aainstitutetext: Faculty of Physics, Shahrood University of Technology, P.O.
Box 3619995161, Shahrood, Iranbbinstitutetext: School of Physics, Damghan
University, Damghan, 41167-36716, Iran

# Probing inside a charged hairy black hole in massive gravity

Mirmani Mirjalali a , Seyed Ali Hosseini Mansoori b , Leila Shahkarami a , and
Morteza Rafiee<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper, we investigate the internal structure of a charged hairy black
hole solution in the non-linear massive gravity. We first consider the impact
of various configurations of massive gravity on the condensate operator and
then probe the black hole interior dynamics. Like a standard holographic
superconductor system, just below the critical temperature, the interior
evolves through several distinct epochs, including a collapse of the Einstein-
Rosen bridge, Josephson oscillations of the scalar field, and finally a Kasner
(or Kasner inversion) cosmology. However, for the large massive gravity
parameter, we see distinguishing features for the interior dynamics. In this
regime, at a given temperature, the Einstein-Rosen bridge collapse and
subsequent Josephson oscillations epochs completely disappear from the
interior dynamics and the final Kasner cosmology epoch starts exactly after
the would-be inner horizon and the system does not experience the Kasner
inversion epoch.

##  1 Introduction

In recent years there has been growing interest in understanding the internal
structure of a black hole. Take for instance, a Reisser-Nordstrom (RN) black
hole reissner1916eigengravitation ; nordstrom1918energy  or a Kerr black hole
kerr1963gravitational . After crossing the event horizon, there is an inner
Cauchy horizon before reaching near the singularity where the spacetime
curvature becomes infinite. The appearance of Cauchy horizon results in
breaking down the predictability of the classical dynamics geroch1970domain
and appear to violate the strong cosmic censorship (SCC) conjecture
Penrose:1969pc .

On the other hand, holographic arguments suggest that Cauchy horizons cannot
survive in the full quantum gravity theories Papadodimas:2019msp ;
Balasubramanian:2019qwk . In this respect, recently in Refs. Frenkel:2020ysx ;
Hartnoll:2020rwq  the authors considered a deformation of a thermal CFT state
by a relevant scalar operator, which in the bulk results in a deformation of
the black hole singularity, at late interior times, into a more general Kasner
geometry from which it can be concluded that there is no Cauchy horizon as
supported by the SCC conjecture. The study has been soon extended to the self-
interacting $\phi^{4}$ scalar theory Wang:2020nkd  and holographic axionic
models with a neutral scalar hair Mansoori:2021wxf .

A similar situation happens for the asymptotically Anti-de Sitter (AdS)
solutions known as holographic superconductors in the presence of a massive
scalar and vector hair charged under a Maxwell field Hartnoll:2020fhc ;
Cai:2020wrp ; Henneaux:2022ijt ; Cai:2021obq . See also Refs. Sword:2021pfm ;
Liu:2021hap ; Devecioglu:2021xug ; VandeMoortel:2021gsp ; Auzzi:2022bfd ;
An:2022lvo  for further theoretical developments in the no inner-horizon
theorem. Moreover, it has been shown that the Null Energy Condition (NEC)
necessarily removes the Cauchy horizon of black holes in the presence of the
charged hairs Yang:2021civ ; An:2021plu ; Caceres:2022smh .

In addition, there is an interesting dynamics inside the horizon of such
holographic superconductor models Hartnoll:2020fhc . Just below the critical
temperature $T_{c}$ when the scalar field is very small, the interior evolves
through several dynamical epochs inside the event horizon, including a
collapse of the Einstein-Rosen bridge, the Josephson oscillations of the
scalar field, and the final Kasner singularity. Notice that the Kasner
geometry exponent is extremely sensitive to the black hole temperature near
the critical temperature so that at a discrete set of temperatures, an
infinite number of Kasner inversions is seen numerically Hartnoll:2020fhc .
The same behavior has been reported in an asymptotically flat hairy black hole
Dias:2021afz  and a holographic axionic model including Einstein-Maxwell-
scalar (EMS) coupling term Sword:2021pfm .

In this paper, we extend the no inner-horizon theorem to charged black holes
in the context of the non-linear massive gravity theory and study the interior
dynamics of the corresponding hairy black holes. In fact, here we build our
setup on a recent holographic model of non-linear massive gravity in which, by
giving a mass to the graviton in the bulk, the momentum is no longer conserved
at the boundary Vegh:2013sk . Indeed, the bulk graviton mass breaks the
diffeomorphism invariance of the gravitational action and therefore, within
the holographic framework, the stress-energy tensor of the dual field theory
is not conserved and in turn the momentum can be dissipated. One reason that
makes this model interesting is that the black brane solutions of this model,
in the absence of any scalar hair and even Maxwell tensor, have an inner
horizon due to the finite graviton mass. In addition to this fact, we expect
the extra bulk degrees of freedom produced by including massive gravity terms
introduce distinguishing features for the interior dynamics. The main goal of
the present work is to consider this issue.

The remainder of the paper is organized as follows. In Section 2, we first
introduce the gravitational model used in this work, a massive gravity theory
coupled to a charged scalar field $\phi$ in analogy with a holographic
superconductor. Then we investigate the condensation operator as a feature of
the black hole exterior beyond the event horizon. In Section 3, we discuss
dynamical epochs inside the horizon like the collapse of the Einstein-Rosen
bridge associated with the instability of the inner Cauchy horizon triggered
by the charged scalar hair. Sections 4 and 5 are devoted to study Josephson
oscillations epoch and the Kasner form near the spacelike singularity. In
Section 6, we investigate the interior dynamics of the massive gravity black
holes at the large massive parameter limit. Finally, we conclude with some
discussions in Section 7.

