Yan Liu${}^{\,a,b}$111Email<EMAIL_ADDRESS>and Hong-Da
Lyu${}^{\,a}$222Email<EMAIL_ADDRESS>
aCenter for Gravitational Physics, Department of Space Science,
and International Research Institute of Multidisciplinary Science,
Beihang University, Beijing 100191, China

bPeng Huanwu Collaborative Center for Research and Education,
Beihang University, Beijing 100191, China

###### Contents

  1. 1 Introduction
  2. 2 Neutral helical black holes
    1. 2.1 The interior structure
    2. 2.2 Kasner exponents
  3. 3 Charged helical black holes
    1. 3.1 Non-existence of inner horizon
    2. 3.2 The interior structure
    3. 3.3 Kasner exponents
  4. 4 Conclusion and discussion
  5. A Expansions in neutral black holes
  6. B Expansions in charged black holes

##  1 Introduction

Holography has established deep connections between black holes and quantum
many-body systems. During the last two decades, motivated by the problems in
quantum many-body physics, lots of interesting black hole solutions have been
constructed, such as hairy black holes which are dual to strongly correlated
superconductors [1], asymptotic Lifshitz black holes which are dual to quantum
critical states with Lifshitz scaling [2, 3], inhomogeneous black holes which
are dual to systems with translational symmetry breaking [4] etc. Suggestive
insights to the quantum many-body problems have been provided from black hole
physics, see e.g. [5] for a recent summary.

Most of the studies on the gravitational side have been focusing on the
exterior dynamics of black holes to make connections to the dual field theory.
This is because of the fact that at classical level the interior dynamics is
causally disconnected with the physics outside. However, the interior physics
is believed to be crucial for understanding the whole dynamics of the dual
quantum many-body system, for example, the island formula for calculating the
Page curve involves the interior dynamics [6]. Therefore it is important to
study the interior structure of black holes, in order to investigate the
fundamental relationship between the interior dynamics and the dual quantum
many-body physics.

Recently, the rich interior dynamics for black holes in applied holography
have been actively explored in [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. For
neutral black holes deformed by constant scalar sources in the dual theory,
their interior geometries evolve smoothly from the horizon to Kasner
singularities at late interior time [7].333Similar behavior was also found for
a large class of neutral black holes [11, 13, 14]. Later it was shown that the
interior dynamics of charged black holes with scalar or vector hair becomes
much richer. In [8] it was found that inside the charged black hole with
neutral scalar hair, the inner horizon is destroyed by a collapse of the
Einstein-Rosen bridge and the geometry smoothly evolves from the horizon to
the Kasner singularity. When the scalar hair is charged, i.e. in the
holographic superconductor model, close to the transition temperature the
interior geometry evolves from the horizon to the singularity, which typically
includes the collapse of the Einstein-Rosen bridge, Josephson oscillation and
the (successive) Kasner epochs with Kasner inversion or transition [9]. Quite
similar behaviors were also found in the generalized holographic
superconductor with an axion field [16] and the holographic p-wave
superconductor [17], while special behavior was also found, such as the
absence of the Einstein-Rosen bridge collapse close to the would-be inner
horizon for special parameters of the holographic superconductors with axion
fields.

It is well known that the generic asymptotic solutions near spacelike
singularities in Einstein gravity with generic initial conditions have been
studied in the seminal work by Belinski, Khalatnikov and Lifshitz (BKL) [18,
19]. The asymptotic AdS black holes with scalar or vector hair have special
singularity structure since there are some symmetries in the system. For these
black holes the universal dynamics of Kasner epochs close to the singularity
were further analyzed based on the billiard approach [20] and were shown to be
consistent with the BKL analysis. Nevertheless, the numerical investigation of
particular black holes with scalar or vector hair uncovers all the interior
structure from the horizon to the singularity and might provide further
fundamental connection to the dual field theory. Therefore, it is interesting
to explore the interior dynamics for more explicit examples of black holes,
including the ones dual to deformations with the energy-momentum tensor.

In this paper we will study the interior dynamics of five dimensional helical
black holes. In five dimensional helical black holes, the three dimensional
spatial Euclidean symmetry is explicitly or spontaneously broken to Bianchi
VII0 symmetry, which are dual to spatially modulated phases. Helical black
holes have been widely studied in the framework of AdS/CMT to describe helical
deformed four dimensional CFTs [21], interaction-driven insulating phases
along helical direction [22], helical current phases in Einstein-Maxwell-
Chern-Simons theory [23], helical superconductors with helical order [24] and
so on. Here we will be interested in the interior dynamics of neutral helical
black holes in Einstein gravity [21] and charged helical black holes in
Einstein-Maxwell theory respectively.

Different from previous studies on the interior dynamics of the scalar or
vector hairy black holes, for the helical black holes in our study the
interior dynamics is induced by the metric field related to the helical
deformation. For the neutral helical black holes, we will study the typical
profiles of the metric fields of the helical black holes from the horizon to
the spacelike singularity. Remarkably, we will show that at low temperature
and small helical deformation strength there is an oscillatory regime close to
the horizon and before reaching the Kasner epoch. There is no such oscillation
behavior for all the neutral black holes studied in the literature [7, 11, 13,
14]. For the charged helical black holes, we will first prove that there is no
inner Cauchy horizon and then present the typical profiles with oscillations
from the horizon to the Kasner singularity. Intriguingly, the oscillation
regime exists and it is located near the horizon while before the collapse of
Einstein-Rosen bridge, in contrast to the behavior found in the literature
where the oscillations occur after the collapse [9]. In both these helical
black holes, the Kasner geometry is stable.

The paper is organized as follows. In Sec. 2, we first present the setups of
neutral helical black holes in Einstein gravity and then study their interior
structure. In Sec. 3, we study the interior structure of charged helical black
holes in Einstein-Maxwell gravity. We conclude and discuss open questions in
Sec. 4. Some details of the calculations are collected in the appendices.

##  2 Neutral helical black holes

Neutral helical black holes exist in Einstein gravity with a negative
cosmological constant by tuning a specific helical source for the energy-
momentum tensor of the dual field theory [21]. The deformation is encoded in
the metric field and in this sense the interior dynamics is driven by the
“deformed” metric field. We first shortly review the setup in [21] where the
exterior geometry of neutral helical black hole solutions were explored, and
then study the interior geometry of these black holes.

We consider five-dimensional Einstein gravity with the action

$\displaystyle S=\frac{1}{16\pi G}\int d^{5}x\sqrt{-g}\left(R+12\right)$ (2.1)

where the cosmological constant of AdS5 is $\Lambda=-6$. The equations of
motion are

$\displaystyle R_{ab}+4g_{ab}=0\,.$ (2.2)

The ansatz of metric is

$\displaystyle
ds^{2}=-gf^{2}dt^{2}+\frac{dr^{2}}{g}+h^{2}\omega_{1}^{2}+r^{2}(e^{2\alpha}\omega_{2}^{2}+e^{-2\alpha}\omega_{3}^{2})\,,$
(2.3)

where $g,f,h,\alpha$ are functions of radial coordinate $r$. Here $\omega_{i}$
are one-forms

$\displaystyle\omega_{1}=dx_{1}\,,~{}~{}~{}\omega_{2}=\cos{kx_{1}}dx_{2}-\sin{kx_{1}}dx_{3}\,,~{}~{}~{}\omega_{3}=\sin{kx_{1}}dx_{2}+\cos{kx_{1}}dx_{3}\,$
(2.4)

with constant wave-number $k$, satisfying Bianchi VII0 algebra

$d\omega_{1}=0\,,~{}~{}~{}d\omega_{2}=k\omega_{3}\wedge\omega_{1}\,,~{}~{}d\omega_{3}=k\omega_{1}\wedge\omega_{2}\,.$
(2.5)

The AdS boundary is located at $r\to\infty$. The horizon is at $r=r_{h}$ while
the singularity is at $r\to 0$. Note that the translational symmetry along
$x_{1}$ direction is broken by the helical structure. The system is
anisotropic and the boundary value of the field $\alpha$ characterizes the
helical source for the energy-momentum tensor of the dual field theory.

The ansatz (2.3) has Bianchi VII0 symmetry and the solution describes a static
neutral helical black hole. Note that (2.3) is invariant under
$k\to-k,x_{1}\to-x_{1}$ and here we will consider the cases with $k>0$.

Substituting (2.3) into (2.2) we obtain

$\displaystyle\begin{split}\frac{f^{\prime}}{f}+\frac{(-2rh^{2}(g\alpha^{\prime
2}+2)+r(2k^{2}\sinh^{2}{2\alpha}-gh^{\prime
2})+(g+4r^{2})hh^{\prime})}{gh(rh^{\prime}+2h)}&=0\,,\\\
g^{\prime}+\frac{2[h^{2}(r^{2}g\alpha^{\prime 2}+g-2r^{2})+r^{2}(gh^{\prime
2}-k^{2}\sinh^{2}{2\alpha})+r(g-4r^{2})hh^{\prime}]}{rh(rh^{\prime}+2h)}&=0\,,\\\
h^{\prime\prime}+\frac{4rh^{\prime}}{g}-\frac{4h}{g}+\frac{h^{\prime}}{r}-\frac{h^{\prime
2}}{h}+\frac{2k^{2}\sinh^{2}{2\alpha}}{gh}&=0\,,\\\
\alpha^{\prime\prime}+\frac{4r\alpha^{\prime}}{g}+\frac{\alpha^{\prime}}{r}-\frac{k^{2}\sinh{4\alpha}}{gh^{2}}&=0\,,\end{split}$
(2.6)

where primes are derivatives with respect to $r$. We have four independent
ODE’s for four fields.

The ansatz and EOM have three scaling symmetries

$\displaystyle r\rightarrow\lambda
r\,,~{}~{}~{}\\{t,x_{2},x_{3}\\}\rightarrow\lambda^{-1}\\{t,x_{2},x_{3}\\}\,,~{}~{}~{}g\rightarrow\lambda^{2}g\,;$
(2.7) $\displaystyle t\rightarrow\lambda
t\,,~{}~{}~{}f\rightarrow\lambda^{-1}f\,;$ (2.8) $\displaystyle
x_{1}\rightarrow\lambda^{-1}x_{1}\,,~{}~{}~{}h\rightarrow\lambda
h\,,~{}~{}~{}k\rightarrow\lambda k\,.$ (2.9)

These symmetries are important for numerical calculations.

There is a useful radially conserved quantity associate with the above
symmetries,

$\displaystyle
Q=\frac{2r^{4}h}{f}\left(\frac{f^{2}g}{r^{2}}\right)^{\prime}\,.$ (2.10)

It can be used to check our numerical computations.

The near horizon and near boundary expansions of the metric fields can be
found in appendix A. From these expansions and the above scaling symmetries,
we find that the whole bulk geometry from the singularity to the boundary is
completely determined by two444We have fixed the nontrivial dynamical scale
due to the conformal anomaly which is related to the log terms for the
expansions near the boundary [21]. dimensionless free parameters $\alpha_{b}$
and $T/k$ of the boundary field theory, where $\alpha_{b}$ characterizes the
strength of the helical deformation, $T$ is the temperature and $k$
characterizes the pitch of the helical structure.

The thermodynamic quantities are

$T=\frac{r_{h}f_{h}}{\pi f_{b}}\,,~{}~{}~{}~{}s=\frac{4\pi
r_{h}^{2}h_{h}}{h_{b}}\,$ (2.11)

where $r_{h}$ is the location of horizon, $f_{h},h_{h}$ and $f_{b},h_{b}$ are
values of metric fields $f,h$ evaluating at the horizon and the boundary
respectively. Since the metric fields $f,g,h,\alpha$ are smooth along the
radial direction, one can integrate the system from boundary towards to the
singularity to obtain the full bulk geometry.

Note that the system has the AdS-Schwarzschild solution when $\alpha=0$

$g=r^{2}-\frac{r_{h}^{4}}{r^{2}}\,,~{}~{}f=1\,,~{}~{}~{}h=r\,.$ (2.12)

When $\alpha_{b}\neq 0$ and $k\neq 0$, we have a neutral helical black hole
solution with nontrivial profile of $\alpha$. In this sense we state that the
interior dynamics of the helical black hole is induced by the metric field
$\alpha$.

The UV field theory is specified by the deformations of wave-number $k$ and
the strength of the deformation $\alpha_{b}$. Equivalently, the dual field
theory lives on the spacetime [21]

$ds^{2}=-dt^{2}+\omega_{1}^{2}+e^{2\alpha_{b}}\omega_{2}^{2}+e^{-2\alpha_{b}}\omega_{3}^{2}\,$
(2.13)

at finite temperature with $\omega_{i}$ defined in (2.4).

