# Biorthogonal Dynamical Quantum Phase Transitions in Non-Hermitian Systems

Yecheng Jing  Department of Physics and Chongqing Key Laboratory for Strongly
Coupled Physics, Chongqing University, Chongqing 401331, China  Jian-Jun Dong
<EMAIL_ADDRESS>Department of Physics and Chongqing Key Laboratory for
Strongly Coupled Physics, Chongqing University, Chongqing 401331, China  Yu-Yu
Zhang<EMAIL_ADDRESS>Department of Physics and Chongqing Key Laboratory
for Strongly Coupled Physics, Chongqing University, Chongqing 401331, China
Zi-Xiang Hu<EMAIL_ADDRESS>Department of Physics and Chongqing Key
Laboratory for Strongly Coupled Physics, Chongqing University, Chongqing
401331, China

###### Abstract

By utilizing biorthogonal bases, we develop a comprehensive framework for
studying biorthogonal dynamical quantum phase transitions in non-Hermitian
systems. With the help of the previously overlooked associated state, we
define the automatically normalized biorthogonal Loschmidt echo. This approach
is capable of handling arbitrary non-Hermitian systems with complex
eigenvalues and naturally eliminates the negative value of Loschmidt rate
obtained without the biorthogonal bases. Taking the non-Hermitian Su-
Schrieffer-Heeger model as a concrete example, a $1/2$ change of dynamical
topological order parameter in biorthogonal bases is observed which is not
shown in self-normal bases. Furthermore, we discover that the periodicity of
biorthogonal dynamical quantum phase transitions depends on whether the two-
level subsystem at the critical momentum oscillates or reaches a steady state.

The past decades have witnessed the flourishing of non-Hermitian physics in
nonconservative systems as found in a variety of physical realms including
open quantum systems Rotter2009JPA , electronic systems with interactions
Yoshida2018PRB ; Shen2018PRL ; Fu2020PRL , and classical systems with gain or
loss Makris2008PRL ; Klaiman2008PRL ; Malzard2015PRL . In these systems
Bender1998PRL ; Helbig2020NatPhys ; Liu2021Research ; Weidemann2020Science ;
Brandenbourger2019NatCommun ; Ghatak2020PNAS ; XZhang2021NatCommun ;
LZhang2021NatCommun ; Xiao2020NatPhys ; Wang2021JOpt ; ZhongWang2018PRL ;
Sato2019PRX ; Yokomizo2019PRL ; Shen2018PRL2 ; Balabanov2021PRR ; Guo2023PRL ,
many novel physics and unprecedented phenomena have been explored recently,
such as the exceptional points Hodaei2017Nature ; Bergholtz2021RMP , the non-
Hermitian skin effects Zhang2020PRL ; XiujuanZhang2022AdvPX , the bulk Fermi
arcs Zhou2018Science , and so on. In contrast to the Hermitian systems with
real eigenvalues and orthogonal eigenstates, the eigenvalues and eigenstates
in a general non-Hermitian Hamiltonian are not necessarily real and orthogonal
Ueda2020AdvPhys . To be more precise, the orthogonality of eigenstates is
replaced by the notion of biorthogonality that defines the relation between
the Hilbert space of states and its dual space, leading to the so-called
“biorthogonal quantum mechanics” Brody2014JPA ; Emil2018PRL ; Emil2019PRBR ;
Emil2020PRR . A direct consequence of the biorthogonality is that the
transition probability between one state $\left|\phi\right\rangle$ to its
time-evolved state $\left|\phi\left(t\right)\right\rangle$ should be carefully
defined. The traditional viewpoint
$p=\left|\left\langle\phi\left(t\right)|\phi\right\rangle\right|^{2}$ used in
Hermitian systems cannot be applied to a general non-Hermitian Hamiltonian. A
schematic framework to deal with this becomes an urgent topic due to the rapid
development of nonequilibrium studys in non-Hermitian systems ShuChen2021PRA ;
HuiZhai2020NatPhys ; Lin2022NPJQI ; Liu2023PRA ; Hauke2022PRXQ ;
Kawabata2023PRX ; Yoshimura2020PRB ; Mcdonald2022PRB ; Yin2022PRB ;
Agarwal2022arXiv ; Agarwal2023arXiv ; Roubeas2023JHEP .

Dynamical quantum phase transition (DQPT) is arguably one of the most
important nonequilibrium phenomena in modern many-body physics and has also
been extensively studied in past decades Heyl2013PRL ; Budich2016PRB ;
Dong2019PRB ; Nie2020PRL ; ShuChen2023PRB . It was first introduced in the
Hermitian transverse field Ising model Heyl2013PRL  and was generalized to
mixed state Heyl2017PRB ; Bhattacharya2017PRB , finite temperature
Abeling2016PRB ; Sedlmayr2018PRB ; Halimeh2018PRL ; Halimeh2018PRB ;
Mera2018PRB , Floquet systems Yang2019PRB ; Naji2022PRA ; Jafari2021PRA ;
Jafari2022PRB , and slow quench process Sharma2016PRB . It was also observed
experimentally with trapped ions Shen2017PRL ; Zhang2017Nature , Rydberg atoms
Bernien2017Nature , ultracold atoms Sengstock2018NatPhys , superconducting
qubits HengFan2019PRAppl , nanomechanical and photonic systems
JiangfengDu2019PRB ; PengXue2019PRL ; GuangcanGuo2020LSA . The key quantity
characterizing DQPT is the Loschmidt echo or dynamical fidelity,
$\mathcal{L}\left(t\right)\equiv\left|\left\langle\Psi\left(0\right)|\Psi\left(t\right)\right\rangle\right|^{2}$,
quantifying the time-dependent deviation from an arbitrary initial state
$\left|\Psi\left(0\right)\right\rangle$. In Hermitian systems, DQPTs occur
whenever the time-evolved state $\left|\Psi\left(t\right)\right\rangle$
becomes orthogonal to the initial state
$\left|\Psi\left(0\right)\right\rangle$ and the critical time $t_{c}$ is
defined as $\mathcal{L}\left(t_{c}\right)=0$. Efforts have been made to
generalize this concept to non-Hermitian systems, leading to many interesting
predictions such as the half-integer jumps in dynamical topological order
parameter (DTOP) Mondal2022PRB , but the special biorthogonality has been
ignored and an enforced normalized factor has been used Mondal2022PRB ;
Zhou2018PRA ; Zhou2021NJP ; Mondal2022arxiv . Very recently, it has been shown
that the non-Hermitian systems should be described by the biorthogonal
fidelity and the biorthogonal Loschmidt echo instead of the conventional
counterparts in Hermitian systems ShuChen2018PRA ; Huang2021PRR , but it is
limited to the parity-time symmetry cases Sun2022FP ; Tang2022EPL . A natural
treatment for a general non-Hermitian Hamiltonian is still lacking, which
severely limits our exploration of the richness of DQPTs in these systems.

In this Letter, we address this issue by proposing a new theoretical framework
to deal with these nonequilibrium phenomena in general non-Hermitian systems.
Based on biorthogonal quantum mechanics, we reformulate the transition
probability between $\left|\Psi\left(0\right)\right\rangle$ and
$\left|\Psi\left(t\right)\right\rangle$ with the biorthogonal bases and the
associated states. We introduce the concept of biorthogonal dynamical quantum
phase transitions and compare it with its self-normal counterpart, utilizing
the non-Hermitian Su-Schrieffer-Heeger model as a concrete example. We conduct
a comprehensive investigation of the biorthogonal Loschmidt rate, biorthogonal
DTOP, Fisher zeros, and the transition probability in momentum space during a
sudden quench. Our calculations reveal a half-integer jump in the biorthogonal
DTOP and demonstrate that the periodicity of biorthogonal DQPTs depends on
whether the two-level subsystem at the critical momentum undergoes oscillation
or settles into a steady state. Our theory establishes a general framework for
studying nonequilibrium DQPTs in non-Hermitian systems.

