# Deep Probabilistic Matrix Factorization on Graphs: Application to Drug Repositioning in Antimicrobial Resistance

Sayantika Chatterjee, Stuti Jain, Kriti Kumar, Emilie Chouzenoux, Angshul Majumdar

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## To cite this version:

Sayantika Chatterjee, Stuti Jain, Kriti Kumar, Emilie Chouzenoux, Angshul Majumdar. Deep Probabilistic Matrix Factorization on Graphs: Application to Drug Repositioning in Antimicrobial Resistance. IEEE/ACM Transactions on Computational Biology and Bioinformatics, In press, 10.1109/TCBBIO.2025.3570865. hal-05093953

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Submitted on 2 Jun 2025

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# Deep Probabilistic Matrix Factorization on Graphs: Application to Drug Repositioning in Antimicrobial Resistance

Sayantika Chatterjee, Stuti Jain, Kriti Kumar, Member, IEEE, Emilie Chouzenoux, Senior Member, IEEE, and Angshul Majumdar, Senior Member, IEEE

**Abstract**—Antimicrobial resistance (AMR) is a significant global health challenge caused by the misuse and overuse of antibiotics in various sectors, leading to the development of resistant bacteria. In such infections, the first-line antibiotics intended for specific diseases become ineffective, necessitating the repurposing of other antibiotics for treatment. To address this, we have developed a new algorithm for general-purpose drug repositioning based on a matrix completion framework on graphs. Our probabilistic approach combines deep matrix factorization with graph learning to achieve precise drug repurposing. In this study, we curated a new dataset on antibiotic-bacteria associations. Applying our proposed method to this dataset demonstrates that our approach outperforms benchmarks in both general-purpose drug repositioning and three specific AMR case studies.

**Index Terms**—Antimicrobial resistance, Drug repurposing, Matrix completion

## I. INTRODUCTION

ANTIMICROBIAL resistance (AMR) is one of the most significant public health challenges of the modern era, arising primarily due to the misuse and overuse of antibiotics in both human healthcare and livestock industries [1], [2]. The rapid proliferation of resistant bacterial strains has led to the failure of conventional first-line antibiotics, necessitating alternative strategies for treatment. WHO estimated that 1.27 million deaths were directly attributable to antibiotic-resistant infections in 2019, with a total of 4.95 million deaths associated with AMR in the same year [3]. The situation is projected to worsen, with a forecast of 5.2 million deaths in the western Pacific alone by 2030 [4].

Compounding this crisis is the sharp decline in the discovery of new antibiotics; only 22 new antibiotics were approved by the USFDA between 2012 and 2022 [5]. As a result, clinicians increasingly rely on drug combination therapies (drug cocktails) to counter AMR infections [6]. This underscores the importance of drug repositioning or drug repurposing, wherein existing drugs are identified for new therapeutic applications [7]. Drug repurposing offers several advantages over new drug

discovery, including significantly reduced development time, lower costs, and established safety profiles [8], [9]. Notable examples include auranofin, an antirheumatoid arthritis drug that has been successfully repurposed as an antibiotic for gastrointestinal protozoa [8], and sildenafil, initially developed for hypertension but later found effective for erectile dysfunction [10].

The urgency of drug repositioning was exemplified during the COVID-19 pandemic, when researchers rapidly explored existing drugs for potential antiviral activity, leading to a surge in research publications on the topic [11]. AI-driven computational models have played a crucial role in this domain, expediting drug repurposing by screening large molecular libraries to identify promising candidates [12], [13]. For instance, a study by Mongia et al. [14] used AI models to systematically analyze antiviral drugs and predict their effectiveness against novel viruses. However, antibiotic repurposing presents unique challenges compared to antiviral treatments, as antibiotic resistance varies geographically, requiring localized datasets and adaptive methodologies. In this work, we propose a novel computational approach for antibiotic repositioning by leveraging deep probabilistic matrix factorization on graphs.

## II. BACKGROUND

Existing drug repositioning methods can be broadly categorized into network-based models, which employ matrix completion, and classification-based models, which utilize machine learning techniques. While deep learning models have demonstrated high accuracy in data-rich environments, they often struggle in drug repositioning due to the limited availability of labeled data, typically comprising only a few hundred antibiotics and bacterial strains. As a result, hybrid approaches that integrate deep learning with network-based techniques have gained traction in the field [15], [16], [17].

Our approach builds upon prior work in matrix completion, which formulates drug-bacteria association prediction as a low-rank approximation problem. In matrix completion, a partially observed matrix is approximated by decomposing it into two or more factor matrices. Probabilistic matrix factorization (PMF) extends this concept by incorporating Gaussian priors, allowing for more robust inference even in sparse datasets [18]. Furthermore, recent advancements have introduced graph regularization to matrix factorization, where similarity graphs encode structural and genomic relationships between drugs and bacteria, thereby improving prediction accuracy [19].

S. Jain* is with Med-El, Austria. This work was done when she was with CVN, Inria Saclay, Univ. Paris Saclay, 91190 Gif-sur-Yvette, France. e-mail: (stuti.jain@inria.fr).

E. Chouzenoux is with CVN, Inria Saclay, Univ. Paris Saclay, 91190 Gif-sur-Yvette, France. e-mail: (emilie.chouzenoux@centralesupelec.fr).

A. Majumdar is with IAI TCG CREST, Kolkata, India, 700091. e-mail: (angshul.majumdar@tcgcrest.org).

S. Chatterjee* and K. Kumar are with IIIT - Delhi, India, 110020. e-mail: (sayantika21175@iiitd.ac.in, kritik@iiitd.ac.in).

<footnote>*S. Chatterjee and S. Jain are co-first authors.</footnote>
Traditional graph-based matrix factorization methods, however, suffer from a critical limitation: the graph structures used for regularization are predefined and non-adaptive. This rigidity can lead to suboptimal predictions when the underlying biological interactions deviate from the assumed similarity structure. To address this, we propose a deep probabilistic matrix factorization on graphs (DPMFG) framework, which extends previous graph-regularized models [17] by incorporating a probabilistic formulation that allows the graph to update dynamically. By learning an adaptive graph structure, our model enhances drug repositioning performance in the face of uncertain or incomplete biological data.

### III. DATASET

The information about bacterial infections and the corresponding effective drugs are taken from the antibiotic guidelines 2018 of Christian Medical College, Vellore and the antibiotic policy manual of AIIMS (All India Institute of Medical Sciences). These two manuals specify the bacterial infections, including the causative organisms (bacteria) and the effective drugs or antibiotics.

Our approach for drug-bacteria association is based on matrix completion. In this approach, antibiotics are represented along the rows, while bacteria are represented along the columns (or vice versa). Following the provided guidelines, we assigned a value of 1 to indicate the effectiveness of a drug against a particular bacterial infection, thus representing true positives in the matrix. However, it is important to acknowledge that information on drugs known to be ineffective against specific bacterial infections was not available in the guidelines. This limitation is not exclusive to our data set, as it is a common challenge in various biological interaction problems, including drug-target interaction [20], [21], drug-disease interaction [22], [23], drug-drug interaction [24], [25], drug-virus interaction [14], and others. The resulting matrix has dimensions of 53×61, with 53 antibiotics and 61 bacteria.

Standard matrix completion cannot incorporate metadata in its framework. However, a host of information is available for both drugs and bacteria. Past research in biological interaction where metadata is available indicates that graphical matrix completion methods yield very good results. In this class of techniques, the graphs incorporate the associated metadata.

To incorporate relevant drug and bacteria metadata, we obtain all the necessary information on drugs from the DrugBank vocabulary. From there, we extracted the unique DrugBank ID for antibiotics. However, instead of utilizing the raw data for the drugs, we focus on the structural similarity between them. This follows from a prior work on drug virus association [14], [17]. This similarity is computed using the SIMCOMP score, which requires access to the Kyoto Encyclopedia of Genes and Genomes (KEGG) [26]. Each KEGG object is assigned a unique identifier known as the KEGG ID (kid). The linking between KEGG and DrugBank can be found in the linking file<fnref n="1" /> [27]. In cases where certain drugs were missing from the file, we manually supplemented the information to establish the connection between each antibiotic and its corresponding

DrugBank ID with the appropriate KEGG ID in the KEGG Compound/KEGG Drug databases<fnref n="2" />, <fnref n="3" /> of the KEGG.