###  2.1 The interior structure

With the above setups, we can numerically solve the system for given $T/k$ and
$\alpha_{b}$ from the AdS boundary to the black hole singularity. In Fig. 1,
we show three typical configurations of the full dynamical evolution for the
metric fields inside the horizon, where we have fixed $\alpha_{b}=0.1$ and
increase the temperature $T/k$ from $0.05$ in the left plot to $0.1$ in the
middle plot, to $0.3$ in the right plot. Note that $r/r_{h}=1$ is the horizon
and $r/r_{h}\to 0$ is the singularity. We have numerically solved the system
from the horizon to $r/r_{h}\sim 10^{-90}$ although we only show part of them
in the figure since the configurations from $10^{-4}$ to $10^{-90}$ are
trivial constants. We have used the radial conserved quantity (2.10) to check
the accuracy of our numerics. We find that at all the different temperatures,
evolving from the horizon to the singularity, the fields evolve after an
oscillate regime or smoothly to the stable Kasner epoch. When the temperature
is low enough, the curves related to fields $\alpha,f,h$ has an oscillation
regime close to the horizon.555This reminds us the Josephson oscillation for
the scalar field in the interior of holographic superconductor [9]. For the
neutral helical black holes there is no background phase winding phenomenon
and we have not been aware of any analogous Josephson effect. Therefore we
call this just oscillation. At higher temperature, the oscillation behavior
disappears. Note that in previous studies, no oscillation behavior was found
for neutral black holes with scalar deformations [7]. In addition, the Kasner
exponents are stable and we have not found any Kasner inversion or Kasner
transition for these neutral helical black holes.

Figure 1: Three typical examples of the configuration in the neutral helical
black hole interior with fixing $\alpha_{b}=0.1$ at different temperatures
$T/k=0.05$ (left), $0.1$ (middle) and $0.3$ (right). The functions we plot are
$rf^{\prime}/(\alpha_{h}^{2}f)$ (purple), $r\alpha^{\prime}/\alpha_{h}$
(blue), $(rh^{\prime}/h-1)/(4\alpha_{h}^{2})$ (red), $rg^{\prime}/g$ (black).

In Fig. 2, we show the oscillation regime of the field $\alpha$ (which has
been rescaled according to $\tilde{\alpha}=\alpha/\alpha_{h}$) as a function
of $r/r_{h}$ for fixing $\alpha_{b}$ (left) or $T/k$ (right) respectively. The
left plot in Fig. 2 shows that for fixing $\alpha_{b}$, the field $\alpha$
oscillates more dramatic at lower temperature, and the right plot shows that
for fixing temperature the smaller $\alpha_{b}$ is, the more times $\alpha$
oscillates. We also find similar oscillation behavior at low temperature for
the fields $rf^{\prime}/f$ and $rh^{\prime}/h$ which can be seen from Fig.
1.666We have checked that there is no oscillation behavior for the metric
fields outside the horizon for the parameters. Additionally, the frequency of
oscillations for $rf^{\prime}/f$ and $rh^{\prime}/h$ are roughly two times of
$r\alpha^{\prime}$. It is interesting to note that at low temperature inside
the horizon, the spatial components of the metric $e^{\pm 2\alpha}$ in (2.3)
show oscillation behavior before reaching Kasner regime. Our results also
suggest that the oscillation behavior near the horizon is not necessarily to
be in contact with the collapse of the Einstein-Rosen bridge. It remains to be
seen that if there exists an analytical way to calculate the fields in the
oscillation regime and the holographic signature of the oscillation behavior
in the dual field theory.

Figure 2: The plots of $\tilde{\alpha}=\alpha/\alpha_{h}$ along radial
direction in the oscillation region for fixing $\alpha_{b}=0.1$ (left) while
$T/k=0.015$ (purple), $0.05$ (blue), $0.1$ (red), $0.3$ (black), as well as
for fixing $T/k=0.05$ (right) while $\alpha_{b}=0.1$ (purple), $0.3$ (blue),
$0.5$ (red), $1$(black). Here $\alpha_{h}$ is the horizon value of $\alpha$.

###  2.2 Kasner exponents

After the oscillation regime, the fields evolve to the Kasner epoch which can
be studied analytically. Near the singularity, i.e. $r\to 0$, we assume

$\displaystyle\begin{split}g=-\frac{g_{0}}{r^{n_{g}}}+\dots\,,~{}~{}~{}f=f_{0}r^{n_{f}}+\dots\,,~{}~{}~{}h=h_{0}r^{n_{h}}+\dots\,,~{}~{}~{}\alpha=n_{\alpha}\log
r+\alpha_{0}+\dots\,\end{split}$ (2.14)

where $g_{0},n_{g},f_{0},n_{f},h_{0},n_{h},n_{\alpha},\alpha_{0}$ are all
constants, and the “$\dots$” are subleading in $r$ compared to the first
terms.

When $r\to 0$, the equations of motion (2.6) can be simplified as

$\displaystyle\begin{split}f^{\prime}+\frac{f(hh^{\prime}-rh^{\prime
2}-2rh^{2}\alpha^{\prime 2})}{h(2h+rh^{\prime})}&=0\,,\\\
g^{\prime}+\frac{2g(rhh^{\prime}+r^{2}h^{\prime 2}+h^{2}(1+r^{2}\alpha^{\prime
2}))}{rh(2h+rh^{\prime})}&=0\,,\\\ h^{\prime\prime}-\frac{h^{\prime
2}}{h}+\frac{h^{\prime}}{r}&=0\,,\\\
\alpha^{\prime\prime}+\frac{\alpha^{\prime}}{r}&=0\,,\end{split}$ (2.15)

from which we obtain

$\displaystyle
n_{f}=\frac{n_{h}(n_{h}-1)+2n_{\alpha}^{2}}{n_{h}+2}\,,~{}~{}~{}~{}~{}n_{g}=\frac{2(n_{h}^{2}+n_{h}+n_{\alpha}^{2}+1)}{n_{h}+2}\,.$
(2.16)

Note that in above equations (2.15), we have assumed that the terms ignored
should be subleading. More explicitely, we have assumed

$\displaystyle
n_{g}>-2\,,~{}~{}~{}n_{g}-2n_{h}+2>0\,,~{}~{}~{}n_{g}-2n_{h}+2\pm
4n_{\alpha}>0\,.$ (2.17)

Combing (2.17) and (2.16), we have

$n_{h}>-2,~{}~{}~{}~{}n_{g}\geq 4\sqrt{3}-6\,,~{}~{}~{}~{}n_{\alpha}^{2}+3\pm
2n_{\alpha}(n_{h}+2)>0\,.$ (2.18)

Evaluate the conserved charge (2.10) at the singularity and the horizon, we
obtain

$\displaystyle\begin{split}&\frac{4f_{0}g_{0}h_{0}(3n_{h}-n_{\alpha}^{2}+3)}{n_{h}+2}=8f_{h}h_{h}r_{h}^{3}=2f_{b}h_{b}Ts\,.\end{split}$
(2.19)

Given the fact that the term on the right hand side is positive and
$f_{0},g_{0},h_{0}$ are all positive, we have constraint

$3n_{h}+3>n_{\alpha}^{2}\,,$ (2.20)

from which we further have $n_{h}>-1$.

Although we could not further constrain these parameters, it is necessarily
that all the inequalities should be satisfied. It is easy to see there always
exist a range of values $n_{\alpha},n_{h}$ for which the constraints in (2.18)
and (2.20) are all satisfied. Numerically we have checked that the values we
obtained satisfy the constraints. This indicates the Kasner regime is stable
and we have not found any Kasner inversion or transition in this model. The
stability of Kasner geometry is quite similar to the neutral black holes with
scalar hair in [7, 11, 13, 14].

Starting from (2.14) and performing the coordinate transformation
$\tau=\frac{2}{\sqrt{g_{0}}(n_{g}+2)}r^{(n_{g}+2)/2}$, the metric near the
singularity becomes the Kasner form [25]

$ds^{2}=-d\tau^{2}+c_{t}\tau^{2p_{t}}dt^{2}+c_{1}\tau^{2p_{1}}\omega_{1}^{2}+c_{2}\tau^{2p_{2}}\omega_{2}^{2}+c_{3}\tau^{2p_{3}}\omega_{3}^{2}$
(2.21)

with the Kasner exponents

$\displaystyle
p_{t}=\frac{2n_{f}-n_{g}}{n_{g}+2}\,,~{}~{}p_{1}=\frac{2n_{h}}{n_{g}+2}\,,~{}~{}p_{2}=\frac{2(n_{\alpha}+1)}{n_{g}+2}\,,~{}~{}p_{3}=\frac{2(1-n_{\alpha})}{n_{g}+2}\,.$
(2.22)

Substituting (2.16) into the above equations, we find

$\displaystyle\begin{split}p_{t}=\frac{n_{\alpha}^{2}-2n_{h}-1}{n_{\alpha}^{2}+n_{h}(n_{h}+2)+3}&\,,~{}~{}~{}p_{1}=\frac{n_{h}(n_{h}+2)}{n_{\alpha}^{2}+n_{h}(n_{h}+2)+3}\,,\\\
p_{2}=\frac{(n_{h}+2)(n_{\alpha}+1)}{n_{\alpha}^{2}+n_{h}(n_{h}+2)+3}&\,,~{}~{}~{}p_{3}=\frac{(n_{h}+2)(1-n_{\alpha})}{n_{\alpha}^{2}+n_{h}(n_{h}+2)+3}\,.\end{split}$
(2.23)

From the above equations, one can easily check

$\displaystyle
p_{t}+p_{1}+p_{2}+p_{3}=p_{t}^{2}+p_{1}^{2}+p_{2}^{2}+p_{3}^{2}=1\,.$ (2.24)

Note that different from the neutral black holes in four dimensions [7], here
we have two independent Kasner exponents for the singularity inside the five
dimensional helical black holes. An immediate question is how to describe
these two independent Kasner exponents from the dual field theory. It seems
that we need two different observables at the same time. We leave this
interesting question for future study.

The Kasner exponents can be obtained numerically. In Fig. 3, we have shown the
four Kasner exponents in (2.21) as functions of $T/k$ at different
$\alpha_{b}$.

Figure 3: The Kasner exponents as a function of $T/k$ for $\alpha_{b}=1$
(red), $0.5$ (green), $0.1$ (blue). The gray dotted line is for Schwarzschild
black hole with $p_{t}=-0.5=-p_{1}=-p_{2}=-p_{3}$.

In Fig. 4, we show the dependence of Kasner exponents as functions of
$\alpha_{b}$ for fixed temperature $T/k=0.1$. We have numerically check that
the Kasner relations (2.24) are satisfied for all the configurations we have
considered. Furthermore, numerically for neutral helical black holes we only
find the case with $p_{t}<0$ while the other Kanser exponents are positive.

Figure 4: The Kasner exponents as functions of $\alpha_{b}$ at $T/k=0.1$. The
gray dotted line is corresponding value of Schwarzschild black hole.

##  3 Charged helical black holes

Comparing to the neutral Schwarzschild black holes with scalar deformation,
the interior dynamics of charged Reinssner-Nordström black holes with scalar
deformation is much more fruitful [7, 8, 9]. Similarly, one might expect that
the charged helical black holes contain quite interesting interior dynamics.

In this section we consider the interior structure of five dimensional helical
black holes at finite density in Einstein-Maxwell gravity. The action is

$\displaystyle\begin{split}S=\int
d^{5}x\sqrt{-g}\bigg{(}R+12-\frac{1}{4}F^{2}\bigg{)}\,.\end{split}$ (3.1)

The equations of motion are

$\displaystyle R_{ab}+4g_{ab}$ $\displaystyle=\frac{1}{2}F_{ac}F_{b}^{\
c}-\frac{1}{12}g_{ab}F_{cd}F^{cd}\,,$ (3.2) $\displaystyle\nabla_{b}F^{ba}$
$\displaystyle=0\,.$ (3.3)

The ansatz for the background fields are

$\displaystyle\begin{split}ds^{2}&=-gf^{2}dt^{2}+\frac{dr^{2}}{g}+h^{2}\omega_{1}^{2}+r^{2}e^{2\alpha}\omega_{2}^{2}+r^{2}e^{-2\alpha}\omega_{3}^{2}\,,\\\
A&=\psi dt\,,\end{split}$ (3.4)

where $f,g,h,\alpha,\psi$ are functions of $r$. The metric part is the same as
(2.3) for neutral helical black holes and here we have an additional gauge
potential. $\omega_{i}$ are one-forms defined in (2.4) satisfying Bianchi VII0
algebra. The dual field theory is located at $r\to\infty$. The horizon and the
singularity are located at $r=r_{h}$ and $r\to 0$ respectively.

