# Erasing the past of photons

Jinxian Guo1,4, Qizhang Yuan2,∗, Yuan Wu3,4, Weiping Zhang1,4,5 1School of
Physics and Astronomy, and Tsung-Dao Lee Institute, Shanghai Jiao Tong
University, Shanghai 200240, P. R. China  2Department of Physics, Mathematics
and Science College of Shanghai Normal University, Shanghai 200234, P. R.
China  3State Key Laboratory of Precision Spectroscopy, Quantum Institute of
Atom and Light, Department of Physics, East China Normal University, Shanghai
200062, P. R. China  4Shanghai Research Center for Quantum Sciences, Shanghai
201315, P. R. China  5Collaborative Innovation Center of Extreme Optics,
Shanxi University, Taiyuan, Shanxi 030006, P. R. China<EMAIL_ADDRESS>
###### Abstract

Photons have been proved to hide their past, especially in interferometers.
Recently, several surprising results have been shown in nested Mach-Zehnder
interferometer(NMZI), proving that the ‘weak trace’ of photons can be hidden.
This paper rearranges the three path interference problem which gives the
general result of its output and the conditions for eliminating the weak
trace. Surprisingly, such conditions strongly depend on the detection method
of the output photon, as well as the phase differences and splitting ratios in
the NMZI. Further analysis gives a new weak trace elimination phenomenon which
is independent of the position sensitive detector. Moreover, we setup an
experiment to verify the weak trace elimination conditions and demonstrate the
new weak trace elimination phenomenon.

  * May 2021

##  1 Introduction

The past of a quantum particle has been discussed for decades [1, 2, 3, 4, 5,
6, 7, 8, 9, 10]. And yet, physicists are still not sure the trajectory a
quantum particle took from the source by only measuring it in a detector [11,
12, 13, 14, 15]. Such discussions mostly focus on interferometers, such as
double slit interferometer [16, 1, 17, 11, 18] and Mach-Zehnder interferometer
(MZI) [19, 3, 4, 10, 20, 21, 5, 6, 7, 8, 9]. For example, Wheler et al. [1, 2]
tried to infer the past path of photons through the output of photon detector
in a double-slit interferometer and found a counter-factual result: we can’t
infer the past of photons, because photons can hide their past.

Such phenomenon was further discussed by Vaidman [3, 4, 10] in a nested Mach-
Zehnder interferometer (NMZI). They proposed to use the weak trace as the
criterion of whether the light passes through a mirror, then the photon will
pass through discontinuous paths in NMZI. This counter-factual result was
explained by the two-state vector formalism[22], which has aroused extensive
theoretical discussions [23, 24, 25, 26, 15, 14, 13, 20, 6, 27, 9] and
experimental demonstrations [16, 8] in recent years. Danan et al. [8] used a
position resolved photon detector in NMZI to find out the weak trace caused by
the vibration of mirrors along the trajectory. Alonso et al. [6] found that
using dove prism in NMZI can change the weak trace of photons which making the
photons’ past change. Yuan et al. [27, 9] interpreted such weak trace problem
with three path interference and shows a new condition to hide the past of
photons. However, these researches focus on the weak trace of photons under
certain conditions and the corresponding photon states.

In this article, we focus on the properties of NMZI which gives a general
expression of the output and general conditions to erase the past of photons.
A detail discussion shows that the weak trace elimination depends on the
detection method of photons, as well as the phase differences and the
splitting ratios of beam splitters (BSs) in NMZI. Our results can not only
explain the experimental results in Ref. [8], but also predict a new method to
eliminate the weak trace. In addition, we experimentally demonstrated such new
effects without any position sensitive detector.

##  2 Theoretical model

Figure 1: Scheme of NMZI. All the splitting ratios of the beam splitters used
in this NMZI are adjustable. The mirrors A, B, C, E, F are vibrating with
different frequencies.

Our model is shown in the Fig. 1, where the beam-splitting ratios of the BSs
are all arbitrarily adjustable. When a photon is emitted from the light
source, the photon state passing through the BS1 is

$|\psi_{BS2}\rangle=t_{1}|U\rangle+ir_{1}|L\rangle,$ (1)

where $t_{1}$ is amplitude transmissivity of the BS1 and
$r_{1}=\sqrt{1-t_{1}^{2}}$ is the corresponding amplitude reflectivity. Here
$|L\rangle$ is the photon state that is reflected by BS1 and goes through the
lower arm of the outer MZI, while $|U\rangle$ is the photon state propagates
along the upper arm of MZI through BS1.

Similarly, when photons pass through the mirror E and BS2 of the inner MZI,
the state changes to

$|\psi_{BS2}\rangle=t_{1}|U\rangle+ir_{1}t_{2}e^{i\varphi_{e}}|u\rangle-
r_{1}r_{2}e^{i\varphi_{e}}|l\rangle,$ (2)

where $\varphi_{e}$ is the complex phase caused by the vibration of mirror E.
The real part and the imaginary part represent the change of the optical phase
and intensity of light caused by the vibration, respectively. The vibration of
mirror causes a tiny change of optical path when photons pass through the
mirror, which induces the phase change of the photons. Meanwhile, the
reflection angle of light also changes due to the vibration of the mirror,
which makes the intensity of the interference change. Therefore, both tiny
changes exist in the final output of interferometer, acting as the weak trace
of the photon. State $|l\rangle$ is the state that the photon reflected by BS2
with amplitude reflectivity $r_{2}$ and propagates along the lower arm of the
internal MZI. State $|u\rangle$ is transmitted by BS2 with amplitude
transmissivity $t_{2}=\sqrt{1-r_{2}^{2}}$ and propagates along the upper arm
of the inner MZI.

Similar to mirror E, mirrors A, B and C also bring tiny complex phases to
photons which are

$|\psi_{ABC}\rangle=t_{1}e^{i\varphi_{a}}|U\rangle+ir_{1}t_{2}e^{i(\varphi_{b}+\varphi_{e})}|u\rangle-
r_{1}r_{2}e^{i(\varphi_{c}+\varphi_{e})}|l\rangle.$ (3)

When the photons pass through the beam combiner BS3 of the internal MZI, the
state of the photons becomes

$\displaystyle|\psi_{BS3}\rangle=t_{1}e^{i\varphi_{a}}|U\rangle-(r_{1}t_{2}r_{3}e^{i\left(\varphi_{e}+\varphi_{b}\right)}e^{i\phi}+r_{1}r_{2}t_{3}e^{i\left(\varphi_{e}+\varphi_{c}\right)})|out1\rangle$
$\displaystyle-i(r_{1}r_{2}r_{3}e^{i\left(\varphi_{e}+\varphi_{c}\right)}-r_{1}t_{2}t_{3}e^{i\left(\varphi_{e}+\varphi_{b}\right)}e^{i\phi})|L^{\prime}\rangle,$
(4)

where $t_{3}$ and $r_{3}=\sqrt{1-t_{3}^{2}}$ are the amplitude transmissivity
and amplitude reflectivity of BS3, respectively. State $|L^{\prime}\rangle$
refers to the state that the photons propagate along the lower arm of the
external MZI and reach the mirror F after the beam combination. While the
state $|out1\rangle$ denotes that the photons leave NMZI without being
detected. Here we introduce a real phase difference $\phi$ between the two
arms of the internal interferometer $|u\rangle$ and $|l\rangle$ since the
output of the interferometer may be different due to the phase.

