63</u></td><td>12.05</td><td><b>8.74</b></td><td>13.72</td><td><u>10.08</u></td><td><b>9.26</b></td><td><b>7.14</b></td></tr><tr><td>+PERSPAGE RANK</td><td>13.39</td><td>11.31</td><td><u>29.33</u></td><td>21.60</td><td>8.86</td><td>6.59</td><td><b>12.13</b></td><td><b>8.73</b></td><td>13.72</td><td>10.01</td><td>9.22</td><td>7.10</td></tr><tr><td>+HEAT</td><td>13.48</td><td>11.29</td><td><b>29.35</b></td><td><u>21.67</u></td><td>8.80</td><td>6.55</td><td>12.04</td><td>8.69</td><td><b>13.77</b></td><td><b>10.09</b></td><td><b>9.27</b></td><td><b>7.14</b></td></tr><tr><td rowspan="4">DiffMM</td><td>+ZEROS</td><td><b>13.34</b></td><td><u>7.36</u></td><td>27.83</td><td>15.97</td><td>8.69</td><td>4.52</td><td>11.59</td><td>6.23</td><td>13.04</td><td>7.03</td><td>9.23</td><td>4.89</td></tr><tr><td>+RANDOM</td><td>12.85</td><td>7.34</td><td>28.33</td><td><b>16.61</b></td><td>8.65</td><td><b>4.53</b></td><td>11.77</td><td>6.23</td><td>13.27</td><td>7.12</td><td>9.06</td><td>4.82</td></tr><tr><td>+GLOBALMEAN</td><td><u>13.20</u></td><td><b>7.42</b></td><td>28.02</td><td>16.17</td><td>8.61</td><td>4.47</td><td>11.42</td><td>6.10</><td><b>13.35</b></td><td><b>7.32</b></td><td>8.92</td><td>4.68</td></tr><tr><td>+AUTOEnc</td><td>13.17</td><td>7.31</td><td>28.04</td><td>16.17</td><td>8.71</td><td>4.51</td><td>11.65</td><td>6.19</td><td>13.25</td><td>7.16</td><td>9.26</td><td>4.88</td></tr><tr><td rowspan="4">DiffMM</td><td>+GRAPHAUTOEnc</td><td>12.79</td><td>7.19</td><td>28.37</td><td>16.27</td><td>8.67</td><td>4.50</td><td><b>12.00</b></td><td><b>6.37</b></td><td>13.08</td><td>7.03</td><td>9.18</td><td>4.82</td></tr><tr><td>+NEIGHMEAN</td><td>12.89</td><td>7.04</td><td><u>28.50</u></td><td>16.31</td><td>8.68</td><td><u>4.52</u></td><td>11.77</td><td>6.25</><td><u>13.31</u></td><td><u>7.19</u></td><td>9.25</td><td><b>4.90</b></td></tr><tr><td>+MULTIHOP</td><td>12.69</td><td>6.98</td><td><b>28.73</b></td><td>16.38</td><td><u>8.72</u></td><td>4.50</td><td>11.82</td><td>6.30</td><td>13.24</td><td>7.12</td><td><b>9.29</b></td><td><b>4.90</b></td></tr><tr><td>+PERSPAGE RANK</td><td>12.83</td><td>6.99</td><td>28.25</td><td>16.30</td><td><b>8.81</b></td><td>4.49</td><td>11.93</td><td><u>6.34</u></td><td>13.25</td><td>7.11</td><td>9.26</td><td>4.88</td></tr><tr><td></td><td>+HEAT</td><td>12.60</td><td>6.93</td><td>28.49</td><td><u>16.46</u></td><td>8.66</td><td>4.50</td><td>11.92</td><td>6.33</td><td>13.27</td><td>7.13</td><td><u>9.28</u></td><td><b>4.90</b></td></tr></tbody></table>

homophily proposed in [38]. Specifically, we adapt its formulation to our case, where it is calculated for each modality in the system. Note that we consider the feature homophily only

in the *dropped* setting as we need all available multimodal features to calculate it. The measure is high when close nodes in the graph tend to have similar features, and low otherwise.
<figure><img src="image_4.png" /><figcaption>Fig. 3: Performance variation for (a) VBPR and (b) MGCN with the HEAT imputation method by considering different sparsification rates and propagation hops on the Sports dataset.</figcaption></figure>

Figure 4 depicts the performance variation for VBPR and MGCN on Sports dataset, equipped with HEAT imputation, considering various levels of feature homophily (x-axis) and TopN sparsification (y-axis). The study is conducted on the Visual and Textual modalities separately, as we calculate the feature homophily on each of them.