Substituting the ansatz (3.4) into EOM, we obtain

$\displaystyle\begin{split}\frac{f^{\prime}}{f}+\frac{(4r^{2}+g)hh^{\prime}+r(2k^{2}\sinh^{2}{2\alpha}-gh^{\prime
2})-2rh^{2}(g\alpha^{\prime 2}+2)}{gh(2h+rh^{\prime})}+\frac{r\psi^{\prime
2}(h-rh^{\prime})}{6f^{2}g(2h+rh^{\prime})}&=0\,,\\\
\frac{g^{\prime}}{g}+\frac{2\left(r(g-4r^{2})hh^{\prime}-r^{2}(k^{2}\sinh^{2}{2\alpha}-gh^{\prime
2})+h^{2}(g+r^{2}g\alpha^{\prime
2}-2r^{2})\right)}{rgh(2h+rh^{\prime})}+\frac{r\psi^{\prime
2}(h+2rh^{\prime})}{6f^{2}g(2h+rh^{\prime})}&=0\,,\\\
h^{\prime\prime}+\frac{h^{\prime}}{r}-\frac{h^{\prime
2}}{h}+\frac{2k^{2}\sinh^{2}{2\alpha}}{gh}+\frac{1}{g}\left(\frac{\psi^{\prime
2}}{6f^{2}}-4\right)(h-rh^{\prime})&=0\,,\\\
\alpha^{\prime\prime}+\frac{\alpha^{\prime}}{r}+\frac{4r\alpha^{\prime}}{g}-\frac{r\alpha^{\prime}\psi^{\prime
2}}{6f^{2}g}-\frac{k^{2}\sinh{4\alpha}}{gh^{2}}&=0\,,\\\
\psi^{\prime\prime}-\psi^{\prime}\left(\frac{f^{\prime}}{f}-\frac{h^{\prime}}{h}-\frac{2}{r}\right)&=0\,.\end{split}$
(3.5)

When $\psi=0$, the above equations reduce to the neutral case with equations
(2.6). Furthermore, the last equation can be simplified to
$\left(\frac{r^{2}h\psi^{\prime}}{f}\right)^{\prime}=0.$ This implies

$\displaystyle\psi^{\prime}=\frac{2f}{r^{2}h}\psi_{0}\,,$ (3.6)

where $\psi_{0}$ is the integration constant which could be interpreted as the
charge density of the dual field theory. Plugging (3.6) into equations of
motion (3.5), we obtain four equations for functions $f,g,h,\alpha$.

Three different sets of scaling symmetries of the ansatz (3.4) are

$\displaystyle r\rightarrow\lambda
r\,,~{}~{}~{}\\{t,x_{2},x_{3}\\}\rightarrow\lambda^{-1}\\{t,x_{2},x_{3}\\}\,,~{}~{}~{}g\rightarrow\lambda^{2}g\,,~{}~{}~{}\psi\rightarrow\lambda\psi\,;$
(3.7) $\displaystyle t\rightarrow\lambda
t\,,~{}~{}~{}f\rightarrow\lambda^{-1}f\,,~{}~{}~{}\psi\rightarrow\lambda^{-1}\psi\,;$
(3.8) $\displaystyle
x_{1}\rightarrow\lambda^{-1}x_{1}\,,~{}~{}~{}h\rightarrow\lambda
h\,,~{}~{}~{}k\rightarrow\lambda k\,.$ (3.9)

These scaling symmetries are important to numerically solve the system. A
useful radial conserved quantity associated to the above symmetries is

$\displaystyle
Q=\frac{2r^{4}h}{f}\left(\left(\frac{f^{2}g}{r^{2}}\right)^{\prime}-\frac{\psi\psi^{\prime}}{r^{2}}\right)\,.$
(3.10)

This conserved quantity can be used to check our numerical calculations.

The near horizon and near boundary expansions of the fields can be found in
appendix B. It turns out the bulk geometry is completely determined by three
dimensionless parameters $T/\mu$, $k/\mu$ and $\alpha_{b}$, where $T$ and
$\mu$ are the temperature and the chemical potential of the charged helical
black hole, the wave-number $k$ characterizes the pitch of the helical
structure which is equal to $2\pi/k$ and $\alpha_{b}$ is the strength of the
helical deformation.

The thermodynamic quantities of the system are

$\displaystyle T=\frac{r_{h}(24f_{h}^{2}-\psi_{1}^{2})}{24\pi
f_{b}f_{h}}=\frac{f_{h}(6h_{h}^{2}r_{h}^{4}-\psi_{0}^{2})}{6f_{b}h_{h}^{2}r_{h}^{3}}\,,~{}~{}~{}~{}s=\frac{4\pi
r_{h}^{2}h_{h}}{h_{b}}\,.$ (3.11)

One can check that the equations (3.5) have solutions as Reissner–Nordström
black hole when $\alpha=0$,

$\displaystyle\begin{split}f&=1\,,~{}~{}~{}~{}h=r\,,\\\
g&=r^{2}-\frac{r_{h}^{2}(3r_{h}^{2}+\mu^{2})}{3r^{2}}+\frac{\mu^{2}r_{h}^{4}}{3r^{4}}\,,\\\
\psi&=\mu\left(1-\frac{r_{h}^{2}}{r^{2}}\right)\,.\end{split}$ (3.12)

When we have a non-trivial source $\alpha_{b}$, the field $\alpha$ has non-
trivial profile.

In the following we will numerically solve the system and study the interior
structure of the charged helical black holes.

###  3.1 Non-existence of inner horizon

We first give a proof that there could not be any inner horizon for the
charged helical black holes (3.4). The proof of no-inner horizon theorem for
charged black hole in presence of charged scalar field or vector field can be
found in [9, 10, 17]. Although our setup is different since the deformation is
driven by the metric field $\alpha$, we could use similar strategy to prove
it.

The proof of no inner horizon can be obtained from the equation of motion for
$\alpha$. First, we note that a compact equation of $\alpha$ can be obtained
by a linear combination of the first, second and fourth equations in (3.5),

$\displaystyle\left(r^{2}fgh\alpha^{\prime}\right)^{\prime}=\frac{k^{2}r^{2}f}{h}\sinh{4\alpha}\,.$
(3.13)

If there were two horizons $g(r_{h})=g(r_{i})=0$ with outter horizon $r_{h}$
and inner horizon $r_{i}$, from (3.13) we would have

$\displaystyle
0=\int_{r_{i}}^{r_{h}}dr\left(\frac{r^{2}fgh\alpha^{\prime}}{\alpha}\right)^{\prime}=\int_{r_{i}}^{r_{h}}dr\left(\frac{k^{2}r^{2}f}{h}\frac{\sinh{4\alpha}}{\alpha}-\frac{r^{2}fhg\alpha^{\prime
2}}{\alpha^{2}}\right)\,.$ (3.14)

The first equality is because of $g(r_{h})=g(r_{i})=0$. However, since we have
$g(r)<0$ between the two horizons777Note that $f,h$ are set to be positive
from the boundary value, and should be always positive along the radial
direction otherwise the Kretschmann scalar is divergent at the location where
$f=0~{}\text{or}~{}h=0$. which implies the integrand of right hand side is
positive. Therefore there can not be an inner horizon for the charged helical
black holes we considered.

###  3.2 The interior structure

We can solve the systems of charged helical black holes numerically from the
boundary to the singularity for given $\alpha_{b},k/\mu,T/\mu$ and we focus on
the interior structure. In Fig. 5 we show one example of the plot with the
existence of oscillation regimes. Note that the horizon is located at
$r/r_{h}=1$ and the singularity is at $r=0$. We find that from the horizon to
the singularity, it is clear that there are three distinct dynamical regimes:
including firstly the oscillations of the metric fields, secondly the collapse
of the Einstien-Rosen bridge, and finally a stable Kasner regime. In the left
plot, the Kanser regime clearly exists and we have checked until $r/r_{h}\sim
10^{-100}$ to confirm that the Kasner geometry is stable.888We have checked
that the configurations from numerics are smooth. The right plot (with proper
rescales) is an enlarged version of the oscillation regime and the collapse of
the Einstein-Rosen bridge. We identify the would-be inner horizon as the
location of the Reissner-Nordström black hole with same $T/\mu$. For the
parameter we considered here $p_{t}$ is close to 1 and we find that from the
horizon to the would-be inner horizon, $g_{tt}$ behaves quite similar to the
Reissner-Nordström black hole. Close to the would-be inner horizon,
$rf^{\prime}/f$ (or $g_{tt}$) has a rapid “jump” and we call this the collapse
of the Einstein-Rosen bridge following [8, 9]. Note that intriguingly the
rapid collapse of the Einstein-Rosen bridge occurs after the oscillations in
the interior time, which is completely different from the previous studies on
the interior dynamics driven by the scalar or vector field. It should be
emphasized that this behavior is quite general for any configurations with
oscillation regime in the charged helical black holes.999Similar to the
neutral case, the frequency of oscillations for $rf^{\prime}/f$ and
$rh^{\prime}/h$ are roughly two times of $r\alpha^{\prime}$. Similar to the
neutral helical black holes, here the spatial components of the metric
$e^{2\alpha}$ and $e^{-2\alpha}$ oscillate. The oscillations of the metric
field with spatial components have also been found in the example of black
holes with vector hair [17]. Beside the relative location of oscillations and
collapse in the interior time is different here, the other difference is that
here the non-trivial interior structure is induced by the “deformed” metric
field $\alpha$.

Figure 5: The interior configuration of charged helical black hole with
$\alpha_{b}=0.01\,,~{}T/\mu=0.1\,,~{}k/\mu=7$. Left: The plots of
$10^{-27}rf^{\prime}/f$ (purple), $10^{-27}rg^{\prime}/g$ (black),
$rh^{\prime}/h$ (red) and $10^{-12}r\alpha^{\prime}$ (blue) as functions of
$r/r_{h}$. Note that the corners of these plots are smooth. Right: The plots
in the region between the event horizon and the would-be inner horizon
$r/r_{h}=0.7696$ (the orange vertical line) of
$rf^{\prime}/(10\alpha_{h}^{2}f)$ (purple), $rg^{\prime}/g$ (black),
$(rh^{\prime}/h-1)/(4\alpha_{h}^{2})$ (red) and
$r\alpha^{\prime}/(4\alpha_{h})$ (blue) where $\alpha_{h}$ is the horizon
value of $\alpha$.

In Fig. 6, we show the oscillation of the metric field $\alpha$ along the
radial direction in different situations. Here we rescale
$\tilde{\alpha}=\alpha/\alpha_{h}$ with $\alpha_{h}$ the value of $\alpha$ at
the horizon. In the left plot, we fix $\alpha_{b}=0.01,k/\mu=5$ and tune the
temperature $T/\mu$ from $0.05$ to $1$, and find that qualitatively the lower
the temperature is, the more times the field $\alpha$ oscillates. This
behavior is the same as the neutral case which is shown in the left plot of
Fig. 2. Note that for the purple and blue lines in the left figure, the system
quickly enters into the Kasner regime and $\alpha$ behaves like
$n_{\alpha}\log r$ afterwards with constant $n_{\alpha}$. In the middle plot,
we fix $T/\mu=0.1,k/\mu=5$ and tune $\alpha_{b}$. The smaller $\alpha_{b}$ is,
the more times $\alpha$ oscillates, which is also similar to the neutral case.
In the right plot, we fix $\alpha_{b}=0.01,T/\mu=0.1$ and tune the wave-number
$k/\mu$. We find that the larger value of $k/\mu$, i.e. smaller pitch of the
helical structure, the more times $\alpha$ oscillates.

Figure 6: The plots of $\tilde{\alpha}=\alpha/\alpha_{h}$ along radial
direction in the oscillation region for fixing $\alpha_{b}=0.01\,,k/\mu=5$
(left) while $T/\mu=0.05$ (purple), $0.1$ (blue), $0.3$ (red), $1$ (black),
for fixing $T/\mu=0.1\,,k/\mu=5$ (middle) while $\alpha_{b}=0.01$ (purple),
$0.5$ (blue), $1$ (red), $2$ (black), and for fixing
$\alpha_{b}=0.01\,,T/\mu=0.1$ (right) while $k/\mu=8$ (purple), $5$ (blue),
$3$ (red), $1$ (black). Note that $\alpha_{h}$ is the horizon value of
$\alpha$.