After BS4, the final photon state from the interferometer is

$\displaystyle|\psi_{BS4}\rangle=i\left[t_{1}r_{4}e^{i\chi}e^{i\varphi_{a}}-(r_{1}r_{2}r_{3}e^{i\left(\varphi_{e}+\varphi_{c}\right)}-r_{1}t_{2}t_{3}e^{i\left(\varphi_{e}+\varphi_{b}\right)}e^{i\phi})e^{i\varphi_{f}}t_{4}\right]|D\rangle$
$\displaystyle+\left[t_{1}t_{4}e^{i\varphi_{a}}e^{i\chi}+(r_{1}r_{2}r_{3}e^{i\left(\varphi_{e}+\varphi_{c}\right)}-r_{1}t_{2}t_{3}e^{i\left(\varphi_{e}+\varphi_{b}\right)}e^{i\phi})e^{i\varphi_{f}}r_{4}\right]|out2\rangle$
$\displaystyle-(r_{1}t_{2}r_{3}e^{i\left(\varphi_{e}+\varphi_{b}\right)}e^{i\phi}+r_{1}r_{2}t_{3}e^{i\left(\varphi_{e}+\varphi_{c}\right)})|out1\rangle,$
(5)

where $t_{4}$ and $r_{4}=\sqrt{1-t_{4}^{2}}$ are the amplitude transmissivity
and reflectivity of BS4. Here we introduce another real phase $\chi$ to
represent a phase difference between the two arms of the external
interferometer $|L\rightarrow l\rightarrow L^{\prime}\rangle$ and $|U\rangle$.
The state $|D\rangle$ means that the photons are finally received by the
detector D, while the other state $|out2\rangle$ means that the photons are
not detected after passing through the NMZI.

Now we can directly get the signal of detector D as

$P_{D}=\left|ae^{i\varphi_{a}}e^{i\chi}+be^{i\left(\varphi_{e}+\varphi_{b}+\varphi_{f}\right)}-ce^{i\left(\varphi_{e}+\varphi_{c}+\varphi_{f}\right)}e^{i\phi}\right|^{2},$
(6)

where $a=t_{1}r_{4},b=r_{1}t_{2}t_{3}t_{4},c=r_{1}r_{2}r_{3}t_{4}$ are the
amplitudes of lights inside the NMZI.

In the weak trace experiment, the complex phases caused by mirror vibrations
are usually very small. That is to say, it has little influence on the output
of the interference results. Under such condition, we can make a linear
approximation $e^{i\varphi_{k}}\approx 1+i\varphi_{k}$, where $k=a,b,c,e,f$.
Then the Eq. 6 can be simplified to

$\displaystyle P_{D}$ $\displaystyle\approx$
$\displaystyle\left|ae^{i\chi}+b-ce^{i\phi}\right|^{2}-2Im[\left(ae^{i\chi}+b-ce^{i\phi}\right)\left(b-ce^{i\phi}\right)\left(\varphi_{e}+\varphi_{f}\right)]$
(7)
$\displaystyle-2Im[\left(ae^{i\chi}+b-ce^{i\phi}\right)\left(a\varphi_{a}e^{i\chi}+b\varphi_{b}-c\varphi_{c}e^{i\phi}\right)].$

Here we only keep the linear terms and neglect the high order terms. This is
the general output of the NMZI. With different splitting ratios and relative
phases $\phi$ and $\chi$, we can obtain different results.

##  3 Analysis and Discussion

According to the general expression in Eq. 7, in principle, we can find and
eliminate the weak traces of photons. For example, to eliminate the weak trace
caused by mirror A, we can extract the term containing $\varphi_{a}$ in Eq. 7
as

$-2aIm\left\\{\left[a+be^{i\chi}-ce^{-i\left(\phi-\chi\right)}\right]\varphi_{a}\right\\}=0.$
(8)

Here we neglect the high order terms due to the tiny phase approximation.

As we know, the amplitude $a$ and the phase $\varphi_{a}$ are not always zero.
To eliminate the weak trace caused by mirror A, Eq. 8 should be expanded which
gives

$\cases{a+bcos(\chi)-ccos(\phi-\chi)=0,&eliminating $Re(\varphi_{a})$,\\\
bsin(\chi)-csin(\phi-\chi)=0,&eliminating $Im(\varphi_{a})$.\\\ }$ (9)

In this case, the output light does not contain any phase terms in first
order. However, such conditions make the output of the NMZI becomes zero, i.e.
$P_{D}\approx 0$. Obviously, the photon can not reach the detector which makes
this condition useless.

But interestingly, the weak trace has been eliminated in Danan et al’s
experiment while the photons actually reached the detector. In such
experiment, they used a position sensitive quad-cell photodetector with
differential detection method [8] to obtain signals from NMZI. Such detection
method erases the interference fringes caused by $Re(\varphi_{k}),k=a,b,c,e,f$
while the intensity variances caused by $Im(\varphi_{k})$ remain. That is to
say, they eliminate the real part of complex phase when doing detection while
the imaginary part exist in the outputs. And they tuned the phases of NMZI to
cancel the imaginary part.

If a similar position sensitive detection is used in our case, we can keep the
term of $Re(\varphi_{a})$ and only eliminate the term of $Im(\varphi_{a})$ as

$\cases{a+bcos(\chi)-ccos(\phi-\chi)=0,&eliminating $Re(\varphi_{a})$,\\\
bsin(\chi)-csin(\phi-\chi)\neq 0,&remaining $Im(\varphi_{a})$.\\\ }$ (10)

Then the output of NMZI contains no weak trace of mirror A. This is only true
when $Re(\varphi_{a})=0$ or the detector can not get the information of
$Re(\varphi_{a})$. In such case, the weak trace caused by mirror A can be
eliminated in the output of NMZI when the beam splitting ratios of BSs and the
phases of the interferometer satisfy the relations of Eq. 10.

Even more interesting is that there is another way to eliminate the weak
traces of photons. When $Im(\varphi_{a})=0$ or the variance of intensity can
not be detected, the weak trace caused by mirror A can also be eliminated when

$\cases{a+bcos(\chi)-ccos(\phi-\chi)\neq 0,&remaining $Re(\varphi_{a})$,\\\
bsin(\chi)-csin(\phi-\chi)=0,&eliminating $Im(\varphi_{a})$.\\\ }$ (11)

Through these two cases, it is easy to find that the condition of eliminating
weak trace of photons completely depends on the way of detection. The weak
traces of photons vary with different detection methods, as well as the
conditions of eliminating weak traces. Moreover, due to the simultaneous
interference of three paths in NMZI, the weak trace elimination effect is much
easier to achieve than the cases mentioned in previous literatures [3, 8, 27].
Actually, by adjusting the beam splitting ratios of BSs and the phases of
interferometer, eliminating the weak traces of photons has become a completely
controllable effect.

According to the discussion above, we can further obtain the conditions for
eliminating weak traces caused by mirrors B, C, E and F from Eq. 8 which shows
in table 1

Table 1: The conditions for eliminating weak traces of photons Weak trace to be eliminated | Condition
---|---
$Im(\varphi_{a})$ | $a+b\cos\chi-c\cos\left(\phi-\chi\right)=0$
$Re(\varphi_{a})$ | $b\sin\chi-c\sin\left(\phi-\chi\right)=0$
$Im(\varphi_{b})$ | $a\cos\chi+b-c\cos\phi=0$
$Re(\varphi_{b})$ | $a\sin\chi-b-c\sin\phi=0$
$Im(\varphi_{c})$ | $a\cos\left(\phi-\chi\right)+b\cos\phi-c=0$
$Re(\varphi_{c})$ | $a\sin\left(\phi-\chi\right)+b\sin\phi-c=0$
$Im(\varphi_{e})$ & $Im(\varphi_{f})$ | $b^{2}+c^{2}-2bc\cos\phi+ab\cos\chi-ac\cos\left(\phi-\chi\right)=0$
$Re(\varphi_{e})$ & $Re(\varphi_{f})$ | $ab\sin\chi+ac\sin\left(\phi-\chi\right)=0$

It can be seen from the table that the weak traces caused by any mirrors can
be eliminated under certain conditions, but all weak traces cannot be
eliminated at the same time. In the previous work [3, 4, 27], the weak trace
caused by mirror E and mirror F can be surprisingly eliminated together.
However, in the table we can see that the weak traces caused by any two
mirrors can be eliminated at the same time, not only by mirror E and mirror F.
What we need to do is carefully adjusted the phase differences and the beam
splitting ratios of BSs. We can even eliminate the weak traces caused by three
mirrors which is hardly seen in the previous work.