We first review some basic properties of biorthogonal quantum mechanics in
non-Hermitian systems and reformulate the probability assignment rules between
two states with biorthogonal bases Brody2014JPA . For a general non-Hermitian
Hamiltonian $H\neq H^{{\dagger}}$, the eigenvalue equations of $H$ and
$H^{{\dagger}}$ are given by

$\displaystyle H|u_{n}\rangle$
$\displaystyle=\epsilon_{n}|u_{n}\rangle,\quad\langle
u_{n}|H^{\dagger}=\epsilon_{n}^{\ast}\langle u_{n}|,$ $\displaystyle
H^{{\dagger}}\left|\widetilde{u}_{n}\right\rangle$
$\displaystyle=\epsilon_{n}^{\ast}\left|\widetilde{u}_{n}\right\rangle,\quad\left\langle\widetilde{u}_{n}\right|H=\epsilon_{n}\left\langle\widetilde{u}_{n}\right|,$
(1)

where $\epsilon_{n}$ is the nth eigenvalue, and $|u_{n}\rangle$ and
$\left|\widetilde{u}_{n}\right\rangle$ are the right and left eigenstates that
satisfy the completeness relation $\sum_{n}|\widetilde{u}_{n}\rangle\langle
u_{n}|=1$ and the biorthonormal relation
$\langle\widetilde{u}_{m}|u_{n}\rangle=\delta_{m,n}$. Note that under this
condition $\langle\widetilde{u}_{n}|u_{n}\rangle=1$, eigenstates are no longer
normalized. In particular, we have $\langle u_{n}|u_{n}\rangle\geq 1$ and
$\langle u_{n}|u_{m}\rangle\neq 0$ if $n\neq m$. It challenges the traditional
probabilistic interpretation used in Hermitian quantum mechanics. For
instance, there cannot be a “transition” from one eigenstate $|u_{m}\rangle$
to another eigenstate $|u_{n}\rangle$ due to the orthonormal relation $\langle
u_{n}|u_{m}\rangle=0$ if $n\neq m$ in Hermitian systems. To reconcile these
apparent contradictions we need the introduction of the so-called associated
state and the redefinition of the inner product. For an arbitrary state
$|\psi\rangle$, its associated state $|\widetilde{\psi}\rangle$ is defined
according to the following relation Brody2014JPA :

$|\psi\rangle=\sum_{n}c_{n}|u_{n}\rangle\longleftrightarrow|\widetilde{\psi}\rangle=\sum_{n}c_{n}|\widetilde{u}_{n}\rangle,$
(2)

while the dual state
$\langle\widetilde{\psi}|=\sum_{n}c_{n}^{\ast}\left\langle\widetilde{u}_{n}\right|$
is given by the Hermitian conjugate of $|\widetilde{\psi}\rangle$. The inner
product between $|\psi\rangle$ and another state
$|\phi\rangle=\sum_{m}d_{m}|u_{m}\rangle$ is thus defined as

$\langle\phi,\psi\rangle\equiv\langle\widetilde{\phi}|\psi\rangle=\sum_{m,n}\langle\widetilde{u}_{m}|d_{m}^{\ast}c_{n}|u_{n}\rangle=\sum_{n}d_{n}^{\ast}c_{n}.$
(3)

It is easy to show that the norm of a state $|\psi\rangle$ is
$\sqrt{\langle\widetilde{\psi}|\psi\rangle}$. With these new definitions, the
transition probability between $|\psi\rangle$ and $|\phi\rangle$ for a
biorthogonal system is given by

$p=\frac{\langle\widetilde{\psi}|\phi\rangle\langle\widetilde{\phi}|\psi\rangle}{\langle\widetilde{\psi}|\psi\rangle\langle\widetilde{\phi}|\phi\rangle},$
(4)

where the denominator acts as a natural normalizing factor. In this case, $p$
is a real number ranging from $0$ to $1$, which meets the requirements of
probability interpretation satisfactorily. If the Hamiltonian is Hermitian
$H=H^{{\dagger}}$, we have $|\widetilde{u}_{n}\rangle=|u_{n}\rangle$ and
$|\widetilde{\psi}\rangle=|\psi\rangle$. Then Eq. (4) reduces to the
conventional definition of the transition probability in Hermitian quantum
mechanics
$p=\langle\psi|\phi\rangle\langle\phi|\psi\rangle/\left(\langle\psi|\psi\rangle\langle\phi|\phi\rangle\right)$.
Another important consequence of Eq. (4) is that the projection from
$|\psi\rangle=\sum_{n}c_{n}|u_{n}\rangle$ to $|u_{n}\rangle$ becomes

$p_{n}=\frac{\langle\widetilde{\psi}|u_{n}\rangle\langle\widetilde{u}_{n}|\psi\rangle}{\langle\widetilde{\psi}|\psi\rangle\langle\widetilde{u}_{n}|u_{n}\rangle}=\frac{c_{n}^{\ast}c_{n}}{\sum_{m}c_{m}^{\ast}c_{m}},$
(5)

satisfying the normalization condition $\sum_{n}p_{n}=1$.

By now the static aspects of non-Hermitian systems are clear, but the
dynamical problems remain controversial. Although the time evolution of an
arbitrary initial state $|\Psi\left(0\right)\rangle$ under a time-independent
Hamiltonian $H$ is given by the well-known Schrödinger equation
$|\Psi\left(t\right)\rangle=\operatorname{e}^{-\operatorname*{i}Ht}\left|\Psi\left(0\right)\right\rangle$,
the direct generalization
$|\widetilde{\Psi}\left(t\right)\rangle=\operatorname{e}^{-\operatorname*{i}H^{{\dagger}}t}|\widetilde{\Psi}\left(0\right)\rangle$
Tang2022EPL  may lead to unreasonable complex probabilities if there is no
parity-time symmetry SM . To establish a general framework for non-Hermitian
Hamiltonian with complex eigenvalues, we use the associated state
$|\widetilde{\Psi}\left(t\right)\rangle=\sum_{n}c_{n}|\widetilde{u}_{n}\rangle$
where $c_{n}=\langle\widetilde{u}_{n}|\Psi\left(t\right)\rangle$ to deal with
the dynamical problem. The overlap between $|\Psi\left(0\right)\rangle$ and
$|\Psi\left(t\right)\rangle$ is characterized by the biorthogonal Loschmidt
echo,

$\mathcal{L}\left(t\right)=\frac{\langle\widetilde{\Psi}\left(0\right)|\Psi\left(t\right)\rangle\langle\widetilde{\Psi}\left(t\right)|\Psi\left(0\right)\rangle}{\langle\widetilde{\Psi}\left(t\right)|\Psi\left(t\right)\rangle\langle\widetilde{\Psi}\left(0\right)|\Psi\left(0\right)\rangle}.$
(6)

As in the Hermitian case, the biorthogonal DQPTs can be defined as
$\mathcal{L}\left(t_{c}\right)=0$ with the critical time $t_{c}$.