###  3.3 Kasner exponents

Similar to the neutral helical black hole, we assume that when $r\to 0$,
metric fields have the form

$\displaystyle\begin{split}g=-\frac{g_{0}}{r^{n_{g}}}+\dots\,,~{}~{}~{}f=f_{0}r^{n_{f}}+\dots\,,~{}~{}~{}h=h_{0}r^{n_{h}}+\dots\,,~{}~{}~{}\alpha=n_{\alpha}\log
r+\alpha_{0}+\dots\,,\end{split}$ (3.15)

then the gauge field $\psi$ can be obtained by integrating (3.6)

$\displaystyle\psi=\psi_{i}-\frac{2f_{0}(n_{h}+2)\psi_{0}}{2h_{0}(2n_{h}-n_{\alpha}^{2}+1)}r^{\frac{2(n_{\alpha}^{2}-2n_{h}-1)}{n_{h}+2}}+\dots\,,$
(3.16)

where $\psi_{i}$ is an integration constant which is determined by the
condition $\psi_{h}=0$.

With above ansatz, the equations (3.5) can be simplified to

$\displaystyle\begin{split}\frac{f^{\prime}}{f}+\frac{hh^{\prime}-rh^{\prime
2}-2rh^{2}\alpha^{\prime 2}}{h(2h+rh^{\prime})}&=0\,,\\\
\frac{g^{\prime}}{g}+\frac{2(rhh^{\prime}+r^{2}h^{\prime
2}+h^{2}(1+r^{2}\alpha^{\prime 2}))}{rh(2h+rh^{\prime})}&=0\,,\\\
h^{\prime\prime}-h^{\prime}(\frac{h^{\prime}}{h}-\frac{1}{r})&=0\,,\\\
\alpha^{\prime\prime}+\frac{\alpha^{\prime}}{r}&=0\,,\end{split}$ (3.17)

which are the same as the neutral case. The analytical solution of (3.17) have
the same form as (2.16) with

$\displaystyle
n_{f}=\frac{n_{h}(n_{h}-1)+2n_{\alpha}^{2}}{n_{h}+2}\,,~{}~{}~{}~{}~{}n_{g}=\frac{2(n_{h}^{2}+n_{h}+n_{\alpha}^{2}+1)}{n_{h}+2}\,.$
(3.18)

We have used the assumption that the terms ignored in the above equations
should be sub-leading which constrains

$\displaystyle\begin{split}n_{g}>-2\,,~{}n_{g}-2n_{h}-2>0\,,~{}n_{g}-2n_{h}+2>0\,,~{}n_{g}-2n_{h}+2\pm
4n_{\alpha}>0\,.\end{split}$ (3.19)

The second inequality, which has not presented in neutral case, comes from the
terms of Maxwell field in equations (3.5). From (3.18) and (3.19) we obtain

$\displaystyle
n_{h}>-2\,,~{}~{}~{}n_{g}>1\,,~{}~{}~{}n_{\alpha}^{2}>2n_{h}+1\,,~{}~{}n_{\alpha}^{2}+3\pm
2n_{\alpha}(n_{h}+2)>0\,.$ (3.20)

Obviously the power exponent of (3.16) is positive and the leading term is the
constant term.

Evaluating the conserved charge (3.10) at the singularity and the horizon, we
obtain

$\displaystyle\frac{4f_{0}g_{0}h_{0}(3n_{h}-n_{\alpha}^{2}+3)}{n_{h}+2}-4\psi_{0}\psi_{i}=\frac{h_{h}r_{h}^{3}}{3f_{h}}(24f_{h}^{2}-\psi_{1}^{2})=2f_{b}h_{b}Ts\,,$
(3.21)

where the first term should be non-negative because we set $f_{b}>0,h_{b}>0$
at boundary and the temperature and entropy density must be equal or greater
than $0$. We have numerically check that all the above inequalities (3.20) are
satisfied and this indicates the Kasner geometry is stable. This is quite
similar to the charged black holes with neutral scalar field where Kasner
geometry is stable [8].

Since the ansatz (3.15) of the solution near the singularity and the equations
(3.17) are the same as neutral case, we can perform the coordinate
transformation $\tau=\frac{2}{\sqrt{g_{0}}(n_{g}+2)}r^{(n_{g}+2)/2}$ to obtain
the Kasner form for (3.15),

$ds^{2}=-d\tau^{2}+c_{t}\tau^{2p_{t}}dt^{2}+c_{1}\tau^{2p_{1}}\omega_{1}^{2}+c_{2}\tau^{2p_{2}}\omega_{2}^{2}+c_{3}\tau^{2p_{3}}\omega_{3}^{2}$
(3.22)

with the Kasner exponents

$\displaystyle
p_{t}=\frac{2n_{f}-n_{g}}{n_{g}+2}\,,~{}~{}p_{1}=\frac{2n_{h}}{n_{g}+2}\,,~{}~{}p_{2}=\frac{2(n_{\alpha}+1)}{n_{g}+2}\,,~{}~{}p_{3}=\frac{2(1-n_{\alpha})}{n_{g}+2}\,.$
(3.23)

From (3.18), we have

$\displaystyle\begin{split}p_{t}=\frac{n_{\alpha}^{2}-2n_{h}-1}{n_{\alpha}^{2}+n_{h}(n_{h}+2)+3}&\,,~{}~{}~{}p_{1}=\frac{n_{h}(n_{h}+2)}{n_{\alpha}^{2}+n_{h}(n_{h}+2)+3}\,,\\\
p_{2}=\frac{(n_{h}+2)(n_{\alpha}+1)}{n_{\alpha}^{2}+n_{h}(n_{h}+2)+3}&\,,~{}~{}~{}p_{3}=\frac{(n_{h}+2)(1-n_{\alpha})}{n_{\alpha}^{2}+n_{h}(n_{h}+2)+3}\,.\end{split}$
(3.24)

The same as neutral case, from the above relations we have

$\displaystyle
p_{t}+p_{1}+p_{2}+p_{3}=p_{t}^{2}+p_{1}^{2}+p_{2}^{2}+p_{3}^{2}=1\,.$ (3.25)

There is a constraint on $p_{t}$ from (3.20)

$\displaystyle 0<p_{t}<1\,.$ (3.26)

The Kasner exponents can be obtained from the numerical solutions. In Fig. 7,
we show the dependence of the Kasner exponents as functions of $T/\mu$ at
$\alpha_{b}=0.5,k/\mu=5$. One important message here is that $0<p_{t}<1$. When
$T/\mu$ is small or very large, $p_{t}\to 1$.

Figure 7: The Kasner exponents as functions of $T/\mu$ at
$\alpha_{b}=0.5\,,k/{\mu}=5$. The gray dotted line is corresponding value of
regular inner horizon of Reissner–Nordström black hole with
$p_{t}=1\,,p_{1}=p_{2}=p_{3}=0$.

In Fig. 8, we show the dependence of the Kasner exponents as functions of
$\alpha_{b}$ at $T/\mu=0.5,k/\mu=5$. The qualitative behaviors of
$-p_{t},p_{i},~{}(i=1,2,3)$ depending on $T/\mu$ or $\alpha_{b}$ are quite
similar to $p_{t},p_{i},~{}(i=1,2,3)$ in the case of neutral helical black
holes in Fig. 4.

Figure 8: The Kasner exponents as functions of $\alpha_{b}$ at
$T/{\mu}=0.5\,,k/{\mu}=5$. The gray dotted line is corresponding value of
regular inner horizon of Reissner–Nordström black hole.

In Fig. 9, we show the dependence of the Kasner exponents as functions of
$k/\mu$ at $\alpha_{b}=0.5\,,T/{\mu}=0.5$. In all these examples, we find that
$0<p_{t}<1$. We have checked that the Kasner relations (3.25) are always
satisfied for all the configurations we have considered.

Figure 9: The Kasner exponents as functions of $k/\mu$ at
$\alpha_{b}=0.5\,,T/{\mu}=0.5$. The gray dotted line is corresponding value of
regular inner horizon of Reissner–Nordström black hole.

##  4 Conclusion and discussion

We have studied the interior structures of neutral and charged helical black
holes. One different aspect compared to previous studies in the literature is
that the structure of helical black holes is induced by the deformation which
is related to the metric field while not the scalar or vector hair. For
neutral helical black holes in Einstein gravity, we found that the interior
geometry flows from the horizon to the spacelike Kasner singularity. At low
temperature and small helical deformation strength, the metric field related
to the deformation oscillates close to the horizon. This behavior is quite
different from the previous examples of neutral black holes. For charged
helical black holes in Einstein-Maxwell gravity, we have shown that the inner
Cauchy horizon is removed by the helical deformation. The interior geometry
also evolves from the horizon to a stable Kasner singularity. There is an
oscillation regime at low temperatures, small helical deformation strength and
small pitch of helical structure. Remarkably, the oscillation occurs near the
horizon and before the collapses of the Einstein-Rosen bridge, in contrast to
the known results in the literature for charged black holes with scalar or
vector hair.

For charged black holes with scalar or vector hair, the Kasner regime might
not be stable [9, 17, 20]. However, for charged helical black holes we have
not found any example with Kasner inversion or transition. It would be
extremely interesting to study other helical black holes [22, 23, 24], or to
consider (neutral or charged) black holes with spatial Euclidean group broken
down to symmetries of other Bianchi types different from Bianchi VII0 [26,
27], in order to further understand the dynamical chaotic behavior of Kasner
geometry near the singularity for black holes induced by deformations related
to the metric field.

From the holographic point of view, the interesting internal structure is
expected to have corresponding descriptions in the dual field theory. Some
observables have been explored to study the physics inside the horizon,
including the correlation functions [28, 29, 30, 7, 14], entanglement entropy
[31, 7], and complexity [32, 33, 34, 35] etc. We have shown that in the
interior there are oscillation regimes for both neutral and charged helical
black holes, and the collapse of the Einstein-Rosen bridge for charged helical
black holes. It was pointed out in [8, 9] that since the collapse of Einstein-
Rosen bridge indicates the existence of interior extreme value of $g_{tt}$,
the geodesics approximation therefore gives rise to a purely damped
quasinormal mode in the dual field theory. When the metric field with the
components of the spatial directions $\omega_{2}$ and $\omega_{3}$ oscillates,
the properties of geodesics with conserved momentum along $x_{2}$ or $x_{3}$
directions might be nontrivial and subtle, thus interesting behavior might
arise for the dual Green’s function at large momentum. It would be interesting
to determine the precise holographic signature for the oscillation behavior of
the metric field. Finally, in the helical black holes, there are two
independent Kasner exponents for the spacelike singularity. It would be quite
worthwhile to explore how to characterize them using the physical quantities
in the dual field theory.

## Acknowledgments

We would like to thank Ling-Long Gao for discussions. This work is supported
by the National Natural Science Foundation of China grant No.11875083.

##  Appendix A Expansions in neutral black holes

Here we show the details of the expansions of fields near horizon and near
boundary for the neutral helical black holes discussed in Sec. 2. Close to the
black hole horizon $r\to r_{h}$, we obtain the series solutions

$\displaystyle\begin{split}g&=4r_{h}(r-r_{h})+\frac{k^{2}\cosh{4\alpha_{h}}-k^{2}-4h_{h}^{2}}{2h_{h}^{2}}\,(r-r_{h})^{2}+\cdots\,,\\\
f&=f_{h}\left(1-\frac{k^{2}(-4h_{h}^{2}+k^{2}+3k^{2}\cosh{4\alpha_{h}})\sinh^{2}{2\alpha_{h}}}{48h_{h}^{4}r_{h}^{2}}\,(r-r_{h})^{2}+\cdots\right)\,,\\\
h&=h_{h}+\frac{2h_{h}^{2}-k^{2}\sinh^{2}{2\alpha_{h}}}{2h_{h}r_{h}}\,(r-r_{h})-\frac{k^{4}\sinh^{2}{4\alpha_{h}}}{16h_{h}^{3}r_{h}^{2}}\,(r-r_{h})^{2}+\cdots\,,\\\
\alpha&=\alpha_{h}+\frac{k^{2}\sinh{4\alpha_{h}}}{4h_{h}^{2}r_{h}}\,(r-r_{h})-\frac{k^{2}(8h_{h}^{2}+k^{2}-3k^{2}\cosh{4\alpha_{h}})\sinh{4\alpha_{h}}}{32h_{h}^{4}r_{h}^{2}}\,(r-r_{h})^{2}+\cdots\,.\end{split}$
(A.1)

Among five independent parameters $r_{h},f_{h},h_{h},\alpha_{h},k$ at the
horizon, three of them could be removed by the scaling symmetries (2.7, 2.8,
2.9). More precisely, we can use (2.7) to set $r_{h}=1$, (2.8) to set
$f_{h}=1$ and use (2.9) to set $k=1$. Then we have only two free parameters
$h_{h},\alpha_{h}$ at the horizon as shooting parameters. In the dual field
theory, these two parameters are related to the two scale invariant free
parameters $\alpha_{b},T/k$.