Similarly, we also compute the similarity between bacteria, focusing on their genomic similarity. To accomplish this, we acquired the full genome sequences in FASTA format from the National Center for Biotechnology Information (NCBI) genome database<fnref n="4" />. It is important to note that we specifically considered the primary strain for each bacteria, resulting in a total of 61 genome sequences. These sequences serve as input for calculating the similarity among bacteria.

#### A. Structural Similarity for Drugs

The drug data includes DrugBank IDs and their corresponding KEGG IDs. These DrugBank IDs are then combined with the corresponding KEGG Compound IDs to create a new dataset that includes the DrugBank ID, KEGG Compound ID, KEGG Drug ID, and the Drug Name. Each drug in the dataset is associated with either a KEGG Compound ID or a KEGG Drug ID that matches its DrugBank ID.

To calculate the similarity among the drugs, we consider their chemical significance or chemical structure. For this purpose, we utilize the SIMCOMP (SIMilar COMPound) server, which allows us to perform a chemical similarity search. The SIMCOMP score is employed to measure the chemical structure similarity between two drugs [22]. This score is obtained using a graph-based method that identifies the most extensive common elements shared by the chemical structures of the compounds.

To perform sequence analysis, we utilize GenomeNet Tools API14. We set a cut-off value of 0.001 for the SIMCOMP score calculation. If the SIMCOMP score for a drug falls below this cut-off or if a drug does not have any available KEGG IDs, we assign a similarity score of 1 to itself and a score of 0 to all remaining pharmaceutical in the dataset. Note that the initial SIMCOMP matrix is sparsified to retain only the *pp*-nearest neighbours of each drug to preserve the local geometry of the original data.

#### B. Bacteria Genomic Similarity

To determine the similarity between two bacteria based on their genome sequences, we employ the Genome-Genome Distance Calculator (GGDC) [23]. This method utilizes advanced in silico techniques to compare the genomes of different organisms, mimicking the conventional DNA-DNA hybridization (DDH) method with certain improvements and avoiding its limitations. The GGDC tool offers significantly faster performance compared to other available techniques.

We also explored using Stretcher, a global alignment tool from the Emboss suite, to determine the similarity between two genome sequences. However, this approach proved to be time-consuming, requiring approximately 10 hours to compute the sequence similarity. In contrast, GGDC delivers the most accurate and efficient results in a considerably shorter time frame.

<footnote n="1">¹https://www.drugbank.ca/releases/latest#external-links</footnote>

<footnote n="2">²https://www.genome.jp/kegg/compound/</footnote>

<footnote n="3">³https://www.genome.jp/kegg/drug/</footnote>

<footnote n="4">⁴https://www.ncbi.nlm.nih.gov/genome/</footnote>
For GGDC analysis, we submit multiple query genome sequences in FASTA format representing various bacteria, along with a single reference genome sequence in FASTA format. These query genomes and the reference genome are treated as inputs for the GGDC web server. The result of the comparison can be obtained by emailing the first author; it will contain the relevant information derived from the analysis. Here, again, the similarity matrix is sparsified to retain only the *pp*-nearest neighbours of each bacteria.

## IV. DEEP PROBABILISTIC MATRIX FACTORIZATION ON GRAPH

### A. Background

This section provides an introduction to matrix factorization approaches for matrix completion.

1) **Matrix Completion:** In this problem, we have a partially known matrix $Y \in \mathbb{R}^{N \times M}$, and we have to recover a full matrix $R \in \mathbb{R}^{N \times M}$ from it. Let us define the set $\mathcal{D}$ as the set of indices belonging to the partially observed matrix:

$$ \mathcal{D} = \{i \in \{1, \dots, N\}, j \in \{1, \dots, M\} \text{ s.t. } (i, j) \text{ is observed}\}. $$

The partially known matrix $Y$ can be expressed as:

$$ Y = B \odot R, \qquad (1) $$

where $\odot$ is the Hadamard product and matrix $B \in \{0, 1\}^{N \times M}$ masks the indexes outside the set $\mathcal{D}$, and is defined such that $B_{ij} = 1$ if $(i, j) \in \mathcal{D}$, and $B_{ij} = 0$ otherwise.

The task of matrix completion (MC) is to recover the entries of $R$ (or equivalently, of $Y$) that do not belong to the set of observed indices $\mathcal{D}$.

2) **Matrix Factorization (MF):** In MF [28], the missing entries in matrix $R$ are recovered by minimizing a simple least-squares function under some structural prior constraints on $R$. Specifically, $R$ is assumed to be the product of two matrices $U_1 \in \mathbb{R}^{N \times K}$ and $U_2 \in \mathbb{R}^{K \times M}$, where $K \ge 1$ defines a latent space dimension (typically low compared to $(N, M)$). The matrices $U_1$ and $U_2$ are inferred by solving the following:

$$ \min_{U_1, U_2} \|Y - B \odot (U_1 U_2)\|_F^2. \qquad (2) $$

Then, the full matrix is recovered as $R = U_1 U_2$. However, the problem (2) is still ill-posed and additional priors are frequently added to produce better solutions.

The NMF (non-negative MF) formulation, created by imposing a positivity constraint on the entries of the latent components $(U_1, U_2)$, is probably one of the most well-known formulations. Nuclear norm minimization [29] is another strategy that imposes a low rank on $R$. In such approaches, additional regularization procedures based on graph modeling have been considered in [30] and [31], respectively. The so-called deep MF method, which is studied, for example, in the context of MC in [32], relies on MF models with more than two components (i.e. factors). The recent study [17] has proposed a graph regularized variant for deep MC.

3) **Probabilistic Matrix Factorization (PMF):** PMF [18] introduces probabilistic models of latent factors $U_1$ and $U_2$ in the aforementioned MF formulation. Precisely:

* Each observed entry $(Y_{ij})_{(i,j) \in \mathcal{D}}$ is assumed to be the realization of a Gaussian distribution, with mean $[U_1 U_2]_{i,j}$ and variance $\sigma^2$ (positive scalar assumed to be known).

* Each observed entry $([U_1]_{ik})_{1 \le i \le N, 1 \le k \le K}$ is assumed to be the realization of a Gaussian distribution with zero mean and variance $\sigma^2_{U_1}$ (positive scalar assumed to be known).

* Each observed entry $([U_2]_{kj})_{1 \le k \le K, 1 \le j \le M}$ is assumed to be the realization of a Gaussian distribution with zero mean and variance $\sigma^2_{U_2}$ (positive scalar assumed to be known).

The maximum a posteriori (MAP) estimator of $(U_1, U_2)$ given $Y$, associated with the above model is obtained by solving:

$$ \min_{U_1, U_2} \frac{1}{2\sigma^2} \|Y - B \odot (U_1 U_2)\|_F^2 + \frac{1}{2\sigma_{U_1}^2} \|U_1\|_F^2 + \frac{1}{2\sigma_{U_2}^2} \|U_2\|_F^2. \quad (3) $$

The minimization of the above cost function with respect to $U_1$ (resp. $U_2$) amounts to inverting a linear system, which can be performed using a conjugate gradient solver (for instance). The PMF formulation can be enhanced by incorporating correlated Gaussian distributions, which results in the PMFG formulation described below.

4) **Probabilistic Matrix Factorization with Graph regularization (PMFG):** A graph regularization approach, PMFG, is introduced in [19], where the prior distributions of $U_1$ and $U_2$ include a graph regularization strategy. This strategy entails jointly inferring correlations along the rows (resp. columns) of $U_1$ (resp. $U_2$), and estimating the factors $U_1$ and $U_2$. Two precision matrices, $\Gamma_{U_1} \in S_N^{++}$ and $\Gamma_{U_2} \in S_M^{++}$, are used to model these correlations. Here, we denote $S_N$ (resp. $S_N^{++}$) as the set of symmetric (resp. symmetric positive definite matrices) of size $N \times N$. Hence, the PMFG prior can be explained as:

* The columns $([U_1]_{:,k})_{1 \le k \le K} \in \mathbb{R}^N$ of $U_1$ are independent realizations of a multivariate Gaussian distribution with zero mean and covariance $C_{U_1} = \Gamma_{U_1}^{-1}$;

* The rows $([U_2]_{k,:})_{1 \le k \le K} \in \mathbb{R}^M$ of $U_2$, are independent realizations of a multivariate Gaussian distribution with zero mean and covariance $C_{U_2} = \Gamma_{U_2}^{-1}$.