The first two observable trends suggest that: (i) in general, feature homophily is very low and limited to the range [0, 0.20]; (ii) TopN sparsification and feature homophily are indirectly correspondent, as higher TopN values (so retaining more and low-quality neighbors of each node) inevitably lead to lower feature homophily and vice-versa. Indeed, observation (i) is confirmed by the outcomes from [38], highlighting that the user-item graph in recommendation shows, in most cases, low levels of feature homophily.

Then, by considering the overall plots, we see again different trends between VBPR and MGCN. Regarding VBPR, we recognize higher recommendation performance coinciding with high levels of feature homophily, which seems to be aligned with what observed by the authors in [36]. Differently, on MGCN, we clearly notice higher recommendation performance for lower levels of homophily. Once again, this behavior might be ascribed to MGCN working with the user-item graph, whose perspective is strictly related to that of the item-item co-purchase graph we use for the graph-based imputations. Thus, as such imputations work by supporting the graph-based multimodal recommender systems in exploring vast portions of the user-item graph, they benefit from higher values of TopN and, consequently, lower levels of feature homophily. That is, feature homophily might be less impacting than for other multimodal recommender systems.

In conclusion, this study confirms the importance of hyper-parameter selection for the proposed graph-based approaches

<figure><img src="image_5.png" /><figcaption>Fig. 4: Performance variation (Recall) on Sports for (a) VBPR and (b) MGCN and with HEAT imputation. Specifically, we display how the Recall changes with different levels of feature homophily (x-axis) and item-item sparsification (y-axis).</figcaption></figure>

depending on the specific setting (as reported in the previous RQ). Moreover, regarding the generally low values of feature homophily, it underlines the necessity of better understanding and assessing this aspect, especially in the case this is preventing our solutions from providing even better performance. Thus, we will devote future work to the outlined direction.

### E. RQ5) Impact of missing features percentage for traditional, autoencoder-, and graph-based imputations

This analysis is devoted to assessing the recommendation performance variation when considering selected imputation methods with different percentages of missing modalities. Specifically, we select FREEDOM and LGMRec as multimodal recommendation models, equipped with GLOBALMEAN, AUTOENC, and HEAT imputation methods, on the Baby recommendation dataset. Then, we simulate five scenarios where, either on the Visual, the Textual, or the Visual + Textual modalities, the 10%, 50%, or 90% of items have missing features. For each of these configurations, we repeat the items sampling five times to ensure independence from the single-conducted sampling. Results are reported in Table V in terms of Recall@20, where the first performance column refers to the Recall in the setting with all complete modalities, corresponding to our *dropped* setting (where no item has missing multimodal features).