We get the series solution near UV boundary $r\to\infty$

$\displaystyle\begin{split}f&=f_{b}\bigg{(}1-\frac{k^{2}\sinh^{2}{2\alpha_{b}}}{6h_{b}^{2}r^{2}}-\frac{c_{h}}{h_{b}r^{4}}-\frac{k^{4}\sinh^{2}{2\alpha_{b}}}{24h_{b}^{4}r^{4}}\,(5+7\cosh{4\alpha_{b}})\\\
&~{}~{}~{}~{}-\frac{\log
r}{r^{4}}\frac{k^{4}}{6h_{b}^{4}}\,(\cosh{4\alpha_{b}}-\cosh{8\alpha_{b}})+\cdots\bigg{)}\,,\\\
g&=r^{2}\bigg{(}1+\frac{k^{2}\sinh^{2}{2\alpha_{b}}}{3h_{b}^{2}r^{2}}+\frac{M}{r^{4}}+\frac{\log
r}{r^{4}}\frac{k^{4}}{3h_{b}^{4}}\,(\cosh{4\alpha_{b}}-\cosh{8\alpha_{b}})+\cdots\bigg{)}\,,\\\
h&=r\bigg{(}h_{b}+\frac{k^{2}\sinh^{2}{2\alpha_{b}}}{2h_{b}^{2}r^{2}}+\frac{c_{h}}{r^{4}}+\frac{\log
r}{r^{4}}\frac{k^{4}}{6h_{b}^{4}}\,(\cosh{4\alpha_{b}}-\cosh{8\alpha_{b}})+\cdots\bigg{)}\,,\\\
\alpha&=\alpha_{b}-\frac{k^{2}\sinh{4\alpha_{b}}}{4h_{b}^{2}r^{2}}+\frac{c_{\alpha}}{r^{4}}+\frac{\log
r}{r^{4}}\frac{k^{4}\sinh{4\alpha_{b}}}{12h_{b}^{4}}\,\left(4\cosh{4\alpha_{b}}-1\right)+\cdots\,.\end{split}$
(A.2)

An asymptotic AdS black hole solution should have $f_{b}=1\,,h_{b}=1$. After
integrating the EOM from horizon to boundary, we can read the value of $f\,,h$
at boundary to obtain $f_{b}$ and $h_{b}$ then we use (2.8) to set $f_{b}=1$
and use (2.9) to set $h_{b}=1$. With the data at the horizon, we can further
obtain the interior configurations by integrating the system from the horizon
to the singularity. Equivalently, with the boundary values of $\alpha_{b}$ and
$T/\mu$, the whole system for the boundary to the singularity are completely
determined.

##  Appendix B Expansions in charged black holes

In this appendix, we show the calculation details of the series solutions near
the horizon and near the boundary for the charged helical black holes studied
in Sec. 3. We obtain the IR series solutions near the horizon $r\to r_{h}$

$\displaystyle\begin{split}g&=r_{h}\left(4-\frac{\psi_{1}^{2}}{6f_{h}^{2}}\right)(r-r_{h})+\cdots\,,\\\
f&=f_{h}+\frac{12f_{h}^{3}k^{2}\psi_{1}^{2}\sinh^{2}{2\alpha_{h}}}{r_{h}h_{h}^{2}(\psi_{1}^{2}-24f_{h}^{2})^{2}}(r-r_{h})+\cdots\,,\\\
h&=h_{h}+\left(\frac{h_{h}}{r_{h}}+\frac{12f_{h}^{2}k^{2}\sinh^{2}{2\alpha_{h}}}{r_{h}h_{h}(\psi_{1}^{2}-24f_{h}^{2})}\right)(r-r_{h})+\cdots\,,\\\
\alpha&=\alpha_{h}+\frac{6f_{h}^{2}k^{2}\sinh{4\alpha_{h}}}{r_{h}h_{h}^{2}(\psi_{1}^{2}-24f_{h}^{2})}(r-r_{h})+\cdots\,,\\\
\psi&=\psi_{1}(r-r_{h})+\cdots\,.\end{split}$ (B.1)

where we choose the gauge $\psi(r_{h})=0$. Similar to the neutral case, we
will use (3.7) to set $r_{h}=1$, (3.8) to set $f_{h}=1$ and use (3.9) to set
$k=1$. Then the only three free shooting parameters at the horizon are
$h_{h},\alpha_{h},\psi_{1}$, which are related to the scale invariant
parameters $T/{\mu},k/{\mu},\alpha_{b}$ in the dual field theory.

The UV expansions near the boundary $r\to\infty$ are

$\displaystyle\begin{split}f&=f_{b}\bigg{(}1-\frac{k^{2}\sinh^{2}{2\alpha_{b}}}{6h_{b}^{2}r^{2}}-\frac{c_{h}}{h_{b}r^{4}}-\frac{k^{4}\sinh^{2}{2\alpha_{b}}}{24h_{b}^{4}r^{4}}(5+7\cosh{4\alpha_{b}})\\\
&~{}~{}~{}~{}~{}~{}~{}-\frac{\log
r}{r^{4}}\frac{k^{4}}{6h_{b}^{4}}(\cosh{4\alpha_{b}}-\cosh{8\alpha_{b}})+\cdots\bigg{)}\,,\\\
g&=r^{2}\bigg{(}1+\frac{k^{2}\sinh^{2}{2\alpha_{b}}}{3h_{b}^{2}r^{2}}+\frac{M}{r^{4}}+\frac{\log
r}{r^{4}}\frac{k^{4}}{3h_{b}^{4}}(\cosh{4\alpha_{b}}-\cosh{8\alpha_{b}})+\cdots\bigg{)}\,,\\\
h&=r\bigg{(}h_{b}+\frac{k^{2}\sinh^{2}{2\alpha_{b}}}{2h_{b}r^{2}}+\frac{c_{h}}{r^{4}}+\frac{\log
r}{r^{4}}\frac{k^{4}}{6h_{b}^{3}}(\cosh{4\alpha_{b}}-\cosh{8\alpha_{b}})+\cdots\bigg{)}\,,\\\
\alpha&=\alpha_{b}-\frac{k^{2}\sinh{4\alpha_{b}}}{4h_{b}^{2}r^{2}}+\frac{c_{\alpha}}{r^{4}}+\frac{\log
r}{r^{4}}\frac{k^{4}}{12h_{b}^{4}}\sinh{4\alpha_{b}}(4\cosh{4\alpha_{b}}-1)+\cdots\,,\\\
\psi&=\psi_{b}+\frac{\rho}{r^{2}}-\frac{k^{2}\rho\sinh^{2}{2\alpha_{b}}}{3h_{b}^{2}r^{4}}+\cdots\,.\end{split}$
(B.2)

Similarly when we obtain a numerical solution, we need to use (3.8) and (3.9)
to set $f_{b}=1$ and $h_{b}=1$ respectively.

## References

  * [1] S. A. Hartnoll, C. P. Herzog and G. T. Horowitz, Holographic Superconductors, JHEP 12 (2008), 015 [arXiv:0810.1563].
  * [2] S. Kachru, X. Liu and M. Mulligan, Gravity duals of Lifshitz-like fixed points, Phys. Rev. D 78 (2008), 106005 [arXiv:0808.1725].
  * [3] M. Taylor, Lifshitz holography, Class. Quant. Grav. 33 (2016) no.3, 033001 [arXiv:1512.0355].
  * [4] G. T. Horowitz, J. E. Santos and D. Tong, Optical Conductivity with Holographic Lattices, JHEP 07 (2012), 168 [arXiv:1204.0519].
  * [5] M. Blake, Y. Gu, S. A. Hartnoll, H. Liu, A. Lucas, K. Rajagopal, B. Swingle and B. Yoshida, Snowmass White Paper: New ideas for many-body quantum systems from string theory and black holes, [arXiv:2203.04718].
  * [6] A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, The entropy of Hawking radiation, [arXiv:2006.06872].
  * [7] A. Frenkel, S. A. Hartnoll, J. Kruthoff and Z. D. Shi, Holographic flows from CFT to the Kasner universe, JHEP 08, 003 (2020) [arXiv:2004.01192].
  * [8] S. A. Hartnoll, G. T. Horowitz, J. Kruthoff and J. E. Santos, Gravitational duals to the grand canonical ensemble abhor Cauchy horizons, JHEP 10 (2020), 102 [arXiv:2006.10056].
  * [9] S. A. Hartnoll, G. T. Horowitz, J. Kruthoff and J. E. Santos, Diving into a holographic superconductor, SciPost Phys. 10 (2021), 009 [arXiv:2008.12786].
  * [10] R. G. Cai, L. Li and R. Q. Yang, No Inner-Horizon Theorem for Black Holes with Charged Scalar Hair, [arXiv:2009.05520].
  * [11] Y. Q. Wang, Y. Song, Q. Xiang, S. W. Wei, T. Zhu and Y. X. Liu, Holographic flows with scalar self-interaction toward the Kasner universe, [arXiv:2009.06277].
  * [12] N. Grandi and I. Salazar Landea, Diving inside a hairy black hole, JHEP 05 (2021), 152 [arXiv:2102.02707].
  * [13] S. A. H. Mansoori, L. Li, M. Rafiee and M. Baggioli, What’s inside a hairy black hole in massive gravity? [arXiv:2108.01471].
  * [14] Y. Liu, H. D. Lyu and A. Raju, Black hole singularities across phase transitions, JHEP 10, 140 (2021) [arXiv:2108.04554].
  * [15] O. J. C. Dias, G. T. Horowitz and J. E. Santos, Inside an asymptotically flat hairy black hole, JHEP 12, 179 (2021) [arXiv:2110.06225].
  * [16] L. Sword and D. Vegh, Kasner geometries inside holographic superconductors, [arXiv:2112.14177].
  * [17] R. G. Cai, C. Ge, L. Li and R. Q. Yang, Inside anisotropic black hole with vector hair, JHEP 02 (2022), 139 [arXiv:2112.04206].
  * [18] V. A. Belinsky, I. M. Khalatnikov, and E. M. Lifshitz, Oscillatory approach to a singular point in the relativistic cosmology, Adv. Phys. 19 (1970) 525; A general solution of the Einstein equations with a time singularity, Adv. Phys. 31, 639 (1982).
  * [19] V. Belinski and M. Henneaux, The Cosmological Singularity, Cambridge Universe Press.
  * [20] M. Henneaux, The final Kasner regime inside black holes with scalar or vector hair, [arXiv:2202.04155].
  * [21] A. Donos, J. P. Gauntlett and C. Pantelidou, Conformal field theories in $d=4$ with a helical twist, Phys. Rev. D 91 (2015), 066003 [arXiv:1412.3446].
  * [22] A. Donos and S. A. Hartnoll, Interaction-driven localization in holography, Nature Phys. 9, 649-655 (2013) [arXiv:1212.2998].
  * [23] A. Donos and J. P. Gauntlett, Black holes dual to helical current phases, Phys. Rev. D 86, 064010 (2012) [arXiv:1204.1734].
  * [24] A. Donos and J. P. Gauntlett, Helical superconducting black holes, Phys. Rev. Lett. 108, 211601 (2012) [arXiv:1203.0533].
  * [25] E. Kasner, Geometrical theorems on Einstein’s cosmological equations, Am. J. Math. 43, 217-221 (1921).
  * [26] C. W. Misner, Mixmaster universe, Phys. Rev. Lett. 22, 1071-1074 (1969).
  * [27] N. Iizuka, S. Kachru, N. Kundu, P. Narayan, N. Sircar and S. P. Trivedi, Bianchi Attractors: A Classification of Extremal Black Brane Geometries, JHEP 07 (2012), 193 [arXiv:1201.4861].
  * [28] L. Fidkowski, V. Hubeny, M. Kleban and S. Shenker, The Black hole singularity in AdS / CFT, JHEP 02, 014 (2004) [arXiv:hep-th/0306170].
  * [29] G. Festuccia and H. Liu, Excursions beyond the horizon: Black hole singularities in Yang-Mills theories. I., JHEP 04 (2006), 044 [arXiv:hep-th/0506202].
  * [30] M. Grinberg and J. Maldacena, Proper time to the black hole singularity from thermal one-point functions, JHEP 03 (2021), 131, [arXiv:2011.01004].
  * [31] T. Hartman and J. Maldacena, Time Evolution of Entanglement Entropy from Black Hole Interiors, JHEP 05 (2013), 014 [arXiv:1303.1080].
  * [32] E. Caceres, A. Kundu, A. K. Patra and S. Shashi, Trans-IR Flows to Black Hole Singularities, [arXiv:2201.06579].
  * [33] A. Bhattacharya, A. Bhattacharyya, P. Nandy and A. K. Patra, Bath deformations, islands, and holographic complexity, Phys. Rev. D 105 (2022) no.6, 066019 [arXiv:2112.06967].
  * [34] Y. S. An, L. Li, F. G. Yang and R. Q. Yang, Interior Structure and Complexity Growth Rate of Holographic Superconductor from M-Theory, [arXiv:2205.02442].
  * [35] R. Auzzi, S. Bolognesi, E. Rabinovici, F. I. S. Massolo and G. Tallarita, On the time dependence of holographic complexity for charged AdS black holes with scalar hair, [arXiv:2205.03365].