The precision matrices $\Gamma_{U_1}$ and $\Gamma_{U_2}$ are related to Gaussian graphical models. It is assumed that $\Gamma_{U_1}$, $\Gamma_{U_2}$ are associated with two underlying Gaussian distributions [33]. Here, the matrix $\Gamma_{U_1}$ (resp. $\Gamma_{U_2}$) can be interpreted as the adjacency matrix of an undirected graph where each edge (i.e., a non-zero entry in the precision matrix) identifies the two entries of $[U_1]_{:,k}$ (resp. $[U_2]_{k, :}$) being correlated, given all the others.

The MAP estimate can now be obtained by solving:

$$ \min_{U_1, U_2, \Gamma_{U_1}, \Gamma_{U_2}} \frac{1}{2\sigma^2} \|Y - B \odot (U_1 U_2)\|_F^2 + \frac{1}{2} \text{tr}(U_1^\top \Gamma_{U_1} U_1) + \frac{1}{2} \mathcal{L}(\Gamma_{U_1}) + \frac{1}{2} \text{tr}(U_2 \Gamma_{U_2} U_2^\top) + \frac{1}{2} \mathcal{L}(\Gamma_{U_2}), \quad (4) $$
where tr(·) denotes the trace operation and

$$ (\forall \Gamma_{U_1} \in S_N) \quad \mathcal{L}(\Gamma_{U_1}) = \begin{cases} -\ln \det(\Gamma_{U_1}) & \text{if } \Gamma_{U_1} \in S_N^{++} \\ +\infty & \text{otherwise} \end{cases} \quad (5) $$

with det(·) the determinant operation.

The PMFG formulation does not include any physical information for the factors ($U_1, U_2$). However, such expert knowledge is present in many applications that can be exploited to determine the more or less likely correlations between the variables. Extending the PMFG paradigm, the current work aims to propose a novel formulation to account for such prior knowledge.

## B. Proposed Deep Probabilistic Matrix Factorization on Graph (DPMFG)

1) Incorporating the prior knowledge as regularization: It is expected that one expert has access to some prior information about the position of the graph edges (i.e. the non-zero elements in the precision matrices $\Gamma_{U_1}$ and $\Gamma_{U_2}$). Following our previous work on drug-drug interaction [24], we propose to incorporate such expert knowledge through an extra regularization term in the loss function. Two symmetric adjacency matrices, $A_{U_1} \in [0, +\infty)^{N \times N}$ and $A_{U_2} \in [0, +\infty)^{M \times M}$, play a fundamental role in this new prior. These matrices, set by the user, are used to eliminate erroneous edges and favor anticipated ones in the estimated graph:

* $[\Gamma_{U_1}]_{ij} = [\Gamma_{U_1}]_{ji}$ should be encouraged to be high if $[A_{U_1}]_{ij}$ is large for some $(i, j)$ with $i \neq j$.

* $[\Gamma_{U_1}]_{ij} = [\Gamma_{U_1}]_{ji}$ should be encouraged to be low if $[A_{U_1}]_{ij}$ is small for some $(i, j)$ with $i \neq j$.

Similar prior is enforced on the second pair ($A_{U_2}, \Gamma_{U_2}$).

In order to build a suitable regularization function, we introduce the following sets:

$$ \mathcal{E}_{U_1} = \{(i, j) \in \{1, \dots, N\}^2, i \neq j \text{ and } [A_{U_1}]_{ij} > \tau\} \quad (6) $$

$$ \bar{\mathcal{E}}_{U_1} = \{(i, j) \in \{1, \dots, N\}^2, i \neq j \text{ and } [A_{U_1}]_{ij} \le \tau\} \quad (7) $$

$$ \mathcal{E}_{U_2} = \{(i, j) \in \{1, \dots, M\}^2, i \neq j \text{ and } [A_{U_2}]_{ij} > \tau\} \quad (8) $$

$$ \bar{\mathcal{E}}_{U_2} = \{(i, j) \in \{1, \dots, M\}^2, i \neq j \text{ and } [A_{U_2}]_{ij} \le \tau\} \quad (9) $$

with $\tau \ge 0$ a given detection threshold. We propose to use an $\ell_1$ term to promote sparsity in the regions where edges should not appear (i.e., $\bar{\mathcal{E}}_{U_1}$ and $\bar{\mathcal{E}}_{U_2}$), while we introduce a log-barrier term, smoothed by $\delta > 0$, in regions where edges should be promoted (i.e., $\mathcal{E}_{U_1}$ and $\mathcal{E}_{U_2}$). A quadratic term that can be viewed as an elastic-net penalty is added to avoid unbounded values in the entries of the sought covariance matrices.

In previously introduced MF-based formulations, for instance, in PMFG, the non-linearity introduced by the factor $B \odot U_1 U_2$ inside the least-squares data fidelity term adds instability to the minimization problem. To stabilize convergence, we finally propose to introduce an auxiliary variable $X \in \mathbb{R}^{N \times M}$, that is penalized to be close to the product $U_1 U_2$. As a result, the binary mask $B$ and the factorized expression $U_1 U_2$ are split, simplifying the solution using an alternating minimization method. In essence, we propose the following:

$$ \begin{aligned} & \min_{X, U_1, U_2, \Gamma_{U_1}, \Gamma_{U_2}} \frac{1}{2\sigma^2} \|Y - B \odot X\|_F^2 + \frac{1}{2} \mathcal{L}(\Gamma_{U_1}) \\ & + \frac{1}{2} \operatorname{tr}(U_1^\top \Gamma_{U_1} U_1) + \frac{1}{2} \operatorname{tr}(U_2 \Gamma_{U_2} U_2^\top) + \frac{1}{2} \mathcal{L}(\Gamma_{U_2}) \\ & + \lambda_{U_1} \sum_{i,j \in \bar{\mathcal{E}}_{U_1}} |[\Gamma_{U_1}]_{ij}| + \lambda_{U_2} \sum_{i,j \in \bar{\mathcal{E}}_{U_2}} |[\Gamma_{U_2}]_{ij}| \\ & \quad - \lambda_{U_1} \sum_{i,j \in \mathcal{E}_{U_1}} \ln(|[\Gamma_{U_1}]_{ij}| + \delta) \\ & \quad - \lambda_{U_2} \sum_{i,j \in \mathcal{E}_{U_2}} \ln(|[\Gamma_{U_2}]_{ij}| + \delta) \\ & \quad + \frac{\lambda_{U_1}}{2} \|\Gamma_{U_1}\|_F^2 + \frac{\lambda_{U_2}}{2} \|\Gamma_{U_2}\|_F^2 \\ & \quad + \frac{\lambda_R}{2} \|X - U_1 U_2\|_F^2. \end{aligned} \quad (10) $$

Parameter $\lambda_R > 0$ controls the fulfillment of the equality constraint $X = U_1 U_2$ while parameters $(\lambda_{U_1}, \lambda_{U_2}) > 0$ control the regularization imposed on both precision matrices. Kindly note that all terms apart from the data fidelity term in (10) are encoding 'prior knowledge'. This is our shallow formulation with two factors. This can be made deeper by allowing for more factors.