We may examine the biorthogonal DQPT by considering a general non-Hermitian
Hamiltonian $H=\sum_{k}\psi_{k}^{{\dagger}}H_{k}\psi_{k}$ with
$H_{k}=\bm{d}_{k}\cdot\bm{\sigma}$. Here, $\bm{\sigma}$ is the vector of the
Pauli matrices, $\bm{d}_{k}=\left(x_{k},y_{k},z_{k}\right)$ represents the
vector of expansion coefficients, and at least one component of $\bm{d}_{k}$
is complex in non-Hermitian systems. The eigenenergies of $H_{k}$ are given by
$\pm\epsilon_{k}=\pm\sqrt{\bm{d}_{k}^{2}}$, and the corresponding eigenstates
are $|u_{k\pm}\rangle$. Considering a quench process where the model
Hamiltonian changes from $\bm{d}_{k}^{i}$ at time $t=0^{-}$ to
$\bm{d}_{k}^{f}$ at time $t=0^{+}$, the initial state
$|\Psi\left(0\right)\rangle={\textstyle\bigotimes\nolimits_{k}}|u_{k-}^{i}\rangle$
which is defined as the tensor product of all $|u_{k-}^{i}\rangle$ is evolved
under the postquench Hamiltonian $\bm{d}_{k}^{f}$. The biorthogonal Loschmidt
echo can be expressed as $\mathcal{L}(t)=\prod_{k}g_{k}(t)$ with SM

$\displaystyle
g_{k}(t)=\frac{|\cos(\epsilon_{k}^{f}t)-\operatorname*{i}\sin(\epsilon_{k}^{f}t)\langle\widetilde{u}_{k-}^{i}|\frac{H_{k}^{f}}{\epsilon_{k}^{f}}|u_{k-}^{i}\rangle|^{2}}{\langle\widetilde{u}_{k-}^{i}(t)|u_{k-}^{i}(t)\rangle},$
(7)

where
$|u_{k-}^{i}(t)\rangle=\text{e}^{-\operatorname*{i}H^{f}_{k}t}|u_{k-}^{i}\rangle$.
To obtain a nonzero and well-defined quantity in the thermodynamic limit it is
useful to consider the biorthogonal Loschmidt rate

$\displaystyle\text{LR}(t)$
$\displaystyle=-\displaystyle\lim_{N\to\infty}\frac{1}{N}\ln{\mathcal{L}(t)},$
(8)

where $N$ is the system size. Zeros in $\mathcal{L}(t)$ at critical times
$t_{c}$ correspond to nonanalyticities (cusps or divergencies) in
$\text{LR}(t)$. If there is at least one pair of critical parameters $k_{c}$
and $t_{c}$ such that $g_{k_{c}}(t_{c})=0$, then $\mathcal{L}(t_{c})=0$. The
solution of $g_{k_{c}}(t_{c})=0$ is

$\displaystyle
t_{c}=\frac{\pi}{2\epsilon^{f}_{k_{c}}}(2n+1)-\frac{\operatorname*{i}}{\epsilon^{f}_{k_{c}}}\tanh^{-1}\langle\widetilde{u}_{k_{c}-}^{i}|\frac{H_{k_{c}}^{f}}{\epsilon_{k_{c}}^{f}}|u_{{k}_{c}-}^{i}\rangle,$
(9)

where $n$ is an integer. If we can obtain a positive real solution $t_{c}$,
the system will undergo a biorthogonal DQPT. In general, it is difficult to
access $k_{c}$ in a finite-size system because momentum takes quantized
values. Thus the divergence of $\text{LR}(t)$ in the thermodynamic limit
becomes a cusp in a finite-size system except for some fine-tuned quench
parameters or twist boundary conditions ShuChen2023PRB . For the same reason,
it is also difficult to obtain a positive real solution $t_{c}$ in a finite
system. To address this issue, we can solve the equation $\mathcal{L}(t)=0$
inversely and study the distribution of the roots in the complex time plane.
The solution of $\mathcal{L}(t_{c})=0$ depends on the system size $N$. If for
all $\epsilon>0$, there exists a natural number $N_{0}$ such that
$\operatorname{Im}[t_{c}(N)]<\epsilon$ holds for all $N>N_{0}$, a real
solution
$\lim_{N\rightarrow\infty}\operatorname{Im}\left[t_{c}\left(N\right)\right]=0$
can be obtained, indicating a biorthogonal DQPT.

Figure 1: (a) The sketch of non-Hermitian Su-Schrieffer-Heeger model. The box
indicates the unit cell. (b) The $\gamma$-$\eta$ phase diagram. Each region
filled with a different color corresponds to a phase with a topological
winding number $w$. The boundaries between the regions are determined using
$\epsilon_{k}=0$. The black points represent different parameters ($\eta$,
$\gamma$) which will be used below. Specifically, the points A to G represent
($-2$, $5$), ($-0.2$, $5$), ($0.2$, $5$), ($2$, $5$), ($-2$, $1$), ($-0.2$,
$1$), and ($0.2$, $1$), respectively.

Furthermore, similar to the DTOP in Hermitian systems Vajna2015PRB , we can
introduce a biorthogonal DTOP to describe biorthogonal DQPT. The biorthogonal
DTOP is defined as

$\displaystyle\nu(t)=\frac{1}{2\pi}\int_{0}^{2\pi}dk\partial_{k}\phi_{k}^{G}(t),$
(10)

where the biorthogonal geometrical phase is
$\phi^{G}_{k}(t)=\phi_{k}(t)-\phi_{k}^{\text{dyn}}(t)$, with $\phi_{k}(t)$
being the phase of $g_{k}(t)$ and the biorthogonal dynamical phase is given by
SM

$\displaystyle\phi_{k}^{\text{dyn}}(t)=$
$\displaystyle-\int_{0}^{t}ds\frac{\langle\widetilde{u}_{k-}^{i}(s)|H_{k}^{f}|u_{k-}^{i}(s)\rangle}{\langle\widetilde{u}_{k-}^{i}(s)|u_{k-}^{i}(s)\rangle}$
$\displaystyle+\frac{\text{i}}{2}\text{ln}\langle\widetilde{u}_{k-}^{i}(t)|u_{k-}^{i}(t)\rangle.$
(11)

To demonstrate the efficacy of our novel theoretical framework in dealing with
non-Hermitian Hamiltonians featuring complex eigenvalues, we present a
comprehensive examination of the non-Hermitian Su-Schrieffer-Heeger model as
outlined below. The Hamiltonian is

$\displaystyle H$
$\displaystyle=\sum_{j}\left[\left(1+\eta+\frac{\gamma}{2}\right)c_{j,b}^{{\dagger}}c_{j,a}+\left(1+\eta-\frac{\gamma}{2}\right)c_{j,a}^{{\dagger}}c_{j,b}\right.$
$\displaystyle\left.\quad+\left(1-\eta\right)c_{j,a}^{{\dagger}}c_{j+1,b}+\left(1-\eta\right)c_{j+1,b}^{{\dagger}}c_{j,a}\right],$
(12)

where $\eta$ determines the strength of intracell and intercell hopping and
$\gamma$ controls the degree of non-Hermiticity, as shown in Fig. 1(a). For
the periodic boundary condition, the bulk Hamiltonian gets the standard
bilinear form
$H=\sum_{k}\psi_{k}^{\dagger}\left(\bm{d}_{k}\cdot\bm{\sigma}\right)\psi_{k}$
in momentum space, where
$\psi_{k}^{\dagger}=(c_{k,a}^{{\dagger}},c_{k,b}^{{\dagger}})$ and

$\bm{d}_{k}=\left(\left(1+\eta\right)+\left(1-\eta\right)\cos
k,\left(1-\eta\right)\sin k-\operatorname*{i}\frac{\gamma}{2},0\right).$ (13)

The dispersion is $\pm\epsilon_{k}=\pm\sqrt{x_{k}^{2}+y_{k}^{2}}$. It becomes
gapless at the exceptional points. Thus the solution of $\epsilon_{k}=0$
determines the phase boundary, i.e., $k_{c}=0,\;\gamma=\pm 4$ and
$k_{c}=\pi,\;\gamma=\pm 4\eta$. Combining with the winding number
$w=\frac{1}{2\pi}\int_{0}^{2\pi}dk\partial_{k}\left(\arctan\frac{y_{k}}{x_{k}}\right)$
Yin2018PRA , we present the phase diagram in Fig. 1(b) for convenience, where
we only consider the case $\gamma>0$ due to the symmetry of the phase diagram
with respect to $\gamma$.