11institutetext: Software Engineering and IT Security, Fraunhofer IEM,
Paderborn, Germany 22institutetext: Department of Computer Science, Paderborn
University, Paderborn, Germany 33institutetext: MCML, LMU Munich, Munich,
Germany

# Meta-Learning for Automated Selection of Anomaly Detectors for Semi-
Supervised Datasets

David Schubert 11 Pritha Gupta 22 Marcel Wever 33

###### Abstract

In anomaly detection, a prominent task is to induce a model to identify
anomalies learned solely based on normal data. Generally, one is interested in
finding an anomaly detector that correctly identifies anomalies, i.e., data
points that do not belong to the normal class, without raising too many false
alarms. Which anomaly detector is best suited depends on the dataset at hand
and thus needs to be tailored. The quality of an anomaly detector may be
assessed via confusion-based metrics such as the Matthews correlation
coefficient (MCC). However, since during training only normal data is
available in a semi-supervised setting, such metrics are not accessible. To
facilitate automated machine learning for anomaly detectors, we propose to
employ meta-learning to predict MCC scores based on metrics that can be
computed with normal data only. First promising results can be obtained
considering the hypervolume and the false positive rate as meta-features.

###### Keywords:

Anomaly Detection Meta-learning AutoML

##  1 Introduction

automated machine learning (AutoML) [1] refers to the vision of automating the
data science process emerging from the unmatched demand for expertise in data
science and machine learning in particular. A substantial part of AutoML
literature is concerned with the selection, configuration, and composition of
machine learning algorithms tailored for a certain task, consisting of a
dataset and a performance measure [2]. After first promising results for
standard (binary or multi-class) classification and regression tasks on
tabular data could be achieved [3, 4, 5], AutoML quickly spread to further
data types, learning problems, and tasks, such as image classification [6],
multi-label classification [7], remaining useful lifetime estimation in
predictive maintenance [8], online learning [9], natural language processing
[10], and multi-modal data [11]. While the aforementioned extensions of AutoML
require supervision in terms of labels being provided for fitting the models,
more recently, AutoML methods for dealing with unsupervised or semi-supervised
machine learning tasks are proposed as well.

One such problem class deals with the detection of outliers, sometimes also
referred to as anomalies. While performance complementarity can also be
observed for anomaly detectors, i.e., for different datasets, different
anomaly detection methods perform best, this setting imposes severe challenges
for adopting AutoML methods to determine the most suitable anomaly detection
method. First, if the provided dataset contains both classes at all, i.e.,
normal and anomalous, their frequency is typically unbalanced. More
specifically, it is usually assumed that anomalies occur only with very low
frequency. Second, the dataset at hand may comprise no anomalies at all, and
the anomaly detection method is meant to detect any new data points that
deviate from the training data and thus represent anomalies [12, 13], e.g., in
intrusion detection. In the following, we refer to this setting as being semi-
supervised. For commonly available sampling-based AutoML approaches, the
latter data situation is difficult to handle since they rely on probing
sampled algorithms and hyperparameter values for the given data. For example,
a solution candidate is evaluated via cross-validation and the average
performance for a performance measure of interest is observed, e.g., error
rate for classification or mean squared error for regression. Obviously, in
the domain of anomaly detection, we are interested in finding a model that can
detect anomalies reliably but also classifies normal data points as such.
Performance measures such as MCC or the area under the ROC curve are
considered to account for the imbalance between normal and anomalous data
points. However, these performance measures require the presence of both types
of data points normal _and_ anomalous data points such that they cannot be
computed in the semi-supervised setting mentioned above.

In this paper we investigate the use of meta-learning to overcome the issue of
evaluating algorithms and their hyperparameter values for a performance
measure of interest, which may require data points of both classes. To this
end, we assess the predictive power of two metrics that can be computed with
normal data points only: hypervolume of anomaly detectors and the false
positive rate (FPR). In our experimental section, we find promising results
when utilizing these metrics in terms of meta-features for landmarking and
feature descriptions of anomaly detectors.

##  2 CASH for Semi-Supervised Anomaly Detection

Let $\mathcal{X}$ and $\mathcal{Y}=\\{0,1\\}$ be an instance space and a
target space respectively. Furthermore, we assume instances $x\in\mathcal{X}$
to be (non-deterministically) associated with a label $y\in\mathcal{Y}$. In
the setting of anomaly detection, we associate instances $x$ with $y=1$ in
case it is an anomaly and $y=0$ if it is normal. Moreover, anomalies are
assumed to occur only seldomly, i.e., $y=1$ is a minority class, and during
training only normal data points are provided. This setting is also
occasionally referred to as semi-supervised anomaly detection [12]. For
convenience, let $\\{0\\}=:\mathcal{Y}_{0}\subset\mathcal{Y}$. We seek to
learn a hypothesis $h:\mathcal{X}\rightarrow\mathcal{Y}$ from training data
given in the form of
$\mathcal{D}=\\{(x_{i},y_{i})\in\mathcal{X}\times\mathcal{Y}_{0}\mid 1\leq
i\leq n\\}$ that is able to discriminate well between normal and anomalous
data points.

Let be $\mathcal{A}:=\\{A^{(1)},A^{(2)},\ldots A^{(m)}\\}$ a set of anomaly
detectors with corresponding hyperparameter spaces
$\Lambda^{(1)},\Lambda^{(2)},\ldots\Lambda^{(m)}$. Additionally, a dataset of
training instances
$\mathcal{D}:=\\{(\mathcal{X},\mathcal{Y}_{0})\\}_{i=1}^{n}$, and a
performance measure $m:\mathcal{Y}\times\mathcal{Y}\rightarrow\mathbb{R}$, we
aim to find the most suitable anomaly detector $A^{\ast}$ together with its
hyperparameter setting $\lambda^{\ast}$ with respect to $m$:

$A^{\ast}_{\lambda^{\ast}}\in\underset{A^{(i)}\in\mathcal{A},\lambda\in\Lambda^{(i)}}{\arg\max}\mathbb{E}_{(x,y)\sim
P}\,\,\left[m(A^{(i)}_{\lambda}(\mathcal{D},x),y)\right]\,\,\,,$ (1)

where $A_{\lambda}^{(i)}$ is trained on training data $\mathcal{D}$ and makes
a prediction on $x$ which is then compared to the ground truth $y$ and without
loss of generality $m$ is to be maximized. This problem is better known as the
combined algorithm selection and hyperparameter optimization (CASH) problem as
initially formalized in [3].

To find such an $A^{\ast}_{\lambda^{\ast}}$, in sampling-based AutoML systems,
one would split the dataset $\mathcal{D}$ into subsets of training and
validation data $\mathcal{D}_{train}$ and $\mathcal{D}_{val}$ respectively and
use the performance on the validation data as an estimate of the true
generalization performance:

$\widehat{A}^{\ast}_{\widehat{\lambda}^{\ast}}\in\underset{A^{(i)}\in\mathcal{A},\lambda\in\Lambda^{(i)}}{\arg\max}\mathbb{E}_{\mathcal{D}_{train},\mathcal{D}_{val}=(x_{j},y_{j})_{j=1}^{s}}\,\,\left[\frac{1}{s}\sum_{j=1}^{s}m(A^{(i)}_{\lambda}(\mathcal{D}_{train},x_{j}),y_{j})\right]\,\,\,.$
(2)

##  3 Related Work

To the best of our knowledge, there are no other publications that focus on
combined algorithm selection and hyperparameter optimization (CASH) in a semi-
supervised setting. Therefore, we refer to unsupervised approaches and semi-
supervised hyperparameter optimization approaches in this section.

In [14], Zhao et al. present MetaOD, an approach to unsupervised outlier model
selection. It is based on the construction of meta-features for a corpus of
training datasets with outlier labels and the performance of over $300$ models
for each dataset. Similarly to our evaluation, Zhao et al. use the meta-
features and performance values to train a performance predictor, which is
used to choose high performing models for new datasets. However, we focus on
the semi-supervised setting of outlier detection rendering landmark features
meant to capture information regarding potential outliers not applicable.

Putina et al. introduce AutoAD [15], a framework with the same purpose as
MetaOD but without utilizing meta-learning. They measure the performance of an
anomaly detector via metrics applied to the data before and after removing the
top-ranked anomalies predicted by the detector in question. Again, this
approach is not applicable to our semi-supervised setting as the employed
metrics imply the existence of anomalies in the tails of the data
distribution. We do not assume any anomalies being present in the datasets for
which an anomaly detector is to be optimized. Other approaches like PyODDS
[16], TODS [17], AutoOD [18], and LSCP [19] utilize supervised performance
metrics rendering them not applicable to our semi-supervised setting too.

Tax et al. have a body of works evolving from the idea of the _hypervolume_ of
one-class classifiers as being the hypervolume of the subspace that a one-
class classifier predicts to belong to the target class [20, 21, 22]. In [21],
they aim at optimizing the hyperparameters of the support vector data
description. For this purpose, they estimate the hypervolume of the support
vector data description and use a linear combination of this estimation and
the FPR (error on the target class) as performance metric for optimization.
The linear combination encompasses a trade-off parameter that has to be set
manually.

Our approach is motivated by this idea to a large extend. However, we focus on
the more general CASH problem. To this end, we evaluate the utilization of the
hypervolume and the FPR as features description of anomaly detectors and for
landmarking purposes. Furthermore, we train a meta-model for the prediction of
the quality of anomaly detectors on unseen datasets.

##  4 Meta-Learning for Selecting Anomaly Detectors

According to the assumption that $\mathcal{D}$ only contains normal data,
i.e., data points of the form $(x_{j},y_{j})$ with $y_{j}\in\mathcal{Y}_{0}$,
performance measures that quantify how accurately an anomaly detector may
identify anomalies, e.g., via MCC, precision, recall, or AUC, cannot be
assessed for evaluation. Hence, such measures can neither be employed by
AutoML systems to search for the most appropriate anomaly detector for a given
data set.

To overcome this issue, we aim to substitute the performance measure $m$ in
Eq. 2 by a surrogate model $\widehat{m}$, which provided a feature description
$f_{A}\in\mathbb{R}^{k}$ of algorithm $A^{(i)}$ together with its
hyperparameter setting $\lambda$ and a feature description
$f_{\mathcal{D}}\in\mathbb{R}^{l}$ of the dataset $\mathcal{D}$ in question
predicts the measure of interest. To this end, the surrogate is build on data
sets for which anomalies are actually known and hence a performance value can
be computed. Then, we can substitute $m$ by $\widehat{m}$ in Eq. 2 to obtain

$\widehat{A}^{\ast}_{\widehat{\lambda}^{\ast}}\in\underset{A^{(i)}\in\mathcal{A},\lambda\in\Lambda^{(i)}}{\arg\max}\,\,\widehat{m}(f_{\mathcal{A}}(A^{(i)},\lambda),f_{\mathcal{D}}(\mathcal{D}))\,\,\,.$

In the following we consider mainly two types of features as to describe
algorithms as well as datasets (in terms of landmarking features): hypervolume
and FPR.