##  2 Model

We work here with a new class of non-linear massive gravity theories presented
for the first time in Vegh:2013sk  in which the ghost field is absent. In such
a theory, there is a special coupling between the metric tensor $g_{\mu\nu}$
and a fixed reference metric $f_{\mu\nu}$, giving a mass to $g_{\mu\nu}$ and
breaking diffeomorphism invariance. From the holographic point of view, the
violation of the diffeomorphism in the bulk gravity side results in the
dissipation of the momentum in the boundary field theory. By including a
negative cosmological constant and a minimally coupled Maxwell field
$A_{\mu}$, the gravitational action of such a massive gravity is given by

$S=\int dx^{4}\sqrt{-g}\Big{[}R+6-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\alpha{\rm
Tr}\mathcal{K}+\beta\Big{[}\Big{(}{\rm Tr}\mathcal{K}\Big{)}^{2}-{\rm
Tr}\mathcal{K}^{2}\Big{]}\Big{]}\,,$ (1)

where $\alpha$ and $\beta$ are arbitrary dimensionless constants111Here, we
have chosen units in such a way that the gravitational constant $16\pi G=1$
and the AdS radius $L$ is set to one.. In addition, the matrix $\mathcal{K}$
is defined as

$\mathcal{K}^{\mu}_{\rho}\mathcal{K}_{\nu}^{\rho}=g^{\mu\rho}\gamma_{\rho\nu},\hskip
14.22636pt\text{with}\hskip
14.22636pt\gamma_{\mu\nu}=\text{diag}\Big{(}0,0,\kappa,\kappa\Big{)},$ (2)

in which $\kappa$ is a constant. Note that all indices are raised and lowered
with the dynamical metric $g_{\mu\nu}$ in the coordinate ($t,r,x,y$) where $r$
is the holographic radial direction with the UV boundary at $r=0$. Clearly,
this choice of the reference metric explicitly breaks the bulk differmorphism
along the spatial directions $x$ and $y$. Correspondingly, the boundary theory
has no conserved momentum currents. The action (1) admits black brane
solutions corresponding to the following metric ansatz Vegh:2013sk

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{r^{2}}\Big{(}-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+dx^{2}+dy^{2}\Big{)},$
(3) A $\displaystyle=$
$\displaystyle\Phi(r)dt\,,\quad\Phi(r)=\rho\Big{(}r_{+}-r\Big{)},$ (4)
$\displaystyle f(r)$ $\displaystyle=$ $\displaystyle
1-\frac{r^{3}}{r_{+}^{3}}+\frac{\rho^{2}}{4}\Big{(}r^{4}-r^{3}r_{+}\Big{)}+\alpha\frac{\kappa}{2r_{+}}\Big{(}rr_{+}-\frac{r^{3}}{r_{+}}\Big{)}+\beta\kappa^{2}\Big{(}r^{2}-\frac{r^{3}}{r_{+}}\Big{)},$
(5)

where $\rho$ is the charge density. As long as
$\frac{r_{+}^{4}\rho^{2}}{4}-\kappa r_{+}\Big{(}\alpha+\beta\kappa
r_{+}\Big{)}\leq 3$ (due to the positiveness of the black hole temperature)
and $\rho>0$, a smooth Cauchy horizon exists in the interior of such a black
hole. Moreover, for stability of both the bulk and boundary theory, one
requires Vegh:2013sk ; Davison:2013jba

$M^{2}(r)=-\beta-\frac{\alpha}{2\kappa r}\geq 0,$ (6)

for all $r$. It means that at the IR horizon ($r_{+}$), we also have
$M^{2}(r_{+})\geq 0$. Now, we are interested in the black hole solutions that
can develop scalar hair through adding a particular coupling of a massive
charged scalar field via

$S^{\phi}=\int d^{4}x\sqrt{-g}\left[-g^{\mu\nu}\Big{(}\partial_{\mu}\phi-
iqA_{\mu}\phi\Big{)}\Big{(}\partial_{\nu}\phi-
iqA_{\nu}\phi\Big{)}-m^{2}\phi^{2}\right],$ (7)

to the Maxwell term of the action (1). According to the gauge/gravity duality,
the Maxwell field $A_{\mu}$ is dual to the current of a global $U(1)$ symmetry
in the boundary field theory. In addition, the mass square $m^{2}$ of the bulk
scalar field $\phi$ determines the scaling dimension
$\Delta=3/2+\sqrt{9/2+m^{2}}$ of the dual operator $O$ with charge $q$ under
this global symmetry222Note that the negative value of $m^{2}$ corresponds to
a relevant operator with $\Delta<3$ in the boundary field theory.
Hartnoll:2008vx . In order to find static black hole solutions in the new
setup, we take the following ansatz,

$ds^{2}=\frac{1}{r^{2}}\Big{(}-f(r)e^{-\chi(r)}dt^{2}+\frac{dr^{2}}{f(r)}+dx^{2}+dy^{2}\Big{)},\quad\phi=\phi(r)\,,\quad\textbf{A}=\Phi(r)dt\,,$
(8)

where, in our choice of coordinates, the AdS boundary is at $r=0$ and the
black hole singularity locates at $r\to\infty$. In addition, at the event
horizon $r_{+}$, the blackening function $f(r_{+})$ vanishes, while we require
$\Phi(r_{+})=0$ in order to get a finite norm of the gauge field at the
horizon. Moreover, in terms of the metric functions the temperature is defined
as

$T=\frac{1}{4\pi}|f^{\prime}(r_{+})|e^{-\chi(r_{+})/2}.$ (9)

