ISCB/ISMB 2022

SPARSE: a sparse hypergraph neural network forlearning multiple types of latent combinations toaccurately predict drug–drug interactions
Duc Anh Nguyen1,*, Canh Hao Nguyen1, Peter Petschner1,2 and Hiroshi Mamitsuka1,3
1

Bioinformatics Center, Institute for Chemical Research, Kyoto University, Uji, Japan, 2Department of Pharmacodynamics, Semmelweis
University, Budapest, Hungary and 3Department of Computer Science, Aalto University, Espoo, Finland*To whom correspondence should be addressed.

Abstract
Motivation: Predicting side effects of drug–drug interactions (DDIs) is an important task in pharmacology. The stateof-the-art methods for DDI prediction use hypergraph neural networks to learn latent representations of drugs andside effects to express high-order relationships among two interacting drugs and a side effect. The idea of thesemethods is that each side effect is caused by a unique combination of latent features of the corresponding interacting drugs. However, in reality, a side effect might have multiple, different mechanisms that cannot be represented
by a single combination of latent features of drugs. Moreover, DDI data are sparse, suggesting that using a sparsity
regularization would help to learn better latent representations to improve prediction performances.
Results: We propose SPARSE, which encodes the DDI hypergraph and drug features to latent spaces to learn
multiple types of combinations of latent features of drugs and side effects, controlling the model sparsity by a sparseprior. Our extensive experiments using both synthetic and three real-world DDI datasets showed the clear predictive
performance advantage of SPARSE over cutting-edge competing methods. Also, latent feature analysis over unknown top predictions by SPARSE demonstrated the interpretability advantage contributed by the model sparsity.
Availability and implementation: Code and data can be accessed at https://github.com/anhnda/SPARSE.
Contact: ducanh@kuicr.kyoto-u.ac.jp
Supplementary information: Supplementary data are available at Bioinformatics online.

1 Introduction
A drug-drug interaction (DDI) is a reaction between two drugs, whereby the effects of one drug are modified by the concomitant use of thesecond drug. A DDI might cause side effects, which are unwanted
effects and are responsible for significant patient morbidity and mortality (Magro et al., 2012). Hence, predicting side effects of a DDI, i.e.
DDI prediction, is a very important task to guarantee drug safety.
Using machine learning has emerged as a prominent approachfor DDI prediction, making the prediction fast and highly accurate(Xu et al., 2019; Zitnik et al., 2018). The traditional machine learning methods such as support-vector machines (Kastrin et al., 2018),
logistic regression (Mei and Zhang, 2021) or feedforward neuralnetworks (Wang et al., 2019) use predefined drug features to predict
side effects as labels. However, DDI data have more information.
Particularly, DDIs can be represented by a graph, called a DDIgraph, where nodes are drugs and edges are interacting drugs. The
DDI graph can be learned with graph neural networks (Zitnik et al.,2018). Nonetheless, DDI graphs are only limited to pairwise relationships of drug pairs while there still exist many side effects, which
can be represented by other relationships, such as co-occurrence.
Then, a state-of-the-art generalization of a DDI graph can be a DDI
hypergraph, which can capture higher-order relationships, where
drugs and side effects are both nodes, and each hyperedge is a tripleof a side effect with two interacting drugs.

C The Author(s) 2022. Published by Oxford University Press.
V

On the DDI hypergraph, hypergraph neural networks can beapplied to learn the representations of drugs and side effects altogether. In DDIs, two drugs with totally different properties canstill interact with each other, hence the traditional hypergraph neural networks using similarity assumption on node representations are
not suitable (Feng et al., 2019). Instead, CentSmoothie, a current
cutting-edge hypergraph neural network for DDIs (Nguyen et al.,2021), assumes that each side effect is caused by a unique combination of latent features of the corresponding interacting drugs.
However, in real life, each side effect might have many different
mechanisms (Suleyman et al., 2010) that cannot be reflected in a single combination of drug latent features. Hence, it is necessary tolearn different types of combinations of drug latent features for eachside effect. This is the first problem (P1), which we address in thisarticle.
To solve P1, we borrow one idea of stochastic block models(SBMs) on hypergraphs such that each node (e.g. drug or side effect)
has one or several latent features (Anandkumar et al., 2013; Pal andZhu, 2021) and there exist interactions (associations) of latent features. This method can learn different types of combinations of drug
latent features for each side effect, at once. In addition, to improve
the quality of learned latent features, input node features also can beused (Zhang et al., 2019). However, transformations from input
node features and node relationships in the hypergraphs to latentfeatures might be complex and, especially, non-linear. This is thei333

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Bioinformatics, 38, 2022, i333–i341
https://doi.org/10.1093/bioinformatics/btac250