Overall, results show that the graph-based imputation method (HEAT) is the approach that performs the best in most cases; this becomes especially evident on LGMRec, where
<table><caption>TABLE V: Performance variation (Recall) of FREEDOM and LGMRec (equipped with GLOBALMEAN, AUTOENC, and HEAT imputations) on the Baby dataset, with different percentages of missing features (i.e., Complete, 10%, 50%, and 90%) on the Visual, Textual, and Visual + Textual modalities.</caption><thead><tr><th colspan="5">FREEDOM</th><th colspan="5">LGMRec</th></tr><tr><th rowspan="2">Imputation</th><th colspan="4">Visual</th><th rowspan="2">Imputation</th><th colspan="4">Visual</th></tr><tr><th>Complete</th><th>10%</th><th>50%</th><th>90%</th><th>Complete</th><th>10%</th><th>50%</th><th>90%</th></tr></thead><tbody><tr><td>+GLOBALMEAN</td><td>8.95</td><td>8.54±0.10</td><td><strong>8.60±0.09</strong></td><td>8.47±0.09</td><td>+GLOBALMEAN</td><td>8.87</td><td>8.83±0.11</td><td>8.54±0.20</td><td>8.18±0.05</td></tr><tr><td>+AUTOENC</td><td>8.95</td><td>8.60±0.15</td><td>8.50±0.11</td><td><strong>8.50±0.12</strong></td><td>+AUTOENC</td><td>8.87</td><td>8.85±0.16</td><td>8.76±0.16</td><td><strong>8.49±0.14</strong></td></tr><tr><td>+HEAT</td><td>8.95</td><td><strong>8.63±0.10</strong></td><td>8.50±0.07</td><td><strong>8.50±0.05</strong></td><td>+HEAT</td><td>8.87</td><td><strong>8.91±0.07</strong></td><td><strong>8.82±0.06</strong></td><td>8.48±0.04</td></tr><tr><th rowspan="2">Imputation</th><th colspan="4">Textual</th><th rowspan="2">Imputation</th><th colspan="4">Textual</th></tr><tr><th>Complete</th><th>10%</th><th>50%</th><th>90%</th><th>Complete</th><th>10%</th><th>50%</th><th>90%</th></tr><tr><td>+GLOBALMEAN</td><td>8.95</td><td>8.25±0.05</td><td>7.46±0.20</td><td>7.43±0.12</td><td>+GLOBALMEAN</td><td>8.87</td><td>8.74±0.12</td><td>8.19±0.15</td><td>7.59±0.17</td></tr><tr><td>+AUTOENC</td><td>8.95</td><td><strong>8.46±0.21</strong></td><td><strong>7.92±0.08</strong></td><td><strong>7.70±0.11</strong></td><td>+AUTOENC</td><td>8.87</td><td>8.81±0.07</td><td>8.25±0.18</td><td>7.40±0.05</td></tr><tr><td>+HEAT</td><td>8.95</td><td>8.33±0.11</td><td>7.59±0.10</td><td>7.19±0.17</td><td>+HEAT</td><td>8.87</td><td><strong>8.82±0.05</strong></td><td><strong>8.36±0.14</strong></td><td><strong>7.96±0.15</strong></td></tr><tr><th rowspan="2">Imputation</th><th colspan="4">Visual + Textual</th><th rowspan="2">Imputation</th><th colspan="4">Visual + Textual</th></tr><tr><th>Complete</th><th>10%</th><th>50%</th><th>90%</th><th>Complete</th><th>10%</th><th>50%</th><th>90%</th></tr><tr><td>+GLOBALMEAN</td><td>8.95</td><td>8.23±0.10</td><td>7.24±0.18</td><td>6.88±0.09</td><td>+GLOBALMEAN</td><td>8.87</td><td>8.52±0.14</td><td>7.49±0.21</td><td><strong>6.36±0.28</strong></td></tr><tr><td>+AUTOENC</td><td>8.95</td><td>7.16±0.15</td><td>6.96±0.14</td><td>6.88±0.13</td><td>+AUTOENC</td><td>8.87</td><td>8.62±0.09</td><td>7.63±0.37</td><td>5.80±0.13</td></tr><tr><td>+HEAT</td><td>8.95</td><td><strong>8.30±0.07</strong></td><td><strong>7.48±0.12</strong></td><td><strong>6.99±0.08</strong></td><td>+HEAT</td><td>8.87</td><td><strong>8.68±0.16</strong></td><td><strong>7.83±0.21</strong></td><td>6.24±0.14</td></tr></tbody></table>

HEAT is able to preserve recommendation performance even for higher percentages of missing modalities (e.g., see the Textual modality setting for 90% of missing information, or the Visual + Textual modality setting for 50% of missing information). The exception to the observed trend is the Textual modality for FREEDOM, where the autoencoder-based imputation (AUTOENC) is steadily the best. However, if we consider those scenarios with the percentage of missingness limited to 50% (which are the most realistic ones according to Table I, see the last column), our method settles as the best or second-to-best solution even in the Textual modality.

To summarise, graph-based imputation approaches demonstrate very high performance in most settings with various percentages of missing modalities, and in realistic scenarios, they still provide sufficiently improved results. Nevertheless, we will devote future directions of our work to delving into the results from the Textual missingness setting, which could represent an area for further investigation.

