JournalTitle DOI HERE Advance Access Publication Date: Day Month Year Paper

Lloyd et al.

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# Tensor Factorisation for Polypharmacy Side Effect Prediction

Oliver Lloyd  Yi Liu  Tom R. Gaunt  MRC Integrative Epidemiology Unit, Bristol
Medical School, University of Bristol, United Kingdom

(2024; 2024; Date; Date; Date)

###### Abstract

Adverse reactions caused by drug combinations are an increasingly common
phenomenon, making their accurate prediction an important challenge in modern
medicine. However, the polynomial nature of this problem renders lab-based
identification of adverse reactions insufficient. Dozens of computational
approaches have therefore been proposed for the task in recent years, with
varying degrees of success. One group of methods that has seemingly been
under-utilised in this area is tensor factorisation, despite their clear
applicability to this type of data. In this work, we apply three such models
to a benchmark dataset in order to compare them against established
techniques. We find, in contrast to previous reports, that for this task
tensor factorisation models are competitive with state-of-the-art graph neural
network models and we recommend that future work in this field considers
cheaper methods with linear complexity before running costly deep learning
processes.

###### keywords:

polypharmacy side effect prediction, adverse drug reaction, knowledge graph
embeddings, tensor factorisation

##  1 Introduction

The comforts of modern life are causing a demographic shift in populations
globally. By 2050, the number of people aged 60+ is projected to more than
double (World Health Organisation, 2022), compared to an overall population
increase of around 21% (United Nations, 2022). With this increase, medical
events that primarily affect older people will become increasingly important
areas of research. Multimorbidity is one such phenomenon, whose prevalence
among elderly populations may range from 55-98% (Marengoni et al., 2011).
Closely associated with multimorbidity is another issue: polypharmacy, the
taking of two or more medications simultaneously by the same individual.
Polypharmacy can be usefully employed in some contexts, for example in order
to achieve drug synergism (Tallarida, 2011), where two medications are
combined to produce an effect greater than the sum of their individual
effects. However, the practise of polypharmacy can also lead to the emergence,
via chemical interactions, of adverse drug reactions (ADRs) that are not
associated with either drug individually (Ahmed et al., 2014). ADRs put a
large strain on healthcare systems, with a systematic review from 2002 putting
their annual cost to the UK National Health Service at £380 million (Wiffen et
al., 2002). Given the general population increase and demographic shift
mentioned above, plus inflation, it is very plausible that this annual cost
may soon approach £1 billion.

Correct prediction of ADRs related to polypharmacy is a challenging task,
owing mainly to the extremely large numbers involved. In England, an estimated
8.4 million people are taking at least 5 prescribed medications, and a quarter
of those are taking 10 or more (UK Government, 2021). The combinatorial nature
of the problem means that testing every possible n-combination of the
thousands of commercially available drugs quickly becomes infeasible in wet-
lab experiments or clinical trials, even with smaller values of n than are
commonly found in real populations. As a result, pre-clinical screening with
statistical/computational methods is a necessity for identifying drug
combinations of interest (Ryall and Tan, 2015). Recently, there has been an
explosion of interest in graph-embedding methods for solving the problem of
polypharmacy side effect (PSE) prediction. This can largely be traced back to
the work of Zitnik et al., in which the authors construct a knowledge graph
from the following data: drug-target, protein-protein interaction,
monopharmacy side effects, and drug-pair side effects (Zitnik et al., 2018). A
portion of the drug-pair side effects are left out to enable out-of-sample
prediction and assessment. By constructing the data in this way, the problem
is cast as a multirelational link prediction (LP) problem, which, the authors
claim, makes theirs the first technique that allows prediction of the type of
side effect that will occur, rather than a simple binary categorisation or
magnitude of effect. The Decagon model itself consists of two components.
Firstly, node embeddings for the network are encoded using a graph
convolutional model. These embeddings are then passed to a tensor
factorisation (TF) decoder which produces a score for a particular side effect
$r$ between a drug pair ($v_{i}$, $v_{j}$). Finally, that score is passed to a
sigmoid function which outputs a probability that the given triple ($v_{i}$,
$r$, $v_{j}$) is true.