second problem (P2), which has not been addressed in existing
SBMs and we address in this article.
Moreover, DDI data are sparse (e.g. in the largest DDI dataset,
97.6% of all triples of drug–drug-side effects are not a DDI), suggesting that the model for learning DDIs also should be sparse.
However, recent work on DDIs has not used this sparsity of the data
(Nguyen et al., 2021; Zitnik et al., 2018), which might potentially
impair model performance. This is the third problem (P3), which we
address in this article.
We propose SPARSE, a new model for DDI prediction, to solve
the above three problems. For P1, we assume that there exist drug
and side effect latent features with latent interactions so that eachside effect latent feature interacts with several pairs of drug latentfeatures. For P2, we encode drug features and the DDI hypergraphaltogether in the latent representations using a suitable hypergraphneural network. For P3, we guide the model to preserve the sparsityof the data using a suitable sparsity control. Figure 1 schematicallyillustrates these ideas of our model. That is, the model consists of
two parts: (i) an encoder and (ii) a decoder. The encoder encodes the
input of the DDI hypergraph (e.g. three hyperedges in Fig. 1) withdrug features into latent spaces of drug and side effect latent representations, and interactions of latent features. The decoder reconstructs from the latent spaces the DDI hypergraph with new DDI
predictions (e.g. the dotted hyperedge in Fig. 1). Finally, a sparsity
prior (horseshoe priors in our model) is used to control the sparsityof the latent interactions.
Our extensive experiments first validated the advantage of
SPARSE in terms of prediction performance by using both synthetic
and real-world datasets. Throughout all experiments on prediction
performance, SPARSE achieved better prediction performances thancompeting methods, such as CentSmoothie and SBM. For example,
in the experiment of using the largest real DDI dataset, called
TWOSIDES, SPARSE achieved area under the ROC curve (AUC) of
0.9524 and (area under the precision-recall curve (AUPR) of 0.882,
while CentSmoothie achieved AUC of 0.9348 and AUPR of 0.8749
and SBM achieved AUC of 0.9337 and AUPR of 0.8583. Similarly
when using JADERDDI, another DDI dataset, SPARSE achieved
AUC of 0.9698 and AUPR of 0.7348, while CentSmoothie was
AUC of 0.9684 and AUPR of 0.6044 and SBM was AUC of 0.9428
and AUPR of 0.5963.
We then examined the top prediction obtained by SPARSE,
which is trained by using the whole TWOSIDES. That is, wechecked the number of overlaps between the top 400 predictions by
one method and DDIs in drugs.com (Drugs.com, 2021; Thelwall
et al., 2017), which is a commonly used online web checker for
DDI. We found 98 DDIs in drugs.com out of the top 400

Fig. 1. A schematic illustration of the procedure in the proposed model, SPARSE

D.A.Nguyen et al.
predictions, while by using the same procedure, CentSmoothie
found only 71 DDIs out of the top 400 predictions, implying thatSPARSE can find new DDIs more than competing methods.
Finally, we validated the prediction results by characterizing the
top predictions obtained by SPARSE. In more detail, we checked thebiological properties, such as target proteins, of the top 10 triples ofdrug–drug-side effect, predicted by SPARSE, by using latent featuresconnected to these top 10 predictions. We then found that top predictions can be associated with some biological mechanisms andparticularly with responsible proteins/pathways. These results indicate that our model, SPARSE, can provide high predictive performances as well as latent biological knowledge beneficial to understandthe background behind predicted DDIs.

2 Related work
Machine-learning models for DDI prediction can be divided into
non-graph-based and graph-based ones. For non-graph-based models, the inputs are the predefined feature vectors of pairs of drugs,
the outputs are the corresponding side effects, and the models are
multi-label classifiers, e.g. support-vector machines (Kastrin et al.,2018) or a multilayer feedforward neural network (Wang et al.,2019). Instead of only using predefined drug feature vectors, graphbased methods for DDI use graph neural networks to learn new latent representations of drugs from molecular graphs or DDI graphs.
In molecular graphs, each drug is considered as a graph that nodes
are atoms and edges are connections of atoms (Harada et al., 2020;
Xu et al., 2019). In DDI graphs, DDIs are considered as pairwise
relationships and formulated in the form of a graph where nodes are
drugs and edges are drug interactions with side effects as labels(Zitnik et al., 2018). The latter one has shown to be more effective
for DDI prediction since it can use both pharmacological information and biological information rather than only molecular graphs(Zitnik et al., 2018).
However, one drawback of using graph neural networks on DDI
graphs is that it does not use multiple relationships (labels) at thesame time. Side effects themselves have relationships with eachother, e.g. co-occurrences. Existing work often fixes them as onehot vectors to indicate the presence of the side effects. This representation considers side effects independently, potentially making themodels under-utilize the side effect relationships.
Hypergraph neural networks on DDI overcome the above drawback by learning representations of drug and side effect nodes altogether in latent spaces (Nguyen et al., 2021). DDI is considered as
high-order relationships of drug–drug-side effects in the form of a

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hypergraph where nodes are both drugs and side effects, and eachhyperedge is a triple of two interacting drugs and a side effect causedby the drugs. There are two types of hypergraph neural networks
models on the DDI: similarity based and non-similarity based. The
similarity-based models, e.g. traditional spectral-based hypergraphneural networks, assume that interacting drugs should have similar
representations (Fan et al., 2021; Feng et al., 2019). However, in
DDI, two interacting drugs are not necessarily similar. For nonsimilarity models, the current state-of-the-art method is
CentSmoothie (Nguyen et al., 2021) that assumes that the representation of a side effect can be represented by a combination of latentfeatures of two drugs causing the side effect. However,
CentSmoothie cannot deal with multiple combinations of latent features at the same time.
In order to deal with multiple combinations of latent features,
one possible approach is to use the idea of SBMs, which can beapplied to hypergraphs, with each node belonging to several latentfeatures (groups) and associations of latent features (groups)(Anandkumar et al., 2013). However this has not been applied to
DDI hypergraphs, and more importantly, SBM is based on linear assumption, while DDI can be generated through more complex relations to be represented by non-linearity.
Many studies have shown the benefits of sparsity regularization,
which is a commonly used method to achieve sparsity of models, especially on noisy and sparse data (Carvalho et al., 2009; Tibshirani,1996). In a Bayesian viewpoint, sparsity regularization can beunderstood as a result of using sparse prior distributions. A state-ofthe-art method for sparsity regularization is to use horseshoe priors(Carvalho et al., 2009; Piironen and Vehtari, 2017). It shows an advantage in comparison with traditional Laplace prior (Lasso regularization) (Tibshirani, 1996) in that the horseshoe prior allows toshrink in both directions: no shrinkage for important features andcomplete shrinkage for non-important (noise) features. A comparable shrinkage prior with the horseshoe prior is the spike-and-slab
prior (Hoeting et al., 1999). However, the spike-and-slab prior is adiscrete prior that requires the Markov chain Monte Carlo samplingfor optimization, which is not effective for large-scale datasets like
DDI.

near 1 and disappears near 0, meaning that it always shrinks all features, containing important ones. In contrast, the density of ji;j;kwith the horseshoe prior has two peaks at 0 and 1, meaning that thehorseshoe prior allows two kinds of shrinkage: no shrinkage tomaintain important features and complete shrinkage to remove unimportant features.