### F. RQ6) Impact of missing features on multimodal scenarios with more than two modalities

We ultimately examine a multimodal recommendation scenario involving three modalities. In particular, we select the recent MicroLens dataset<fnref n="2" /> [72]. Unlike the Amazon datasets we used in the previous RQs, MicroLens comes with 3 modalities: visual, textual, and video. Starting from the original MicroLens-100K version, we apply an iterative k-core algorithm (with k = 7) to retain users and items with the most interactions, resulting in the following statistics: 7,556 users, 4,628 items, and 81,505 interactions. Then, we simulate the missingness of modalities for 30% of the items in the catalog.

Table VI reports results of MGCN when imputing missing modalities with two traditional strategies (i.e., RANDOM and GLOBALMEAN) and two graph-based techniques (i.e., MULTIHOP and PERSPAGERANK). As previously observed in our

<table><caption>TABLE VI: Impact of traditional and graph-based imputation methods on the performance of MGCN for the MicroLens dataset having 3 modalities with 30% of missing modalities.</caption><thead><tr><th>Imputation</th><th>Recall</th><th>nDCG</th></tr></thead><tbody><tr><td>+RANDOM</td><td>23.28±0.28</td><td>10.48±0.14</td></tr><tr><td>+GLOBALMEAN</td><td>23.38±0.35</td><td>10.55±0.18</td></tr><tr><td>+MULTIHOP</td><td>23.41±0.37</td><td>10.45±0.22</td></tr><tr><td>+PERSPAGERANK</td><td><strong>23.45±0.46</strong></td><td><strong>10.60±0.25</strong></td></tr></tbody></table>

analyses, this further confirms the efficacy of our proposed graph-based imputations, which can also effectively operate in multimodal scenarios with more than two modalities.

## VII. CONCLUSION AND FUTURE WORK

In this paper, we tackled the problem of missing modalities in multimodal recommendation, which has found very limited attention in the recent literature. First, we formalised (for the first time) the issue by outlining differences with the problem of missing information in machine learning. Then, we proposed to leverage the item-item co-purchase graph to re-sketch the problem of missing modalities as a task of graph feature interpolation. Our solutions act as model-agnostic techniques applicable to any existing multimodal recommender systems without affecting their training (they run during the pre-processing phase), and are completely training-free. Extensive experiments on 7 datasets (with either 2 or 3 modalities), 14 (multimodal) recommenders, and against 5 imputation baselines (traditional and autoencoder-based imputation) outlined important aspects of our approaches. First, the graph-based imputations can preserve or even widen the performance gap between traditional and multimodal recommender systems. Second, the graph-based solutions largely outperform the other imputation strategies. Moreover, further analyses demonstrate that our approaches need careful hyperparameter selection to adapt to the specific multimodal recommender system. Despite their largely-demonstrated efficacy, final results regarding the impact of feature homophily and the percentage of missing

<footnote n="2">²https://github.com/westlake-repl/MicroLens.</footnote>
features on each modality underline some performance limitations in our approaches, paving the way for interesting future directions of the current work. Among others, we plan to redesign our graph-based imputation strategies with more refined sparsifications [39] and integrate them into the end-to-end recommendation pipeline with the multimodal recommendation model. This would ensure that the framework exploits the information-rich item-item similarity graph as done in the related literature [17], [21], [73], along with ad-hoc modality alignment constraints to provide further guidance signal to the imputation of missing modalities [18], [19], [21], [47], and works effectively in cold-start/noisy scenarios.

## ACKNOWLEDGMENTS

D.M. and F.D.M. acknowledge the support of the Innov4-ePiK project managed by the French National Research Agency under the 4th PIA, integrated into France2030 (ANR-23-RHUS-0002). C.P. and T.D.N. acknowledge the support of the P+ARTS project funded by the European Union Next-GenerationEU (NRRP – M4C1, Investment 3.4, INTAFAM00037, CUP: G43C24000640006). The authors acknowledge Inria, Mesocentre, and the CINECA (ISCRA initiative) for high-performance computing resources.

## REFERENCES

[1] R. He and J. J. McAuley, "VBPR: visual bayesian personalized ranking from implicit feedback," in AAAI, 2016.

[2] D. Malitesta, G. Cornacchia, C. Pomo, F. A. Merra, T. D. Noia, and E. D. Sciascio, "Formalizing multimedia recommendation through multimodal deep learning," *Trans. Recomm. Syst.*, vol. 3, no. 3, pp. 37:1-37:33, 2025.