2) Integrating deep structure within PMF: The widespread use of deep learning has led to the development of deeper versions of shallow models [34]–[36]. Recently, [37] proposed a matrix completion problem involving deep factorization. Following the recent trend, we propose to generalize the previous model by considering that $X$ can be factorized into more than 2 factors. Without losing in generality, we explicit our methodology in the case of three factors (i.e., 2 layers), denoted $U_1 \in \mathbb{R}^{N \times k_1}$, $U_2 \in \mathbb{R}^{k_1 \times k_2}$, $U_3 \in \mathbb{R}^{k_2 \times M}$. Parameters $k_1$ and $k_2$ represent the (usually small) latent factor dimensions, typically fine-tuned through cross-validation over a training set. Here, we preserve the graphical priors on the first and last factors, $U_1$ and $U_3$, related to precision matrices $\Gamma_{U_1}$ and $\Gamma_{U_3}$, respectively. No expert knowledge is introduced for the middle factors (here, $U_2$). The minimization problem results in:

$$ \begin{aligned} & \min_{X \in \mathbb{R}^{N \times M}, U_1 \in \mathbb{R}^{N \times k_1}, U_2 \in \mathbb{R}^{k_1 \times k_2}, U_3 \in \mathbb{R}^{k_2 \times M}, \Gamma_{U_1} \in S_N, \Gamma_{U_3} \in S_M} \frac{1}{2\sigma^2} \|Y - B \odot X\|_F^2 \\ & + \frac{1}{2} \operatorname{tr}(U_1^\top \Gamma_{U_1} U_1) + \frac{1}{2} \mathcal{L}(\Gamma_{U_1}) + \frac{1}{2} \operatorname{tr}(U_3 \Gamma_{U_3} U_3^\top) \\ & + \frac{1}{2} \mathcal{L}(\Gamma_{U_3}) + \lambda_{U_1} \sum_{i,j \in \bar{\mathcal{E}}_{U_1}} |[\Gamma_{U_1}]_{ij}| + \lambda_{U_3} \sum_{i,j \in \bar{\mathcal{E}}_{U_3}} |[\Gamma_{U_3}]_{ij}| \\ & \quad - \lambda_{U_1} \sum_{i,j \in \mathcal{E}_{U_1}} \ln([\Gamma_{U_1}]_{ij} + \delta) - \lambda_{U_3} \sum_{i,j \in \mathcal{E}_{U_3}} \ln([\Gamma_{U_3}]_{ij} + \delta) \\ & \quad + \frac{\lambda_{U_1}}{2} \|\Gamma_{U_1}\|_F^2 + \frac{\lambda_{U_3}}{2} \|\Gamma_{U_3}\|_F^2 + \frac{\lambda_R}{2} \|X - U_1 U_2 U_3\|_F^2. \end{aligned} \quad (11) $$

Note that, in the case of two factors, we retrieve the model from Section IV-B1. Here, we have shown the deeper model with three factors; the depth can be increased further if required.

3) Optimization Algorithm: We propose to use an alternating minimization strategy to solve (11). Here again, without loss of generality, we consider the case of three factors.

Let us denote $F(X, U_1, U_2, U_3, \Gamma_{U_1}, \Gamma_{U_3})$ the loss function presented in (11). The initial matrix $U_1^{(0)}$ is obtained through the singular-valued decomposition of $Y = SDZ^\top$. We further decompose the right part of the decomposition, $DZ^\top$, to
initialize $U_2^{(0)}$ and $U_3^{(0)}$, respectively (for the case with 4 factor matrices, $U_2^{(0)}$ and $U_4^{(0)}$ would be obtained by decomposing the right part of the decomposition again). Then, starting with an initialization $(X^{(0)}, U_1^{(0)}, U_2^{(0)}, U_3^{(0)}, \Gamma_{U_1}^{(0)}, \Gamma_{U_3}^{(0)})$, for every iteration $p \in \mathbb{N}$, the algorithm updates following an alternating minimization scheme. The explicit update rules are as follows:

**Update of X:** The update reads:

$$X^{(p+1)} = \operatorname{argmin}_{X \in \mathbb{R}^{N \times M}} \frac{1}{2\sigma^2} \|Y - B \odot X\|_F^2 + \frac{\lambda_R}{2} \|X - U_1^{(p)} U_2^{(p)} U_3^{(p)}\|_F^2. \quad (12)$$

This is a strictly convex quadratic problem whose solution satisfies the following optimality condition:

$$\frac{1}{\sigma^2} Y + \lambda_R U_1^{(p)} U_2^{(p)} U_3^{(p)} = \frac{1}{\sigma^2} B \odot X^{(p+1)} + \lambda_R X^{(p+1)}. \quad (13)$$

The above equation is a linear system that can be solved efficiently by linear conjugate gradient solver.

**Update of $U_1$:** The update reads:

$$U_1^{(p+1)} = \operatorname{argmin}_{U_1 \in \mathbb{R}^{N \times k_1}} \frac{1}{2} \operatorname{tr}(U_1^\top \Gamma_{U_1}^{(p)} U_1) + \frac{\lambda_R}{2} \|X^{(p+1)} - U_1 U_2^{(p)} U_3^{(p)}\|_F^2. \quad (14)$$

This amounts to search for $U^{(p+1)}$ satisfying

$$\Gamma_{U_1}^{(p)} U_1 + \lambda_R (U_1 U_2^{(p)} U_3^{(p)} - X^{(p+1)})(U_2^{(p)} U_3^{(p)})^\top = 0, \quad (15)$$

or, equivalently,

$$\Gamma_{U_1}^{(p)} U_1 + U_1 (\lambda_R U_2^{(p)} U_3^{(p)} (U_2^{(p)} U_3^{(p)})^\top) = \lambda_R X^{(p+1)} (U_2^{(p)} U_3^{(p)})^\top. \quad (16)$$

Solving the above Sylvester equation, we get $U_1^{(p+1)}$.

**Update of $U_2$:** The update is:

$$U_2 = (U_1^{(p+1)})^\dagger X^{(p+1)} (U_3^{(p)})^\dagger \quad (17)$$

**Update of $U_3$:** The update reads:

$$U_3^{(p+1)} = \operatorname{argmin}_{U_3 \in \mathbb{R}^{k_2 \times M}} \frac{\lambda_R}{2} \|X^{(p+1)} - U_1^{(p+1)} U_2^{(p+1)} U_3\|_F^2 + \frac{1}{2} \operatorname{tr}(U_3 \Gamma_{U_3}^{(p)} U_3^{(p)}) \quad (18)$$

This amounts to search for $U_3^{(p+1)}$ satisfying

$$\lambda_R (U_1^{(p+1)} U_2^{(p+1)} U_3 - X^{(p+1)}) U_1^{(p+1)} U_2^{(p+1)} + U_3 \Gamma_{U_3}^{(p)} = 0, \quad (19)$$

or, equivalently,

$$\lambda_R (U_1^{(p+1)} U_2^{(p+1)})^\top U_1^{(p+1)} U_2^{(p+1)} U_3 + U_3 \Gamma_{U_3}^{(p)} = \lambda_R X^{(p+1)} U_1^{(p+1)} U_2^{(p+1)} \quad (20)$$

Solving the above Sylvester equation, we get $U_3^{(p+1)}$.

**Update of $\Gamma_{U_1}$:** We must solve:

$$\Gamma_{U_1}^{(p+1)} = \operatorname{argmin}_{\Gamma_{U_1} \in S_M} \operatorname{tr}(U_1^{(p+1)} (U_1^{(p+1)})^\top \Gamma_{U_1}) + \mathcal{L}(\Gamma_{U_1}) + 2\lambda_U \sum_{i,j \in \bar{\mathcal{E}}_{U_1}} |[\Gamma_{U_1}]_{ij}| - 2\lambda_U \sum_{i,j \in \mathcal{E}_{U_1}} \ln([\Gamma_{U_1}]_{ij} + \delta) + \lambda_U \|\Gamma_{U_1}\|_F^2 \quad (21)$$

Let us split the cost function involved above into two terms, so that

$$\Gamma_{U_1}^{(p+1)} = \operatorname{argmin}_{\Gamma_{U_1} \in S_M} f(\Gamma_{U_1}) + g(\Gamma_{U_1}), \quad (22)$$

$$\text{with} \quad (\forall U_1 \in S_M) \quad f(\Gamma_{U_1}) = \operatorname{tr}(U_1^{(p+1)} (U_1^{(p+1)})^\top \Gamma_{U_1}) + \mathcal{L}(\Gamma_{U_1}), \quad (23)$$

and

$$(\forall U_1 \in S_M) \quad g(\Gamma_{U_1}) = 2\lambda_U \sum_{i,j \in \bar{\mathcal{E}}_{U_1}} |[\Gamma_{U_1}]_{ij}| - 2\lambda_{U_1} \sum_{i,j \in \mathcal{E}_{U_1}} \ln([\Gamma_{U_1}]_{ij} + \delta) + \lambda_{U_1} \|\Gamma_{U_1}\|_F^2. \quad (24)$$