##  4 Experimental Results

To verify the above results, we set up an experiment as shown in Fig. 2. We
use a half wave plate and a PBS to realize an adjustable BS. A normal detector
is used to detect the output of the interferometer directly, so as to avoid
detecting the signal caused by the change of light intensity and obtain the
interference signal caused by phase. As shown in Fig. 2, each mirror vibrates
with different frequencies. The final output signal of the NMZI is detected by
D1 and analyzed by the power spectrum analyzer to obtain the weak traces of
photons.

Figure 2: Experimental setup of NMZI. The mirror A, B, C, E, F are vibrating
with frequency $f_{A}=1.1kHz$, $f_{B}=1.2kHz$, $f_{C}=1.3kHz$,
$f_{E}=1.4kHz$,$f_{F}=1.5kHz$, respectively.

According to table 1, the two relative phase $\phi$ and $\chi$ plays a central
role of eliminating the weak traces. Therefore, we need to simultaneous detect
the two phases of the NMZI accurately. As shown in Fig, 2, D2 is used to
measure the photon state $|out1\rangle$, the interference fringes of the
internal interferometer. And D3 is used to measure the photon state
$|out2\rangle$, another output of NMZI. Such two detectors give the
interference fringes which can be used to obtain the two phases in real time.
According to Eq. 5 we can get the two phase as

$\displaystyle\phi$ $\displaystyle=$
$\displaystyle\cos^{-1}\left[\frac{P_{D2}-r_{1}^{2}\left(t_{2}^{2}r_{3}^{2}+r_{2}^{2}t_{3}^{2}\right)}{2r_{1}^{2}t_{2}r_{2}t_{3}r_{3}}\right],$
(12) $\displaystyle\chi$ $\displaystyle=$
$\displaystyle\cos^{-1}\frac{P_{D3}-\left(t_{1}^{4}t_{4}^{4}+r_{1}^{2}r_{2}^{2}r_{3}^{2}r_{4}^{2}+r_{1}^{2}t_{2}^{2}t_{3}^{2}r_{4}^{2}-2r_{1}^{2}r_{2}t_{2}r_{3}t_{3}r_{4}^{2}\cos\phi\right)}{2r_{1}t_{1}r_{4}t_{4}\sqrt{r_{2}^{2}r_{3}^{2}+t_{2}^{2}t_{3}^{2}-2r_{2}r_{3}t_{2}t_{3}\cos\phi}}-\beta,$
(13)

where $P_{D2}$ and $P_{D3}$ are the optical power detected by D2 and D3, and

$\beta=\cos^{-1}\frac{\left(r_{2}r_{3}-t_{2}t_{3}\cos\phi\right)}{\sqrt{r_{2}^{2}r_{3}^{2}+t_{2}^{2}t_{3}^{2}-2r_{2}r_{3}t_{2}t_{3}\cos\phi}}.$
(14)

Figure 3: The power spectrum of the photon current from D1 shows the the
elimination of weak trace from (a) mirror A, (b) mirror B, (c) mirror E and F.

When we set $r_{1}=r_{4}=\sqrt{2/3},r_{2}=r_{3}=\sqrt{1/2}$, the intensity of
the light fields in the three paths of NMZI are the same. Then the output of
NMZI varies with the two phases $\phi$ and $\chi$ as shown in Fig. 3. Since
the detector D1 is a normal detector which can only detect the phase sensitive
interference fringes, the weak traces we can detect is only the real part of
complex phases. As we can see in Fig. 3 (a), the weak trace of mirror A is
eliminated when $\phi=0.223rad,\chi=0.11rad$ which is consistent with Eq. 11.
When the phase of the internal and external interferometer becomes
$\phi=-0.33rad$ and $\chi=0.74rad$, the weak trace caused by mirror B is
eliminated as shown in Fig. 3 (b). Moreover, the weak traces of mirror E and
mirror F are eliminated as shown in Fig. 3 (c) if we set the phases to
$\phi=-0.45rad$ and $\chi=0.44rad$. Such results are all consist with the
predictions of table 1.

Figure 4: The experimental conditions and the corresponding fittings of
eliminating the weak trace of mirror A. As we can see, the relation between
the two phases $\phi$ and $\chi$ varies with the splitting ratio of BS2
$r_{2}$.

Furthermore, by tuning the half wave plates before each PBS we change the beam
splitting ratios of BSs. Then we can still eliminate the weak trace caused by
mirrors through adjusting the phases $\phi$ and $\chi$. Fig. 4 shows the
relationship between the phases and $r_{2}$ when eliminating the weak trace of
mirror A. As the phase difference $\chi$ changing, the phase $\phi$ varies and
fits well with the theoretical simulation. Also, due to the variance of the
splitting ratio $r_{2}$, the relation between the two phases changes which is
also consistent with the theoretical simulation.

##  5 Conclusion

We rearrange the three path interference problem which gives the general
conditions for eliminating weak traces. This general result can not only
explain the previous experimental results but also predicted a new weak trace
elimination effect. Further discussion shows that the elimination of photon’s
weak trace depends on both photon detection method and the output photon state
from the interferometer while the photon state depends on the phase
differences and beam splitting ratio of BSs. Since the photon detection method
determines the manifestation of the weak trace, we discover a new method to
eliminate the weak trace. In summarize, we can change the way of photon
detection, control the beam splitting ratio and phase to eliminate any weak
trace. Moreover, these results have potential applications in quantum
information and quantum precision measurement.

##  6 Acknowledgment

We acknowledge financial support from the National Science Foundation of China
(Grant NO. 11904227, 11654005, 11804225), the Sailing Program of Shanghai
Science and Technology Committee under Grant 19YF1421800, 19YF1414300,
Shanghai Municipal Science and Technology Major Project (Grant NO.
2019SHZDZX01) and the Startup Fund for Youngman Research at SJTU, W. Z. also
acknowledges additional support from the Shanghai talent program.

## References

## References

  * [1] Wheeler J A 1978 The “past” and the “delayed-choice” double-slit experiment Mathematical Foundations of Quantum Theory ed Marlow A (Academic Press) pp 9–48 ISBN 978-0-12-473250-6 URL https://www.sciencedirect.com/science/article/pii/B9780124732506500066
  * [2] Wheeler J A 1984 Quantum Theory and Measurement vol 1 (Princeton: Princeton University Press) p 182–213
  * [3] Vaidman L 2013 Phys. Rev. A 87(5) 052104 URL https://link.aps.org/doi/10.1103/PhysRevA.87.052104
  * [4] Vaidman L 2014 Phys. Rev. A 89(2) 024102 URL https://link.aps.org/doi/10.1103/PhysRevA.89.024102
  * [5] Englert B G, Horia K, Dai J, Len Y L and Ng H K 2017 Phys. Rev. A 96(2) 022126 URL https://link.aps.org/doi/10.1103/PhysRevA.96.022126
  * [6] Alonso M A and Jordan A N 2015 Quantum Stud.: Math. and Found. 2 255–261 URL https://doi.org/10.1007/s40509-015-0044-8
  * [7] Sokolovski D 2017 Physics Letters A 381 227–232 ISSN 0375-9601 URL https://www.sciencedirect.com/science/article/pii/S0375960116316590
  * [8] Danan A, Farfurnik D, Bar-Ad S and Vaidman L 2013 Phys. Rev. Lett. 111(24) 240402 URL https://link.aps.org/doi/10.1103/PhysRevLett.111.240402
  * [9] Yuan Q and Feng X 2021 Chin. Opt. Lett. 19 012701 URL http://col.osa.org/abstract.cfm?URI=col-19-1-012701
  * [10] Vaidman L and Tsutsui I 2018 Entropy 20 ISSN 1099-4300 URL https://www.mdpi.com/1099-4300/20/7/538
  * [11] P Storey S Tan M C and Walls D 1994 Nature 367
  * [12] S Durr T N and Rempe G 1998 Nature 395
  * [13] Potoček V and Ferenczi G 2015 Phys. Rev. A 92(2) 023829 URL https://link.aps.org/doi/10.1103/PhysRevA.92.023829
  * [14] Griffiths R B 2016 Phys. Rev. A 94(3) 032115 URL https://link.aps.org/doi/10.1103/PhysRevA.94.032115
  * [15] Geppert-Kleinrath H, Denkmayr T, Sponar S, Lemmel H, Jenke T and Hasegawa Y 2018 Phys. Rev. A 97(5) 052111 URL https://link.aps.org/doi/10.1103/PhysRevA.97.052111
  * [16] Zhou Z Y, Zhu Z H, Liu S L, Li Y H, Shi S, Ding D S, Chen L X, Gao W, Guo G C and Shi B S 2017 Science Bulletin 62 1185–1192 ISSN 2095-9273 URL https://www.sciencedirect.com/science/article/pii/S2095927317304358
  * [17] M Scully B E and Walter H 1991 Nature 351
  * [18] Leggett A J 2009 Compendium of Quantum Physics vol 1 (Berlin: Springer) p 161–166
  * [19] A Peruzzo P Shadbolt N B S P and Brien J L O 2012 Science 338
  * [20] Nikolaev G N 2017 JETP Letters 105(3) 152–157 URL https://doi.org/10.1134/S0021364017030110
  * [21] T Hellmuth HWalther A Z a S 1987 Phys. Rev. A 35
  * [22] Aharonov Y and Vaidman L 1990 Phys. Rev. A 41(1) 11–20 URL https://link.aps.org/doi/10.1103/PhysRevA.41.11
  * [23] Vaidman L 2016 Phys. Rev. A 93(1) 017801 URL https://link.aps.org/doi/10.1103/PhysRevA.93.017801
  * [24] Vaidman L 2017 Phys. Rev. A 95(6) 066101 URL https://link.aps.org/doi/10.1103/PhysRevA.95.066101
  * [25] Vaidman L 2018 Comment on ”multifold paths of neutrons in the three-beam interferometer detected by a tiny energy kick” (Preprint 1806.01774)
  * [26] Vaidman L 2017 JETP Letters 105(7) 473–474 URL https://doi.org/10.1134/S0021364017070037
  * [27] Yuan Q and Feng X 2019 Phys. Rev. A 99(5) 053805 URL https://link.aps.org/doi/10.1103/PhysRevA.99.053805