[3] Q. Liu, J. Hu, Y. Xiao, X. Zhao, J. Gao, W. Wang, Q. Li, and J. Tang, "Multimodal recommender systems: A survey," *ACM Comput. Surv.*, vol. 57, no. 2, pp. 26:1-26:17, 2025.

[4] H. Zhou, X. Zhou, Z. Zeng, L. Zhang, and Z. Shen, "A comprehensive survey on multimodal recommender systems: Taxonomy, evaluation, and future directions," *CoRR*, vol. abs/2302.04473, 2023.

[5] J. Xu, Z. Chen, S. Yang, J. Li, W. Wang, X. Hu, S. Hoi, and E. C. H. Ngai, "A survey on multimodal recommender systems: Recent advances and future directions," *CoRR*, vol. abs/2502.15711, 2025.

[6] Q. Liu, J. Zhu, Y. Yang, Q. Dai, Z. Du, X. Wu, Z. Zhao, R. Zhang, and Z. Dong, "Multimodal pretraining, adaptation, and generation for recommendation: A survey," in *KDD*, 2024.

[7] L. Pan, Z. Pan, F. Cai, and H. Chen, "Multimodal recommender systems: A survey of representation, modeling, and optimization," *Inf. Fusion*, vol. 128, p. 103991, 2026.

[8] X. Chen, H. Chen, H. Xu, Y. Zhang, Y. Cao, Z. Qin, and H. Zha, "Personalized fashion recommendation with visual explanations based on multimodal attention network: Towards visually explainable recommendation," in *SIGIR*, 2019.

[9] S. Oramas, O. Nieto, M. Sordo, and X. Serra, "A deep multimodal approach for cold-start music recommendation," in *DLRS@RecSys*, 2017.

[10] W. Min, S. Jiang, and R. C. Jain, "Food recommendation: Framework, existing solutions, and challenges," *IEEE Trans. Multim.*, 2020.

[11] D. Cai, S. Qian, Q. Fang, and C. Xu, "Heterogeneous hierarchical feature aggregation network for personalized micro-video recommendation," *IEEE Trans. Multim.*, vol. 24, pp. 805-818, 2022.

[12] D. Malitesta, E. Rossi, C. Pomo, T. D. Noia, and F. D. Malliaros, "Do we really need to drop items with missing modalities in multimodal recommendation?" in *CIKM*, 2024.

[13] X. Wang, X. He, M. Wang, F. Feng, and T. Chua, "Neural graph collaborative filtering," in *SIGIR*, 2019.

[14] Y. Wei, X. Wang, L. Nie, X. He, R. Hong, and T. Chua, "MMGCN: multi-modal graph convolution network for personalized recommendation of micro-video," in *ACM Multimedia*, 2019.

[15] Y. Wei, X. Wang, L. Nie, X. He, and T. Chua, "Graph-refined convolutional network for multimedia recommendation with implicit feedback," in *ACM Multimedia*, 2020.

[16] J. Zhang, Y. Zhu, Q. Liu, S. Wu, S. Wang, and L. Wang, "Mining latent structures for multimedia recommendation," in *ACM Multimedia*, 2021.

[17] X. Zhou and Z. Shen, "A tale of two graphs: Freezing and denoising graph structures for multimodal recommendation," in *ACM Multimedia*, 2023.

[18] X. Zhou, H. Zhou, Y. Liu, Z. Zeng, C. Miao, P. Wang, Y. You, and F. Jiang, "Bootstrap latent representations for multi-modal recommendation," in *WWW*, 2023.

[19] W. Wei, C. Huang, L. Xia, and C. Zhang, "Multi-modal self-supervised learning for recommendation," in *WWW*, 2023.

[20] Z. Guo, J. Li, G. Li, C. Wang, S. Shi, and B. Ruan, "Lgmrec: Local and global graph learning for multimodal recommendation," in *AAAI*, 2024.

[21] J. Xu, Z. Chen, S. Yang, J. Li, H. Wang, and E. C. H. Ngai, "MENTOR: multi-level self-supervised learning for multimodal recommendation," in *AAAI*, 2025.