Hypervolume

The hypervolume can be considered as a means to describe how tightly an
anomaly detector fits the normal data points. With a smaller hypervolume,
chances are low that anomalies are missed, whereas a larger hypervolume may
lead to anomalies not being identified as such.

False Positive Rate

The FPR assesses how many data points in the training data are falsely
classified as anomalies, i.e., the anomaly detector would raise a false alarm
in this case.

It is easy to see that minimizing both the hypervolume and the FPR is
conflicting with each other. Therefore, we seek to find a method yielding a
suitable tradeoff between the two measures.

##  5 Experimental Evaluation

In the following we assess the predictive power of the hypervolume and the FPR
for the task of selecting a suitable anomaly detector for an unseen dataset.
First, we consider a linear combination of the features to order anomaly
detectors and evaluate the general usefulness of these metrics in terms of a
feature description of the detectors. Second, we train a meta-model to predict
the MCC score of anomaly detectors. This model gets as an input landmarking
features resulting from the hypervolume and FPR of a fixed portfolio of
anomaly detectors and the hypervolume and FPR of the anomaly detector in
question.

Figure 1: BPMN Diagram [23] of the Experiments

Figure 1 shows the process that we follow for our experiments. We continuously
refer to this figure in the course of this section. All experiments of the
evaluation were executed on virtual machines with Ubuntu 20.04.4, 16 cores
(Intel Xeon E5-2695 v3), and 128 GB RAM. Sections 5.1 and 5.2 elaborate on the
assimilation of the meta-learning dataset and a correlation analysis
respectively. Thereafter, Section 5.3 discusses the results of the evaluation.

###  5.1 Meta-Learning Dataset Assimilation

We base our meta-learning datasets on a number of datasets for outlier
detection. The source of these datasets is the collection of Tax et al. [24].
The estimation of the hypervolume of anomaly detectors, which we explain in
the context of the corresponding activity, scales poorly with an increasing
number of dimensions. Thus, we only select those datasets that have 8 features
or less, resulting in 15 Base Datasets, which can be inferred from Table 1. We
execute the Meta-learning Dataset Assimilation activity iteratively for all 15
Base Datasets.

One of our basic assumptions is that outliers are the minority class (cf.
Section 2). Not all of the datasets reflect this assumption. Thus, we Sub-
sample Outliers to ensure that anomalies constitute between 5% and 10% of the
total dataset. We do this only for those datasets, where outliers originally
amount to more than 10% of the dataset. To Create the Train and Test Datasets,
we shuffle the Base Dataset and perform a stratified split into Train (70%)
and Test Datasets (20%). The remaining 10% are kept for future experiments
that are not in the focus of this publication. Furthermore, we remove all
outliers from the Train dataset to match our semi-supervised setting. Lastly,
we scale the Train and Test Datasets, subtracting the mean and scale the
features independently to the interquartile range with respect to the Train
Dataset.

Figure 1 shows, that the control flow forks at this point into the Construct
Landmark Dataset and Construct Detector Dataset activities. These activities
extract our meta-features, which are meant to characterize datasets and
anomaly detectors, respectively.

The landmark features are the hypervolume and the FPR of the Anomaly Detectors
provided by PyOD111https://pyod.readthedocs.io/ in version 0.9.7 [25] in their
default configuration. Here, we exclude those algorithms that do not finish
after one week of runtime for each dataset resulting in at most 18 Anomaly
Detectors per Base Dataset.

The estimation of the hypervolume that a model spans with respect to a certain
dataset follows the approach of Tax et al. [22]. Correspondingly, we fit the
smallest hyper-sphere around the train data and generate 35 million uniformly
distributed data instances within this sphere. We train the Anomaly Detectors
on the Train Dataset. The fraction of instances classified as belonging to the
target class is an estimate of the hypervolume of the corresponding anomaly
detector. Please note, that we inferred the number of generated data instances
from initial experiments. Generally, the number of necessary data instances
grows exponentially with the number of dimensions of the Train Dataset. This
is the reason for the aforementioned bad scaling of the hypervolume
estimation. To Compute the FPR, we execute a monte-carlo cross validation with
a 30% test-size and ten repetitions on the Train Dataset.

The parallel activity Construct Detector Dataset extracts the hypervolume, the
FPR, and the MCC of 50 randomly configured Anomaly Detectors of PyOD per Train
Dataset. The estimation of the hypervolume of these 50 Anomaly Detectors and
the FPR follows the computations in the context of the Landmark Dataset
explained above. We use the MCC of these anomaly detectors as a performance
measure, which we compute on the corresponding Test Dataset. If the feature
extraction via a particular anomaly detector does not terminate within 5 hours
(while executing 4 anomaly detectors in parallel), we replace it with a newly
configured anomaly detector.

The last activity is Combine Datasets, which uses the Landmark Dataset and the
Detector Dataset to create the Meta-learning Dataset. We combine the features
of the former datasets such that each instance of the resulting Meta-learning
Dataset encompasses all landmarking features of the Train Dataset and the
detector features of one particular anomaly detector.

###  5.2 Correlation Analysis

We execute the Correlation Analysis iteratively for all 15 Meta-learning
Datasets. The Meta-train and Meta-test Datasets are created in a leave-one-out
fashion. Meaning, in each iteration of the Correlation Analysis one Meta-
learning Dataset is used as Meta-test Dataset and the remainder is merged and
used as Meta-train Dataset. Since not all of the 18 Anomaly Detectors
terminate for all datasets in the context of the Compute Landmark Dataset
activity, we remove all landmark features that have no values in the Meta-test
Dataset and scale the remaining features individually to be in $[0,1]$.

After the creation of these datasets, the control flow forks again. The lower
control flow ranks the Anomaly Detectors covered by the Meta-test Dataset
according to the simple linear combination of their detector features as a
score:

$lc(A_{\lambda},\mathcal{D})=1-\frac{hypervolume(A_{\lambda},\mathcal{D})+FPR(A_{\lambda},\mathcal{D})}{2}\,\,\,.$
(3)

The upper control flow ranks the Anomaly Detectors via a meta-model that
predicts the MCC of the Anomaly Detectors covered by the Meta-test Dataset
based on the landmarking and detector hypervolume and FPR features. We opt for
a mean strategy to Impute Missing Values. Furthermore, we use the Meta-train
Dataset to train a random forest regressor (Meta-Model) in default
parameterization222https://scikit-
learn.org/1.1/modules/generated/sklearn.ensemble.RandomForestRegressor.html.
Thereafter, we rank the Anomaly Detectors of the Meta-test Dataset using the
predicted MCC of the Meta-Model as a score.

We execute the Compute Metrics activity for the Linear Ranking and the Meta-
learn Ranking separately. At the beginning of the activity, we scale the MCC
to the $[0,1]$ interval to avoid coping with negative scores. As metrics, we
use the regret@k, Kendalls rank correlation coefficient ($\tau$) [26], and the
normalized discounted cumulative gain (NDCG) [27], which are metrics that
refer to rankings. We base our evaluation on ranking metrics because a correct
ranking of anomaly detectors is more important to guide an AutoML system than
precise prediction of an anomaly detectors performance, e.g., in terms of the
mean squared error. While a high precision in the predicted quality is
certainly desirable, it is not necessary for choosing the most promising out
of a set of anomaly detectors. Here, a precise ranking is sufficient.

The regret@k compares the performance of the best model within a top-k ranking
with the actual best model known for a dataset and reports the absolute
difference. In our case it refers to the scaled MCC and gives an intuition
about the performance of the top-ranked anomaly detectors.

$\tau$ measures the correspondence between two rankings, which are the ranking
given by the method in question and the optimal ranking given by the true MCC
values. $\tau$ ranges in the interval $[-1,1]$ where negative values indicate
a negative correlation of the rankings and positive values a positive
correlation. We utilize the b-version of $\tau$, which accounts for ties.

Similarly, the NDCG is a measure to compare a predicted ranking to an optimal
one. In comparison to $\tau$, the actual scores (in our case the scaled MCC
values) influence the NDCG and not only the ranking inferred from those
scores. The NDCG ranges in the $[0,1]$ interval where values close to 1 denote
a high quality of the predicted ranking.

###  5.3 Results

In this section, we discuss the performance of the Linear Ranking and Meta-
learn Ranking approaches discussed in Section 5.2. Due to the lack of directly
comparable approaches (cf. Section 3), we introduce three different baselines.
Table 1 reports on the performance of the approaches with respect to the
metrics introduced in Section 5.2. Additionally, it reports the maximum, mean,
and minimum of the MCC of the anomaly detectors considered for prediction
scaled to $[0,1]$ to give an impression of the corresponding distribution.
Please note that the NDCG is generally not particularly insightful and
reported for completeness.

Table 1: Correlation analysis results. Small arrows indicate whether a metric is to be maximized or minimized where the best performance per dataset and metric is bolded. For datasets marked with a †, anomaly detectors considered for prediction have a mean scaled MCC score of at least 0.6, meaning they actually are capable of identifying anomalies reasonably. We use the following abbreviations for the ranking approaches: R = random, FPR = false positive rate, HV = hypervolume, L = linear combination, M = meta-model, and Mc = cherry picked variant of the meta-model. Meta-test | scaled MCC | regret@1 $\downarrow$ | regret@5 $\downarrow$ |  NDCG $\uparrow$ |  $\tau$ $\uparrow$
---|---|---|---|---|---
Dataset | Max | Mean | Min | R | FPR | HV | L | M | Mc | R | FPR | HV | L | M | Mc | R | FPR | HV | L | M | Mc | R | FPR | HV | L | M | Mc
Balance Middle | .59 | .47 | .40 | .12 | .11 | .07 | .12 | .12 |  | .05 | .10 | .05 | .03 | .04 |  | .96 | .97 | .97 | .96 | .96 |  | -.01 | .15 | -.16 | -.01 | -.14 |
Biomed Healthy | .84 | .45 | .31 | .35 | .38 | .30 | .42 | .38 |  | .22 | .38 | .30 | .42 | .38 |  | .89 | .89 | .89 | .86 | .89 |  | .01 | .22 | -.26 | -.22 | .06 |
Diabetes Absent | .67 | .56 | .45 | .07 | .18 | .13 | .19 | .04 |  | .03 | .12 | .02 | .12 | .03 |  | .96 | .93 | .97 | .94 | .97 |  | .01 | -.38 | .26 | -.23 | -.06 |
Diabetes Present | .67 | .51 | .41 | .16 | .19 | .17 | .22 | .14 |  | .09 | .19 | .07 | .22 | .14 |  | .95 | .94 | .95 | .93 | .96 |  | 0 | -.11 | -.26 | -.36 | .03 |
Ecoli Periplasm | .73 | .57 | .40 | .15 | .01 | .17 | .17 | .20 |  | .07 | .01 | .15 | .11 | .05 |  | .95 | .96 | .95 | .95 | .94 |  | -.01 | .03 | .13 | .16 | -.01 |
Iris Versicolor | .73 | .49 | .36 | .26 | .26 | .14 | .26 | .30 |  | .13 | 0 | .10 | .12 | .12 |  | .93 | .93 | .96 | .94 | .92 |  | -.02 | -.19 | .28 | .13 | -.06 |
Iris Virginica | .84 | .53 | .35 | .24 | .37 | .21 | 0 | .37 |  | .14 | 0 | .13 | 0 | 0 |  | .91 | .89 | .96 | .96 | .90 |  | .01 | -.11 | .45 | .39 | .17 |
Liver 1 | .64 | .49 | .40 | .14 | .16 | .16 | .19 | .16 |  | .08 | .16 | .10 | .10 | .03 |  | .95 | .95 | .94 | .94 | .96 |  | 0 | 0 | -.20 | -.09 | .04 |
Survival G5 | .85 | .54 | .40 | .31 | .39 | .39 | .19 | .34 |  | .22 | 0 | .25 | 0 | .23 |  | .94 | .95 | .93 | .97 | .93 |  | -.02 | .30 | .05 | .41 | -.11 |
Survival S5 | .72 | .50 | .37 | .21 | .26 | .27 | .27 | .27 |  | .11 | .24 | .08 | .17 | .26 |  | .94 | .92 | .94 | .92 | .92 |  | .03 | -.15 | .18 | 0 | -.05 |
Biomed Diseased† | 1 | .64 | .31 | .15 | 0 | .50 | 0 | 0 | 0 | .09 | 0 | .37 | 0 | 0 | 0 | .90 | .97 | .89 | .96 | .98 | .98 | 0 | .44 | -.02 | .41 | .54 | .54
Balance Left† | .92 | .62 | .44 | .30 | .33 | .31 | 0 | .33 | .13 | .18 | .13 | .30 | 0 | .16 | .13 | .94 | .94 | .94 | .98 | .95 | .96 | 0 | .07 | .24 | .48 | .29 | .39
Balance Right† | .86 | .62 | .42 | .25 | .39 | .23 | 0 | .25 | .08 | .11 | .08 | .23 | 0 | .23 | 0 | .94 | .92 | .94 | .98 | .93 | .99 | 0 | .12 | .12 | .53 | .01 | .59
Iris Setosa† | 1 | .68 | .41 | .22 | 0 | .42 | .27 | .35 | .27 | .16 | 0 | .34 | 0 | .27 | 0 | .93 | .98 | .91 | .95 | .92 | .97 | 0 | .46 | -.24 | .28 | -.18 | .28
Liver 2† | .85 | .60 | .35 | .12 | 0 | .32 | 0 | .32 | 0 | .05 | 0 | .18 | 0 | .18 | 0 | .93 | .99 | .91 | .96 | .91 | .97 | .02 | .49 | -.17 | .21 | -.18 | .23
Mean | .79 | .55 | .39 | .20 | .20 | .25 | .15 | .24 | .10 | .12 | .09 | .18 | .09 | .14 | .03 | .93 | .94 | .94 | .95 | .94 | .97 | 0 | .09 | .03 | .14 | .02 | .41
Median | .84 | .54 | .40 | .21 | .19 | .23 | .19 | .27 | .08 | .11 | .08 | .15 | .03 | .14 | 0 | .94 | .94 | .94 | .95 | .93 | .97 | 0 | .07 | .05 | .16 | -.01 | .39