From the actions (1) and (7), the independent equations of motion read

$\displaystyle
r^{2}e^{-\frac{\chi}{2}}\left(e^{\frac{\chi}{2}}\Phi^{\prime}\right)^{\prime}$
$\displaystyle=$ $\displaystyle\frac{2q^{2}\phi^{2}}{f}\Phi\,,$ (10)
$\displaystyle
r^{4}e^{\frac{\chi}{2}}\left(\frac{e^{-\frac{\chi}{2}}f\phi^{\prime}}{r^{2}}\right)^{\prime}$
$\displaystyle=$
$\displaystyle\left(m^{2}-\frac{q^{2}r^{2}e^{\chi}\Phi^{2}}{f}\right)\phi\,,$
(11) $\displaystyle\chi^{\prime}$ $\displaystyle=$
$\displaystyle\frac{rq^{2}e^{\chi}}{f^{2}}\phi^{2}\Phi^{2}+r(\phi^{\prime})^{2}\,,$
(12) $\displaystyle
4e^{\frac{\chi}{2}}r^{4}\Big{(}\frac{e^{-\frac{\chi}{2}}}{r^{3}}f\Big{)}^{\prime}$
$\displaystyle=$ $\displaystyle 2m^{2}\phi^{2}+r^{4}e^{\chi}\Phi^{\prime
2}-4\kappa r(\alpha+\beta\kappa r)-12\,,$ (13)

with a prime denoting the derivative with respect to $r$. In general, these
coupled differential equations do not allow analytical solutions, thus we
solve them numerically by integrating out from the horizon to the boundary at
$r=0$. It turns out that, by the regularity condition at the horizon, one
finds that the above equations are fully determined by three independent
parameters at the horizon, $\phi_{+}=\phi(r_{+})$, $\Phi_{+}=\Phi(r_{+})$, and
$\chi_{+}=\chi(r_{+})$ for given $r_{+}$, $\alpha$, $\beta$, $q$ and $m$. From
the regularity of the Euclidean on-shell action, the temperature (9) is
obtained in terms of these quantities as follows:

$T=\frac{e^{-\frac{\chi_{+}}{2}}}{16\pi
r_{+}}\Big{[}12+4\phi_{+}^{2}-e^{\chi_{+}}\Phi_{+}^{2}r_{+}^{4}+4\kappa
r_{+}\Big{(}\alpha+\beta\kappa r_{+}\Big{)}\Big{]}.$ (14)

On the other hand, near the AdS boundary $r\to 0$, the equations (10)–(13) are
solved by the following series expansions:

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle\phi_{0}\,r+\left\langle
O\right\rangle r^{2}+\dots\,,$ (15) $\displaystyle\Phi$ $\displaystyle=$
$\displaystyle\mu-\rho r+\dots\,,$ (16) $\displaystyle\chi$ $\displaystyle=$
$\displaystyle\frac{\phi_{0}^{2}}{2}\,r^{2}+\frac{4\phi_{0}\left\langle
O\right\rangle}{3}r^{3}+\dots\,,$ (17) $\displaystyle f$ $\displaystyle=$
$\displaystyle
1+\frac{\alpha\kappa}{2}r+\beta\kappa^{2}r^{2}+\phi_{0}^{2}r^{2}-\left\langle
T_{tt}\right\rangle r^{3}+\dots\,,$ (18)

where we have considered $m^{2}=-2$ to be concrete, and have taken the
normalization of the time coordinate of the metric at the boundary such that
$\chi(r=0)=0$ in order for the temperature of the black hole to be the
temperature of the boundary field theory. Here, $\phi_{0}$ is the source of
the scalar operator $O$ of the boundary field theory333The general asymptotic
behavior of the scalar field $\phi$ as $r\to 0$ is
$\phi=\phi_{0}r+\phi_{1}r^{2}.$ In the framework of holographic
superconductors, both of these falloffs are normalizable, so one can choose
boundary conditions in which either one vanishes. As a result, the condensate
of the scalar operator $O$ in the boundary field theory dual to the field
$\phi$ is given by Hartnoll:2008vx  $\left\langle
O\right\rangle=\begin{cases}\sqrt{2}\phi_{0}&\text{as}\ \phi_{1}=0,\\\
\sqrt{2}\phi_{1}&\text{as}\ \phi_{0}=0.\end{cases}$ Here we are interested in
working with the first branch of the above relation.. Moreover, $\mu$, $\rho$,
and $\left\langle T_{tt}\right\rangle$ denote chemical potential, charge
density and the energy density in the boundary field theory, respectively.
After imposing regularity at the horizon $r_{+}$, the boundary quantities
$\\{\rho,\mu,\left\langle O\right\rangle,\left\langle T_{tt}\right\rangle\\}$
can be written in terms of three independent parameters at the horizon, i.e.,
$\phi_{+}=\phi(r_{+})$, $\Phi_{+}=\Phi(r_{+})$, and $\chi_{+}=\chi(r_{+})$.
Nevertheless, the system allows the following scaling symmetry:

$r\rightarrow
r/\lambda,\quad(\phi_{0},T,\kappa,\mu)\rightarrow\lambda(\phi_{0},T,\kappa,\mu),\quad(\rho,\left\langle
O\right\rangle)\rightarrow\lambda^{2}(\rho,\left\langle
O\right\rangle),\quad\left\langle
T_{tt}\right\rangle\rightarrow\lambda^{3}\left\langle T_{tt}\right\rangle,$
(19)

with $\lambda$ a constant as we set $q=r_{+}=1$. However, to find the
solutions numerically, it is more convenient to use a different set of
dimensionless quantities such as $\langle O\rangle/\rho^{2}$, $T/\sqrt{\rho}$,
and $\kappa/\mu$.

Figure 1:  The value of the condensate as a function of the temperature in
various configurations of the massive gravity.

In Fig. 1, we have depicted the condensate for the charged scalar field as a
function of the temperature for various values of $\kappa/\mu=\\{0,1,2\\}$ in
different configurations of the massive gravity background. These plots have
been achieved by numerically solving the differential equations by using a
shooting method.

As shown in Fig. 1, in all cases, below a critical temperature $(T<T_{c})$, a
charged condensate forms. In other words, as temperature decreases below a
critical value, the system undergoes a phase transition from the normal phase
to the superconducting phase. Note that in the normal phase for $T>T_{c}$
where the charged scalar hair vanishes ($\phi=0$), the solution becomes (3).
It is clear that our setup also possesses a second order transition from the
normal state to the superconducting state close to the critical point, which
is a square root behavior predicted by the mean field theory, i.e., $\langle
O\rangle\propto T_{c}^{2}(T-T_{c})^{1/2}$. Moreover, for the configuration
$\beta=3\alpha=-0.6$ and $\alpha=0,\beta=-2$, the condensate increases as
$\kappa/\mu$ enhances, while for the case $\alpha=-2,\beta=0$ this behavior is
inverse. These results are in agreement with those reported in Zeng:2014uoa .