[22] T. Emmanuel, T. M. Maupong, D. Mpoeleng, T. Semong, B. Mphago, and O. Tabona, "A survey on missing data in machine learning," *J. Big Data*, vol. 8, no. 1, p. 140, 2021.

[23] H. Steck, "Training and testing of recommender systems on data missing not at random," in *KDD*, 2010.

[24] M. Wang, M. Gong, X. Zheng, and K. Zhang, "Modeling dynamic missingness of implicit feedback for recommendation," in *NeurIPS*, 2018.

[25] X. Zheng, M. Wang, R. Xu, J. Li, and Y. Wang, "Modeling dynamic missingness of implicit feedback for sequential recommendation," *IEEE Trans. Knowl. Data Eng.*, vol. 34, no. 1, pp. 405-418, 2022.

[26] S. Shi, M. Zhang, X. Yu, Y. Zhang, B. Hao, Y. Liu, and S. Ma, "Adaptive feature sampling for recommendation with missing content feature values," in *CIKM*, 2019.

[27] F. Liu, Z. Cheng, L. Zhu, C. Liu, and L. Nie, "An attribute-aware attentive GCN model for attribute missing in recommendation," *IEEE Trans. Knowl. Data Eng.*, vol. 34, no. 9, pp. 4077-4088, 2022.

[28] M. Ma, J. Ren, L. Zhao, S. Tulyakov, C. Wu, and X. Peng, "SMIL: multimodal learning with severely missing modality," in *AAAI*, 2021.

[29] Y. Lee, Y. Tsai, W. Chiu, and C. Lee, "Multimodal prompting with missing modalities for visual recognition," in *CVPR*, IEEE, 2023.

[30] C. Zhang, X. Chu, L. Ma, Y. Zhu, Y. Wang, J. Wang, and J. Zhao, "M3care: Learning with missing modalities in multimodal healthcare data," in *KDD*, 2022.

[31] C. Wang, M. Niepert, and H. Li, "LRMM: learning to recommend with missing modalities," in *EMNLP*, 2018.

[32] J. Yi and Z. Chen, "Multi-modal variational graph auto-encoder for recommendation systems," *IEEE Trans. Multim.*, 2022.

[33] Z. Lin, Y. Tan, Y. Zhan, W. Liu, F. Wang, C. Chen, S. Wang, and C. Yang, "Contrastive intra- and inter-modality generation for enhancing incomplete multimedia recommendation," in *ACM Multimedia*, 2023.

[34] C. Ganhör, M. Moscati, A. Hausberger, S. Nawaz, and M. Schedl, "A multimodal single-branch embedding network for recommendation in cold-start and missing modality scenarios," in *RecSys*, 2024.

[35] H. Bai, L. Wu, M. Hou, M. Cai, Z. He, Y. Zhou, R. Hong, and M. Wang, "Multimodality invariant learning for multimedia-based new item recommendation," in *SIGIR*, 2024.

[36] E. Rossi, H. Kenlay, M. I. Gorinova, B. P. Chamberlain, X. Dong, and M. M. Bronstein, "On the unreasonable effectiveness of feature propagation in learning on graphs with missing node features," in *LoG*, 2022.

[37] X. Zhou, "Mmrec: Simplifying multimodal recommendation," in *MMA-sia (Workshops)*, 2023.

[38] J. Zhu, X. Song, V. N. Ioannidis, D. Koura, and C. Faloutsos, "Touchupg: Improving feature representation through graph-centric finetuning," in *SIGIR*, 2024.

[39] G. Xv, C. Lin, W. Guan, J. Gou, X. Li, H. Deng, J. Xu, and B. Zheng, "E-commerce search via content collaborative graph neural network," in *KDD*, 2023.

[40] T. Baltrusaitis, C. Ahuja, and L. Morency, "Challenges and applications in multimodal machine learning," in *The Handbook of Multimodal-Multisensor Interfaces, Volume 2 (2)*, 2018.

[41] X. He, K. Deng, X. Wang, Y. Li, Y. Zhang, and M. Wang, "Lightgcn: Simplifying and powering graph convolution network for recommendation," in *SIGIR*, 2020.

[42] Q. Wang, Y. Wei, J. Yin, J. Wu, X. Song, and L. Nie, "Dualgnn: Dual graph neural network for multimedia recommendation," *IEEE Trans. Multim.*, vol. 25, pp. 1074-1084, 2023.