# Predicting Potential Drug Targets Using Tensor Factorisation and Knowledge
Graph Embeddings

Cheng Ye  Data Sciences and Quantitative Biology, Discovery Sciences
R&D, AstraZenecaCambridgeUK ,  Rowan Swiers  Data Sciences and Quantitative
Biology, Discovery Sciences
R&D, AstraZenecaCambridgeUK ,  Stephen Bonner  Data Sciences and Quantitative
Biology, Discovery Sciences
R&D, AstraZenecaCambridgeUK and  Ian Barrett  Data Sciences and Quantitative
Biology, Discovery Sciences
R&D, AstraZenecaCambridgeUK

(2018)

###### Abstract.

The drug discovery and development process is a long and expensive one,
costing over 1 billion USD on average per drug and taking 10-15 years. To
reduce the high levels of attrition throughout the process, there has been a
growing interest in applying machine learning methodologies to various stages
of drug discovery process in the recent decade, including at the earliest
stage – identification of druggable disease genes. In this paper, we have
developed a new tensor factorisation model to predict potential drug targets
(i.e., genes or proteins) for diseases. We created a three-dimensional tensor
which consists of 1,048 targets, 860 diseases and 230,011 evidence attributes
and clinical outcomes connecting them, using data extracted from the Open
Targets and PharmaProjects databases. We enriched the data with gene
representations learned from a drug discovery-oriented knowledge graph and
applied our proposed method to predict the clinical outcomes for unseen target
and disease pairs. We designed three evaluation strategies to measure the
prediction performance and benchmarked several commonly used machine learning
classifiers together with matrix and tensor factorisation methods. The result
shows that incorporating knowledge graph embeddings significantly improves the
prediction accuracy and that training tensor factorisation alongside a dense
neural network outperforms other methods. In summary, our framework combines
two actively studied machine learning approaches to disease target
identification, tensor factorisation and knowledge graph representation
learning, which could be a promising avenue for further exploration in data-
driven drug discovery.

drug discovery, tensor factorisation, knowledge graph

††copyright: acmcopyright††journalyear: 2018††doi:
10.1145/1122445.1122456††conference: Woodstock ’18: ACM Symposium on Neural
Gaze Detection; June 03–05, 2018; Woodstock, NY††booktitle: Woodstock ’18: ACM
Symposium on Neural Gaze Detection, June 03–05, 2018, Woodstock, NY††price:
15.00††isbn: 978-1-4503-XXXX-X/18/06††ccs: Computing methodologies
Factorization methods††ccs: Computing methodologies Knowledge representation
and reasoning

##  1\. Introduction

The discovery and development of drugs is a complex process that aims to
identify pharmalogical agents that are therapeutically effective for curing or
treating diseases. According to the US Food and Drug Administration (FDA), on
average, it takes approximately 12 years to get a newly discovered drug into
the market and the cost is 1.3 billion USD (fda, [n.d.]). There are many
factors that contribute to the high cost of developing a drug: significant
time and resources invested in target identification and the validation
process, multiple early tests for a considerable number of promising compounds
and complex in vitro and in vivo studies to understand the drug toxicity, to
just name a few. Among all these factors, clinical research is arguably the
most critical and costly process. According to AstraZeneca, between the year
2005 and 2010, the success rate of preclinical projects in the company was 66%
while for Phase I, II, and III clinical projects, the rates dropped to 59%,
15% and 60%, respectively (Cook et al., 2014). To raise the success rates of
clinical projects and consequently, reduce the time and cost needed for
developing a drug, it is imperative to develop an effective method to predict
whether a candidate drug target is clinically promising for treating a disease
on a large scale.

Traditionally, scientists identify new drug targets by discovering novel
insights into a disease mechanism from research publications and lab
experiments, which in turn allows scientists to design a drug molecule to
activate or inhibit the target in order to cease or reverse the disease
effects (Hughes et al., 2011) and by repositioning existing drugs to find
unanticipated but beneficial effects against other diseases (Pushpakom et al.,
2019). Together, these processes have generated a large number of therapeutic
hypotheses which are further supported by an enormous amount of both
structured and unstructured evidence data. It is then of great interest to
explore how we can better exploit such a wealth of information to identify and
prioritise drug targets for diseases in an accurate and computationally
efficient manner, using state-of-the-art data science technologies.

Machine learning is a natural candidate for the challenge. The target-disease
pairs can be viewed as data points and their clinical trial outcomes as binary
labels – success or failure. The biological evidence, from research
publications, lab experiments, etc., that associates targets with diseases are
used as features. However, there are several major difficulties in framing
this as a standard binary classification problem. First, there are very few
true positives – in Open Targets and PharmaProjects databases, less than 0.01%
target-disease pairs have approved drugs (Pha, [n.d.]; Ochoa et al., 2021).
Second, the feature space (biological evidence attributes) for pairs of
targets and diseases is extremely sparse. In the latest version of Open
Targets (September, 2020), there are in total 6,551,303 associations, which
are extracted from 20 data sources, connecting 27,610 targets and 13,944
diseases. In other words, as high as 98.3% of the target-disease pairs have no
biological evidence at all. As a consequence, the traditional feature-in,
prediction-out machine learning models might be incompetent in this context.

In this paper, motivated by the recent work (Yao et al., 2019), we explore
whether tensor factorisation can be used as a powerful tool for addressing the
aforementioned challenges. Mathematically, a tensor is a higher order
generalisation of a vector (first order tensor) and a matrix (second order
tensor) in which each dimension corresponds to an axis or a mode. A 3-mode
tensor is therefore a natural representation to host the biological data in
our analysis where the modes correspond to targets, diseases and their
association evidence, respectively. The clinical outcomes are represented as
an additional slice along the evidence mode. To predict the missing values in
the clinical outcome slice, we have developed a novel tensor factorisation
method that utilises target representations learned from a drug discovery-
oriented knowledge graph as side information. To the best of our knowledge,
this is the first tensor factorisation framework that incorporates biomedical
knowledge graphs for the purpose of predicting potential drug targets for
diseases. To aid in reproducibility, we release all the processed public data
sets and model scripts 111The data and code will be available soon.