Recently, in Hartnoll:2020fhc  it has been shown that there is an interesting
dynamics inside the horizon of a holographic superconductor. Below the
critical temperature, the interior evolves through several distinct epochs,
including the collapse of the Einstein-Rosen bridge, Josephson oscillations of
the scalar field, and Kasner geometry. After investigating the condensate as a
feature of the black hole exterior, in the next sections we study the interior
of the black brane solution associated with our model in order to see whether
the same dynamical epochs are observed inside it.

.

##  3 Collapse of the Einstein-Rosen bridge

Just like in recent studies Hartnoll:2020fhc ; Hartnoll:2020rwq ;
Frenkel:2020ysx ; Dias:2021afz , we expect that the black hole interior ends
at a spacelike singularity at $r\to\infty$ after passing the collapse of the
Einstein-Rosen (ER) bridge and the Josephson oscillations epochs. Moreover,
one can observe a crossover that happens at the location of the would-be inner
horizon $r_{\mathcal{I}}$. Particularly, there is typically an inner horizon
$r_{\mathcal{I}}$ for the black hole solution (3) in the absence of the
charged scalar hair $\phi=0$. In practice, by increasing the black hole
temperature slightly below the critical temperature, the solution resembles
(3) until we go near the would-be inner Cauchy horizon. At this point, the
scalar field undergoes an instability very much. This instability is so fast
for small values of the scalar field, which indicates the nonlinear nature of
the dynamics in this regime. In addition, near the would-be inner horizon, the
$tt$ element of metric, i.e., $g_{tt}$ approaches its would-be zero value, it
suddenly suffers a very rapid collapse and becomes exponentially small. This
fact is associated with the collapse of the Einstein-Rosen bridge
Hartnoll:2020rwq . Therefore, in this section, we describe ER bridge for our
case of study.

During the ER collapse and oscillatory epochs, the mass of the scalar field in
Eqs. (11) and (13) and the charge term in the Maxwell equation (10) can be
dropped Hartnoll:2020fhc . By taking these approximations, the equations of
motion read

$\displaystyle\Phi^{\prime}$ $\displaystyle=$ $\displaystyle
E_{0}e^{-\chi/2}\,,$ (20) $\displaystyle
r^{2}e^{-\frac{\chi}{2}}\left(\frac{e^{-\frac{\chi}{2}}f\phi^{\prime}}{r^{2}}\right)^{\prime}$
$\displaystyle=$ $\displaystyle-\frac{q^{2}\Phi^{2}}{f}\phi\,,$ (21)
$\displaystyle\chi^{\prime}$ $\displaystyle=$
$\displaystyle\frac{rq^{2}e^{\chi}}{f^{2}}\phi^{2}\Phi^{2}+r(\phi^{\prime})^{2}\,,$
(22) $\displaystyle
4e^{\frac{\chi}{2}}r^{4}\Big{(}\frac{e^{-\frac{\chi}{2}}}{r^{3}}f\Big{)}^{\prime}$
$\displaystyle=$ $\displaystyle r^{4}E_{0}^{2}-4\kappa r(\alpha+\beta\kappa
r)-12\,,$ (23)

where $E_{0}$ is the constant electric field.

Figure 2:  Metric component $g_{tt}$ and scalar $\phi$ as a function of $r$
near the collapse of the ER bridge. The solid blue curves are the numerical
data and the red dotted curves are the fits to the relations (24) and (25).
Here we take $T/T_{c}=0.995$ and $\kappa/\mu=1$. The dashed vertical lines
show the location of the would-be inner horizon $r_{\mathcal{I}}$.

Following a methodology similar to Hartnoll:2020fhc ; Hartnoll:2020rwq , just
below the critical temperature $T_{c}$ when the scalar field is very small,
the instability is so fast that we can essentially keep the $r$ coordinate
fixed. Let us set $r=r_{\mathcal{I}}+\delta r$, so that $f$, $\chi$, $\Phi$
and $\phi$ are now functions of $\delta r$, while any explicit factors of $r$
in Eqs. (11)–(13) are set to $r_{\mathcal{I}}$. In this limit that $r_{I}$ is
close to the inner horizon of the black hole solution (3), the potential
function $\Phi$ is large compared to its derivative. Thus, in Eqs. (21) and
(22) we can set $\Phi=\Phi_{0}$, in which $\Phi_{0}$ is a constant. Using this
approximation, Eq. (21) can be solved explicitly yielding

$\phi(r)=\phi_{0}\cos\left(q\Phi_{0}\int_{r_{\cal{I}}}^{r}dr\frac{e^{\chi(r)/2}}{f(r)}+\varphi_{0}\right),$
(24)

in which $\phi_{0}$ and $\varphi_{0}$ are two integration constants.
Substituting this relation into Eq. (22), we arrive at
$q^{2}\phi^{2}\Phi_{0}^{2}+e^{-\chi}f^{2}\left(\phi^{\prime}\right)^{2}=q^{2}\phi_{0}^{2}\Phi_{0}^{2}$.
After some algebraic manipulations, we find that the metric component
$g_{tt}=-fe^{-\chi}/r_{\mathcal{I}}^{2}$ must obey Hartnoll:2020rwq

$c_{1}^{2}\log(g_{tt})+g_{tt}=-\,c_{2}^{2}(r-c_{3})\,,\quad\text{with}\quad
c_{1}^{2}=\frac{2q^{2}\phi_{0}^{2}\Phi_{0}^{2}}{r_{\mathcal{I}}^{4}E_{0}^{2}-4\kappa
r_{\mathcal{I}}\Big{(}\alpha+\beta\kappa r_{\mathcal{I}}\Big{)}-12},$ (25)

and $c_{2}>0$ and $c_{3}$ are further constants of integration. Close to the
would-be inner horizon, for $r<c_{3}$, $g_{tt}\propto(c_{3}-r)$ vanishes
linearly disappearing towards $r_{\mathcal{I}}$, while for $r>c_{3}$ one
observes a rapid collapse of $g_{tt}$ to an exponentially small value, i.e.,
$g_{tt}\propto e^{-(c_{2}/c_{1})^{2}(r-c_{3})}$. Given $g_{tt}$, one can now
obtain the solution for $\chi$ and $f$ as follows:

$e^{-\chi}=\frac{c_{2}^{2}g_{tt}^{2}}{\left(c_{1}^{2}+g_{tt}\right)^{2}}\frac{r_{\mathcal{I}}^{3}}{E_{0}^{2}r_{\mathcal{I}}^{4}-4\kappa
r_{\mathcal{I}}\Big{(}\alpha+\beta\kappa r_{\mathcal{I}}\Big{)}-12}\,,\qquad
f=-r_{\mathcal{I}}^{2}e^{\chi}g_{tt}\,.$ (26)

For small values of the scalar field at the horizon, i.e., as $T\to T_{c}$,
these solutions agree well with the full numerical evolution as shown in Fig.
2. As pointed by Hartnoll:2020rwq , the value of $c_{1}/c_{2}$ will be large
as $T\to T_{c}$. We check numerically in Fig. 3 that indeed the ratio
$c_{1}/c_{2}\approx a(1-T/T_{c})^{1/2}+b$ near the critical temperature.
Evidently, the slope of straight lines increases by growing the massive
parameter $\kappa/\mu$. In the limit $T\to T_{c}$, $(c_{2}/c_{1})^{2}\delta r$
in (25) can be very large, which in turn allows the metric component $g_{tt}$
to undergo a sudden change in the vicinity of $r_{\mathcal{I}}$. From the
straight lines depicted in Fig. 3 it can be seen that the slope of these
straight lines increases by enhancing the $\kappa/\mu$ values. It means that
the collapse of the metric element $g_{tt}$ becomes faster at smaller values
of $\kappa/\mu$ parameter. This is illustrated in Fig. 4.

Figure 3:  The variation of $c_{1}/c_{2}$ as a function of the temperature in
various configurations of the massive gravity. The fitted straight lines for
the data are determined by $c_{1}/c_{2}=a(1-T/T_{c})^{1/2}+b$.

We present the collapse of the ER bridge for various values of $\kappa/\mu$ at
$T/T_{c}=0.995$ for different configurations of the massive gravity in Fig. 4.
As can be seen, in the pure non-linear massive gravity configuration the
collapse is shifted outward to a large radial coordinate $r$ by increasing
$\kappa/\mu$ parameter, while for the other configurations, the ER collapse
occurs approximately at the same would-be inner horizon with different slopes.
Therefore, it makes sense that at this temperature the ER collapse could be
completely disappeared at the large massive parameter $\kappa/\mu$ in the pure
non-linear massive configuration with $\alpha\neq 0$ and $\beta=0$. We deal
with this point in Section (6).

Figure 4:  Metric component $g_{tt}$ as a function of $r$ near the collapse of
the ER bridge for various values of $\kappa/\mu$. Here we take
$T/T_{c}=0.995$.

##  4 Josephson oscillation epoch

Since inside the horizon, $r$ is a timelike coordinate while $t$ is spacelike,
the cosine form of the scalar function in (24) can be written as $q\int
A_{\hat{t}}\,d\tau$, where $d\tau=\sqrt{g_{rr}}dr$ is the proper time and
$A_{\hat{t}}=A_{t}/\sqrt{g_{tt}}$ is the vector potential in locally flat
coordinates. Therefore, a nonzero $A_{\hat{t}}$ determines a phase winding in
the $t$ direction. In this respect, the scalar condensate $\phi$ indicates the
superfluid stiffness. As a result, Eq. (24) determines the oscillations in
time of the superfluid stiffness sourced by a background phase winding. This
phenomenon is dubbed as the Josephson effect. By the end of the collapse of
the ER bridge epoch, these Josephson oscillations can be dominated near the
critical temperature Hartnoll:2020fhc .

Just after the end of the collapse of the ER bridge described in the previous
section, the derivative of the Maxwell field $\Phi^{\prime}\propto
e^{-\chi/2}$ is still very small which leads us to ignore the
$(E_{0}^{2}-\frac{4}{r^{3}}\kappa(\alpha+\beta\kappa
r)-\frac{12}{r^{4}})e^{-\frac{\chi}{2}}$ contribution in (23). It follows that
the Maxwell field $\Phi$ and the gravitational field $f$ are given by

$\Phi\simeq\Phi_{o}\,,\qquad\frac{fe^{-\chi/2}}{r^{3}}\simeq-\frac{1}{d_{1}}\,,$
(27)

where $d_{1}$ is constant. To fix it, one needs to match the Josephson
oscillation solution (27) with the $r>c_{3}$ solution (26) of the collapse of
the ER bridge. We thus obtain

$d_{1}\simeq\frac{c_{2}}{c_{1}^{2}}\sqrt{\frac{r_{\mathcal{I}}^{5}}{E_{0}^{2}r_{\mathcal{I}}^{4}-4\kappa
r_{\mathcal{I}}\Big{(}\alpha+\beta\kappa r_{\mathcal{I}}\Big{)}-12}}\,.$ (28)

Because $g_{tt}$ is very small near the inner horizon, here we have used the
appropriate approximation $c_{1}^{2}+g_{tt}^{2}(r_{\mathcal{I}})\simeq
c_{1}^{2}$. The above relation confirms that $d_{1}$ goes to large values as
$T\to T_{c}$.