Commendably, Zitnik et al. made their data publicly available and since then
it has been used as a common benchmark in PSE modelling research. Early
adopters of the dataset found success with a wide variety of techniques,
including kernel ridge regression (Dewulf et al., 2021), semantic predication
embedding (Burkhardt et al., 2019), and product of experts models (Malone et
al., 2019). Each of these approaches improved upon the reported scores of
Decagon while also being more efficient and easier to interpret. Despite this,
as time has progressed we have seen an increasingly homogenised set of methods
being applied to the Decagon dataset, with graph neural networks (GNNs)
becoming dominant from 2022 onwards. GNNs have been reported to perform well
on PSE prediction (Carletti et al., 2021; Zhuang et al., 2023; Saifuddin et
al., 2023), particularly when information pertaining to the chemical structure
of drugs is incorporated into the training process (Li et al., 2023).
Unfortunately, there is no avoiding the fact that they are expensive to run
(Wu et al., 2019), and this problem is set to only get worse for drug-
structural models given that larger biologic drugs are becoming more prevalent
over time (Senior, 2023).

Knowledge graph embedding (KGE) methods are a broad family of techniques that
take heterogenous graphs as input and cast the comprising nodes and relations
into a low-dimensional vector space. They have proved useful in several real-
world tasks, such as item recommendation (Catherine and Cohen, 2016), drug
development and repurposing (Geleta et al., 2021), and knowledge base
completion (Bordes et al., 2013). The application of these techniques to the
PSE modelling problem, however, has been somewhat scarce considering their
relevance for this type of data, alongside their relatively low cost to run
and optimise. This is even more surprising in the light of research which has
suggested that simpler models can in fact outperform more complex ones when
correctly tuned (Ali et al., 2022). One possible explanation for this scarcity
can be found in the original work of Zitnik et al. \- they employed two such
methods (RESCAL and DEDICOM) alongside their graph convolutional approach,
reporting that these two scored the lowest out of all the methods tested. Even
predictions based on the simple method of concatenating drug features achieved
substantially higher scores as measured by the three test metrics. Subsequent
work has suggested that this might be an overly pessimistic view of the
capabilities of KGE methods for this task, with models such as DistMult,
ComplEx, SimplE, and RotatE all matching or surpassing the performance of
Decagon (Malone et al., 2019; Kim and Shin, 2023; Dai et al., 2021). Despite
this, the upper limit for such methods (achieved by properly optimising
hyperparameters) has yet to be tested. Without knowledge of this limit, we
cannot say definitively whether the current state-of-the-art graph neural
network methods are actually providing a worthwhile predictive improvement
over cheaper KGE methods.

The contributions of this paper are twofold. Primarily, we aim to provide
clarity on the true predictive capabilities of KGE methods on the PSE task by
running a comprehensive hyperparameter optimisation process on selected
models. By putting them through exactly the same assessment process outlined
by Decagon (Zitnik et al., 2018), we enable our results to be compared with
other research in the field. As a secondary goal, we compare options for
incorporating monopharmacy data into the embedding process by repeating the
analysis on two graphs where this data is handled differently.

##  2 Methods

We downloaded the raw data used by Decagon from the Stanford Network Analysis
Project http://snap.stanford.edu/decagon/. We then prepared the data for
LibKGE (Broscheit et al., 2020), which reads graphs as a list of edges in RDF
triple (subject, predicate, object) format. There are several ways to approach
the conversion of the data to RDF format. In this work we test two such
methods, which we have named ‘Selfloops’ and ‘Non-naive’ – the difference
between them being the way in which monopharmacy side effect data is included
in the resulting graph. The Selfloops approach treats these side effects as
edges from one drug back to itself (hence the name). The Non-naive
construction method is equivalent, but, following the example of Zitnik et
al., monopharmacy data is instead modelled as n-hot node feature-vectors, with
‘hot’ columns indicating which side effects are associated with a given drug.
We performed dimensionality reduction via principal component analysis (PCA)
on these features to create a smaller matrix for each possible dimensional
size of embeddings. We then modified LibKGE to load these vectors from disk,
using them as the starting point for learning node embeddings for this dataset
rather than any of its usual stochastic initialisation techniques. Graph
statistics for the two networks are listed in table 1, and schemata are shown
in figure 1.

Graph | Meta-nodes | Nodes | Meta-edges | edges
---|---|---|---|---
Selfloops | 2 | 19734 | 11149 | 5485566
Non-naive | 2 | 19734 | 964 | 5310589

Table 1: Node and edge counts for our two constructed variations of the
Decagon graph. Meta-edges are the ‘types’ of edges allowed in the graph, such
that each meta-edge can be associated with one adjacency matrix. A meta-node
is a ‘type‘ of node that all follow the same connectivity rules and appear
together as one or both axes of an adjacency matrix. For example, an instance
of the ‘Drug-Target’ meta-edge can only exist between one ‘Drug’ node and one
‘Gene’ node, and can be represented as a $d$ x $g$ adjacency matrix in which a
value of 1 indicates that drug $d_{i}$ targets gene $g_{j}$, otherwise the
value is 0.