The remainder of the paper is organised as follows. In Literature Review
section, we review recent tensor factorisation applications in drug discovery
and development. In Methods section, we first describe our data collection and
processing procedures and introduce how we use knowledge graphs and target
embeddings in the developed framework. We then explain in detail our proposed
tensor factorisation model and describe the benchmarking strategies and
experimental setup. In Results section we show figures summarising our key
findings. We discuss the contribution and limitations of our work, and point
out several future research directions in the Conclusions section.

##  2\. Literature Review

In recent decades, tensor factorisation has found many useful applications in
a wide range of machine learning areas, such as recommender systems, image
processing and computer vision, and natural language processing (Anisimov et
al., 2014; Haeffele et al., 2014; Van de Cruys, 2009; Park et al., 2017; Zheng
et al., 2016). Such success has led to a growth in the use of tensor
factorisation for addressing biomedical challenges in recent years. For
instance, Chen and Li (Chen and Li, 2018) proposed a tensor decomposition
model to predict the therapeutic benefits of drug combinations. In particular,
a 3-mode tensor was constructed to represent comprehensive relationships
between drugs and diseases as well as their similarity information. The
molecular mechanisms of drug synergy can thus be well revealed via
simultaneously factorising the coupled tensor and similarity matrices. The
authors later applied the technique to model relational drug-target-disease
interactions and successfully showed that the model outperforms some
competitive methods at predicting drug mechanism of actions (Chen and Li,
2019). To address the data sparsity and scalability challenges, Macau, a
powerful, flexible and scalable Bayesian multi-relational factorisation method
for heterogeneous data has been developed (Simm et al., 2015). One unique
advantage of Macau is that it enables the incorporation of side information,
specifically entity and relation features, which is essential for complex
tasks such as drug-protein activity prediction. More recently, Yao et al. (Yao
et al., 2019) have applied Macau to the problem of predicting clinical
outcomes of therapeutic hypotheses and have also tested the performance of
several other machine learning models including Logistic Regression, LASSO,
Random Forest, Matrix Factorisation and Gradient Boosting Machine. The result
suggested that tensor factorisation is comparable or better than the baseline
methods in various cross-validation scenarios.

The importance of target side information in matrix and tensor factorisation
has been discussed by Piro and Di Cunto (Piro and Di Cunto, 2012). In the
paper, they have examined a number of disease gene prioritisation tools that
utilise different types of evidence including text-mining of biomedical
literature, functional annotations, pathways and ontologies, phenotype
relationships, intrinsic gene properties, sequence data, protein–protein
interactions, regulatory information, orthologous relationships and gene
expression information. Generally speaking, most existing methods incorporate
target side information by either directly merging data from different sources
or applying techniques such as principal component analysis (PCA) to evidence
data (Natarajan and Dhillon, 2014; Vitsios and Petrovski, 2020). However, such
ways of using side information do not fully exploit the interconnections
between genes and other biological entities and dimensionality reduction
technologies such as PCA cannot capture the essential gene representations in
the high-dimensional, heterogeneous data. To address this challenge, we
utilise a biomedical knowledge graph, Hetionet (Himmelstein et al., 2017),
which is an integrative network of biomedical knowledge centred around genes
and diseases assembled from different databases, and use gene representations
automatically learned from the graph (known as knowledge graph embeddings
(Wang et al., 2017, 2014)) as the target side information in our proposed
model. We show that by connecting different biological resources via a well-
structured knowledge graph and by learning gene representations which contain
essential information not only from gene-gene interactions and gene-disease
associations but the biomedical knowledge field, the prediction performance
can be improved.

##  3\. Method

###  3.1. Data Collection and Processing

We have created a dataset which combines clinical outcomes from the commercial
database CiteLine PharmaProjects (Pha, [n.d.]) and evidence from the open-
source database Open Targets (Ochoa et al., 2021) between targets and diseases
(this is detailed in Table 1). PharmaProjects has curated lists of clinical
trials specifying the drug being tested, the targeted proteins and the disease
being tested against. We joined these two datasets together to create a
mapping of diseases to targets. We defined that the clinical outcome of a
target and a disease is positive if there has been at least one successful
clinical trial for the pair in the PharmaProjects database. On the other hand,
the clinical outcome is negative if there is no clinical success and at least
one failed clinical trial. Open Targets has seven different data types, each
with an association score, for each target-disease pair. We therefore used
Open Targets to create the evidence data between targets and diseases.

Evidence Type |  Association Score Sources |  Description
---|---|---
Genetic Associations |  ClinVar (EVA), PheWAS Catalog, Gene2Phenotype, Genomics England PanelApp, Open Targets Genetics Portal, UniProt literature, ClinGen |  Curated and calculated associations between genetic variants and mendelian-inherited diseases, common diseases and phenotypes/traits.
Somatic Mutations |  Cancer Gene Census, ClinVar somatic (EVA), IntOGen |  Somatic mutations noted in cancers and other diseases.
Drugs |  ChEMBL |  Database of known drugs and other pharmacological agents linked to bioactivity assay data and disease (where applicable).
Pathways and Systems Biology |  Reactome, Sysbio, SLAPenrich, PROGENy, Project Score (CRISPR) |  Gene sets comprised from curated biological pathways, and oncology-centric resources (CRISPR gene function screens and pathway activities inferred from gene expression data).
RNA Expression |  Expression Atlas |  Gene expression data from different biological conditions (e.g. normal and disease).
Text Mining |  EuropePMC |  Co-occurrences between gene target and disease terms, text mined from PubMed and PubMedCentral.
Animal models |  PhenoDigm |  Associations between gene targets and diseases, inferred from similarities between human disease characteristics and phenotypes observed in mouse models where the gene is perturbed.
Table 1. Target-disease evidence types in Open Targets.

One major challenge occurring when combining the two data sets was that
PharmaProjects uses the MeSH Ontology (Lipscomb, 2000) for diseases while Open
Targets uses the Mondo Ontology (Mungall et al., 2017). To address this
problem, we used the EMBL-EBI Ontology matching tool OXO (oxo, [n.d.]) to
connect these MeSH and Mondo terms. We finally obtained a three-dimensional
tensor in which the dimensions correspond to 1,048 targets, 860 diseases and
their evidence and clinical outcomes (230,011 in total), which is depicted in
Figure 1. The density of the tensor is 3.19%.

Figure 1. A tensor representation of the biological dataset

###  3.2. Knowledge Graph and Target Embeddings

Generally speaking, a knowledge graph is an integrative network representation
of structured and unstructured knowledge, which provides an easier way of
capturing complex semantic relationships than conventional databases. For
instance, in a biomedical knowledge graph, various biological entities are
modelled as nodes, whilst millions of complex biomedical relationships are
modelled as edges (links) connecting nodes. In recent years, many researchers
and pharmaceutical companies have been building biomedical knowledge graphs to
address various drug discovery challenges, e.g., Hetionet (Himmelstein et al.,
2017), PharmKG (Zheng et al., 2020) and Rosalind’s knowledge graph (Paliwal et
al., 2020).

In this paper, we make use of Hetionet, which is one of the earliest
integrative biological networks centred around genes and diseases and now
forms the core of many more recently developed disease knowledge graphs, and
learn effective feature representations of targets using various knowledge
graph embedding techniques. Hetionet contains 47,031 biological entities of 11
types (Gene, Disease, Compound, Pathway, Anatomy etc.) and 2,250,197
relationships of 24 types (Disease-associates-Gene, Compound-treats-Disease,
Gene-participates-Pathway, etc.) integrated from 29 public databases. The
graph models the complex interactions between genes and other genes, diseases,
compounds, pathways, biological processes and many other entities, so the
learned embeddings are able to capture interactive characteristics of genes in
the biomedical field (see Figure 2). We therefore used the gene embeddings to
enrich the target information in our data set and treated them as side
information in the tensor factorisation framework to help improve the
prediction accuracy.