By the use of (27) Eqs. (21) and (23) can be solved in terms of Bessel
functions. Thus, the scalar field solution is

$\phi\simeq
d_{2}J_{0}\left(\frac{|q\Phi_{o}|d_{1}}{2r^{2}}\right)+d_{3}Y_{0}\left(\frac{|q\Phi_{o}|d_{1}}{2r^{2}}\right),$
(29)

in which $d_{2}$ and $d_{3}$ are integration constants. In Fig. 5, we compare
the analytical approximation (29) (blue dashed lines) with the numerical data
for the scalar field (solid lines) for $T/T_{c}=0.995$ and $\kappa/\mu=1$ in
different configurations of massive gravity. Clearly, the analytical
expression (29) of the Josephson oscillations in the limit $T\to T_{c}$ is in
excellent agreement to the numerical solution of the full Eqs. (10)–(12).

Figure 5:  A comparison of the numerical solution (solid blue curves) of the
scalar hair $\phi$ and fits to the analytic relation (29)(red dashed curves)
describing the Josephson oscillations. Here $T/T_{c}=0.995$ and
$\kappa/\mu=1$.

It is worth mentioning that the oscillations of the scalar field start in the
collapse of the ER bridge regime and propagate continuously onto the Josephson
oscillation epoch. Therefore, these constants can be found through matching
the Josephson solution (29) and its derivative for the small $r$ with the ER
bridge collapse solution (24) and its derivative at the large $r$ behavior in
the overlapping region at $r_{\mathcal{I}}$. In this way, one finds

$\displaystyle
d_{2}\simeq\left(\frac{r_{\mathcal{I}}\pi^{2}c_{2}^{2}}{8}\,\frac{\phi_{o}^{2}}{c_{1}^{2}}\right)^{1/4}\sin\left(\frac{c_{2}\sqrt{r_{\mathcal{I}}}}{\sqrt{2}}\,\frac{1}{\phi_{o}c_{1}}-\varphi_{o}+\frac{\pi}{4}\right),$
(30a) $\displaystyle
d_{3}\simeq\left(\frac{r_{\mathcal{I}}\pi^{2}c_{2}^{2}}{8}\,\frac{\phi_{o}^{2}}{c_{1}^{2}}\right)^{1/4}\sin\left(\frac{c_{2}\sqrt{r_{\mathcal{I}}}}{\sqrt{2}}\,\frac{1}{\phi_{o}c_{1}}-\varphi_{o}-\frac{\pi}{4}\right).$
(30b)

Interestingly, the relation (30) confirms that $d_{3}$ is strongly oscillating
with a constant amplitude as $T\to T_{c}$, i.e.

$d_{3}=A\sin\Big{(}\frac{B}{1-\frac{T}{T_{c}}}+C\Big{)}.$ (31)

As illustrated in Fig. 6, this behavior agrees well with numerics over many
oscillations. As an example, for the configuration of the massive gravity with
$\beta=3\alpha=-0.6$, we find that the red dashed curve describing (31) with
$A=1.54553$, $B=0.331731$ and $C=1.3231$ which is in excellent agreement with
the numerical data (blue solid line) over many oscillations as $T\to T_{c}$.

Figure 6:  The oscillatory behavior of $c=-\sqrt{8}d_{3}/\pi$ determining the
Kasner exponents as $T\to T_{c}$. The solid blue lines are the numerical data
and the red dashed curves are fits to the relation (31). From top to bottom,
panels are associated with the massive configuration $\beta=3\alpha=-0.2$,
$\beta=0,\alpha=-2$ and $\beta=-2,\alpha=0$, respectively. Here we have taken
$T/T_{c}=0.995$ and $\kappa/\mu=1$.

Near the singularity, the large $r$ behavior of the scalar field (29) is
important. This yields

$\phi|_{\hbox{\tiny
large}\,r}\simeq\frac{2d_{3}}{\pi}\log\left(d_{1}\frac{qe^{\gamma_{E}}\Phi_{o}}{4r^{2}}\right)+d_{2}+\cdots\,,$
(32)

with $\gamma_{E}$ being the Euler-Mascheroni constant. This logarithmic
behavior determines the onset of a Kasner regime which is discussed in the
next section.

By substituting (27) into (22), one arrives at
$\chi^{\prime}=4\left(r{\phi^{\prime}}^{2}+r^{-5}c_{3}^{2}q^{2}\Phi_{o}^{2}\phi^{2}\right)$.
This allows us to obtain $\chi$. By plugging this solution into (27), we
finally obtain the following expression for the metric function $f$:

$\displaystyle f\simeq-
f_{o}r^{3}\exp\left[\frac{1}{2}\int_{r_{\mathcal{I}}}^{r}\left(\tilde{r}\phi^{\prime\,2}+\frac{q^{2}\Phi_{o}^{2}c_{3}^{2}\phi^{2}}{\tilde{r}^{5}}\right)d\tilde{r}\right]\,,$
(33)

with $f_{o}$ being a constant. If we insert the Bessel functions solution (29)
into the above integral, it can be performed analytically in terms of Bessel
functions. These describe the small oscillations seen in $rf^{\prime}/f$ as
shown in Fig. 8.

Figure 7:  A comparison of the numerical solution (solid blue curves) of
$rf^{\prime}/f$ and fits to the analytic relation (33)(red dashed curves)
describing Josephson oscillations. Here we have chosen $T/T_{c}=0.995$ and
$\kappa/\mu=1$.

Before leaving this section, let us examine the influence of the massive
parameter $\kappa/\mu$ on the Josephson oscillation amplitude. In Fig. 8, we
plot numerically the evolution of $\phi$ and $rf^{\prime}/f$ with respect to
$r$ for different configurations of the massive gravity and $T/T_{c}=0.995$.
As can be seen, varying $\kappa/\mu$ can significantly divert the system
behavior from the standard holographic superconductor case (Hartnoll:2020fhc,
). In fact, one can find that the Josephson oscillations can be even
disappeared by the growth of the massive parameter $\kappa/\mu$ (see section
(6)).

Figure 8:  The dynamical evolution of $\phi$ and $rf^{\prime}/f$ as one
changes the amount of the massive parameter $\kappa/\mu$. These graphs are at
$T/T_{c}=0.995$.