Figure 2: (a) The Loschmidt rate and (b) the DTOP with self-normal and
biorthogonal bases for a typical quench process (from G to F).

For comparison, we also calculate the traditional DQPT based on self-normal
Loschmidt echo
$\mathcal{L}\left(t\right)=\left|\left\langle\Psi\left(0\right)|\Psi\left(t\right)\right\rangle\right|^{2}$,
with an enforced normalized factor to avoid the negative value of Loschmidt
rate Zhou2018PRA . Figure 2(a) presents the self-normal and biorthogonal
Loschmidt rate from the same quenching process. The first feature is that the
critical time at which the cusp appears is different from each other. A
similar phenomenon has been observed in equilibrium quantum phase transitions,
where the biorthogonal and self-normal fidelity predict different quantum
critical points Sun2022FP . It turns out that the biorthogonal fidelity
captures the correct critical point due to the special biorthoganality in non-
Hermitian systems Sun2022FP . Similarly, the nonequilibrium quantum phase
transitions should be described by the biorthogonal time-dependent version of
the fidelity, i.e., biorthogonal Loschmidt echo. The second feature is that
there is an additional critical time $t_{c}\approx 0.74$ in biorthogonal
bases. This can be seen more clearly in the DTOP as shown in Fig. 2(b). A
half-integer jump of $\nu(t)$ can be observed in the biorthogonal bases while
there is no such change in the self-normal bases.

Figure 3: Five different types of quenching processes showing a $\frac{1}{2}$
change in the biorthogonal DTOP. (a) and (b) illustrate the biorthogonal
Loschmidt rate and biorthogonal DTOP, respectively. The $\frac{1}{2}$ change
in the biorthogonal DTOP appears if and only if the prequench phases are in
the middle of the phase diagram, with winding numbers of $\frac{1}{2}$ or $0$.

To show that the half-integer jump of $\nu(t)$ is not a fine-tune result
Ding2020PRBR , we study different quenching processes by changing $\eta$ but
with $\gamma$ fixed. Figures 3(a) and 3(b) present five typical behaviors of
the biorthogonal Loschmidt rate $\text{LR}\left(t\right)$ and biorthogonal
DTOP $\nu(t)$, respectively. And more detailed information can be found in the
Supplemental Material SM . The half-integer jump of $\nu(t)$ can appear alone,
periodically, or accompanied by an integer jump, exhibiting rich behavior in a
single non-Hermitian Hamiltonian. We also find that the half-integer jump
phenomenon occurs if and only if the prequench phases are in the middle of the
phase diagram with 0 or 1/2 winding number.

Figure 4: (a), (b) The dynamical Fisher zeros in the complex plane
$z_{n}(k)=\text{i}t_{n}(k)$ with $k$ lies in the first Brillouin zone for two
typical quench processes. (c), (d) The corresponding transition probability
$p(k_{c},t)$ between $|u_{k_{c}-}^{i}(t)\rangle$ and $|u_{k_{c}+}^{i}\rangle$
at the critical momentum $k_{c}$.

To better understand the biorthogonal DQPTs, we study the dynamical
counterpart of Fisher zeros Fisher1967RPP ; Yang1952PR ; Lee1952PR ;
Andraschko2014PRB ; Stauber2017PRE  in the complex time plane
$z_{n}(k)=\text{i}t_{n}(k)$. The lines $z_{n}(k)$ cross the imaginary time
axis at a critical momentum $k_{c}$ and yield a critical time $t_{c,n}$, as
shown in Figs. 4(a) and 4(b). The discontinuity of the dynamical Fisher zeros
over the imaginary axis is associated with the half-integer jump of $\nu(t)$
Mondal2022PRB . Figures 4(a) and 4(b) also represent two typical types of
biorthogonal DQPTs, depending on whether the lines $z_{n}(k)$ cut the time
axis periodically or nonperiodically. In order to further study these two
types of biorthogonal DQPTs, we also investigate the transition probability
$p(k,t)$ between the time-evolved state
$|u_{k-}^{i}(t)\rangle=\text{e}^{-\text{i}H^{f}_{k}t}|u_{k-}^{i}\rangle$ and
another initial eigenstate $|u_{k+}^{i}\rangle$. By expanding
$|u_{k-}^{i}(t)\rangle$ as
$|u_{k-}^{i}(t)\rangle=c_{1}|u_{k-}^{i}\rangle+c_{2}|u_{k+}^{i}\rangle$, we
have $p(k,t)=c_{2}^{\ast}c_{2}/(c_{1}^{\ast}c_{1}+c_{2}^{\ast}c_{2})$ from Eq.
(5). If there exists $p(k,t)=1$, the time-evolved state
$|u_{k-}^{i}(t)\rangle$ is biorthogonal with $|u_{k-}^{i}\rangle$. Then the
biorthogonal Loschmidt echo equals to zero because we can rewrite
$\mathcal{L}(t)$ as

$\displaystyle\mathcal{L}(t)=\prod_{k}\frac{\langle\widetilde{u}_{k-}^{i}|u_{k-}^{i}(t)\rangle\langle\widetilde{u}_{k-}^{i}(t)|u_{k-}^{i}\rangle}{\langle\widetilde{u}_{k-}^{i}(t)|u_{k-}^{i}(t)\rangle}.$
(14)

As shown in Figs. 4(c) and 4(d), the two types of biorthogonal DQPTs exhibit
two distinct behaviors of $p(k,t)$. For the periodic biorthogonal DQPTs,
$p(k_{c},t)$ oscillate periodically between $0$ and $1$ for fixed critical
momenta $k_{c}$. In this situation, the two-level systems of $k_{c}$ dominate.
Thus the periodicity of biorthogonal DQPTs may be related to the oscillations
of the two-level system Zakrzewski2022arXiv ; Damme2023PRR . On the other
hand, $p(k_{c},t)$ exhibit very interesting behavior when the biorthogonal
DQPTs are no longer periodic. There are many critical momenta $k_{c}$, and
each $k_{c}$ corresponds to only one $t_{c,n}$. Before
$t_{c,n}=-\text{i}z_{n}(k_{c})$, there are $n$ local maximum in $p(k_{c},t)$,
while for $t\gg t_{c,n}$, $p(k_{c},t)$ tends to a fixed value, indicating a
steady state in contrast to the oscillation behavior.

In summary, we propose a new theoretical framework to study the biorthogonal
DQPTs in non-Hermitian systems based on the biorthogonal quantum mechanics. We
reformulate the transition probability between one state
$\left|\Psi\left(0\right)\right\rangle$ and its time-evolved state
$\left|\Psi\left(t\right)\right\rangle$ with the concept of the associated
state. Our scheme can handle a general non-Hermitian Hamiltonian with complex
eigenvalues, and the normalization factors can be introduced naturally. To
demonstrate our approach, we use the non-Hermitian Su-Schrieffer-Heeger model
as a concrete example. Comparing with the self-normal cases, we observe a
clear 1/2 change in the biorthogonal DTOP. Furthermore, our results show that
the periodicity of biorthogonal DQPTs depends on whether the two-level
subsystem at the critical momentum oscillates or reaches a steady state. Our
work opens up avenues for exploring the rich biorthogonal DQPTs in non-
Hermitian systems. An interesting topic for further investigation would be the
study of the mechanism behind the 1/2 change in the biorthogonal DTOP.