Figure 2. Schema of knowledge graph embeddings.

###  3.3. Tensor Factorisation with Knowledge Graph Embeddings

To predict the missing values in the clinical outcome matrix in the data
tensor, we adopted a tensor factorisation model which factorises the data
tensor into a core evidence tensor and two latent target and disease matrices
which need to be learned, as depicted in Figure 3. These together with the
target representation matrix learned from the knowledge graph were then fed
into a dense neural network to generate the output, which will be the
predicted value for the clinical outcome of a target-disease pair.

Figure 3. Schema of the proposed model.

We denote $D={d_{1},d_{2},...,d_{n}}$ the set of diseases and
$T={T_{1},T_{2},...,T_{n}}$ the set of targets present in the data set and the
association score of evidence type $k$ between disease $D_{i}$ and target
$T_{j}$ is represented as $E_{ijk}\in[0,1]$. Similarly, the clinical outcome
between disease $D_{i}$ and target $T_{j}$ is denoted as $X_{ij}\in[0,1]$. Let
the learned latent variables for diseases and targets, and the target
knowledge graph embeddings be $U$, $V$, $W$, respectively. Then the
probability of a successful clinical outcome between disease $D_{i}$ and
target $T_{j}$ is calculated as follows:

(1)
$P(X_{ij}=1|\mathbf{e_{ij}},\mathbf{u_{i}},\mathbf{v_{j}},\mathbf{w_{j}})=f(concat(\mathbf{e_{ij}},\mathbf{u_{i}},\mathbf{v_{j}},\mathbf{w_{j}})),$

where $\mathbf{e_{ij}},\mathbf{u_{i}},\mathbf{v_{j}},\mathbf{w_{j}}$ are the
evidence vector between $D_{i}$ and $T_{j}$, latent variables for $D_{i}$,
latent variables for $T_{j}$, and target representations learned from the
knowledge graph for $T_{j}$, respectively. These vectors are concatenated
together before being input into the function. We define the function f to be
a feed-forward neural network with 2 dense hidden layers of 256 and 64
neurons, both using the ReLU activation function. The output layer uses a
linear activation function.

The dimensionality of the target and disease parameter matrices is
$V\in\mathbb{R}^{|T|,d}$ and $U\in\mathbb{R}^{|D|,d}$, where $d$ is the size
of the embedding for each entity – set to 32 for this work. We have also
explored other dimensionality numbers but found no significant difference in
the result. Meaning that together, these matrices contribute 61,056 learnable
parameters to the model. The neural network, if using a 32-dimensional
Hetionet embeddings has 43,240 parameters. This results in final total model
size of just over 100,000 learnable parameters. These parameters were
optimised using the Adagrad (Ward et al., 2019) variant of Stochastic Gradient
Descent (SGD) with Mean Squared Error (MSE) as the loss function. The model
was trained for 100 epochs, using a batch size of 2000 and a learning rate of
0.05.

We argue that the main advantage of this proposed technique is that the dense
neural network allows for complex interactions between the Open Targets
evidence, the latent variables and the pre-trained knowledge graph embeddings.
In this way the embeddings from the knowledge graph can have non-linear
interactions with the prediction outcome and can interact with both the
disease and the target latent variables. This would not be happen if the
embeddings were used as side information in algorithms such as Macau (Simm et
al., 2015).

We used the following techniques to generate the embedding matrix $W$ for
targets from the Hetionet knowledge graph. All the algorithms were set to use
embedding dimension of 32 and were trained using the default parameters
reported in the respective paper. Node2vec, a homogeneous graph embedding
method that groups nodes with similar neighbourhoods together (Grover and
Leskovec, 2016); Metapath2vec, a heterogeneous graph embedding method that
groups nodes of the same type with similar neighbourhoods together (Dong et
al., 2017); Graph Variational Auto-Encoder (GVAE), a homogeneous graph-based
embedding approach that tries to predict links between nodes by applying a
convolutional operation to latent variables of node neighbours (Kipf and
Welling, 2016); ComplEx, a heterogeneous graph-based embedding approach that
tries to predict links between nodes by applying a decoder operation between
latent variables of node pairs (Trouillon et al., 2016); Random embeddings,
which are randomised vectors that can be compared with others to measure the
random effects on the prediction performance.

##  4\. Experimental Setup

###  4.1. Evaluation Strategies

We used three cross-validation strategies to assess the accuracy of predicting
the unseen clinical outcomes of target and disease pairs. The first randomly
split all the target disease pairs into five folds. We trained the models on
four folds and tested on the remaining fold. This is the standard cross-
validation procedure; however, it could result in the case that target disease
pairs that share the same target, or the same disease are assigned to
different folds, and thus introduce bias when measuring the performance. This
is because if one target has been clinically successful for a particular
disease, then it is likely to be successful for another closely related
disease. Similarly, if a disease is treated well by a particular target then
it is likely to be treated well by another similar target.

To overcome this issue, we designed a second strategy where we first randomly
assigned targets into one of five groups. Target disease pairs are then put
into the fold the target is assigned to. In this way, the target disease pairs
that share the same target are guaranteed to be in the same group. We then
again trained the models on four groups and tested on the remaining group.
Lastly and similarly, we split all diseases into several classes depending on
which part of the human body the disease affects. The target disease pairs are
then put into the class the disease is assigned to.

###  4.2. Baseline Models

We applied the following classic machine learning classifiers to the data.
Logistic regression, a simple linear model using the Scikit-Learn
implementation (Pedregosa et al., 2011); k-nearest neighbours classifier using
the Scikit-Learn implementation (Pedregosa et al., 2011); Random forest, an
ensemble method using the Scikit-Learn implementation (Pedregosa et al.,
2011); Gradient boosting machine using the XGBoost implementation (Chen and
Guestrin, 2016). We also applied Macau Bayesian Probabilistic Matrix
Factorisation (BPMF) to the reduced matrix which contains only targets,
diseases and their clinical outcomes, and used Macau tensor factorisation
(Simm et al., 2015) as well.

###  4.3. Performance Metrics

Two metrics have been used to assess the prediction performance of the
evaluated methods: Area Under Receiver Operator Curve (AUROC) and Area Under
Precision-Recall Curve (AUPRC). The former is normally used to measure how
well a classifier can distinguish between a positive sample and a randomly
selected negative sample in a binary classification problem while the latter
assesses the probability that if a positive sample is selected from the ranked
list produced by a method, all samples ranked above it are positive.

##  5\. Results

Figure 4. Prediction result of our method and baseline machine learning
classifiers.

###  5.1. Evaluating Baseline Models

In this experiment, we evaluated the prediction performance of our proposed
tensor factorisation method by comparing with the baseline models. In
particular, we used the following set of features for each of the baseline
classifiers including:

  * •

Control: Here only random numbers are used as target-disease evidence.

  * •

No Embedding: Here no embeddings are used and only the Open Targets target-
disease evidence is present.

  * •

With Embedding: A combination of Open Targets target-disease evidence and
various target graph embeddings.

We used all of the three cross-validation strategies and the result is
reported in Figure 4. The most striking trend in the figure is that in all
three evaluation scenarios, our method outperforms all the baseline models by
a significant margin, demonstrating the superiority of the tensor
factorisation method. Several other important observations can be made. First,
as we suspected, it is indeed much easier to make predictions using the random
split benchmark than the disease or target split benchmark. This is because in
the random split, the algorithms are more likely to have trained on similar
pairs of targets and diseases, which tend to have a higher success rate if one
of them is a success. We believe that the disease split and target split
scenarios may give a more realistic indication of success chance and are thus
more relevant. Second, in the disease and random split cases, there is a
significant improvement in performance by including some type of target-level
embeddings. This could be because the embeddings uniquely identify the targets
and therefore allow the classifiers to “memorise” the successful drug targets
for a particular disease and make accurate prediction for other similar
targets for that disease. On the other hand, in the case of target splits,
adding additional target embeddings does not help much in improving the
prediction performance except the Metapath2vec embeddings.