3 Materials and methods

3.3 Proposed model

3.1 BackgroundWe propose SPARSE: a sparse model for learning multiple types oflatent combinations of side effects and drugs to predict DDIs. Ourmodel follows an auto-encoder framework with two parts: an encoder and a decoder. The encoder encodes the DDI hypergraph withdrug node features to latent spaces with latent representations ofdrugs and side effects (H), and interactions of latent features (B).
The decoder aims to reconstruct the DDI hypergraph with new predicted hyperedges from H and B. In the following parts, we first present our latent interaction assumption with sparsity for theinteractions of drugs and side effects, and then we describe the encoder and decoder.

We recall definitions for horseshoe priors and n-mode tensor product for 3D tensors, which will be used later.
3.1.1 Horseshoe priors
We summarize the horseshoe prior (Carvalho et al., 2009), a stateof-the-art prior for sparsity control, for a non-negative 3D tensor:
1 K2 K3
. The idea of the horseshoe prior is that
B ¼ fBi;j;k g 2 RK
0þeach Bi;j;k follows a normal distribution with the same zero mean
and a different variance. Each variance has two parts: one is a global
parameter sharing among all variances to decide the sparsity of B
and one is a local parameter to decide the magnitude of each variance by using a heavy tail distribution with the half-Cauchy distribution. In more detail:
Bi;j;k  Nð0; s2 K2i;j;k Þ;(1)

Ki;j;k  Cþ ð0; 1Þ;(2)þ

where s is a global parameter for sparsity, and C ð0; 1Þ is a halfCauchy distribution defined by: pðKi;j;k Þ ¼ 2p 1þK1 2 for Ki;j;k  0.
i;j;kBoth the horseshoe prior and Laplace prior (for Lasso regularization) are shrinkage priors such that by using priors, values of fea^ i;j;k betures tend to be shrunk (Piironen and Vehtari, 2017). Let B
the optimal values without priors, then the optimal values having^ i;j;k , where 0  ji;j;k  1 ispriors has the form: B i;j;k ¼ ð1  ji;j;k ÞBa shrinkage factor depending on the priors. With Laplace prior(Lasso regularization), the density of ji;j;k tends to be a constant

3.1.2 N-mode tensor product
The n-mode tensor product can be understood as a generalization of
the matrix dot product in high-dimension that the product is processed at the nth dimension. Considering in the 3D space with a tensor: B 2 RK1 K2 K3 and a matrix H 2 RTKn ; n 2 f1; 2; 3g, the nmode product of B and H is denoted by Bn H and is defined foreach of n ¼ 1, 2 and 3, as follows:ðB1 HÞt;j;k ¼

K1
X

Bi;j;k Ht;i jt ¼ 1 . . .T; j ¼ 1 . . .K2 ; k ¼ 1 . . .K3 ; (3)i¼1ðB2 HÞi;t;k ¼

K2
X

Bi;j;k Ht;j jt ¼ 1 . . .T; i ¼ 1 . . .K1 ; k ¼ 1 . . .K3 ; (4)j¼1ðB3 HÞi;j;t ¼

K3
X

Bi;j;k Ht;k jt ¼ 1 . . .T; i ¼ 1 . . .K1 ; j ¼ 1 . . .K2 :

(5)

k¼1

3.2 Problem formulation: DDI prediction
We formulate the DDI prediction problem as follows.
Input: Given a DDI hypergraph: G ¼ ðV; EÞ; V ¼ VD [ VS ;E  VD  VD  VS , where VD is a set of drug nodes, VS is a set ofside effect nodes (given u; v 2 VD ; t 2 VS , two triples (u, v, t) andjV jK(v, u, t) are the same). The drug node features are FD 2 R0þD 0 andjV jjV j

the side effect node features are one-hot vectors: FS 2 R0þS S :Output: For e ¼ ðu; v; tÞ 2 VD  VD  VS , calculate a prediction
score for interaction m(e).

3.3.1 Latent interaction assumption
To model DDIs, we suppose that there exist latent spaces with drug
latent features and side effect latent features where DDIs occur. The
latent interaction assumption is that two interacting drugs cause aside effect if there exist a pair of drug latent features of the two
drugs that interact with a latent feature of the side effect.
In detail, the formulation for the latent interaction assumption
can be described as follows. Let LD ¼ f1; . . . ; KD g and LS ¼
f1; . . . ; KS g be the sets of indices of latent features of drugs and side
effect with KD and KS be the numbers of latent features. Let B 2
D KD KS
be a 3D tensor representing interactions of latent featuresRK0þ
of drugs and side effects. The set of interacting latent features is:
A ¼ fði; j; kÞ 2 LD  LD  LS jBi;j;k > 0g.
Considering a triple of two drugs and one side effectsKSDe ¼ ðu; v; tÞ 2 VD  VD  VS . Let hd ðuÞ; hd ðvÞ 2 RK
0þ ; h ðtÞ 2 R0þ be
the vectors representing the presence of latent features of the two drugs
and the side effect, respectively. Let gu ¼ fi 2 LD jhd ðuÞi > 0g; gv ¼

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SPARSE

D.A.Nguyen et al.