We would like to acknowledge valuable discussions with S. Chen. This work is
supported by the National Natural Science Foundation of China Grants No.
12204075, No. 11974064, No. 12075040, and No. 12347101; the China Postdoctoral
Science Foundation Grant No. 2023M730420; the fellowship of Chongqing
Postdoctoral Program for Innovative Talents Grant No. CQBX202222; the Natural
Science Foundation of Chongqing Grant No. CSTB2023NSCQ-MSX0953; the Chongqing
Research Program of Basic Research and Frontier Technology Grants No.
cstc2021jcyjmsxmX0081 and No. cstc2020jcyj-msxmX0890; Chongqing Talents:
Exceptional Young Talents Project No. cstc2021ycjh-bgzxm0147; and the
Fundamental Research Funds for the Central Universities Grants No.
2020CDJQY-Z003, No. 2022CDJJCLK001, and No. 2021CDJQY-007.

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# Representation Transfer for Differentially Private Drug Sensitivity
Prediction

Teppo Niinimäki 1 Mikko Heikkilä 2 Antti Honkela 2,3,4,111These authors
jointly supervised the work. and Samuel Kaski 1,∗
1Helsinki Institute for Information Technology HIIT  Department of Computer
Science  Aalto University  Finland
2Helsinki Institute for Information Technology HIIT  Department of Mathematics
and Statistics  University of Helsinki  Finland
3Department of Public Health  University of Helsinki  Finland and
4Helsinki Institute for Information Technology HIIT  Department of Computer
Science  University of Helsinki  Finland

###### Abstract

Motivation: Human genomic datasets often contain sensitive information that
limits use and sharing of the data. In particular, simple anonymisation
strategies fail to provide sufficient level of protection for genomic data,
because the data are inherently identifiable. Differentially private machine
learning can help by guaranteeing that the published results do not leak too
much information about any individual data point. Recent research has reached
promising results on differentially private drug sensitivity prediction using
gene expression data. Differentially private learning with genomic data is
challenging because it is more difficult to guarantee the privacy in high
dimensions. Dimensionality reduction can help, but if the dimension reduction
mapping is learned from the data, then it needs to be differentially private
too, which can carry a significant privacy cost. Furthermore, the selection of
any hyperparameters (such as the target dimensionality) needs to also avoid
leaking private information.
Results: We study an approach that uses a large public dataset of similar type
to learn a compact representation for differentially private learning. We
compare three representation learning methods: variational autoencoders, PCA
and random projection. We solve two machine learning tasks on gene expression
of cancer cell lines: cancer type classification, and drug sensitivity
prediction. The experiments demonstrate significant benefit from all
representation learning methods with variational autoencoders providing the
most accurate predictions most often. Our results significantly improve over
previous state-of-the-art in accuracy of differentially private drug
sensitivity prediction.

##  1 Introduction

Privacy-preserving machine learning has the potential to enable the research
use of many sensitive datasets, that would otherwise be out of reach for the
community. This is especially the case for medical data, which almost always
contain sensitive information traceable back to the data subjects. As an
example, it has been shown that individuals can be identified from genomic
data (Gymrek et al., 2013) including mixtures from several individuals (Homer
et al., 2008). It is likely that functional genomics data such as gene
expression data are also identifiable. While different anonymisation
strategies (Sweeney, 2002; Machanavajjhala et al., 2007; Li et al., 2007) can
protect the privacy of the data subjects to some degree, they do not have
formal guarantees and can fail to provide sufficient protection in practice
(Ganta et al., 2008).

_Differential privacy_ (DP) (Dwork et al., 2006; Dwork and Roth, 2014) is a
framework that guarantees strict bounds for the amount of leaked private
information, even in the presence of arbitrary side information. The
guarantees are obtained by adding specific forms of randomisation to the
computation process. In a machine learning context this usually means adding
noise either directly to the input of the algorithm (_input perturbation_), to
the output (_output perturbation_), or modifying the algorithm itself, for
instance, by perturbing the optimisation objective (_objective perturbation_).

The privacy guarantee is controlled by a “privacy budget” parameter, usually
denoted by $\epsilon>0$; smaller $\epsilon$ means stricter guarantees, and can
be achieved by increasing the amount of noise. Formally, a randomised
mechanism $\mathcal{M}$ is said to be _$\epsilon$ -differentially private_, if
for all pairs of _neighbouring_ datasets $X,X^{\prime}$ differing222There are
two slightly different definitions of neighbouring datasets. In _bounded_
case, the value of one sample is allowed to change. In _unbounded_ case, the
addition or removal of one sample is allowed. Unbounded $\epsilon$-DP
guarantee implies bounded $2\epsilon$-DP guarantee. This article uses the
bounded case. on a single sample and all measurable subsets $S$ of possible
outputs,

$\Pr(\mathcal{M}(X)\in S)\leq e^{\epsilon}\Pr(\mathcal{M}(X^{\prime})\in S).$

Intuitively, this means that changing one sample in the dataset can change the
output distribution only by a factor $e^{\epsilon}$.

As an extension, $\mathcal{M}$ is said to be _( $\epsilon$,
$\delta$)-differentially private_, if

$\Pr(\mathcal{M}(X)\in S)\leq e^{\epsilon}\Pr(\mathcal{M}(X^{\prime})\in
S)+\delta,$

for all measurable $S$ and all neighbouring datasets $X,X^{\prime}$. The
condition with nonzero $\delta>0$ is often easier to achieve than pure
$\epsilon$-DP.

In this article we are interested in DP learning for drug sensitivity
prediction using gene expression data. First proposed by Staunton et al.
(2001), the drug sensitivity prediction problem has attracted significant
attention recently, including from a DREAM challenge in 2012 (Costello et al.,
2014) that provided standardised evaluation metrics. The scale of the
cytotoxicity assays needed has kept the sizes of the available data sets
relatively small from a machine learning perspective. Honkela et al. (2018)
were the first to apply DP learning to this problem. They needed to
specifically limit the sensitivity of the learning and the dimensionality of
the input data to make the learning feasible.

In abstract terms, our goal in this problem is DP learning of predictive
models with high-dimensional input data, where both input and output variables
need DP protection. This is a case where DP methods tend to run into trouble
with moderate dataset sizes: the amount of noise that needs to be added
usually increases quickly with the dimensionality, leading to output that is
dominated by the noise. This warrants the use of dimensionality reducing
methods with the aim of finding a good low-dimensional representation of the
original data. However, unless one uses a random projection or some other
“dummy” method that does not depend on the data, finding a good representation
can also leak private information. For this reason, the dimension reduction
method itself would also need to be made differentially private, which can
completely invalidate the noise magnitude savings obtained in any downstream
task like prediction.

Different solutions have been proposed for various special cases: Kifer et al.
(2012) solve sparse linear regression problems by using an $\epsilon$-DP
feature selection algorithm. Honkela et al. (2018) utilise external knowledge
to select a relevant subset of features. Kasiviswanathan and Jin (2016) show
theoretical results on using random projections to improve DP learning on
high-dimensional problems. Differentially private versions of methods such as
PCA (Chaudhuri et al., 2012; Dwork et al., 2014) or deep learning (Abadi et
al., 2016; Acs et al., 2018) exist and could be used to learn a
representation, but the noise cost can be impractically large for small but
high-dimensional datasets.