A) Selfloops

B) Non-naive

Figure 1: Schemata for the two graph variants. The only difference between the
two graphs relates to the handling of monopharmacy side effect data. The
‘Selfloops’ approach (subfigure A) includes monopharmacy data as self-looping
edges from a given drug back to itself, whereas the ‘Non-naive’ graph
(subfigure B) doesn’t contain these edges. Instead, monopharmacy associations
are modelled as n-hot vectors of length 10,184. At the start of an embedding
trial, these vectors are reduced to the selected embedding sized using
singular value decomposition and then used as the initialisation points for
the drug embeddings. When training on the Selfloops graph, embeddings are
initialised by drawing from a random distribution, as is standard in LibKGE.

All three methods included in this analysis can be classified as some form of
TF. ComplEx (Trouillon et al., 2016) uses the Hermitian product to embed
entities into the complex space. This (ironically) straightforward approach
allows anti-symmetric relations to be modelled with a low-rank decomposition,
which, in the real space, would only be possible for symmetric relations.
DistMult (Yang et al., 2014), also referred to as ‘bilinear-diag’ by its
authors, is similar to the well known TransE model, but uses a multiplicative
operation rather than an additive one to combine dyadic vectors. Lastly,
SimplE (Kazemi and Poole, 2018) is an enhancement of the Canonical Polyadic
(CP) decomposition for embedding KGs. CP itself was used often in early LP
research but has a major shortfall in that it learns head and tail vectors for
a given node independently – SimplE addresses this problem by also considering
inverted relations to create dependency between these vectors.

The hyperparameter optimisation space, established in configuration files, was
the same for all models. Hyperparameter ranges were based on those from the
search performed by the LibKGE developers in their demonstrative paper
(Ruffinelli et al., 2020). We made a few adjustments to the space, notably
expanding the number of available options for optimiser and loss function from
2 and 3, to 5 and 4 respectively. Other alterations included adding possible
embedding sizes of 32 and 64, and reducing the aggressiveness of the learning
rate scheduler. The grid searches took place over 100 trials, with the first
50 having values chosen by Sobol sequence and the remaining 50 chosen by
Bayesian optimisation through the Ax platform. This Bayesian method works by
iteratively fitting surrogate models (Guassian processes) of trial outcomes
against the attempted hyperparameter configurations – these models are used to
determine the ‘value’ of choosing a particular new parameterisation by
considering both the predicted outcome of that configuration, as well as the
corresponding uncertainty. The parameterisation that provides the best value
is then chosen for the next trial, and afterwards the process is repeated. All
configuration files used to create the experiments are available in our
repository https://doi.org/10.5281/zenodo.10684402.

We removed a portion of both test graphs, prior to the main analysis, for
holdout validation. Following the methodology used by Decagon (Zitnik et al.,
2018), the holdout data was created by randomly removing 10% of the edges
belonging to each PSE. Since the two test graphs incorporate the same
polypharmacy data, the same edges were held out from both in order to enable
fair comparison between the two datasets. This holdout set contains 458,061
edges, leaving a total of 5,761,807 edges in the Selfloops graph, and
5,586,830 in the Non-naive graph. During training, performance was measured
using the standard LP metrics employed in LibKGE, namely mean reciprocal rank
(MRR) and hits@k. Performance on the hold-out edges, our measure for overall
quality, was assessed with the same metrics used by the Decagon authors - area
under receiver-operating characteristic curve (AUROC), area under precision-
recall curve (AUPRC), and average precision at 50 (AP@50). These were
calculated individually per side effect type. A flowchart of the methodology
of this paper is displayed in figure 2.

Figure 2: Workflow for the reported experiment. The datasets that are created
from the raw data differ only in their handling of monopharmacy data. The
holdout split is the same for both because they contain the same polypharmacy
data. Both are then put through the workflow on the right hand side of the
‘holdout split’ node, and results compared. The lower diagram offers a ‘zoomed
in’ look at the learning process in LibKGE that is not displayed in the upper
diagram. This learning process happens once for each of the 6 datset-model
combinations. Key: blue = raw data; yellow = software/code; green = processed
data; grey = trained model.