[43] T. Kim, Y. Lee, K. Shin, and S. Kim, "MARIO: modality-aware attention and modality-preserving decoders for multimedia recommendation," in *CIKM*, 2022.
VIII. BIOGRAPHY SECTION

[44] Y. Liu, S. Yang, C. Lei, G. Wang, H. Tang, J. Zhang, A. Sun, and C. Miao, "Pre-training graph transformer with multimodal side information for recommendation," in *ACM Multimedia*, 2021.

[45] H. Zhou, X. Zhou, L. Zhang, and Z. Shen, "Enhancing dyadic relations with homogeneous graphs for multimodal recommendation," in *ECAI*, 2023.

[46] P. Yu, Z. Tan, G. Lu, and B. Bao, "Multi-view graph convolutional network for multimedia recommendation," in *ACM Multimedia*, 2023.

[47] Z. Tao, X. Liu, Y. Xia, X. Wang, L. Yang, X. Huang, and T. Chua, "Self-supervised learning for multimedia recommendation," *IEEE Trans. Multim.*, vol. 25, pp. 5107-5116, 2023.

[48] Z. Yi, X. Wang, I. Ounis, and C. MacDonald, "Multi-modal graph contrastive learning for micro-video recommendation," in *SIGIR*, 2022.

[49] H. Su, J. Li, F. Li, K. Lu, and L. Zhu, "SOIL: contrastive second-order interest learning for multimodal recommendation," in *ACM Multimedia*, 2024.

[50] X. Zhou and C. Miao, "Disentangled graph variational auto-encoder for multimodal recommendation with interpretability," *IEEE Trans. Multim.*, vol. 26, pp. 7543-7554, 2024.

[51] J. Yi and Z. Chen, "Variational mixture of stochastic experts auto-encoder for multi-modal recommendation," *IEEE Trans. Multim.*, vol. 26, pp. 8941-8954, 2024.

[52] Y. Jiang, L. Xia, W. Wei, D. Luo, K. Lin, and C. Huang, "Diffmm: Multi-modal diffusion model for recommendation," in *ACM Multimedia*, 2024.

[53] W. Wei, X. Ren, J. Tang, Q. Wang, L. Su, S. Cheng, J. Wang, D. Yin, and C. Huang, "Llmrec: Large language models with graph augmentation for recommendation," in *WSDM*, 2024.

[54] R. K. Ong and A. W. H. Khong, "Spectrum-based modality representation fusion graph convolutional network for multimodal recommendation," in *WSDM*, 2025.

[55] P. Yu, Z. Tan, G. Lu, and B. Bao, "Mind individual information! principal graph learning for multimedia recommendation," in *AAAI*, 2025.

[56] J. Wang, F. Yuan, M. Cheng, J. M. Jose, C. Yu, B. Kong, Z. Wang, B. Hu, and Z. Li, "Transrec: Learning transferable recommendation from mixture-of-modality feedback," in *APWeb/WAIM (2)*, 2024.

[57] W. Wei, J. Tang, L. Xia, Y. Jiang, and C. Huang, "Promptmm: Multi-modal knowledge distillation for recommendation with prompt-tuning," in *WWW*, 2024.

[58] G. Lin, Z. Meng, D. Wang, Q. Long, Y. Zhou, and M. Xiao, "GUME: graphs and user modalities enhancement for long-tail multimodal recommendation," in *CIKM*, 2024.

[59] W. E. Strawderman, "Statistical analysis with missing data (roderick j. a. little and donald b. rubin)," *SIAM Rev.*, vol. 31, no. 2, pp. 348-349, 1989.

[60] A. Takasu, "A multicriteria recommendation method for data with missing rating scores," in *ICDKE*, 2011.

[61] J. Chen, C. Wang, M. Ester, Q. Shi, Y. Feng, and C. Chen, "Social recommendation with missing not at random data," in *ICDM*, 2018.

[62] C. Vernade and O. Cappé, "Learning from missing data using selection bias in movie recommendation," in *DSAA*, 2015.

[63] K. He, X. Zhang, S. Ren, and J. Sun, "Deep residual learning for image recognition," in *CVPR*, 2016.

[64] J. Klicpera, S. Weißenberger, and S. Günnemann, "Diffusion improves graph learning," in *NeurIPS*, 2019.