#### Baselines:

The first baseline approach is based on randomization. Here, we report the
mean values of 50 randomized rankings. The second approach utilizes the FPR
detector feature introduced in Section 5.1 and ranks the 50 anomaly detectors
per dataset with the inversed FPR ($1-FPR(A_{\lambda})$) as score. The last
baseline approach works analogously to the second one but uses the inverse of
the hypervolume detector feature.

Table 1 shows the conflicting nature of the FPR and hypervolume features
already mentioned in Section 4 as good $\tau$ performances for one approach
are typically accompanied with rather poor ones for the other. Additionally,
the regret and $\tau$ indicate that the FPR is more informative than the
hypervolume. One possible reason for this is that the FPR directly influences
the MCC but the hypervolume only if it correlates with the false negative
rate. This is not necessarily the case, e.g., if an anomaly detector exactly
covers the hypersphere, which in turn does not cover any anomalies.
Furthermore, the hypervolume is very sensitive to noisy features. An anomaly
detector can expand in the direction of such features and drastically increase
its volume without affecting its MCC.

#### Linear Ranking:

Ranking according to the linear combination of hypervolume and FPR detector
features, yields decent results. The linear combination of the hypervolume and
the FPR seems to be a suitable means to compare different anomaly detectors to
each other and rank them accordingly. Particularly when comparing the combined
approach with the separate baselines introduced before, we see that the
combination outperforms the separate approaches. For 5 datasets the linear
rankings have the top detector in the first place and for two more datasets it
is within the top-5 ranked detectors.

Furthermore, we see that the mean and median of $\tau$ indicate a positive
association in relation to optimal rankings. However, investigating $\tau$ for
the datasets separately yields that 5 datasets show negative $\tau$ values,
which is also the reason for the relative low mean and median $\tau$ of $.14$
and $.16$, respectively. Cross-checking these results with the mean MCC values
of the 50 randomly configured anomaly detectors per dataset indicates a strong
relation. If we restrict the evaluation to those datasets with a mean scaled
MCC of at least $.6$, we end up with a mean regret@1 of $0.05$, regret@5 of
$0$, NDCG of $0.97$, and $\tau$ of $0.38$ (not shown in Table 1). Hence, the
features seem to be more informative for those datasets for which we randomly
find better anomaly detectors on average.

#### Meta-learn Ranking:

Using the hypervolume and FPR as meta-features for both datasets and the
anomaly detector is clearly worse than the linear combination of the detector
features alone. Overall, it can be regarded as being on par with the
randomized approach.

However, a leave-one-out evaluation on cherry picked datasets – analogously to
what we describe in the context of the linear ranking – leads to the meta-
model slightly outperforming the cherry picked linear ranking in terms of the
mean NDCG and $\tau$. Thus, we once more see a strong relation to
particularities of the corresponding meta-learning datasets and the
performance of the encompassed anomaly detectors, which – obviously – the
landmarking features cannot sufficiently express. Additionally, the
hypervolume estimation is very sensitive to the data distribution of the
target class. On the one hand, for distributions well approximated by the
sampled hypersphere, a good anomaly detector has a hypervolume close to 1. On
the other hand, for a distribution on a submanifold, the hypervolume of a good
detector tends towards 0. This relation might be hard to learn. Still, we find
the results to be promising, particularly considering that we do not assume
any information regarding the outlier class of the unseen dataset. Thus, they
form a potential base from which interesting future work may emerge.
Especially, whether additional meta-features, other types of meta-models, or
improved sampling strategies for the hypervolume estimation may help to
improve the accuracy of the predictions for the MCC.

##  6 Conclusion

In this paper, we have considered the learning problem of anomaly detection
where during training only a dataset with normal data points is available.
While this impedes the use of performance measures explicitly quantifying how
well an anomaly detector is able to identify anomalies, for the task of
automatically selecting and configuring anomaly detectors, we proposed to
employ meta-learning to predict measures of interest. In this regard, we
considered mainly two types of meta-features. One that is based on the
hypervolume covered by trained anomaly detectors and the other one considering
the FPRs of anomaly detectors. While a lower value for the former seems
favorable as the anomaly detector fits the training data more tightly, this is
usually in conflict with minimizing the FPR since the smaller the hypervolume,
the more training data points may be classified as positive. Used in
combination, the two features have shown promising performance to be used
directly for ranking anomaly detectors and as meta-features for a meta-model
to predict a measure of interest that would actually require anomalies in the
training data for being evaluated. Moreover, results of the corresponding
experiments, which can rather be considered a proof of concept, indicate that
AutoML for anomaly detectors might be feasible using such surrogate measures
for performance evaluation.

Whether AutoML systems can really work well with such surrogate models for
selecting and configuring anomaly detectors, however, is still an open
question and also outlines interesting future work. Thus, we aim to extend our
approach by augmenting the set of meta-features by more types of informative
features describing the data or the anomaly detector, to improve the quality
of the meta-model. Other research directions are too improve the hypervolume
estimation and to formalize the problem directly as ranking problem.

## References

  * [1] F. Hutter, L. Kotthoff, and J. Vanschoren.  Automated machine learning: methods, systems, challenges.  Springer Nature, 2019.
  * [2] M. Zöller and M. Huber.  Benchmark and survey of automated machine learning frameworks.  Journal of artificial intelligence research, 70, 2021.
  * [3] C. Thornton, F. Hutter, H. Hoos, and K. Leyton-Brown.  Auto-weka: Combined selection and hyperparameter optimization of classification algorithms.  In Proc. of the 19th International Conference on Knowledge Discovery and Data Mining, 2013.
  * [4] M. Feurer, A. Klein, K. Eggensperger, J. Springenberg, M. Blum, and F. Hutter.  Efficient and robust automated machine learning.  Advances in neural information processing systems, 28, 2015.
  * [5] F. Mohr, M. Wever, and E. Hüllermeier.  Ml-plan: Automated machine learning via hierarchical planning.  Machine Learning, 107(8), 2018.
  * [6] P. Ren, Y. Xiao, X. Chang, P. Huang, Z. Li, X. Chen, and X. Wang.  A comprehensive survey of neural architecture search: Challenges and solutions.  ACM Computing Surveys, 54(4), 2021.
  * [7] M. Wever, A. Tornede, F. Mohr, and E. Hüllermeier.  Automl for multi-label classification: Overview and empirical evaluation.  IEEE Trans. on Pattern Analysis and Machine Intelligence, 43(9), 2021.
  * [8] T. Tornede, A. Tornede, M. Wever, and E. Hüllermeier.  Coevolution of remaining useful lifetime estimation pipelines for automated predictive maintenance.  In Proc. of the genetic and evolutionary computation conference, 2021.
  * [9] B. Celik, P. Singh, and J. Vanschoren.  Online automl: An adaptive automl framework for online learning.  arXiv preprint arXiv:2201.09750, 2022.
  * [10] E. Bisong.  Google automl: Cloud natural language processing.  In Building Machine Learning and Deep Learning Models on Google Cloud Platform. Springer, 2019.
  * [11] J. Mueller, X. Shi, and A. Smola.  Faster, simpler, more accurate: Practical automated machine learning with tabular, text, and image data.  In Proc. of the 26th International Conference on Knowledge Discovery & Data Mining, 2020.
  * [12] Varun Chandola, Arindam Banerjee, and Vipin Kumar.  Anomaly detection: A survey.  ACM computing surveys, 41(3), 2009.
  * [13] M. Ahmed, Abdun N. M., and J. Hu.  A survey of network anomaly detection techniques.  Journal of Network and Computer Applications, 60, 2016.
  * [14] Yue Zhao, Ryan Rossi, and Leman Akoglu.  Automatic unsupervised outlier model selection.  In Advances in Neural Information Processing Systems, volume 34. Curran Associates, Inc., 2021.
  * [15] A. Putina, M. Bahri, F. Salutari, and M. Sozio.  AutoAD: an Automated Framework for Unsupervised Anomaly Detection.  In IEEE International Conference on Data Science and Advanced Analytics., Paris / Virtual Event, France, 2022. IEEE.
  * [16] Y. Li, D. Zha, P. Venugopal, N. Zou, and X. Hu.  Pyodds: An end-to-end outlier detection system with automated machine learning.  In Companion Proc. of the Web Conference 2020, New York, NY, USA, 2020. Association for Computing Machinery.
  * [17] K. Lai, D. Zha, G. Wang, J. Xu, Y. Zhao, D. Kumar, Y. Chen, P. Zumkhawaka, M. Wan, D. Martinez, and X. Hu.  Tods: An automated time series outlier detection system.  Proc. of the AAAI Conference on Artificial Intelligence, 35(18), 2021.
  * [18] Y. Li, Z. Chen, D. Zha, K. Zhou, H. Jin, H. Chen, and X. Hu.  Automated anomaly detection via curiosity-guided search and self-imitation learning.  IEEE Trans. Neural Networks Learn. Syst., 33(6), 2022.
  * [19] Y. Zhao, Z. Nasrullah, M. Hryniewicki, and Z. Li.  LSCP: locally selective combination in parallel outlier ensembles.  In Proc. of the 2019 International Conference on Data Mining. SIAM, 2019.
  * [20] D. Tax and K. Muller.  A consistency-based model selection for one-class classification.  In Proc. of the 17th International Conference on Pattern Recognition, 2004., volume 3, 2004.
  * [21] D. Tax and R. Duin.  Outliers and data descriptions.  In Proc. of the 7th Annual Conf. of the Advanced School for Computing and Imaging, 2001. ASCI, 2001.
  * [22] D. Tax and R. Duin.  Uniform object generation for optimizing one-class classifiers.  Journal of Machine Learning Research, 2(2), 2001.  Special Issue on Kernel Methods.
  * [23] Object Management Group (OMG).  Business process model and notation (bpmn), version 2.0.2.  Technical report, 2014.
  * [24] Tax - One-class Datasets.  http://homepage.tudelft.nl/n9d04/occ/index.html.  Accessed: 2022-11-16.
  * [25] Y. Zhao, Z. Nasrullah, and Z. Li.  Pyod: A python toolbox for scalable outlier detection.  Journal of Machine Learning Research, 20(96), 2019.
  * [26] W. Knight.  A computer method for calculating kendall’s tau with ungrouped data.  Journal of the American Statistical Association, 61(314), 1966.
  * [27] K. Järvelin and J. Kekäläinen.  Cumulated gain-based evaluation of ir techniques.  ACM Trans. Information Systems, 20(4), 2002.

  *[MCC]: Matthews correlation coefficient
  *[AutoML]: automated machine learning
  *[FPR]: false positive rate
  *[CASH]: combined algorithm selection and hyperparameter optimization
  *[$\tau$]: Kendalls rank correlation coefficient
  *[NDCG]: normalized discounted cumulative gain
  *[FPRs]: false positive rate