This work was carried out using the computational facilities of the Advanced
Computing Research Centre, University of Bristol
http://www.bristol.ac.uk/acrc. The specific environment was CentOS-7 running
Python 3.8.12 with PyTorch 1.7.1, accelerated with CUDA 11.4 on 4× NVIDIA
GeForce RTX 2080 Ti.

##  3 Results

A

B

C

Figure 3: Out-of-sample prediction performance over 963 side effects as
measured by three metrics: A) Area Under Receiver Operating Characteristic; B)
Area Under Precision Recall Curve; C) Average Precision at 50. Results are
presented for each of 6 experiments, comprising 3 models (ComplEx, DistMult,
SimplE) run on two datasets (Non-naive, Selfloops). Decagon’s performance on
the same data is indicated by the horizontal line.

)  )  )

Results of the assessment are shown in figure 3. All three of our tested KGE
methods outperformed Decagon on both of our constructed graphs, with median
scores improving on Decagon’s performance by 7.62 - 12.2% for AUROC, 10.7 -
16.8% for AUPRC, and 16.1 - 24.5% for AP@50. Remarkably, the rankings of our
six configurations by median score do not change regardless of which metric is
used to order them (table 2).

Model | Dataset | AUROC | AUPRC | AP@50
---|---|---|---|---
SimplE | Selfloops | 0.978 | 0.971 | 1.000
SimplE | Non-naive | 0.973 | 0.965 | 0.980
ComplEx | Selfloops | 0.970 | 0.963 | 0.978
ComplEx | Non-naive | 0.967 | 0.960 | 0.978
DistMult | Selfloops | 0.946 | 0.930 | 0.941
DistMult | Non-naive | 0.938 | 0.921 | 0.932

Table 2: Median performance across 963 polypharmacy side effect types as
measured by three metrics for each of our six experiments.

To provide some context to the performance of our best model, SimplE on the
Selfloops graph, we gathered information about the highest performing methods
that have been presented in the literature. The top 10 of these, as ranked by
AUPRC, are shown in table 3.

Model | AUPRC | Mechanism | Extra data | Citation
---|---|---|---|---
Carletti’s | 0.998 | GAT | None | (Carletti et al., 2021)
MS-ADR | 0.983 | GCN | Enzyme and transporter | (Zhuang et al., 2022)
ADGCL | 0.980 | GNN | Enzyme and transporter | (Zhuang et al., 2023)
GS-ADR | 0.972 | GAT | Enzyme and transporter | (Zhuang and Wang, 2021)
SimplE (this work) | 0.971 | Tensor factorisation | None | -
SimVec | 0.968 | Auto-Encoder | Chemical substructure | (Lukashina et al., 2022)
HyGNN | 0.965 | GCN | Chemical substructure | (Saifuddin et al., 2023)
HLP | 0.965 | GNN | Chemical substructure | (Vaida and Purcell, 2019)
DeepDrug | 0.960 | GNN | Chemical/protein substructure | (Yin et al., 2023)
M2GCN | 0.953 | GNN | None | (Liu et al., 2023)

Table 3: Top 10 performing models by AUPRC on the Decagon dataset. GAT = Graph
Attention Network, GCN = Graph Convolutional Network, GNN = Graph Neural
Network.

On our investigation of the mechanism for including monopharmacy data into the
embedding pipeline, we find that the Selfloops graph achieves better median
results than the Non-naive graph for all three embedding methods. In all
cases, however, this improvement is $\leq$ 0.02, and in the case of the
ComplEx model measured by AP@50, the difference is as low as 0.0004.

Each trial, of the 100 per experiment, ran for a variable number of epochs
that was determined by an early stopping procedure. This procedure checks the
performance of the trial every five epochs starting from 50, ending the trial
if the performance has not improved after two consecutive checks.
Consequently, to compare the speed of computation between experiments we first
standardized the time by dividing it by the total number of epochs run. Figure
4.A shows these values. Although DistMult is the fastest, with a mean time-
per-epoch of 165 seconds on Selfloops and 183 on Non-naive, we observe
surprising consistency overall with only a small gap from DistMult to the
other models. The slowest overall was SimplE on Non-naive, achieving an
average epoch completion time of 219 seconds. Figure 4.B shows the mean number
of epochs (plus standard error) that the trials ran for in each experiment.
The DistMult and SimplE models had comparable epochs-per-trial between the two
datasets, with the former always stopping at the minimum count of 55.
Interestingly, ComplEx was about 30% faster on Non-naive, with a mean epoch
time of 81 seconds versus 115 for the slightly larger graph Selfloops.