Comparing the performance of the various graph embedding approaches more
closely; some interesting observations can be made. For example, on the
disease and random splits, it can be seen that the choice of the graph
embedding has little impact on the over predictive performance – apart from
the RGCN embeddings, which results in significantly worse performance overall.
On the target split, as previous mentioned, both the Metapath2vec and Node2vec
embeddings outperform the others.

###  5.2. Evaluating Matrix and Tensor Factorisation Methods

In this experiment, we compared our proposed tensor factorisation method with
Macau Bayesian probabilistic matrix factorisation and Macau tensor
factorisation (Simm et al., 2015). The target representations in our method
are the Metapath2vec embeddings. We again used all three evaluation
strategies, and the results are highlighted in Figure 5. The figure
demonstrates that in the disease and random split scenarios, our model clearly
outperforms both the Bayesian matrix and tensor factorisation methods, which
shows how incorporating knowledge graph embeddings into the tensor
factorisation can lead to better overall predicative performance. However, our
approach is outperformed by the Macau matrix factorisation on the random data
split.

Figure 5. Prediction result of our method and Macau matrix and tensor
factorisation method.

###  5.3. Evaluating top target-disease predictions

In order to gain some subjective insight into our model predictions, we
examined the top 7 target-disease predictions (ranked by prediction score)
that were not present in the PharmaProjects data set, and that were present in
all 3 data splits (random, disease and protein) (2). Interestingly, we see a
mixture of disease types, and different subtypes of some of the diseases (e.g.
primary and secondary hypertension). This suggests that the predictions aren’t
simply dominated by prominent information types expected in biomedical
literature (e.g. many different subtypes of solid tumour). On closer
examination, some of the predicted relationships would appear to have some
initial plausibility, for example the gene CASR is involved in calcium
homeostasis (Hendy and Canaff, 2016; Pu et al., 2016), an important aspect of
bone biology, and SCN5A is noted to be involved in cardiac function with a
noted link to heart disease (Zaklyazminskaya and Dzemeshkevich, 2016; Remme,
2013). Of course this initial superficial assessment of the predictions
highlights the challenge of interpretability of such approaches, in which
improvements will help us better understand the value and risks in such
predictions. Further work is planned in this area.

Rank | Target-Disease Pair
---|---
1 | CASR - osteoporosis
2 | CASR - congestive heart failure
3 | SCN5A - secondary hypertension
4 | SCN5A - primary hypertension
5 | IFNAR2 - macular degeneration
6 | CHRM2 - lymphocytic colitis
7 | ADCY9 - irritable bowel syndrome
Table 2. Common predicted target-disease pairs in all three evaluation methods

##  6\. Discussion and Conclusion

To help facilitate the identification of potential drug targets at the early
stage of drug discovery and development, in this paper we processed target-
disease evidence data from the Open Targets platform as well as clinical trial
data from PharmaProjects database and employed standard machine learning
classification algorithms and both matrix and tensor factorisation techniques
to predict the clinical outcomes of unseen target-disease pairs. We adopted a
new scheme when training the tensor factorisation method, i.e., alongside with
a deep neural network, and showed that this outperforms many baseline models.
Another main finding is that by using target representations learned by
biomedical knowledge graph embedding algorithms as side information, the
tensor factorisation performance can be dramatically improved, compared with
traditional approaches where the side information only contains limited gene
expression or protein-protein interaction data.

There are a number of limitations of the work reported. First, we did not
distinguish between various types of negative samples (i.e., failed target-
disease clinical trials) in the collected data. In reality, the data
negativeness is highly variable – multiple reasons could be responsible for
the failure of a clinical trial: drug safety (toxicology or clinical safety),
drug efficacy (failure to achieve sufficient efficacy),
pharmacokinetics/pharmacodynamics (PK/PD), commercial benefits, company
strategy, etc (Morgan et al., 2018). These complex scientific and non-
scientific factors make it difficult to obtain the true negative samples. In
the future it would therefore be helpful to take this into consideration and
classify the negative samples in the training set. Second, we treated all the
positive samples, i.e., successful clinical trials, equally important in the
training, however in practice, there exists some data that is more trustworthy
than others. Hence, it would be important to take this into account when
training the model by assigning different weights to different positive data
points and penalise more if the model predicts the highly weighted data
incorrectly. Third, in our framework we only used side information for
targets, it would be helpful in the future to exploit disease and evidence
side information such as disease similarities, via embeddings learned from
knowledge graphs.

Our work could be extended in several ways. For instance, it would be
interesting to take into consideration more biological data sets such as
ChEMBL to further enrich the target-disease-evidence tensor, which will help
reduce the data sparsity. Another interesting line of investigation would be
to design a more rigorous benchmarking strategy such as using well-curated
disease databases for a handful of diseases of interest. Finally, the
prediction improvements shown by utilising graph embeddings suggests there is
still much to explore at the intersection between data modelling and graph
composition, and algorithm application.

###### Acknowledgements.

The authors would like to thank Ufuk Kirik, Manasa Ramakrishna, Elizaveta
Semenova and Claus Bendtsen for help and feedback throughout the preparation
of this manuscript. Stephen Bonner is a fellow of the AstraZeneca post-
doctoral program.