Table 1. Statistics of three real datasets
DatasetTWOSIDESCADDDIJADERDDI

No. ofDrugs

No. of Side
effects

557
587
545

964
969
922

No. of Drug–drugpairs

No. of Drug–drug-side effects(DDIs)

49,677
21,91836,9293 606 046
373 976
222 081fi 2 LD jhd ðvÞi > 0g and gt ¼ fi 2 LS jhs ðtÞi > 0g be the sets of latentfeatures of u, v and t, respectively.
Under the latent interaction assumption, u interacts with v tocause t if:
gu  gv  gt \ A 6¼ 1;(6)or with tensor product formulation:B1 hd ðuÞ2 hd ðvÞ3 hs ðtÞ > 0:

(7)

In practice, we can change the value 0 on the right side ofEquation (7) to a positive threshold. Equation (6) will be used togenerate synthetic data in the experimental section. Equation (7)will be used in the decoder of the model.
Sparsity property
We first define formulations for sparsity measures of the DDI
data and the latent interactions using the percentages of noninteractions. Let sd be the sparsity of the hypergraph G:
sd ¼ 1 

2jEj:
jVD jðjVD j  1ÞjVS j(8)

The sparsity of the latent interactions sl is defined as the percentage of the number of non-interacting triples of the latent features per
the total number of all triples of the latent features.
sl ¼ 1 

2jAj
jLD j2 jLS j

Avg. no. of side effects/No. ofdrug–drug Sparsity
pairs
(%)
72.58
17.06
6.01

Since entries of B are assumed to be randomly sampled according to a
uniform distribution, the number of interactions when B have jBj0 ¼
ð1  sl ÞK2D KS non-zero entries follows a binomial distribution
BinomialðjBj0 ; p1 Þ.

With the assumption that the hypergraph is generated from thisgenerative process, the expected number of non-zero data points(the number of hyperedges) becomes jBj0 :p1 ¼ ð1  sl Þ:n2u nt .
The
expected 2 sparsityofthehypergraphbecomes
ð1s Þn nEðsd Þ ¼ 1  K2l K u t ¼ 1  ð1  sl Þp1 .
D

SThis result leads to Eðsd Þ > sl p1 . It shows a relationship between
the sparsity of the model (sl) and the expected sparsity of the data
generated by the model (Eðsd Þ). It shows that the model can besparse but cannot be as sparse as we want. It can be a hint on setting
sparsity of the model in learning processes.
3.3.2 Encoder
For the encoder, we use a hypergraph neural network with message
passing (Yadati, 2020) to encode the input hypergraph and nodefeatures into latent spaces with node latent representations H and latent interactions B (for simplicity, B can be considered as a free parameter to learn).
jV jKD

H ¼ ðHd ; Hs Þ ¼ gw0 ðG; FÞ 2 R0þD

DDI data are sparse as per statistics in Table 1. It is shown that
97.6% and 99.87% of all triples are non-interacting in TWOSIDES
and JADERDDI, respectively.
The motivation for us to use sparse models is that sparse models,according to statistical learning theory, are usually more reliable
models if they could fit the training data well (Hastie, 2015). As oursparse models have sparse interactions among latent features, wewill prove that they tend to generate sparse data and are suitable for
DDI data. We show a relationship between sparsity of the models
and sparsity of data generated by the models, which are the ones
that best fit the models, as follows.
Property 1:Assume that the DDI data are generated from the true generation model according to formula (7). Assuming that each drug and side
effect has exactly nu and nt non-zero latent features, respectively. Then,
there exists a relationship between the sparsity of the model and the
expected sparsity of the generated data as follows:n2 ntEðsd Þ ¼ 1  ð1  sl Þ 2u :KD KS

jV jKS R0þS;D KD KS
B ¼ fw1 ðG; FÞ 2 RK
;
0þ

(9)

:

97.6
99.77
99.83

(11)
(12)

where gw0 and fw1 are hypergraph neural networks based on message passing (Yadati, 2020) with parameters to learn w0, w1, Hd ¼
sD
fhd ðuÞ 2 RK
0þ ju 2 VD g (node representations of drugs) and H ¼
S
jt
2
V
g
(node
representationsofsideeffects).
The
forfhs ðtÞ 2 RKS
0þ
mulation of each message passing layer has the following form:
n
o 


 
; (13)
hðlþ1Þ ðaÞ ¼ r TMðlÞ a; hðlÞ ðaÞ; ðb; hðlÞ ðaÞÞ
b2e

e2Na

ðlÞ

where h ðaÞ is the representation of node a 2 VD [ VS at layer (l), r
is an activation function, T is an aggregation function (e.g. an average function), Na ¼ fe 2 Eja 2 eg and MðlÞ is a message passing
function at layer (l) to pass information from neighbor nodes inhyperedge e to a:(14)MðlÞ a; hðlÞ ðaÞ; fðb; hðlÞ ðbÞÞgb2e ¼
XMðlÞ ðcðaÞ; cðbÞ; hðlÞ ðaÞ; hðlÞ ðbÞÞ;

(15)

b2e

(10)

where MðlÞ is a two-layer feedforward neural network, cðbÞ ¼
1 if b 2 VD and cðbÞ ¼ 1 if b 2 VS are the node types.