A

B

Figure 4: Running time information for the six experiments. Subfigure A is a
violin plot showing the distribution of compute time per-epoch. Subfigure B
shows a bar plot of the mean number of epochs per-trial, with a black bar
representing the standard error. An early stopping procedure was used to
determine the maximum number of epochs in a trial, with a minimum of 55 and a
maximum of 500.

)  )

We observed varying hyperparameter configurations of the optimal models from
each experiment. Either 1vsAll or KvsAll was the chosen method in all cases
for the sampling of negative training examples, with the top (SimplE-
Selfloops) and bottom two (DistMult) models using 1vsAll and the others using
KvsAll. The chosen loss functions matched this pattern exactly - Binary Cross-
Entropy (BSE) was used under the KvsAll strategy, and Kullback–Leibler
divergence (KL) used under 1vsAll. Adamax was the modal optimiser, chosen in
four out of six cases. The two exceptions, SimplE-Selfloops and ComplEx-
Selfloops, used Adam and Adadelta respectively. Dimensionality also varied
between the embeddings, with half the models opting for m = 128, two selecting
m = 256, and just one using m = 512. Since Non-naive initialises embedding
values with principal components of the n-hot monopharmacy side effect vector,
the ‘weight initialisation’ hyperparameter was only relevant for models on the
Selfloops dataset. Within these, ComplEx and DistMult simply drew values from
a normal distribution, whereas SimplE subsequently applied the ‘Xavier’
(Glorot and Bengio, 2010) modification to these values.

##  4 Discussion

The predictive performance reported here ranks among the best to date - in
fact, there are only four models that claim to achieve greater AUPRC (table
3). The highest of these four, Carletti et al.’s model, extends Decagon’s
architecture by adding a Relation Attention Module (RAM) which ”increases the
expressiveness of the network by weighting the contribution of the messages
exchanged over different relations”. This relatively simple modification to
the Decagon method reportedly causes a drastic improvement in prediction
performance, raising both AUROC and AUPRC to 0.998 from 0.872 and 0.832
respectively. However, the authors provide no publicly available repository to
enable reproduction of their work and, whilst they did send archived code on
request, the lack of documentation meant we were unable to run it. The absence
of the original coder from their team also meant that they could provide no
technical assistance and as a result we were unable to reproduce their
reported model performance. The 2nd to 4th ranked methods all come from the
same pair of authors, Luhe Zhuang and Hong Wang of Shandong Normal University.
All three of their approaches use the same dataset - a modified version of
Decagon’s graph, with enzyme and transporter data added and 97 drug nodes
removed. It is unclear which drug nodes were removed and no motivation for
doing so is given. The earliest presented of their models, GS-ADR, focuses on
the concept of signed edges in the graph, with the paper reporting that
consideration of this results in ‘more effective’ feature representation of
drugs (Zhuang and Wang, 2021). MS-ADR, published in 2022, also uses signed
edges but additionally considers four different biomedical ‘views’ (enzyme,
indication, side effect, and transporter) when convolving information across
the graph (Zhuang et al., 2022). ADGCL is the latest of their models and is
similar to MS-ADR but makes use of an Implicit GNN rather than a convolutional
encoder, finding that comparable performance is achieved (Zhuang et al.,
2023). As with Carletti et al.’s work, none of these three papers offer a
publicly available repository of the code used to enable replication of their
findings. Although AUPRC was not measured in their paper, it is well worth
mentioning the work of Li et al., who report an AUROC of 0.999 on the Decagon
dataset by incorporating drug atomic structure into the network (Li et al.,
2023). In this case, drugs are viewed as both singular graphs, where edges are
bonds between atomic nodes, and also bipartite graphs, where potentially
interacting drug pairs have edges from each atom to all atoms in the other
molecule. By incorporating both the ‘inter’ and ‘intra’ drug views into a co-
attention decoder, Li et al. achieve what is arguably the current state of the
art when it comes to predictive performance on the Decagon data.