The minimization of $f+g$ does not have a close form solution and an inner solver is required. Function $f$ is convex, differentiable on $S_N^{++}$, while $g$ is non-convex, non-differentiable. Luckily, function $g$ is separable over the entries of $\Gamma_{U_1}$, that is:

$$(\forall U_1 \in S_M) \quad g(\Gamma_{U_1}) = \sum_{1 \le i,j \le N} g_{ij}([\Gamma_{U_1}]_{ij}) \quad (25)$$

with, for every $(i,j) \in \{1, \dots, N\}^2$,

$$(\forall \omega \in \mathbb{R}) g_{ij}(\omega) = \begin{cases} 2\lambda_{U_1}|\omega| + \lambda_{U_1}\omega^2 & \text{if } (i,j) \in \bar{\mathcal{E}}_{U_1}, \\ -2\lambda_{U_1}\ln(|\omega|+\delta) + \lambda_{U_1}\omega^2 & \text{if } (i,j) \in \mathcal{E}_{U_1}. \end{cases} \quad (26)$$

We thus opt for running $L \ge 1$ iterations of a proximal gradient algorithm, initialized in $\operatorname{dom} f = S_N^{++}$. This reads as follows:

$$\begin{align} \Gamma_{U_1}^{0} &= \Gamma_{U_1}^{(p)} \nonumber \\ \text{For } \ell = 1, 2, \dots, L \nonumber \\ & \quad \tilde{\Gamma}_{U_1}^{\ell} = \Gamma_{U_1}^{\ell} - \theta_1 \nabla f(\Gamma_{U_1}^{\ell}) \tag{27} \\ \Gamma_{U_1}^{\ell+1} &= \operatorname{prox}_{\theta_1 g}(\tilde{\Gamma}_{U_1}^{\ell}) \nonumber \\ \Gamma_{U_1}^{(p+1)} &= \Gamma_{U_1}^{L}. \nonumber \end{align}$$

Hereabove, $\theta_1$ is a positive stepsize such that all iterates belong to the domain of $f$. The expression for the gradient of $f$ and the proximity operator of $g$ can be found in [24].

**Update of $\Gamma_{U_3}$:** The update reads:

$$\Gamma_{U_3}^{(p+1)} = \operatorname{argmin}_{\Gamma_{U_3} \in S_M} \frac{1}{2} \operatorname{tr}(U_3^{(p+1)} \Gamma_{U_3} (U_3^{(p+1)})^\top) + \frac{1}{2} \mathcal{L}(\Gamma_{U_3}) + \lambda_{U_3} \sum_{i,j \in \bar{\mathcal{E}}_{U_3}} |[\Gamma_{U_3}]_{ij}| - \lambda_{U_3} \sum_{i,j \in \mathcal{E}_{U_3}} \ln([\Gamma_{U_3}]_{ij} + \delta) + \frac{\lambda_{U_3}}{2} \|\Gamma_{U_3}\|_F^2 \quad (28)$$

Let us split the cost function involved above into two terms, so that

$$\Gamma_{U_3}^{(p+1)} = \operatorname{argmin}_{\Gamma_{U_3} \in S_M} f(\Gamma_{U_3}) + g(\Gamma_{U_3}), \quad (29)$$

with

$$(\forall U_3 \in S_M) \quad f(\Gamma_3) = \operatorname{tr}(U_3^{(p+1)} \Gamma_{U_3} (U_3^{(p+1)})^\top) + \mathcal{L}(\Gamma_{U_3}), \quad (30)$$

and

$$(\forall \Gamma_{U_3} \in S_M) \quad g(\Gamma_{U_3}) = 2\lambda_{U_3} \sum_{i,j \in \bar{\mathcal{E}}_{U_3}} |[\Gamma_{U_3}]_{ij}| - 2\lambda_{U_3} \sum_{i,j \in \mathcal{E}_{U_3}} \ln([\Gamma_{U_3}]_{ij} + \delta) + \lambda_{U_3} \|\Gamma_{U_3}\|_F^2. \quad (31)$$

Hereagain, we can run $L \ge 1$ iterations of a proximal gradient algorithm, initialized in $\operatorname{dom} f = S_M^{++}$, which reads as follows:
$$
\begin{align}
\Gamma_{U_3}^{0} &= \Gamma_{U_3}^{(p)} \nonumber \\
\text{For } \ell = 1, 2, \dots, L \nonumber \\
&\quad \tilde{\Gamma}_{U_3}^{\ell} = \Gamma_{U_3}^{\ell} - \theta_2 \nabla f(\Gamma_{U_3}^{\ell}) \tag{32} \\
&\quad \Gamma_{U_3}^{\ell+1} = \operatorname{prox}_{\theta_2 g}(\tilde{\Gamma}_{U_3}^{\ell}) \nonumber \\
&\quad \Gamma_{U_3}^{(p+1)} = \Gamma_{U_3}^{L} \nonumber
\end{align}
$$

with $\theta_2$ a positive stepsize such that all iterates belong to the domain of $f$. The expression for the gradient of $f$ and the proximity operator of $g$ can be found in [24].

The complete method is summarized in Algorithm 1 where $P \ge 1$ iterations.

<caption>**Algorithm 1 DPMFG (2 layers)**</caption>

1: **Input:** $Y$, $A_{U_1}$, $A_{U_2}$.
2: **Parameters:** $(\sigma, s^{(0)}, \lambda_R, \lambda_{U_1}, \lambda_{U_3}) > 0$, $(k_1, k_2, P, L) \ge 1$.
3: **Initialization:** Set $U_1^{(0)}$, $U_2^{(0)}$, $U_3^{(0)}$ with SVD of $Y$.
4: Set $\Gamma_{U_1}^{(0)} = s^{(0)}\mathrm{Id}_N$ and $\Gamma_{U_3}^{(0)} = s^{(0)}\mathrm{Id}_M$.
5: Compute $\mathcal{E}_{U_1}$ and $\bar{\mathcal{E}}_{U_1}$ using (6) and (7).
6: Compute $\mathcal{E}_{U_3}$ and $\bar{\mathcal{E}}_{U_3}$ using (8) and (9).
7: **for** $p = 1, 2, \dots, P$ **end**
8: Update $X^{(p+1)}$ using (13)
9: Update $U_1^{(p+1)}$ using (16)
10: Update $U_2^{(p+1)}$ using (17)
11: Update $U_3^{(p+1)}$ using (20)
12: Update $\Gamma_{U_1}^{(p+1)}$ using (27)
13: Update $\Gamma_{U_3}^{(p+1)}$ using (32)
14: **end**
15: **Return:** $R = U_1^{(P)}U_2^{(P)}U_3^{(P)}$

V. EXPERIMENTAL RESULTS

A. Benchmarking

In this work, we have concentrated on matrix completion based techniques for drug repurposing. This is driven by our experience in prior studies, where we found that this class of techniques yields better results than others. However, we will briefly mention other classes of techniques for completeness. The simplest method involves the application of neighbourhood models from collaborative filtering to this class of problems [38]. A slightly more sophisticated approach is to use network diffusion based approaches [39]. Such methods are unfortunately not as accurate as matrix completion based techniques. For the same reason, the current network based techniques are formulated as matrix completion, not network diffusion.

Current approaches solve the problem using deep learning [40]. Deep learning based techniques are accurate but require large volumes of data, which is typically absent in a problem like ours. Hence, we have not considered this class of techniques. Moreover, vanilla matrix completion techniques cannot incorporate metadata; hence, we do not consider such techniques for benchmarking. We have compared our method with several graph regularized matrix factorization algorithms.