Summing up the results, it can be concluded that the growth of the massive
parameter $\kappa/\mu$ assists to decrease the number of the Josephson
oscillations epoch. Note that at large $\kappa/\mu$ limit, both ER collapse
and Josephson oscillations epochs can be vanished (see Section (6)).
Interestingly, on the other hands, similar to the standard holographic
superconductor case Hartnoll:2020fhc , at temperatures which are far from
$T_{c}$, the collapse of the Einstein-Rosen bridge and subsequent Josephson
oscillations become less dramatic. For an example in Fig. 9, the number of
Josephson oscillations tends to decrease at smaller temperatures.

Figure 9:  The behavior of $r\phi^{\prime}$ and $-rg^{\prime}_{tt}/g_{tt}$
versus $r$ for the temperatures $T/T_{c}=0.965$ (solid), $0.945$ (dashed),
$0.895$ (dot-dashed) and $0.845$ (dotted). One can see fewer Josephson
oscillations and the jump in the derivative of the metric element as the
temperature is lowered. Here we take $\kappa/\mu=1$.

##  5 Kasner cosmology epoch

At the end of the Josephson oscillations epoch, the scalar field $\phi$ grows
logarithmically (32). It means that the system moves to the Kasner epoch. In
this section, we are interested in finding what happens as the system evolves
further inside the Kasner cosmology epoch. Nevertheless, it might be possible
that after the Josephson oscillations the solution enters a new Kasner epoch
as “Kasner inversion” observed in Hartnoll:2020fhc .

Numerically, it is found that once the solution enters the Kasner epoch, the
massive configuration coefficients $\alpha$ and $\beta$, mass and charge terms
in the equations of motion can be dropped. In this respect, the solutions at
large $r$ take the following form:

$\phi=c\sqrt{2}\log r+\dots,\hskip 14.22636pt\chi=2c^{2}\log r+\dots,\hskip
14.22636ptf=-f_{1}r^{3+2c^{2}}+\dots,$ (34)

where $f_{1}$ is constant and $c=-\sqrt{8}d_{3}/\pi$ according to (32). In
addition, using Eq. (10), the Maxwell potential is

$\Phi=\Phi_{K}+E_{K}\int
dre^{\frac{-\chi}{2}}\approx\Phi_{K}+E_{K}r^{1-c^{2}}+\dots.$ (35)

It is obvious that for $|c|>1$, the Maxwell field remains unimportant at large
$r$, similar to that seen previously for a neutral scalar hair
Hartnoll:2020rwq . In this limit, the geometry near the singularity takes a
Kasner form

$ds^{2}\sim-d\tau^{2}+\tau^{2p_{t}}dt^{2}+\tau^{2p_{x}}\Big{(}dx^{2}+dy^{2}\Big{)}\,,\hskip
14.22636pt\phi\sim-p_{\phi}\log\tau\,,$ (36)

where we have changed the $r$ coordinate to the proper time $\tau$ via
$d\tau=\frac{dr}{r\sqrt{f}}$, and the Kasner exponents are obtained to be

$p_{x}=\frac{2}{3+c^{2}},\hskip 14.22636ptp_{t}=\frac{c^{2}-1}{3+c^{2}},\hskip
14.22636ptp_{\phi}=\frac{2\sqrt{2}c}{3+c^{2}}\,,$ (37)

which obey $p_{t}+2p_{x}=1$ and $p_{\phi}^{2}+p_{t}^{2}+2p_{x}^{2}=1$.

In Fig. 10, an interior evolution is illustrated that does not deviate from
the Kasner cosmology (34), because the intermediate Kasner exponent is already
positive, i.e. $p_{t}>0$. In other words, we observe that when $|c|>1$ or
($p_{t}>0$) at the beginning of the Kasner epoch, the system remains described
by the Kasner geometry (34) with the exponents (37) during a journey to the
actual singularity as $r\to\infty$.

Figure 10:  Typical examples of the interior evolution with no Kasner
inversion in different configurations of the massive gravity. In $1<r<10$
(orange shaded area) the system experiences the ER bridge collapse and
Josephson oscillations epoch, then it stays in Kasner regime for ever.
Clearly, in all panels, the function $\log(g_{tt})$ begins to collapse, while
the other functions $r\phi^{\prime}$ and $-rg^{\prime}_{rr}/g_{tt}$ become
constants in the Kasner epoch. In all these graphs $\kappa/\mu=1$.

On the other hand, when $|c|<1$, the growing Maxwell field (35) results in a
transition to a different Kasner solution at $r_{inv}$ with new exponents.
This is called the Kasner inversion found previously in Hartnoll:2020fhc . As
proved analytically in Hartnoll:2020fhc , near $r_{inv}$, the Maxwell field
growth (35) triggers a transition to a new Kasner regime with exponent $c\to
c_{new}=1/c$, or equivalently $p_{t}\to p_{t}^{new}=-p_{t}/(2p_{t}+1)$. It
should be noted that the effect of the massive gravity contribution is
negligible in the Kasner epoch (we refer to this point at the end of this
section), and the above results are still satisfied numerically for our model.
For the resulting plots, see Figs. 11 and 12.

As follows from all panels shown in Fig. 12, at a value of $r$ near the inner
horizon of the non-hair solution of the massive gravity (3),
$rg^{\prime}_{tt}/g_{tt}$ (black curve) experiences a large kick at which
$g_{tt}$ (red curve) becomes very small. This is the collapse of the Einstein-
Rosen bridge. Subsequently, a series of Josephson oscillations in the scalar
field $\phi$ and corresponding imprint in a series of short steps on the
metric derivative $rg^{\prime}_{tt}/g_{tt}$ (or equivalently $rf^{\prime}/f$)
are observed. These all occur at relatively small $r/r_{+}$ (orange shaded
area). Immediately afterwards, the oscillations settle down to an intermediate
Kasner regime with the Kasner exponent $p_{int}$. For an exponentially long
range of $r/r_{+}$, this Kasner epoch remains just before another kick
(whenever $p^{int}_{t}<0$) happens in $g_{tt}$. After that the system enters
the final Kasner epoch with positive exponent
$p_{t}=-p^{int}_{t}/(2p^{int}_{t}+1)>0$ (blue shaded area). Because $g_{tt}$
is a power law function, $-rg^{\prime}_{tt}/g_{tt}$ is constant in both sides
of the Kasner inversion location.