One large benefit of TF methods is their relative simplicity. ComplEx,
DistMult, and SimplE all have linear complexity with respect to the size of
the embeddings produced (Rossi et al., 2021). Even if we take the results of
the papers mentioned above at face value, they are achieved by employing GNNs
which have at least quadratic complexity, if not cubic or higher (Wu et al.,
2021). This is a disadvantage for two main reasons. Firstly, it hinders
scalability - the inclusion of additional data into the base Decagon graph is
a common theme in the surrounding literature, with seven out of the top ten
performing models (by AUPRC) bringing in at least one more set of edges (3).
One has to imagine that prediction performance could be maximised by including
as many relevant data sources as possible, but as the size of the graph grows
GNNs will become prohibitively expensive in computational terms. Secondly, a
higher cost to training means that optimisation of hyperparameters is more
difficult. Under a grid search scenario, far more hyperparameter combinations
can be tested when the model of interest has reduced complexity, so an optimal
configuration for the model can be found more easily and better prediction can
be achieved at lower cost. This factor may well explain the surprisingly good
performance of the models in the work presented here, as for each model-
dataset pair we were able to choose the best from 100 trials based on the
empirical evidence of their performance on the test split within LibKGE.

On the other hand, GNNs do have a definite advantage when it comes to
inductive reasoning. TF methods, as they currently exist, simply cannot make
inferences about unseen nodes and would require the entire graph to be re-
embedded if predictions about new drugs were required. The Decagon dataset is
far from exhaustive with its 645 drugs, and even if it were, we will continue
to see new drugs developed over time, all of which will need to be assessed
for possible PSEs. This is the largest drawback of the models employed in this
work, and is something that would certainly need to be addressed before they
could be used in any translational sense. Another limitation of both sets of
techniques is the restriction to drug dyads, when, in real life, patients may
often be co-prescribed three or more drugs (UK Government, 2021). Unique
interactions can occur in such cases, e.g. when taking skeletal muscle
relaxants (Chen et al., 2022), so a clinically useful ‘polypharmacy’ side
effect prediction model might need to be able to consider more than just the
‘duopharmacy’ case. Hypergraph embedding models, such as those used by HyGNN
(Saifuddin et al., 2023) and HLP (Vaida and Purcell, 2019), seem the most
promising candidates to solve this issue because, by definition, hyperedges
can connect any number of nodes. However, research in this direction has so
far been scarce, probably due to the lack of available training data available
for such a task.

In this work we observed a slight improvement in predictive performance when
including monopharmacy information as self-looping edges in the graph,
compared to using it to initialise embedding vectors. This improvement held
for all metrics and models, with the exception of ComplEx where an AP@50 score
of 0.978 was achieved on both graphs. The level of improvement was at best
0.02, however, so the self-looping form of the dataset should only be
considered ‘better’ if the overall efficiency of the model is not noticeably
hampered by the addition of the 174,977 extra edges. As shown in figure 4.A,
runtime-per-epoch was not notably different between the two datasets when
using the DistMult or SimplE models. ComplEx did take 34 seconds longer on
average (81 vs 115) when embedding Selfloops, so in this particular case it
may actually be worthwhile to include dimensionally-reduced monopharmacy data
as embedding initialisation points, as was done by Zitnik et al.. Generally
speaking though, we can conclude that the monopharmacy side effect data should
be modelled as graph edges instead when working with TF models.

In summary, we have shown here that TF embedding methods can rival state-of-
the-art GNNs on the task of PSE prediction, even when trained with less data.
Given the nature of the challenge, scalability should be a primary concern for
anyone developing such models, and methods with linear complexity provide a
huge advantage in this regard. They also bring the benefit of lower
electricity consumption and therefore a reduced carbon footprint. This is, of
course, an important consideration in the context of climate change. Zitnik et
al. reported very poor performance of their only tested TF models, with both
RESCAL and DEDICOM being outperformed even by a simple predictor. This may
well explain the corresponding lack of application by later researchers.
However, when set up with optimal hyperparameters and run with modern loss
functions and sampling strategies such as those implemented in LibKGE, we have
demonstrated that they do compete with the state-of-the-art. If the hurdle of
inductive reasoning can be cleared, TF methods could be the most cost-
effective way to tackle PSE prediction.

##  5 Competing interests

T.R.G. receives funding from Biogen and GSK for unrelated research.

##  6 Author contributions statement

Oliver Lloyd: Conceptualization, Methodology, Software, Formal analysis,
Investigation, Data Curation, Writing - Original Draft, Visualization.
Yi Liu: Conceptualization, Writing - Review & Editing, Supervision.
Tom R. Gaunt: Conceptualization, Writing - Review & Editing, Supervision,
Project administration, Funding acquisition

##  7 Acknowledgments

The authors thank Patrick Rubin-Delanchy for his helpful input.

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