* Graph regularized matrix factorization (GRMF) [41]

* Graph regularized matrix completion (GRMC) [42]

* Graph regularized deep matrix factorization (GRDMF) [43]

* Graph regularized binary matrix completion (GRBMC) [22]

* Graph regularized probabilistic matrix factorization (GRPMF) [24] (single-layer configuration of the proposed DPMFG method)

Remark that we have not considered, in this work, link prediction based approaches for drug repurposing since empirical evidence points matrix completion based approaches are usually superior [44]–[46]. For drug repurposing, the experimental evaluation was done considering three protocols (or scenarios), described as follows: CV1:10% of randomly selected drug-bacteria associations are masked or left out for testing, while the remaining are considered for training. This allows us to determine the ability of each algorithm to predict associations between existing drugs and bacteria.

CV2:10% of randomly selected bacteria are masked or left out for testing while the remaining are considered for training. This helps in repurposing drugs for new bacterial infections. CV3:10% of randomly selected drugs are masked or left out for testing while the remaining are considered for training. This helps in finding possible applications of a novel antibiotic.

The hyperparameters, encompassing the step size, regularization parameters, and latent factor dimensions utilized in the algorithms for various scenarios, underwent fine-tuning following the experimental protocols detailed above. Optimal values for these hyperparameters were determined through a grid search conducted on the training data. The hyperparameters obtained for CV2 were utilized to predict the drugs against multidrug-resistant bacteria. Similarly, the hyperparameters obtained for CV3 were utilized to predict the bacteria for any novel drug. Tables I, II, and III present a comprehensive list of optimal hyperparameter values obtained for each algorithm for the different scenarios.

We carry out 10 fold cross-validation for each protocol. A random subset of the data is concealed, and the algorithm predicts these missing values. The performance of all algorithms was evaluated using two standard metrics: AUC (Area Under the Receiver Operating Characteristic curve) and AUPR (Area Under the Precision-Recall curve). Tables IV, V and VI present the results for CV1, CV2 and CV3, respectively, reporting the mean and standard deviation (SD) for the two metrics. Note that in the current context, where the distribution of classes is skewed, AUPR is a more appropriate measure for effective drug repositioning.

The CV1 scenario focuses on the re-utilization of existing drugs to combat current infections. While this scenario may not hold significant practical importance, it serves as a standard benchmark in drug-disease association studies. Our proposed method yields the best results. However, increasing the number of layers from two to three does not give any benefit. The existing techniques report comparatively good results in terms of AUC but are significantly worse in terms of AUPR. This shows that our method is considerably better in terms of drug repositioning.

The CV2 scenario carries substantial practical significance as it simulates a situation where the existing drugs can be employed to treat new bacterial infections. This scenario closely mirrors real-world challenges in addressing novel strains of
<table><caption>TABLE I
PARAMETERS USED IN SEVERAL ALGORITHMS FOR CV1</caption><thead><tr><th>Algorithms</th><th>Hyperparameters Values</th></tr></thead><tbody><tr><td>GRMC</td><td>pp = 7; λ = 0.03; μ<sub>1</sub> = 0.02; μ<sub>2</sub> = 0.005; ν<sub>1</sub> = 0.4; ν<sub>2</sub> = 0.3;</td></tr><tr><td>GRMF</td><td>pp = 7; λ<sub>l</sub> = 0.05; λ<sub>d</sub> = 0.03; λ<sub>t</sub> = 0.03;</td></tr><tr><td>GRDMF</td><td>pp = 2; μ<sub>1</sub> = 0.1; μ<sub>2</sub> = 0.05, α = 1; k<sub>1</sub> = 25; k<sub>2</sub> = 5;</td></tr><tr><td>GRBMC</td><td>pp = 2; λ = 0.8; μ<sub>1</sub> = 0.1; μ<sub>2</sub> = 0.1;</td></tr><tr><td>GRPMF</td><td>pp = 2, k<sub>1</sub> = 15, λ<sub>U<sub>1</sub></sub> = 1, λ<sub>U<sub>2</sub></sub> = 1, λ<sub>R</sub> = 1, σ = 0.5, s<sup>0</sup> = 1, θ<sub>1</sub> = 0.07, θ<sub>2</sub> = 0.01</td></tr><tr><td>Proposed 2 layer</td><td>pp = 5, k<sub>1</sub> = 16, k<sub>2</sub> = 6, λ<sub>U<sub>1</sub></sub> = 0.8, λ<sub>U<sub>3</sub></sub> = 3, λ<sub>R</sub> = 0.9, σ = 0.5, s<sup>0</sup> = 0.001, θ<sub>1</sub> = 0.1, θ<sub>2</sub> = 0.01</td></tr><tr><td>Proposed 3 layer</td><td>pp = 3, k<sub>1</sub> = 38, k<sub>2</sub> = 13, k<sub>3</sub> = 6, λ<sub>U<sub>1</sub></sub> = 1.5, λ<sub>U<sub>4</sub></sub> = 0.5, λ<sub>R</sub> = 1, σ = 0.5, s<sup>0</sup> = 0.9, θ<sub>1</sub> = 0.1, θ<sub>2</sub> = 0.01</td></tr></tbody></table>

<table><caption>TABLE II
PARAMETERS USED IN SEVERAL ALGORITHMS FOR CV2</caption><thead><tr><th>Algorithms</th><th>Hyperparameters Values</th></tr></thead><tbody><tr><td>GRMC</td><td>pp = 2; λ = 0.1; μ<sub>1</sub> = 0.01; μ<sub>2</sub> = 1; ν<sub>1</sub> = 0.4; ν<sub>2</sub> = 0.1;</td></tr><tr><td>GRMF</td><td>pp = 8; λ<sub>l</sub> = 1.5; λ<sub>d</sub> = 0.1; λ<sub>t</sub> = 0.5;</td></tr><tr><td>GRDMF</td><td>pp = 2; μ<sub>1</sub> = 0.06; μ<sub>2</sub> = 1.6, α = 20; k<sub>1</sub> = 25; k<sub>2</sub> = 20;</td></tr><tr><td>GRBMC</td><td>pp = 2; λ = 0.5; μ<sub>1</sub> = 3; μ<sub>2</sub> = 0.02;</td></tr><tr><td>GRPMF</td><td>pp = 2, k<sub>1</sub> = 25, λ<sub>U<sub>1</sub></sub> = 0.01, λ<sub>U<sub>2</sub></sub> = 0.3, λ<sub>R</sub> = 0.9, σ = 0.1, s<sup>0</sup> = 2, θ<sub>1</sub> = 0.01, θ<sub>2</sub> = 0.01</td></tr><tr><td>Proposed 2 layer</td><td>pp = 3, k<sub>1</sub> = 24, k<sub>2</sub> = 10, λ<sub>U<sub>1</sub></sub> = 1, λ<sub>U<sub>3</sub></sub> = 0.02, λ<sub>R</sub> = 0.1, σ = 1, s<sup>0</sup> = 0.1, θ<sub>1</sub> = 0.1, θ<sub>2</sub> = 0.01</td></tr><tr><td>Proposed 3 layer</td><td>pp = 6, k<sub>1</sub> = 23, k<sub>2</sub> = 12, k<sub>3</sub> = 8, λ<sub>U<sub>1</sub></sub> = 0.2, λ<sub>U<sub>4</sub></sub> = 0.01, λ<sub>R</sub> = 0.01, σ = 0.01, s<sup>0</sup> = 1, θ<sub>1</sub> = 0.01, θ<sub>2</sub> = 1.3</td></tr></tbody></table>