Proof:For a pair of drug u, v to cause side effect t, thenB1 hd ðuÞ2 hd ðvÞ3 hs ðtÞ > 0: This means that there is at least one nonzero entry of B corresponding to latent features of u, v and t. Since there
are exactly n2u nt possible entries of B corresponding latent features of u,v and t, then the probability of a uniform sampling of entries of B ton2 ncorresponding to these latent features is p1 ¼ K2u Kt . This is the probabilityD

3.3.3 Decoder
The reconstruction of the hypergraph is from the latent interaction
assumption. The likelihood to reconstruct each triple e ¼ ðu; v; tÞ 2
VD  VD  VE follows a Gaussian distribution:2 !
1
1 iðeÞ  mw0 ;w1 ðeÞÞ;(16)pðejB; HÞ ¼ pﬃﬃﬃﬃﬃﬃ exp 
2
r
r 2p

S

of having an interaction among the features (that generates a side effectdata point).

where iðeÞ ¼ 1 if e 2 E, i(e) ¼ 0 if e 2 E ¼ VD  VD  VS =E, and
mw0 ;w1 ðeÞ is the mean value for the latent interaction of e:

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mw0 ;w1 ðeÞ ¼ B1 hd ðuÞ2 hd ðvÞ3 hs ðtÞ:(17)

Equation (17) is also the score for the interactions of triples (u, v,t) used for prediction. The likelihood for the decoder is:YpðGjB; HÞ ¼
pðejB; HÞ:(18)e¼ðu;v;tÞ2VD VD VE

3.3.4 Objective function
The objective function for our method is to maximize a posterior of
the model. The objective function consists of two parts: one for loglikelihood of the model and one for the prior for sparsity control.
D KD KS
be the horseshoe prior parameter for B and s be
Let K 2 RK0þ
the hyperparameter for the global sparsity of the horseshoe prior.
We have the following objective function:argmax log pðGjB; HÞ þ log pðBjK; sÞ þ log pðKÞ ;|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}
B;H;K0 |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}
log likelihood(19)

log of horseshoe priorwhere log pðGjB; HÞ is the log-likelihood of Equation (18) with H in
Equation (11) and B in Equation (12), and log pðBjK; sÞ þ log pðKÞis the logarithm of the horseshoe prior:log pðBjK; sÞ ¼

X 12

Bi;j;ksKi;j;klog pðKÞ ¼

!2

X

þ

log

X

log K1i;j;k þ const;

1:
1 þ K2i;k;j

(20)

(21)

We then use stochastic gradient descent libraries in the PyTorchframework for optimizing Equation (19).
We also consider two other variants of SPARSE: SPARSEO for notusing any sparsity prior and SPARSEL for using Laplace prior (Lasso
regularization), to examine the effect of using the horseshoe prior.

1. Generating latent interactions. Given sets of indices of drug latent features: LD ¼ f1; 2; . . . ; KD g and side effect latent features:LS ¼ f1; 2; . . . ; KS g.
a. Initialize a set of latent interactions A ¼ 1.
b. For each k 2 LS :1. Sample the number of drug latent feature pairs:
nk ¼ RandomIntegerðMÞ, where M is the maximum
number of pairs.
2. Sample nk pairs ði; jÞ 2 LD  LD . For each pair (i, j):
A ¼ A [ fði; j; kÞg.
2. Generating drug interactions:a. Generate drug and side effect latent features. Assume that
there are VD drugs and VS side effects.
1. For each drug u 2 VD :i. Sample the number of drug latent features:nu ¼ RandomInterðN1 Þ, where N1 is the maximum
number of drug latent features.
ii. Sample gu  LD ; jgu j ¼ nu . For drug feature vectors F:D cmu 2 RK0; mu ½i ¼ 1 if bi=cc 2 gu ;0þ ; mu
fu ¼ Gaussianðmu ; dÞ. F ¼ F [ fu .
2. For each side effect t 2 VS , sample the number of side effect latent feature nt ¼ RandomInterðN2 Þ and Sample
gt  LS ; jgt j ¼ nt .
b. Generating true triples E . Initialize E ! 1. For
ðu; v; tÞ 2 VD  VD  VS , if gu  gv  gt \ A! ¼ 1 then (u, v,t) is a true triple: E ¼ E [ ðu; v; tÞ.
c. Adding noise:
1. For each e 2 E , replace e by a random sample e0 2 E ¼
VD  VD  VS =E with probability r. The final set of triples of drug–drug-side effects is E.
Finally, we have a synthetic dataset with triples of drug–drugside effects E and drug feature vectors F.

4 Experimental results
We validated SPARSE in two scenarios: synthetic data and real data.
On the synthetic data, assuming that the data are generated from thelatent interactions, we examined if SPARSE can recover the latentinteractions under changing hyperparameters of data: the number oflatent features, sparsity and amount of noise. On real data, wechecked the prediction performance of SPARSE in comparison withstate-of-the-art DDI prediction methods by using three real-world
DDI datasets. Additionally, we evaluated if the top unknown predictions by SPARSE can be related to biological phenomena like functions and mechanisms.
For all experiments, we used 20-fold cross-validation by dividing
hyperedges into 20-folds, keeping the same number of hyperedges(side effects) in each fold. We reported the mean and standard deviation of the two commonly used measures AUC and AUPR. Also, allreported results were the highest performances through grid searchesof hyperparameters. There were three hyperparameters for gridsearches for SPARSE: (i) latent feature sizes. The tested values were30, 40, 50 and 60. We set the same size for all layers. (ii) Global
sparsity s. The tested values were 0.01, 0.02, 0.03, 0.05 and 0.1 and(iii) the numbers of neural layers. The tested values were 1, 2 and 3.
The hyperparameter values obtained were 50 for the latent featuresize, s ¼ 0:02 for TWOSIDES and s ¼ 0:01 for CADDDI and
JADERDDI, and the number of neural layers was 2. All experiments
were run in a computer with Intel Core I7-9700 CPU, 8 GB GeForce
RTX 2080 GPU and 32 GB RAM.

4.1 Synthetic data
4.1.1 Data generation
The generation process for synthetic data consists of two steps:(i) generating latent interactions and (ii) generating triples of interacting drug–drug-side effects from the latent interactions, as follows.