## References

  * (1)
  * fda ([n.d.]) [n.d.].  The Drug Development Process.  https://www.fda.gov/patients/learn-about-drug-and-device-approvals/drug-development-process.
  * oxo ([n.d.]) [n.d.].  EMBL-EBI Ontology Xref Service (OxO).  https://www.ebi.ac.uk/spot/oxo/.
  * Pha ([n.d.]) [n.d.].  Informa’s Pharmaprojects.  https://pharmaintelligence.informa.com/products-and-services/data-and-analysis/pharmaprojects.
  * Anisimov et al. (2014) Anatoly Anisimov, Oleksandr Marchenko, Volodymyr Taranukha, and Taras Vozniuk. 2014.  Semantic and syntactic model of natural language based on non-negative matrix and tensor factorization. In _International Conference on Natural Language Processing_. Springer, 177–184.
  * Chen and Li (2018) Huiyuan Chen and Jing Li. 2018.  Drugcom: Synergistic discovery of drug combinations using tensor decomposition. In _2018 IEEE International Conference on Data Mining (ICDM)_. IEEE, 899–904.
  * Chen and Li (2019) Huiyuan Chen and Jing Li. 2019.  Modeling relational drug-target-disease interactions via tensor factorization with multiple web sources. In _The World Wide Web Conference_. 218–227.
  * Chen and Guestrin (2016) Tianqi Chen and Carlos Guestrin. 2016.  Xgboost: A scalable tree boosting system. In _Proceedings of the 22nd acm sigkdd international conference on knowledge discovery and data mining_. 785–794.
  * Cook et al. (2014) David Cook, Dearg Brown, Robert Alexander, Ruth March, Paul Morgan, Gemma Satterthwaite, and Menelas N Pangalos. 2014.  Lessons learned from the fate of AstraZeneca’s drug pipeline: a five-dimensional framework.  _Nature reviews Drug discovery_ 13, 6 (2014), 419–431.
  * Dong et al. (2017) Yuxiao Dong, Nitesh V Chawla, and Ananthram Swami. 2017\.  metapath2vec: Scalable representation learning for heterogeneous networks. In _Proceedings of the 23rd ACM SIGKDD international conference on knowledge discovery and data mining_. 135–144.
  * Grover and Leskovec (2016) Aditya Grover and Jure Leskovec. 2016.  node2vec: Scalable feature learning for networks. In _Proceedings of the 22nd ACM SIGKDD international conference on Knowledge discovery and data mining_. 855–864.
  * Haeffele et al. (2014) Benjamin Haeffele, Eric Young, and Rene Vidal. 2014.  Structured low-rank matrix factorization: Optimality, algorithm, and applications to image processing. In _International conference on machine learning_. 2007–2015.
  * Hendy and Canaff (2016) Geoffrey N Hendy and Lucie Canaff. 2016.  Calcium-sensing receptor, proinflammatory cytokines and calcium homeostasis. In _Seminars in cell & developmental biology_, Vol. 49. Elsevier, 37–43.
  * Himmelstein et al. (2017) Daniel Scott Himmelstein, Antoine Lizee, Christine Hessler, Leo Brueggeman, Sabrina L Chen, Dexter Hadley, Ari Green, Pouya Khankhanian, and Sergio E Baranzini. 2017.  Systematic integration of biomedical knowledge prioritizes drugs for repurposing.  _Elife_ 6 (2017), e26726.
  * Hughes et al. (2011) James P Hughes, Stephen Rees, S Barrett Kalindjian, and Karen L Philpott. 2011.  Principles of early drug discovery.  _British journal of pharmacology_ 162, 6 (2011), 1239–1249.
  * Kipf and Welling (2016) Thomas N Kipf and Max Welling. 2016.  Variational graph auto-encoders.  _arXiv preprint arXiv:1611.07308_ (2016).
  * Lipscomb (2000) Carolyn E Lipscomb. 2000\.  Medical subject headings (MeSH).  _Bulletin of the Medical Library Association_ 88, 3 (2000), 265\.
  * Morgan et al. (2018) Paul Morgan, Dean G Brown, Simon Lennard, Mark J Anderton, J Carl Barrett, Ulf Eriksson, Mark Fidock, Bengt Hamren, Anthony Johnson, Ruth E March, et al. 2018\.  Impact of a five-dimensional framework on R&D productivity at AstraZeneca.  _Nature reviews Drug discovery_ 17, 3 (2018), 167\.
  * Mungall et al. (2017) Christopher J Mungall, Julie A McMurry, Sebastian Köhler, James P Balhoff, Charles Borromeo, Matthew Brush, Seth Carbon, Tom Conlin, Nathan Dunn, Mark Engelstad, et al. 2017\.  The Monarch Initiative: an integrative data and analytic platform connecting phenotypes to genotypes across species.  _Nucleic acids research_ 45, D1 (2017), D712–D722.
  * Natarajan and Dhillon (2014) Nagarajan Natarajan and Inderjit S Dhillon. 2014.  Inductive matrix completion for predicting gene–disease associations.  _Bioinformatics_ 30, 12 (2014), i60–i68.
  * Ochoa et al. (2021) David Ochoa, Andrew Hercules, Miguel Carmona, Daniel Suveges, Asier Gonzalez-Uriarte, Cinzia Malangone, Alfredo Miranda, Luca Fumis, Denise Carvalho-Silva, Michaela Spitzer, et al. 2021\.  Open Targets Platform: supporting systematic drug–target identification and prioritisation.  _Nucleic Acids Research_ 49, D1 (2021), D1302–D1310.
  * Paliwal et al. (2020) Saee Paliwal, Alex de Giorgio, Daniel Neil, Jean-Baptiste Michel, and Alix MB Lacoste. 2020\.  Preclinical validation of therapeutic targets predicted by tensor factorization on heterogeneous graphs.  _Scientific reports_ 10, 1 (2020), 1–19.
  * Park et al. (2017) Yoojin Park, Jinoh Oh, and Hwanjo Yu. 2017.  RecTime: Real-Time recommender system for online broadcasting.  _Information Sciences_ 409 (2017), 1–16.
  * Pedregosa et al. (2011) Fabian Pedregosa, Gaël Varoquaux, Alexandre Gramfort, Vincent Michel, Bertrand Thirion, Olivier Grisel, Mathieu Blondel, Peter Prettenhofer, Ron Weiss, Vincent Dubourg, et al. 2011\.  Scikit-learn: Machine learning in Python.  _the Journal of machine Learning research_ 12 (2011), 2825–2830.
  * Piro and Di Cunto (2012) Rosario M Piro and Ferdinando Di Cunto. 2012.  Computational approaches to disease-gene prediction: rationale, classification and successes.  _The FEBS journal_ 279, 5 (2012), 678–696.
  * Pu et al. (2016) Fan Pu, Ning Chen, and Shenghui Xue. 2016.  Calcium intake, calcium homeostasis and health.  _Food Science and Human Wellness_ 5, 1 (2016), 8 – 16.  https://doi.org/10.1016/j.fshw.2016.01.001
  * Pushpakom et al. (2019) Sudeep Pushpakom, Francesco Iorio, Patrick A Eyers, K Jane Escott, Shirley Hopper, Andrew Wells, Andrew Doig, Tim Guilliams, Joanna Latimer, Christine McNamee, et al. 2019\.  Drug repurposing: progress, challenges and recommendations.  _Nature reviews Drug discovery_ 18, 1 (2019), 41–58.
  * Remme (2013) C. A. Remme. 2013\.  Cardiac sodium channelopathy associated with SCN5A mutations: electrophysiological, molecular and genetic aspects.  _The Journal Physiology_ 591, 17 (2013), 4099–4116.
  * Simm et al. (2015) Jaak Simm, Adam Arany, Pooya Zakeri, Tom Haber, Jörg K Wegner, Vladimir Chupakhin, Hugo Ceulemans, and Yves Moreau. 2015\.  Macau: scalable bayesian multi-relational factorization with side information using MCMC.  _arXiv preprint arXiv:1509.04610_ (2015).
  * Trouillon et al. (2016) Théo Trouillon, Johannes Welbl, Sebastian Riedel, Éric Gaussier, and Guillaume Bouchard. 2016.  Complex embeddings for simple link prediction. International Conference on Machine Learning (ICML).
  * Van de Cruys (2009) Tim Van de Cruys. 2009\.  A non-negative tensor factorization model for selectional preference induction. In _Proceedings of the Workshop on Geometrical Models of Natural Language Semantics_. Association for Computational Linguistics, 83–90.
  * Vitsios and Petrovski (2020) Dimitrios Vitsios and Slavé Petrovski. 2020.  Mantis-ml: Disease-Agnostic Gene Prioritization from High-Throughput Genomic Screens by Stochastic Semi-supervised Learning.  _The American Journal of Human Genetics_ 106, 5 (2020), 659–678.
  * Wang et al. (2017) Quan Wang, Zhendong Mao, Bin Wang, and Li Guo. 2017\.  Knowledge graph embedding: A survey of approaches and applications.  _IEEE Transactions on Knowledge and Data Engineering_ 29, 12 (2017), 2724–2743.
  * Wang et al. (2014) Zhen Wang, Jianwen Zhang, Jianlin Feng, and Zheng Chen. 2014\.  Knowledge graph embedding by translating on hyperplanes. In _Proceedings of the AAAI Conference on Artificial Intelligence_ , Vol. 28.
  * Ward et al. (2019) Rachel Ward, Xiaoxia Wu, and Leon Bottou. 2019.  AdaGrad stepsizes: Sharp convergence over nonconvex landscapes. In _International Conference on Machine Learning_. PMLR, 6677–6686.
  * Yao et al. (2019) Jin Yao, Mark R Hurle, Matthew R Nelson, and Pankaj Agarwal. 2019\.  Predicting clinically promising therapeutic hypotheses using tensor factorization.  _BMC bioinformatics_ 20, 1 (2019), 69.
  * Zaklyazminskaya and Dzemeshkevich (2016) Elena Zaklyazminskaya and Sergei Dzemeshkevich. 2016.  The role of mutations in the SCN5A gene in cardiomyopathies.  _Biochimica et Biophysica Acta (BBA) - Molecular Cell Research_ 1863, 7, Part B (2016), 1799 – 1805.  https://doi.org/10.1016/j.bbamcr.2016.02.014 Cardiomyocyte Biology: Integration of Developmental and Environmental Cues in the Heart.
  * Zheng et al. (2020) Shuangjia Zheng, Jiahua Rao, Ying Song, Jixian Zhang, Xianglu Xiao, Evandro Fei Fang, Yuedong Yang, and Zhangming Niu. 2020\.  PharmKG: a dedicated knowledge graph benchmark for bomedical data mining.  _Briefings in Bioinformatics_ (2020).
  * Zheng et al. (2016) Xiaolin Zheng, Weifeng Ding, Zhen Lin, and Chaochao Chen. 2016\.  Topic tensor factorization for recommender system.  _Information Sciences_ 372 (2016), 276–293.