<table><caption>TABLE III
PARAMETERS USED IN SEVERAL ALGORITHMS FOR CV3</caption><thead><tr><th>Algorithms</th><th>Hyperparameters Values</th></tr></thead><tbody><tr><td>GRMC</td><td>pp = 2; λ = 0.01; μ<sub>1</sub> = 1; μ<sub>2</sub> = 0.1; ν<sub>1</sub> = 0.4; ν<sub>2</sub> = 0.1</td></tr><tr><td>GRMF</td><td>pp = 7; λ<sub>l</sub> = 0.0413; λ<sub>d</sub> = 0.02; λ<sub>t</sub> = 0.03;</td></tr><tr><td>GRDMF</td><td>pp = 2; μ<sub>1</sub> = 0.4; μ<sub>2</sub> = 0.7, α = 1; k<sub>1</sub> = 25; k<sub>2</sub> = 5;</td></tr><tr><td>GRBMC</td><td>pp = 2; λ = 0.8; μ<sub>1</sub> = 0.08; μ<sub>2</sub> = 1;</td></tr><tr><td>GRPMF</td><td>pp = 3, k<sub>1</sub> = 25, λ<sub>U<sub>1</sub></sub> = 2, λ<sub>U<sub>2</sub></sub> = 0.1, λ<sub>R</sub> = 1, σ = 2, s<sup>0</sup> = 1, θ<sub>1</sub> = 0.01, θ<sub>2</sub> = 0.01</td></tr><tr><td>Proposed 2 layer</td><td>pp = 1, k<sub>1</sub> = 21, k<sub>2</sub> = 3, λ<sub>U<sub>1</sub></sub> = 0.6, λ<sub>U<sub>3</sub></sub> = 1.2, λ<sub>R</sub> = 0.9, σ = 1, s<sup>0</sup> = 1, θ<sub>1</sub> = 0.1, θ<sub>2</sub> = 0.01</td></tr><tr><td>Proposed 3 layer</td><td>pp = 3, k<sub>1</sub> = 23, k<sub>2</sub> = 22, k<sub>3</sub> = 1, λ<sub>U<sub>1</sub></sub> = 0.8, λ<sub>U<sub>4</sub></sub> = 0.7, λ<sub>R</sub> = 1, σ = 1, s<sup>0</sup> = 1.6, θ<sub>1</sub> = 0.1, θ<sub>2</sub> = 0.009</td></tr></tbody></table>

<table><caption>TABLE IV
RESULTS FOR CV1</caption><thead><tr><th>Metrics</th><th>Proposed (3 layer)</th><th>Proposed (2 layer)</th><th>GRPMF</th><th>GRMC</th><th>GRBMC</th><th>GRDMF</th><th>GRMF</th></tr></thead><tbody><tr><td>AUC</td><td>0.929 (SD = 0.017)</td><td>0.929 (SD = 0.018)</td><td>0.903 (SD = 0.032)</td><td>0.913 (SD = 0.024)</td><td>0.913 (SD = 0.022)</td><td>0.903 (SD = 0.022)</td><td>0.912 (SD = 0.046)</td></tr><tr><td>AUPR</td><td>0.835 (SD = 0.024)</td><td>0.835 (SD = 0.026)</td><td>0.798 (SD = 0.056)</td><td>0.727 (SD = 0.045)</td><td>0.757 (SD = 0.038)</td><td>0.764 (SD = 0.041)</td><td>0.774 (SD = 0.117)</td></tr></tbody></table>

<table><caption>TABLE V
RESULTS FOR CV2</caption><thead><tr><th>Metrics</th><th>Proposed (3 layer)</th><th>Proposed (2 layer)</th><th>GRPMF</th><th>GRMC</th><th>GRBMC</th><th>GRDMF</th><th>GRMF</th></tr></thead><tbody><tr><td>AUC</td><td>0.697 (SD = 0.078)</td><td>0.681 (SD = 0.083)</td><td>0.644 (SD = 0.108)</td><td>0.686 (SD = 0.101)</td><td>0.683 (SD = 0.103)</td><td>0.672 (SD = 0.112)</td><td>0.667 (SD = 0.068)</td></tr><tr><td>AUPR</td><td>0.386 (SD = 0.178)</td><td>0.374 (SD = 0.171)</td><td>0.363 (SD = 0.179)</td><td>0.374 (SD = 0.189)</td><td>0.376 (SD = 0.194)</td><td>0.369 (SD = 0.183)</td><td>0.349 (SD = 0.183)</td></tr></tbody></table>

<table><caption>TABLE VI
RESULTS FOR CV3</caption><thead><tr><th>Metrics</th><th>Proposed (3 layer)</th><th>Proposed (2 layer)</th><th>GRPMF</th><th>GRMC</th><th>GRBMC</th><th>GRDMF</th><th>GRMF</th></tr></thead><tbody><tr><td>AUC</td><td>0.812 (SD = 0.063)</td><td>0.747 (SD = 0.075)</td><td>0.720 (SD = 0.069)</td><td>0.797 (SD = 0.059)</td><td>0.803 (SD = 0.066)</td><td>0.712 (SD = 0.056)</td><td>0.790 (SD = 0.079)</td></tr><tr><td>AUPR</td><td>0.619 (SD = 0.095)</td><td>0.492 (SD = 0.068)</td><td>0.463 (SD = 0.128)</td><td>0.552 (SD = 0.091)</td><td>0.583 (SD = 0.110)</td><td>0.434 (SD = 0.107)</td><td>0.548 (SD = 0.088)</td></tr></tbody></table>

bacterial infections. Expanding on the insights from CV1, we note similar trends in CV2. However, both AUC and AUPR have reduced significantly compared to CV1. This can be explained from the theory of matrix completion [29], [47]. It says that matrix completion is optimal when none of the rows or columns are completely empty. CV2 violates this condition, and hence the results deteriorate.

In CV3 scenario, the emphasis is on identifying which existing bacterial infections can be effectively treated with new antibiotics, holding practical importance, especially in the context of antibiotic development. The overall trend is consistent with CV1 and CV2. Note that of all the scenarios, CV1 is the simplest. This is because all the antibiotics and bacteria are present in the dataset at all times. In CV2, all the drugs are present and the problem is to treat new (unseen) bacterial infections. This is the most challenging scenario. Hence, all the algorithms show poor performance. CV3 is

a moderately challenging scenario where all the bacteria are present but there are new antibiotics. Consequently, overall the results of CV3 are better than CV2.

Please note we cannot do ablation studies as such for all the terms in the proposed optimization formulation (11) but, we have chosen the benchmarks so that the effect of the different components of the proposed methodology can be studied. The effect of network depth can be analyzed through the results obtained from the proposed DPMFG (3 layers), DPMFG (2 layers), and GRPMF (single-layer configuration of the proposed DPMFG method). The effect of probabilistic elements can be analyzed by studying the results of GRMF (shallow-variant), GRDMF (deep variant) that are non-probabilistic methods against GRPMF (shallow-variant of DPMFG), and DPMFG (deep variant) that are based on probabilistic matrix factorization. Since graph-regularized methods offered superior performance compared to other standard matrix factor-
<table><caption>TABLE VII
KLEBSIELLA PNEUMONIAE U25 (*GREEN – EFFECTIVE, RED – NEFFECTIVE, BLACK – UNKNOWN)</caption><thead><tr><th>GRMF</th><th>GRDMF</th><th>GRBMC</th><th>GRMC</th><th>Proposed</th></tr></thead><tbody><tr><td>Tazobactam</td><td>Piperacillin</td><td>Tazobactam</td><td>Piperacillin</td><td>Amikacin</td></tr><tr><td>Piperacillin</td><td>Tazobactam</td><td>Piperacillin</td><td>Tazobactam</td><td>Piperacillin</td></tr><tr><td>Vancomycin</td><td>Colistin</td><td>Amikacin</td><td>Colistin</td><td>Tazobactam</td></tr><tr><td>Amikacin</td><td>Amikacin</td><td>Colistin</td><td>Amikacin</td><td>Colistin</td></tr><tr><td>Colistin</td><td>Ertapenem</td><td>Meropenem</td><td>Meropenem</td><td>Meropenem</td></tr></tbody></table>

<table><caption>TABLE VIII
NEISSERIA GONORRHOEAE H041 (*GREEN – EFFECTIVE, RED – NEFFECTIVE, BLACK – UNKNOWN, BLUE – EXPERIMENTAL)</caption><thead><tr><th>GRMF</th><th>GRDMF</th><th>GRBMC</th><th>GRMC</th><th>Proposed</th></tr></thead><tbody><tr><td>Ceftriaxone</td><td>Ceftriaxone</td><td>Ceftriaxone</td><td>Ceftriaxone</td><td>Ampicillin</td></tr><tr><td>Clavulanate</td><td>Vancomycin</td><td>Tazobactam</td><td>Tazobactam</td><td>Piperacillin</td></tr><tr><td>Amoxicillin</td><td>Piperacillin</td><td>Piperacillin</td><td>Piperacillin</td><td>Tazobactam</td></tr><tr><td>Vancomycin</td><td>Tazobactam</td><td>Vancomycin</td><td>Vancomycin</td><td>Gentamicin</td></tr><tr><td>Penicillin G</td><td>Ampicillin</td><td>Clavulanate</td><td>Clavulanate</td><td>Cefazolin</td></tr></tbody></table>