4.1.2 Experiments
The synthetic data has five hyperparameters: the number of drugs,
the number of side effects, the number of latent interactions, data
sparsity and the amount of noise (noise rate). We evaluated our
methods by changing one hyperparameter, fixing the other four.
The hyperparameters changed are (i) number of latent features, (ii)data sparsity and (iii) noise rate.
1) Changing the number of latent features.
Setting: VD ¼ 400; VS ¼ 300, noise rate r ¼ 0.01. We changed KD ¼
KS 2 f5; 10; 20; 30; 40; 50g. For each (KD, KS), we selected N1, N2
and M such that the sparsity of the generated data is kept at 0.98.
Compared methods: We compared four methods: SPARSEO (nosparsity control), CentSmoothie (Nguyen et al., 2021), a similaritybased hypergraph neural network, HPNN (Feng et al., 2019) and
SBM on hypergraph (Anandkumar et al., 2013).
Results: Figure 2a shows the results, where SPARSEO achieved
the highest performances among the compared methods in all cases.
We had the following two findings:
1. For the small number of latent features, the performance of
CentSmoothie was close to SPARSEO (both AUC and AUPR
were around 0.99 under KD ¼ KS ¼ 5). However, by increasing
the number of latent features, the performance gap between
SPARSEO and CentSmoothie also increased (gaps in AUC and
AUPR were around 0.01 and 0.03, respectively, when
KD ¼ KS ¼ 50). This result implies that CentSmoothie was unable to distinguish latent interactions clearly for a large numberof latent interactions, while SPARSEO worked better for capturing multiple latent interactions.

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Fig. 2. Performances on synthetic data, when changing (a) #latent features, (b) sparsity and (c) amount of noise

Table 2. Comparison of performances of the methods on the real DDI datasets
MethodMRGNN
Decagon
SpecConv
HPNN
SBM
CentSmoothie
SPARSEOSPARSELSPARSETWOSIDESCADDDIJADERDDI

AUC

AUPR

AUC

AUPR

AUC

AUPR

0.8452 6 0.0036
0.8639 6 0.0029
0.8785 6 0.0025
0.9044 6 0.0003
0.9337 6 0.0002
0.9348 6 0.0002
0.9511 6 0.0002
0.9517 6 0.0001
0.9524 6 0.0001

0.8029 6 0.0039
0.8094 6 0.0024
0.8256 6 0.0022
0.8410 6 0.0007
0.8583 6 0.0004
0.8749 6 0.0013
0.8811 6 0.0001
0.8815 6 0.0002
0.8820 6 0.0002

0.9226 6 0.0015
0.9132 6 0.0014
0.8971 6 0.0055
0.9495 6 0.0004
0.9588 6 0.0006
0.9846 6 0.0001
0.9824 6 0.0009
0.9859 6 0.0007
0.9837 6 0.0010

0.7113 6 0.0031
0.6338 6 0.0029
0.6640 6 0.0014
0.7020 6 0.0018
0.8170 6 0.0008
0.8230 6 0.0019
0.8773 6 0.0014
0.8797 6 0.0010
0.8843 6 0.0012

0.9049 6 0.0009
0.9099 6 0.0012
0.8862 6 0.0025
0.9127 6 0.0004
0.9428 6 0.0006
0.9684 6 0.0004
0.9692 6 0.0007
0.9694 6 0.0011
0.9698 6 0.0008

0.3698 6 0.0019
0.4710 6 0.0027
0.5162 6 0.0047
0.5198 6 0.0016
0.5963 6 0.0018
0.6044 6 0.0025
0.7230 6 0.0008
0.7276 6 0.0017
0.7348 6 0.0018

2. The performances of SBM were lower than both CentSmoothie
and SPARSEO , since SBM did not use the node features, which
decreased the performance. HPNN, a similarity-based hypergraph neural network, had the lowest performance since the two
drugs of a DDI do not necessarily have similarity in the data generated from latent interactions. Overall, these results indicatedthat SPARSEO can recover the latent interactions better than theother methods.
2) Changing data sparsity.
Setting: VD ¼ 400; VS ¼ 300; KD ¼ KS ¼ 50; N1 ¼ N2 ¼ 4 and
r ¼ 0:01. We changed M in f50; 40; 30; 20; 10; 5g, resulting in data
sparsity in f0:6; 0:75; 0:88; 0:92; 0:95; 0:98g, respectively.
Compared methods: Since in the previous experiment, SPARSEO
outperformed the compared methods already, we compared
SPARSE and two variants SPARSEL and SPARSEO (please see theend of Section 3.3.4) to check the effect of sparse priors.
Results: Figure 2b shows the results, where SPARSE achieved thehighest performance, followed by SPARSEL and SPARSEO . In particular, the performance advantage by SPARSE using sparsity control was clearer with higher sparsity. These results indicate that thehorseshoe prior is suitable for learning sparse data.
3) Changing the amount of noise.
Setting: VD ¼ 400; VS ¼ 300; KD ¼ KS ¼ 50; N1 ¼ N2 ¼ 4, M ¼ 1(keeping the data sparsity of 0.98). We changed noise r in
½0; 0:01; 0:05; 0:10; 0:20.
Compared methods: We again compared SPARSE with two variants SPARSEL and SPARSEO to examine the effectiveness of thesparse priors to deal with noise.
Results: Figure 2c shows the results, where again SPARSE
achieved the highest performances among the three methods for all
different amounts of noise. When there are no noises, the performances of the three methods were very close to each other. However, as
the amount of noise is increased, the advantage of SPARSE over theother two methods became clearer. For example, when the amount ofnoise is 20%, the gap between SPARSE and SPARSEL reached around
0.07, and the gap between SPARSE and SPARSEO was around 0.1.

These results suggest that the horseshoe prior could deal with noise
better than the Laplace prior and the case with no sparsity prior.