We study a straightforward solution based on feature representation transfer,
similar to self-taught learning of Raina et al. (2007). By using an additional
non-sensitive dataset to learn the representation, we can apply more advanced
representation learning methods. This approach has many advantages: we do not
need labels for the additional data set, although in our case we make use of
labels for a different task; and only the main learning algorithm needs to be
differentially private, while the representation can be learned using any non-
DP method. Additionally, the public data can also be used for optimising any
hyperparameters for the representation learning. In this article, we consider
PCA and variational autoencoders.

Differentially private transfer learning was recently considered by Wang et
al. (2019) in a hypothesis transfer setting, where models trained on several
related source domains are used to improve learning in the desired target
domain. While related, this approach is not directly applicable to our problem
because the setup is different.

Another related approach was considered by Papernot et al. (2017), who propose
differentially private semi-supervised knowledge transfer that uses an
ensemble of “teacher” models trained on private data to label unlabeled public
data, which is then used to train a “student” model that will be released. The
method is flexible in a sense that it can use any “black-box” model as
teachers and student. However, it is limited to classification tasks. In
addition, it seems to require a rather large private dataset in addition to a
small public dataset—a setting that is somewhat different from what we are
interested in.

Yet another strategy is to learn a differentially private unsupervised
generative model for the data (including the target variable for the
prediction task of interest), use it to generate a synthetic version of the
data, and use a non-DP algorithm for the actual learning task of interest.
Several methods have been proposed for differentially private data sharing
(Zhang et al., 2017; Xie et al., 2018; Acs et al., 2018) that could be used
for generative model learning and data generation. For the problem we are
considering, however, this approach is problematic as it requires solving a
more general and difficult learning task, good solution of which would
typically require orders of magnitude more private data than a direct solution
of the original prediction task.

The rest of this article is organised as follows: In Section 2 we formalise
the problem setting and give an overview of our proposed approach. Section 3
gives more details on the implementation of different parts of the proposed
approach. And finally, in Section 4 we conduct experiments with the approach
on two different prediction tasks on genomic data.

##  2 Approach

We assume a setting where we have a private dataset containing a high-
dimensional $n\times d$ feature matrix $X_{\text{priv}}$ and an $n\times 1$
target vector $Y_{\text{priv}}$, where $n$ is the number of samples and $d$ is
the number of features. The goal is to learn a differentially private
predictor from $X_{\text{priv}}$ to $Y_{\text{priv}}$. As learning to predict
$Y_{\text{priv}}$ from high-dimensional $X_{\text{priv}}$ directly is
typically not feasible, with moderate sample size and a reasonable privacy
budget, we opt for using public data to learn a low-dimensional representation
for $X_{\text{priv}}$. Therefore, we also assume a publicly available dataset
of an $m\times d$ feature matrix $X_{\text{pub}}$ and an $m\times 1$ auxiliary
target vector $Y_{\text{pub}}^{\prime}$ for a related auxiliary prediction
task. While a representation can be learned with $X_{\text{pub}}$ only, the
availability of $Y_{\text{pub}}^{\prime}$ is useful for selecting the size of
the representation and any other hyperparameters.

We make the following informal assumptions about the relation of the public
and the private data: (1) $X_{\text{priv}}$ and $X_{\text{pub}}$ contain the
same set of features and are either draws from the same distribution or
otherwise distributed similarly enough that using the same mapping to compute
a representation is reasonable. (2) $Y_{\text{pub}}^{\prime}$ may or may not
be of the same type as $Y_{\text{priv}}$, but the prediction tasks should
resemble each other enough that the prediction of $Y_{\text{pub}}^{\prime}$
can be used for optimising the hyperparameters for the main task of predicting
$Y_{\text{priv}}$.

Figure 1: The process of learning $f$ and $g$. Since the learning of $g$ is
DP, the leakage of information outside of the “privacy wall” is controlled.

We propose the following procedure:

  1. 1.

Use the public data to learn a dimension-reducing representation mapping
$f:\mathbb{R}^{d}\rightarrow\mathbb{R}^{r}$, where $r\ll d$, such that
$f^{-1}(f(X_{\text{pub}}))\approx X_{\text{pub}}$.

  2. 2.

Obtain a low-dimensional representation $Z_{\text{priv}}$ of the private
feature data by applying $f$ to $X_{\text{priv}}$.

  3. 3.

Learn a differentially private predictor $g$ such that
$g(Z_{\text{priv}})\approx Y_{\text{priv}}$.

  4. 4.

Publish $g\circ f$.

An overview of the learning process is shown in Figure 1.

It is easy to see that the proposed process has the same DP-guarantees as the
learning algorithm of step 3:

######  Theorem 1.

If step 3 is $(\epsilon,\delta)$-DP w.r.t. $Z_{\text{priv}}$ and
$Y_{\text{priv}}$, then the whole process is also $(\epsilon,\delta)$-DP w.r.t
$X_{\text{priv}}$ and $Y_{\text{priv}}$.

###### Proof.

As the learning of $f$ does not use private data, it does not leak any private
information. Since each row of $Z_{\text{priv}}$ depends only on the
corresponding row of $X_{\text{priv}}$, $(\epsilon,\delta)$-guarantees w.r.t.
$Z_{\text{priv}}$ translate directly to guarantees w.r.t. $X_{\text{priv}}$. ∎

In the following section we give some methods that can be used to implement
the DP predictor $g$ and the representation mapping $f$. In addition, we
describe a procedure for tuning the hyperparameters of $f$.

##  3 Methods

###  3.1 Differentially private prediction

Later in Section 4 we will consider prediction tasks that are either real-
valued regression or binary classification tasks. Linear regression will be
applied to the former and logistic regression to the latter. For now, denote
by $X$ the feature matrix and by $y$ the prediction target vector (either
real-valued or binary $\\{-1,1\\}$)

Logistic regression can be made differentially private with objective
perturbation. The usual non-DP version of the problem can be solved by
minimising the regularised negative log-likelihood
$n^{-1}\sum_{i=1}^{n}\log(1+e^{y_{i}w^{T}x_{i}})+\lambda w^{T}w$ with respect
to the weight vector $w$, where $x_{i}$ and $y_{i}$ denote the $i$th sample in
$X$ and $y$ respectively and $\lambda$ controls the strength of $L_{2}$
regularisation. In a method presented by Chaudhuri and Monteleoni (2009),
$\epsilon$-DP privacy is obtained by adding a random bias term $b^{T}w/n$
(where $b$ is a random vector drawn from a distribution with density
proportional to $e^{-\epsilon||b||/2}$) to the optimisation objective. The
method requires that the samples in the input feature data are bounded into a
1-sphere.

Like DP logistic regression, also a DP linear regression algorithm can be
obtained with an analogous objective perturbation method (Kifer et al., 2012).
However, since the underlying model belongs to the exponential family, there
is also an alternative output-perturbation based $\epsilon$-DP method that
does not require iterative optimisation: Compute the sufficient statistics
($X^{T}X$, $X^{T}y$ and $y^{T}y$) and add noise to them via the Laplace-
mechanism (Foulds et al., 2016). We use Bayesian linear regression with
sufficient statistic perturbation and data clipping as described by Honkela et
al. (2018).

###  3.2 Representation learning

_Random projection_ (see e.g. Bingham and Mannila, 2001) projects the
$d$-dimensional data to an $r$-dimensional subspace by multiplying it with a
random $d\times r$ projection matrix. This transformation has been shown to
preserve approximately the distances between data points (Johnson and
Lindenstrauss, 1984), which is often a desired property for dimensionality
reduction methods.