<table><caption>TABLE IX
TUBERCULOSIS (TB) NIRTX011 (*GREEN – EFFECTIVE, RED – NEFFECTIVE, BLACK – UNKNOWN, BLUE – EXPERIMENTAL)</caption><thead><tr><th>GRMF</th><th>GRDMF</th><th>GRBMC</th><th>GRMC</th><th>Proposed</th></tr></thead><tbody><tr><td>Vancomycin</td><td>Ceftriaxone</td><td>Ceftriaxone</td><td>Ceftriaxone</td><td>Levofloxacin</td></tr><tr><td>Ceftriaxone</td><td>Vancomycin</td><td>Vancomycin</td><td>Vancomycin</td><td>Amoxicillin</td></tr><tr><td>Tazobactam</td><td>Doxycycline</td><td>Amoxicillin</td><td>Doxycycline</td><td>Clavulanate</td></tr><tr><td>Piperacillin</td><td>Clavulanate</td><td>Clavulanate</td><td>Clavulanate</td><td>Oseltamivir</td></tr><tr><td>Cefoperazone</td><td>Amoxicillin</td><td>Doxycycline</td><td>Amoxicillin</td><td>Azithromycin</td></tr></tbody></table>

ization/completion methods due to their ability to incorporate metadata, no comparison is provided against them.

## B. Case studies in Drug Resistant Bacteria

Our first case study is the Klebsiella Pneumoniae U25 isolate [48] (results discussed in Table VII). This drug resistant strain found in India is resistant to ciprofloxacin and norfloxacin and other fluoroquinolones. The top-5 recommendations from the different algorithms are shown in the table above. We give the supporting references for the efficacy of different drugs.

**Piperacillin & Tazobactam:** All of the methods predicted this drug; this is a correct prediction that has been supported by prior studies [49], [50].

**Colistin:** Similarly, all methods have correctly predicted colistin; this is used routinely for such drug resistant Klebsiella infections [51].

**Amikacin:** All the methods predict colistin. The original study did not find the strain U25 [51], [52].

**Carbapenems (Ertapenem, Meropenem):** Apart from GRMF, all the method predicted carbapenem (ertapenem or meropenem). This is a correct prediction that has been tested in the original study [48].

GRMF predicted Vancomycin; we could not find literature to support its use in Klebsiella Pneumoniae induced infections.

Overall, one can see that apart from GRMF all the algorithms yield 5 drugs, all of which have the potential to treat the drug resistant strain U25.

The H041 strain of Neisseria Gonorrhoeae was isolated in Japan [53] (results discussed in Table VIII). It was found to be resistant to Ceftriaxone and other cephalosporins. All the algorithms apart from our proposed ones predicted Ceftriaxone; this is in line with empirical practice where Ceftriaxone is the first line of treatment. However, the paper above [53] found this H041 to be resistant to this antibiotic; so the prediction

of this drug is incorrect. Therefore, our algorithm was correct in not predicting the obvious incorrect antibiotic.

GRDM makes all incorrect predictions. All the suggested antibiotics have been found to be ineffective against the strain H041.

**Piperacillin/tazobactam:** All algorithms except GRMF predict these two drugs. These are correct predictions as shown in the original paper that isolated this strain [53].

**Ampicillin:** GRDMF and our proposed method also predict ampicillin. As pointed out in [53], the said strain is not resistant to this antibiotic; therefore, this is a correct prediction.

**Penicillin G:** This drug was predicted by GRMF. H041 was found to be resistant to this antibiotic, as mentioned in [53]. Hence, this prediction is incorrect.

All the algorithms except ours predict Vancomycin/Clavulanate; these were not tested in the aforementioned paper, and therefore, it cannot be said for sure if these drugs would be effective for treating H041. In limited cases Vancomycin was a drug of choice for treating such infections in the 70s and 80s [53], but has not been researched since then for treating multi-drug resistant gonorrhea. Clavulanate, too, has been sparingly used in the past [54], [55] but there is no research to support its efficacy in recent times.

Our proposed method predicts gentamicin. It has been found to be effective for uncomplicated gonorrhea [56]. In vitro studies indicate that the drug has the potential to treat multi-drug resistant gonorrhea [57]–[59]. However, clinical trials are required on this topic.

The NIRTX011 sequence was obtained from the Chennai Bioproject<fnref n="5" /> (discussed in Table IX).

**Amoxycillin/Clavulanate:** All the methods except GRMF predict these two drugs. This has been found to be effective in multi-

<footnote n="5">⁵https://www.ncbi.nlm.nih.gov/bioproject/492975</footnote>
drug resistant TB [60] and also features in the WHO guidelines [61].

**Levofloxacin:** Only our proposed method predicts this antibi-
otic. This is a viable treatment for drug resistant TB and can
be found in the WHO guidelines [61].

Vancomycin: All the methods except ours predicted Van-
comycin. We could not find any definitive research supporting
its usage in clinical settings. However, in vitro studies indicate
its potential in multi-drug resistant tuberculosis [62], [63].

Ceftriaxone: Similarly, Ceftriaxone was found to be effective for multi-drug resistant TB in in vitro studies [64], [65].

Doxycycline: In vitro studies have shown that this antibiotic by itself [66] or in conjunction with amikacin [67] has the potential for use in multi-drug resistant TB.

Other predicted drugs like Piperacillin, Tazobactam and Cefoperazone, although known as a broad-spectrum antibiotics, do not have any established research supporting their use in multi-drug resistant TB. Azithromycin, which has been predicted by our proposed method, has also been discontinued as a treatment option for such extensively drug resistant (XDR) TB in WHO guidelines. Oseltamivir, another prediction of our method, is an antiviral and does not have any effect on any bacterial infections.

For the unknown associations, there is no known research report that proves or disproves the efficacy of Vancomycin for Klebsiella Pneumoniae U25. Similarly, for TB the efficacy of Tazobactam has not been reported in the medical literature. One may simulate the pharmacology by docking. However, these too are computer simulations and the practical efficacy can only be gauranteed after clinical trials since the entries in black (unknown associations) in Tables VII, VIII and IX have not been reported, we cannot analyze them further.

VI. CONCLUSION

This work addresses the problem of antibiotic repositioning for bacterial infections from a computational perspective. This field has garnered interest in the recent past [68]. Such studies mostly concentrate on three areas - 1. predicting AMR given the genomic sequence of the bacterium; 2. new drug (antibiotic) discovery; 3. selecting the best antibiotic (in terms of safety, efficacy, toxicity, etc.) given multiple choices. To the best of our knowledge, there has been no prior study on using computational techniques for actual repurposing of antibiotics. This is the reason why there was no pre-existing universal dataset that accounts for all antibiotics (and their structures) and bacterial infections (and their genomic sequence). This is the first work that curates such a dataset.

Given our experience, such a problem is best modeled as
a graph regularized matrix completion problem. There are
several algorithms that can solve this problem, but all of them
are deterministic. Such an approach does not allow for the
refinement of the graphs. To facilitate this, we have proposed
a new probabilistic approach called deep probabilistic matrix
factorization on graphs. This is the second major contribution
of this work. We show that our proposed method improves
over the benchmarks for the problem of antibiotic bacteria
association prediction on different drug repurposing protocols.
Furthermore, we take case studies in drug resistant bacteria

and show that all the algorithms indeed yield good results in
selecting alternate drugs that the bacteria is not resistant to.

In this work, we have used the graph Laplacian for reg-
ularization. The graph Laplacian has been computed based
on similarities of the antibiotics and the similarities of
the bacteria. In biology, the mutations / changes are more impor-
tant than the similarities hence a potential of improvement of
the results could be based on capturing the mutations instead
of the similarities. As a future work, we could investigate
how to integrate the mutation information into our proposed
framework. Currently, there are hybrid models that combine
network based methods with matrix factorization [69] and
[70]. Motivated by these studies, in the future, we would
like to incorporate some network based techniques into our
formulation to improve the results.

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