4.2 Real data
4.2.1 Data description
We used three real-world datasets for DDI, namely TWOSIDES(Tatonetti et al., 2012), CADDDI and JADERDDI. To our knowledge, TWOSIDES is the largest benchmark dataset for DDI. The
other two datasets, i.e. CADDDI and JADERDDI, were generated
from Canada Vigilance Adverse Reaction Reports and JapaneseAdverse Drug Event Reports, respectively, in the same manner as
the way that TWOSIDES was generated from the adverse events
reported to US Food and Drug Administration (Nguyen et al.,2021). For all datasets, we only chose small molecular drugs, which
can be found in DrugBank. Also, we focused drugs appearing in
more than five interactions (hyperedges) in each dataset. For eachdrug, we used a feature (binary) vector, with the size of 2329, consisting of 881 substructures and 1448 interacting proteins. Table 1
shows a summary statistics of the three real benchmark datasets,
TWOSIDES, CADDDI and JADERDDI.

4.2.2 Predictive performance experiments
Compared methods: For our method, we used SPARSE and two variants SPARSEO and SPARSEL . We further used five methods as competing methods against SPARSE. These competing methods were
CentSmoothie (Nguyen et al., 2021), the traditional similarity-based
hypergraph neural network (HPNN) (Feng et al., 2019), two DDIgraph-based graph neural networks: Decagon (Zitnik et al., 2018)
and SpecConv (Kipf and Welling, 2016) and, a molecular graphbased graph neural network, MRGNN (Xu et al., 2019). Decagon
and CentSmoothie provide available codes, and we ran them withthe recommended settings. For MLNN, MGRNN, SpecConv,
HPNN and SBM, we implemented them and did a grid search forfinding the best hyperparameter values.
Results—Cross-validation predictive performance: Table 2
shows AUC and AUPR results of all competing methods. From thistable, SPARSE and two variants (SPARSEL and SPARSEO ) achieved
the highest performances, followed by CentSmoothie, SBM and

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HPNN. On the other hand, the performances of SpecConv,
Decagon and MRGNN were significantly lower. Amazingly,
SPARSEO (SPARSE without any sparsity prior) achieved still better
performance over CentSmoothie, particularly in AUPR. There wasonly one case (CADDDI), where the AUC of SPARSE was slightly
smaller than that of CentSmoothie. We then ran t-test over the prediction results of these two methods, to examine the significance ofthe difference between CentSmoothie and SPARSE. The resultant Pvalue of t-test was 0.057, indicating that the performance advantage
of CentSmoothie over SPARSE was NOT significant, under the
regular significance level of 0.05. Also, it has to be noted that AUPR
is more useful than AUC for imbalanced data (Saito andRehmsmeier, 2015), which can be often seen practically. We emphasize that DDI is a typical example of this situation. In fact, the
AUPR performance gap between SPARSEO and CentSmoothie
reached around 1%, 5% and 12% in TWOSIDES, CADDDI and
JADERDDI, respectively. The performance gap in JADERDDI is especially sizable. This might be caused by the high sparsity of
JADERDDI (see Table 1).
These results suggest that the latent interaction assumption in
SPARSE is more reasonable and suitable for DDI prediction than
CentSmoothie and the other competing methods. Among SPARSE,SPARSEL and SPARSEO , SPARSE achieved the highest performance. Note that the performance gap between SPARSE andSPARSEL in AUPR became clearer for more sparse data: e.g. onlyaround 0.1% for TWOSIDES, while the gap reached around 1% for
CADDDI and JADERDDI. Hence, we can see that with more sparsedata, the horseshoe prior had advantage over Laplace prior and also
the case with no sparsity prior.
Results—Unknown DDI prediction performance: We evaluatedthe predictive ability of unknown DDIs. That is, we first trained amodel by using the whole TWOSIDES data (the largest dataset),
then predicted the scores of unknown triples (drug–drug-side effect),and finally sorted the predicted triples in the descending order of thescores. We focused on the top 400 predictions of each method andchecked the overlap with the DDIs stored in drugs.com (Drugs.com,
2021; Thelwall et al., 2017), a commonly used web checker for
DDIs. Table 3 shows the number of overlaps between the DDIs in
drugs.com and the top 400 predictions. SPARSE found 98 overlapped DDIs with drugs.com, this number being the highest and followed by CentSmoothie with 71 and HPNN with 48.
4.2.3 Case studies: interpretation of top 10 unknown predictions
SPARSE is an SBM with latent features for drugs, side effects, and interactions. In particular, the model has connections between latent drug
features and latent interactions. Thus from the trained model, we canextract the drug features, which are most associated with each drug latent feature and further extract the drug features most associated witheach latent interaction through the corresponding latent drug feature.
This means that we can retrieve drug features of a DDI if we can connect the DDI with the latent interactions. Algorithm 1 shows thepseudocode of this procedure (with T ¼ 20 in our cases). SPARSE is asparse model, which allows only a limited number of latent interactions
and eventually allows to extract only a limited number of drug features.
This is a sizable advantage of SPARSE for understanding the biological/
chemical background behind predicted DDIs.For case studies, we
extracted drug features (such as protein/pathway names) of the top
unknown DDI predictions by using SPARSE, which was trained by
the entire TWOSIDES. Table 4 shows the top 10 predictions (out of
the 400 predictions in the experiment of the previous section) with theobservable features associated with latent drug features [fifth column
from the right-hand side. In this column, ‘Not clear’ means that to ourTable 3. Number of overlaps with DDIs in drugs.com for the top
400 predictions
MethodSPARSECentSmoothie
HPNN#Overlaps
98
71
48