[65] L. Page, S. Brin, R. Motwani, and T. Winograd, "The pagerank citation ranking: Bringing order to the web." Stanford InfoLab, Technical Report 1999-66, November 1999.

[66] R. Kondor and J. D. Lafferty, "Diffusion kernels on graphs and other discrete input spaces," in *ICML*, 2002.

[67] N. Reimers and I. Gurevych, "Sentence-bert: Sentence embeddings using siamese bert-networks," in *EMNLP/IJCNLP (1)*, 2019.

[68] S. Rendle, C. Freudenthaler, Z. Gantner, and L. Schmidt-Thieme, "BPR: bayesian personalized ranking from implicit feedback," in *UAI*, 2009.

[69] J. Wu, X. Wang, F. Feng, X. He, L. Chen, J. Lian, and X. Xie, "Self-supervised graph learning for recommendation," in *SIGIR*, 2021.

[70] T. N. Kipf and M. Welling, "Semi-supervised classification with graph convolutional networks," in *ICLR (Poster). OpenReview.net*, 2017.

[71] J. Zhu, Y. Yan, L. Zhao, M. Heimann, L. Akoglu, and D. Koutra, "Beyond homophily in graph neural networks: Current limitations and effective designs," in *NeurIPS*, 2020.

[72] Y. Ni, Y. Cheng, X. Liu, J. Fu, Y. Li, X. He, Y. Zhang, and F. Yuan, "A content-driven micro-video recommendation dataset at scale," in *CIKM*, 2025.

[73] H. Zhou, X. Zhou, and Z. Shen, "Enhancing dyadic relations with homogeneous graphs for multimodal recommendation," *ECAI*, 2023.

<figure><img src="image_6.png" /></figure>

**Daniele Malitesta** is a Postdoctoral researcher at Université Paris-Saclay. Previously, he pursued a PhD at Politecnico di Bari (Italy), where he focused on graph-based multimodal recommendation. His works appear in venues such as ECML-PKDD, The Web Conference, SIGIR, and RecSys. Recently, he has given lectures and tutorials on his past and current research topics, which mainly focus on fairness in graph generative models and missing multimodal information in graph machine learning.

<figure><img src="image_7.png" /></figure>

**Emaneuele Rossi** is a Machine Learning Researcher at Vant AI, where he develops generative models at the intersection of machine learning and biology. He holds a Ph.D. from Imperial College London, where he was supervised by Prof. Michael Bronstein and focused on Graph Neural Networks. His research appears at NeurIPS, ICML, ICLR, AAAI, and RecSys. Additionally, he has co-organized the Temporal Graph Learning Workshop at NeurIPS in both 2023 and 2024.

<figure><img src="image_8.png" /></figure>

**Claudio Pomo** is an assistant professor at Politecnico di Bari focused on responsible AI for personalization, reproducibility, and multi-objective evaluation. His work appears at SIGIR, RecSys, ECIR, UMAP, and in Information Science and IP&M; he co-led tutorials (RecSys 2021; LoG 2023), organized the EvalRS workshop at KDD 2023, chaired the RecSys Challenge 2024-2025, and lectured at the RecSys Summer School 2024.

<figure><img src="image_9.png" /></figure>

**Tommaso Di Noia** is a Professor of Computer Science at the Polytechnic University of Bari, Italy. His work spans AI and data management, evolving from knowledge representation and automated reasoning to automated negotiation and recommender systems with knowledge graphs and Linked Open Data. He now focuses on trustworthy AI—adversarial ML, explainability, fairness, and privacy in recommender systems—and served as program co-chair of RecSys 2023 and general co-chair of RecSys 2024.

<figure><img src="image_10.png" /></figure>

**Fragkiskos D. Malliaros** is a Professor at Université Paris-Saclay, CentraleSupélec, Inria. Previously, he was a postdoctoral researcher at UC San Diego (2016-17) and École Polytechnique (2015-16). He is the recipient of the 2012 Google European Doctoral Fellowship in Graph Mining, the 2015 Thesis Prize by École Polytechnique, and best paper awards at TextGraphs-NAACL 2018 and AAAI ICWSM 2020 (honorable mention). His current research interests focus on graph machine learning and applications.