_Principal component analysis_ (PCA) finds an orthogonal linear transformation
that converts the data to coordinates that are uncorrelated and whose variance
decreases from first to last coordinate. When used for dimensionality
reduction, only the first $r$ coordinates are kept—these correspond to the $r$
orthogonal directions in which the variance of the original data is the
highest.

_Variational autoencoder_ (VAE) (Kingma and Welling, 2014) learns a generative
decoder model $p_{\theta}(x|z)$, where $z$ is a latent representation of $x$,
and an encoder model $q_{\xi}(z|x)$ that approximates the posterior
distribution $p_{\theta}(z|x)$. Both $p_{\theta}$ and $q_{\xi}$ are
implemented as neural networks (typically MLPs) and optimised concurrently
with variational inference.

We fix $z$ to be low-dimensional, in which case the learned encoder $q_{\xi}$
can be used for dimensionality reduction by setting $f(x)=\mathrm{E}_{z\sim
q_{\xi}(\cdot|x)}[z]$. (As usual, define $q_{\xi}(\cdot|x)$ as a multivariate
Gaussian distribution parametrised by mean $\mu_{\xi}(x)$ and diagonal
covariance $\Sigma_{\xi}(x)$, in which case $f(x)=\mu_{\xi}(x)$.)

###  3.3 Optimisation of hyperparameters

Figure 2: The process of hyperparameter optimisation. In this example the
auxiliary prediction task is assumed to be classification with multiple
classes (denoted by a, b, $\ldots$, n), which are partitioned into subsets
that consist of two classes each. These are then used in a cross-validation-
like hyperparameter optimisation procedure.

For selecting the dimension of the representation and any other
hyperparameters of the representation-learning algorithm, we propose a
combination of any parameter optimisation approach (such as Bayesian
optimisation, random search or grid search) and a cross-validation-like
procedure for optimising an auxiliary task of predicting
$Y_{\text{pub}}^{\prime}$ from $X_{\text{pub}}$. As no private data are used,
the parameter optimisation phase does not consume any of the available privacy
budget. In addition, if the auxiliary prediction task uses the same method as
the main prediction task, then the hyperparameters could be optimised at the
same time.

First the (public) data are divided into $k$ disjoint subsets. Instead of
using one of the subsets as “validation” data and the rest as “training” data
as in cross-validation, we use one of the subsets to simulate the private data
and the rest to simulate the public data. From now on, these are referred to
as _pseudo-private_ and _pseudo-public_ sets. The proposed framework (from
Section 2) is then applied to these, that is, a representation mapping $f$ is
learned from the pseudo-public data, $f$ is applied to the features of pseudo-
private data, a predictor $g$ is learned for the pseudo-private target
variable and its accuracy is measured. As in $k$-fold cross-validation, this
is repeated for all $k$ possible selections of the pseudo-private subset. For
measuring the accuracy of $g$, (actual) cross-validation can be used, i.e.,
the pseudo-private data can be further divided into different learning and
validation sets.

To mimic the case in which the public and private data do not have exactly the
same distribution, we also want the pseudo-public and pseudo-private data to
be sufficiently different. This guides the optimiser towards selecting
conservative hyperparameters that are more likely to work well on a wide range
of different private datasets. If the auxiliary prediction task is
classification and $Y_{\text{pub}}^{\prime}$ has multiple classes, the subset
division can be based on the classes: Form each subset by selecting the
samples from two (or more) classes. This strategy is based on the assumption
that samples belonging to different classes have different distributions.
Otherwise, for instance clustering (based on either $X_{\text{pub}}$,
$Y_{\text{pub}}^{\prime}$ or both) could be used for finding a good subset
division. An overview of the proposed hyperparameter optimisation method is
shown in Figure 2.

##  4 Results

We conducted experiments with two prediction tasks using cancer cell line gene
expression data: cancer type classification and drug sensitivity prediction.

###  4.1 Representation learning for DP cancer type classification

We first demonstrate the method by classifying TCGA pan-cancer samples
according to the annotated cancer type (e.g. lung squamous cell carcinoma)
using RNA-seq gene expression data. In this task we use the data from The
Cancer Genome Atlas (TCGA) project (The TCGA authors, 2016) as both the
private and public datasets. We use this example because it can be performed
within the large TCGA dataset. Because most cancer type pairs are quite easy
to identify, we focus on a number of most difficult pairs.

We used preprocessed TCGA pan-cancer RNA-seq data available at
https://xenabrowser.net/datapages/. After further preprocessing (filtering out
low-expression genes, applying RLE normalisation) the dataset contains 10534
samples, 14796 genes and 33 distinct cancer types. We pick two cancer types as
private data and the remaining cancer types form the public dataset. The main
and auxiliary prediction tasks are therefore both cancer type classification
tasks, but for distinct classes. For prediction we use the differentially
private logistic regression algorithm by Chaudhuri and Monteleoni (2009).

Table 1: The list of cancer type pairs ordered in descending order by the difficulty of classification. The pairs selected to be tested are numbered. Case | First cancer type | Second cancer type
---|---|---
1 | lung squamous cell carcinoma | head & neck squamous cell carcinoma
2 | bladder urothelial carcinoma | cervical & endocervical cancer
3 | colon adenocarcinoma | rectum adenocarcinoma
4 | stomach adenocarcinoma | esophageal carcinoma
5 | kidney clear cell carcinoma | kidney papillary cell carcinoma
6 | glioblastoma multiforme | sarcoma
- | adrenocortical cancer | uveal melanoma
- | testicular germ cell tumor | uterine carcinosarcoma
- | lung adenocarcinoma | pancreatic adenocarcinoma
7 | ovarian serous cystadenocarcinoma | uterine corpus endometrioid carcinoma
- | brain lower grade glioma | pheochromocytoma & paraganglioma
- | skin cutaneous melanoma | mesothelioma
- | liver hepatocellular carcinoma | kidney chromophobe
8 | breast invasive carcinoma | prostate adenocarcinoma
- | acute myeloid leukemia | diffuse large B-cell lymphoma
- | thyroid carcinoma | cholangiocarcinoma

Figure 3: Logistic regression prediction accuracy (the fraction of correctly classified samples) with $\epsilon=1.0$ in 8 cancer type classification tasks (see Table 1). Data: TCGA. Error bars show the standard deviation of the mean accuracy over 9 independent runs of the testing phase. Table 2: Representation dimensions (repr-dim) and other selected hyperparameters (log learning rate, the number of hidden layers, the size of hidden layers) for different cases on cancer type classification. | RP | PCA | VAE
---|---|---|---
case | repr-dim | log-lr | layers | layer-dim
1 | 7 | 10 | 5 | -3.5 | 1 | 755
2 | 7 | 14 | 5 | -4.8 | 1 | 1925
3 | 8 | 10 | 5 | -5.3 | 2 | 2370
4 | 8 | 14 | 5 | -5.3 | 2 | 1270
5 | 7 | 8 | 4 | -4.3 | 1 | 260
6 | 8 | 10 | 5 | -4.5 | 1 | 330
7 | 7 | 12 | 5 | -3.7 | 2 | 1510
8 | 5 | 7 | 4 | -3.8 | 1 | 88

While the split to private and public data could be done in multiple ways, the
prediction task would be quite easy in many of those. Hence, we use the
following procedure to produce several of these splits: (1) Consider all
$\binom{33}{2}$ possible splits and run a non-DP version of the pipeline (as
in Figure 1) with PCA-based reduction to eight-dimensional space. (2) Build a
sequence of cancer-type pairs by picking the pair that was the hardest to
predict (i.e. has lowest classification accuracy), then from the remaining
cancer types again the pair that was hardest, and so on. The result is a
sequence of 16 pairs ordered by the prediction difficulty (see Table 1). (3)
Of these pairs, select the 6 hardest, as well as those 2 of the remaining
pairs that had at least 200 samples in both classes.