Algorithm 1 Extracting potentially associated drug features
D KD KS
; Hd ¼ fhd ðuÞgInput: Learned parameters B 2 RK
0þjVD jKD
jVS jKSss
2 R0þ
; H ¼ fh ðuÞg 2 R0þ
, drug features matrix
jV jK
Fd ¼ ff d ðuÞg 2 R0þD O , a predicted triple (u, v, t), hyperparameter T
Outout: Associated drug features for the triple//Extract drug features for each latent featurefor k 2 1 . . .KD do
ak ¼ fjjCorrelationðHd:;k ; Fd:;j Þintop Tg
end for//Calculate non-zeros latent interactions.
is the pairwise
dot product,is the outer product.

ss ¼ B ðhd ðuÞ hd ðvÞ
tt ¼ fði; j; kÞjssi;j;k > 0g

hs ðtÞ

//Extract potentially associated drug features for the triple
Re
1
for ði; j; kÞ 2 tt doRe
Re [ fðNon  zero features of f d ðuÞ 2 ai ;Non  zero features of f d ðvÞ 2 ak Þg
end for
Return Re

current understanding of the potential DDI mechanisms, we could not
explain the corresponding low-level (molecular level) background, although our algorithm could find associated drug features], the target
protein of the corresponding drug using DrugBank (sixth column) andthe corresponding reference to each DDI (seventh column). The top
predictions are likely to be similar to each other, since the similar triples are likely to have similar scores. In fact the top predictions inTable 4 have large overlaps, but from the table, we could find the following four points:
1. The fourth and fifth predictions show the cases, where SPARSE
could specify target proteins precisely, confirming the high credibility of these predictions and more importantly, approving thehigh ability of SPARSE for detecting unknown DDIs.
2. The first, second, third and sixth predictions show the cases,
where SPARSE could identify possible interacting protein groups(fourth column), not necessarily directly associated with thedrugs, indicating that SPARSE allows suggesting novel interactions as well as potential target proteins.
3. The validity of the seventh, eighth, ninth and tenth predictions
might be understood by high-level views, like the connection between vision and dizziness/sedation. This result implies thatSPARSE can predict probable interactions, which however cannot be straightforwardly inferred from low-level data.
4. Entirely, we could find relevant references for all top 10 predictions(Baldo, 2018; Fagiolini et al., 2004; Rho et al., 1997;
Venkataraman et al., 2014), giving plausibility of these prediction
and at the same time an additional layer of evidence for the usefulness of SPARSE in practical settings. To facilitate medical researchand confirmation of our findings by subsequent clinical or preclinical studies, we provide the potential mechanisms as a Supplementary
Material for our top predictions. Also, we discuss below the main
biological mechanism for a predicted top 10 interaction:
Naratriptan, Sertraline and abnormal ECG: Sertraline belongs to
the selective serotonin reuptake inhibitor class antidepressants.

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Table 4. Top 10 new (unknown) predictions with potentially associated latent features of proteins and extracted proteins
No.

Drug A

Drug BSide effectObservable features associated with latent features

1

Ciprofloxacin

Mefenamic acid

Abdominal distension

Cytochrome enzymes—

2

Naratriptan

Oxycodone

Abnormal ECG—

3Naratriptan

TramadolAbnormal ECG

4Naratriptan

Sertraline

Abnormal ECG

Serotonin transporters
and receptors
Serotonin transporters
and receptors
Serotonin transporters
and receptors

5

Naratriptan

Paroxetine

Abnormal ECG

Serotonin transporters
and receptors

6
7
8
9
10

Trihexyphenidyl
Carisoprodol
Buspirone
Oxycodone
Carisoprodol

Thiothixene
Orphenadrine
Orphenadrine
Orphenadrine
Zaleplon

Abnormal EEG
Abnormal vision
Abnormal vision
Abnormal vision
Abnormal vision

Dopamine receptors
Not clear
Not clear
Not clear
Not clear

Members of this class inhibit the reuptake of the neurotransmitter serotonin into cells (Ritter et al., 2019). Through this inhibition, sertraline increases serotonin levels outside of the cells andallows serotonin to remain longer at its site of action.
Naratriptan is known to cause heart-related side effects throughserotonin receptor agonism at serotonin type 1 receptors (Dodick
et al., 2004; Ritter et al., 2019). Therefore, the predicted side effect can be a direct consequence of sertraline increasing the levelof endogenous serotonin and naratriptan acting at serotoninreceptors in the heart, with the resulting changes visible in electrocardiogram recordings.

5 Conclusion and discussion
We have proposed SPARSE to learn the latent representations of drugs,
side effects and interactions, through hypergraph neural networks.
SPARSE addresses three important issues of state-of-the-art DDI prediction, which have not been addressed by any other methods. Extensive
empirical validation using both synthetic and real data showed thatSPARSE outperformed all current, cutting-edge methods for DDI prediction, verifying the effectiveness of multiple types of latent interaction
assumptions and the sparsity control setting of SPARSE.
Possible future work is to generalize SPARSE for higher-orderdrug interactions with multiple drugs. Another interesting direction
might be to apply SPARSE to other sparse, high-dimensional data inbioinformatics.

Funding
This work was supported in part by Otsuka Toshimi Scholarship Foundation(to D.A.N.); MEXT KAKENHI [grant number 22K12150] (to C.H.N.); theJapan Society for the Promotion of Science (Postdoctoral Fellowships for
Research in Japan, standard program) [P20809] (to P.P.); and MEXT
KAKENHI [grant numbers: 19H04169, 20F20809, 21H05027 and
22H03645] and the AIPSE program of the Academy of Finland (to H.M.).
Conflict of Interest: none declared.

References
Anandkumar,A. et al. (2014) A tensor approach to learning mixed membership community models. J Mach Learn Res., 1, 2239–2312.

Extracted proteins ofdrugs from DrugBank—
5-Hydroxytryptamine receptor 1B (and 1D) andsodium-dependent
serotonin transporter
5-Hydroxytryptamine receptor 1B (and 1D) andsodium-dependent
serotonin transporter
—
—
—
—
—

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