The full testing pipeline, including the hyperparameter optimisation phase,
was then run separately for each of the 8 selected pairs as a private dataset.
In each case, the remaining 15 pairs form the $k=15$ subsets that were used
for optimising the hyperparameters.

#### Methods

We compare three different representation learning methods: random projection
(RP), principal component analysis (PCA) and variational autoencoder (VAE)
(Kingma and Welling, 2014). VAE was implemented with PyTorch (Paszke et al.,
2017) and uses 1–3 hidden layers with ReLU activation functions for both the
encoder and the decoder. The learning phase uses the Adam optimiser (Kingma
and Ba, 2015) and is given one hour of GPU time with early stopping. The size
of the representation (for RP, PCA and VAE) and other hyperparameters for VAE
(the number of layers, layer sizes, learning rate) are optimised with GPyOpt
(The GPyOpt authors, 2016). We also experimented with optimising a much larger
set of hyperparameters, 12 in total, but GPyOpt had difficulties in obtaining
similar levels of performance.

For each of the 8 test cases we ran the hyperparameter optimisation phase
once, giving it 5 days of time. Then with the best found hyperparameters we
ran the final testing 9 times with different RNG seeds, and report the mean
prediction accuracy as well as the standard deviation of the mean. In
measuring the prediction accuracy (both for hyperparameter optimisation and
for final testing) we use 10-fold cross-validation.

Figure 4: Logistic regression classification accuracy in cancer type classification as a function of $\epsilon$ for case 1. The error bars denote the standard error of the mean when repeating the DP learning but do not cover the uncertainty from hyperparameter selection. Table 3: Selected hyperparameters for different values of $\epsilon$ in the case 1 of cancer type classification. See Table 2 for explanation of the columns. | RP | PCA | VAE
---|---|---|---
$\epsilon$ | repr-dim | log-lr | layers | layer-dim
0.5 | 5 | 5 | 5 | -4.6 | 1 | 725
0.7 | 6 | 14 | 5 | -4.6 | 1 | 880
1 | 14 | 14 | 5 | -4.1 | 1 | 395
1.5 | 9 | 9 | 5 | -4.9 | 1 | 1570
2 | 10 | 11 | 10 | -4.0 | 1 | 680

#### Results

Figure 3 shows the final prediction accuracy in the selected 8 cases for
$\epsilon=1$. While none of the methods fully dominates the others, VAE seems
to get some edge, being clearly the best in about half of the cases and doing
decent job in the rest of the cases too. The selected hyperparameters are
listed in Table 2. Interestingly, VAE seems to always end up with lower
dimensionality of the representation than the other two methods. This could be
due to the fact that VAE allows nonlinear transformations which can help to
compress the relevant information in the data into a smaller number of
dimensions. On the other hand, it is not clear why RP also always chooses
lower dimension than PCA.

The prediction accuracy as a function of $\epsilon$ in the case 1 is shown in
Figure 4 and the corresponding hyperparameters are shown in Table 3. As
expected, larger $\epsilon$ results in better accuracy. There is some
variability compared to case 1 in Figure 3, which is mostly likely due to the
results having been computed with different hyperparameters. Due to the high
computational cost, variability due to hyperparameter adaptation is not
included in the error bars.

Figure 5: The accuracy of drug sensitivity prediction (Spearman’s rank
correlation coefficient between the measured ranking of the cell lines and the
ranking predicted by the models) with differentially private linear regression
($\epsilon=1.0$) on the GDSC data.

###  4.2 Representation learning for DP drug sensitivity prediction

Our main learning task is to predict the sensitivities of cancer cell lines to
certain drugs. In this task we use data from the Genomics of Drug Sensitivity
in Cancer (GDSC) project (Yang et al., 2013) as private data. After
preprocessing the data contains 985 samples, 11714 genes and 265 drugs. The
data are sparse in the sense that not all drugs have been tested on all
samples. For prediction we use the differentially private Bayesian linear
regression algorithm by Honkela et al. (2018). The DP linear regression is
applied for each drug separately, using the full $\epsilon$ budget as if it
was the only drug we are interested in. We then measure and report the average
prediction accuracy over all drugs.

As public data we use the gene expression measurements from the TCGA data with
cancer type classification as the auxiliary prediction task. The private and
public datasets are unified by removing genes not appearing in both datasets.
In addition, since the TCGA gene expression data are RNA-seq-based while GDSC
data are based on microarrays, we apply quantile normalisation to each gene in
the TCGA data to make it match the distribution of the gene in the GDSC data.
(While this operation theoretically breaks the privacy guarantees, in practice
we can avoid the issue by assuming that the expression distributions obtained
with the microarray technology are public knowledge.)

#### Methods

In addition to RP, PCA and VAE, we also compare to DP feature selection by
Sample and Aggregate framework (SAF) as presented by Kifer et al. (2012), as
well as to using a set of 10 preselected genes that were used by Honkela et
al. (2018) in the same prediction task. In the case of SAF half of the privacy
budget is reserved for feature selection.

RP, PCA and VAE learning was performed in a similar manner as in the cancer
type classification task. For selecting the size of the representation of SAF,
we simply ran it with all possible sizes and select the best result (which is
obviously unfair for the other methods and would yield a weaker privacy
guarantee).

#### Results

Figure 5 shows the average prediction performance, measured by Spearman’s rank
correlation. Here PCA and VAE are the best by some margin, both improving
significantly over the previous state-of-the-art with preselected genes. On
the other hand, SAF is clearly the worst as the DP feature selection is
essentially random due to small privacy budget, and since it leaves only half
of the privacy budget for the main prediction task.

##  5 Discussion

Our results clearly demonstrate that representation learning with public data
can significantly improve the accuracy of differentially private learning,
compared to using a set of preselected dimensions or doing differentially
private feature selection. Whether it is beneficial to use more advanced
representation learning methods such as variational autoencoders instead of
simple methods such as PCA or random projections, depends on the task. On some
tasks that certainly seems to be the case.

In our current approach, the representation is learned in an unsupervised
manner and the auxiliary supervised task is only used for hyperparameter
selection. A natural question, that we leave for further work, is whether
representation learning would also benefit from having an integral auxiliary
prediction task that would be learned concurrently with the representation.
The optimisation target would in that case be a combination of unsupervised
reconstruction error and supervised prediction error. This approach would
require an auxiliary target variable, as is the case in this work with
hyperparameter optimisation.

In general, we believe DP learning can be important in opening genomic and
other biomedical datasets to broader use. This can significantly advance open
science and open data, and lead to more accurate models for precision
medicine. So far, the accuracy of DP learning in most practical applications
is not comparable to realistic non-private alternatives. Our present work
makes an important contribution toward making DP learning practical.

In the present work the representation learning was not performed under DP.
This is a clear limitation if the other data set also needs privacy
protection. This can in theory be addressed easily, by simply training the
representation model under DP, but this will likely have an impact on the
accuracy of the final model. Ultimately we believe that a clever combination
of private and non-private data such as in our paper can lead to the best
results.

## Acknowledgements

We thank the reviewers of an earlier workshop version of this article for
helpful comments.

## Funding

This work has been supported by the Academy of Finland [Finnish Center for
Artificial Intelligence FCAI and grants 292334, 294238, 303815, 303816,
313124].

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