To have concrete evidence of the Kasner inversion, we have also plotted the
dynamical evolution of $r\phi^{\prime}$ for a certain range of $c$ in Fig. 12.
As an example, in the massive gravity configuration with $3\alpha=\beta=0.6$,
before inversion $c_{\text{before}}=0.1914$, after inversion
$c_{\text{after}}=5.225$ which is in agreement with the inversion limit
formula, with the error $|c_{\text{after}}-1/c_{\text{before}}|=4.451\times
10^{-9}$.

Figure 11:  An example of an interior evolution with Kasner inversion in
different configurations of the massive gravity. The system evolves from the
outer event horizon at $r_{+}=1$ towards the final singularity at
$r\to\infty$. In the range $1<r<10$ (orange shaded regime) the system goes
quickly through the ER bridge collapse and Josephson oscillations epoch to the
intermediate Kasner regime (white area) and stays relatively steady for an
exponentially long range of $r/r_{+}$ with $|c|>1$ (or $p_{t}^{int}<0$), and
finally there is a Kasner inversion (blue shaded area) with $|c_{new}|>1$ (or
$p_{t}=-p^{int}_{t}/(2p^{int}_{t}+1)>0$). Here $\kappa/\mu=1$ in all panels.

Figure 12:  The value of $r\phi^{\prime}$ near the kasner inversion from $c$
to $c_{new}=1/c$ at $r_{inv}$ which is the location of the transition. Form
the left hand side and top to bottom, panels are drawn for the Kasner
inversion in the range $c\approx 0.1664-0.1914$, $c\approx 0.5792-0.6163$ and
$c\approx 0.4336-0.4653$, respectively.

Figure 13:  The Kasner exponent $p_{t}$ after inversion as a function of the
temperature in various configuration of the massive gravity. Notice that the
accumulation of the oscillations near the critical temperature $T_{c}$ is
described by the analytic expression (31) for $d_{3}$ which is plotted
numerically in Fig. 6. They are for $\kappa/\mu=1$.

We also present the value of the Kasner exponent $p_{t}$ after the inversion
for all temperatures up to $T_{c}$ in Fig. 13. Obviously, close to the
critical temperature $T_{c}$, there are strong oscillations. This is due to an
accumulation of the oscillations in the scalar field just beyond the inner
horizon of the non-hair black hole solution. More precisely, the analytic
result (31) indicates the oscillatory Kasner exponent after a first collapse
of the Einstein- Rosen bridge.

Before ending this section, let us discuss whether or not the massive gravity
contributions can violate the Kasner form (36) near the spacelike singularity
at $r\to\infty$. Generally, a $(d+2)$-dimensional massive gravity is given by
the following action cai2015thermodynamics .

$S_{\text{massive gravity}}=\int
d^{d+2}x\sqrt{-g}\sum_{i=1}^{4}n_{i}U_{i}(g,\gamma),$ (38)

where $n_{i}$ are dimensionless constants444For stability of the theory, all
those coefficients might be required to be negative. and $U_{i}$ are symmetric
polynomials of the eigenvalues of the matrix $\mathcal{K}$ which satisfy the
following relations.

$\displaystyle U_{1}$ $\displaystyle=$ $\displaystyle{\rm Tr}\mathcal{K},$
$\displaystyle U_{2}$ $\displaystyle=$ $\displaystyle\Big{(}{\rm
Tr}\mathcal{K}\Big{)}^{2}-{\rm Tr}\mathcal{K}^{2},$ (39) $\displaystyle U_{3}$
$\displaystyle=$ $\displaystyle\Big{(}{\rm Tr}\mathcal{K}\Big{)}^{3}-3{\rm
Tr}\mathcal{K}{\rm Tr}\mathcal{K}^{2}+2{\rm Tr}\mathcal{K}^{3},$
$\displaystyle U_{4}$ $\displaystyle=$ $\displaystyle\Big{(}{\rm
Tr}\mathcal{K}\Big{)}^{4}-6{\rm Tr}\mathcal{K}^{2}\Big{(}{\rm
Tr}\mathcal{K}\Big{)}^{2}+8{\rm Tr}\mathcal{K}^{3}{\rm
Tr}\mathcal{K}+3\Big{(}{\rm Tr}\mathcal{K}^{2}\Big{)}^{2}-6{\rm
Tr}\mathcal{K}^{4}.$

Now by taking the general reference metric
$f_{\mu\nu}=\text{diag}\Big{(}0,0,\kappa^{2}\delta_{ij}\Big{)}$ in $d+2$
dimensions, one obtains

$\displaystyle U_{1}$ $\displaystyle=$ $\displaystyle d\kappa r,$
$\displaystyle U_{2}$ $\displaystyle=$ $\displaystyle d(d-1)\kappa^{2}r^{2},$
(40) $\displaystyle U_{3}$ $\displaystyle=$ $\displaystyle
d(d-1)(d-2)\kappa^{3}r^{3},$ $\displaystyle U_{4}$ $\displaystyle=$
$\displaystyle d(d-1)(d-2)(d-3)\kappa^{4}r^{4}.$

It is clear that in the 4-dimensional case with $d=2$, we have identically
$U_{3}=U_{4}=0$ as seen in (1), whereas in the 5-dimensional case with $d=3$,
one takes $U_{4}=0$ and so on.

In order to get the Kasner solutions (34), we require the kinetic term of the
scalar field to be dominant with respect to the massive gravity contributions
near the singularity. It means that

$\lim_{r\to\infty}\frac{|\sum_{i=1}^{4}U_{i}|}{r^{d+1+\alpha^{2}}}\ll 1.$ (41)

Clearly, this condition is never violated in any dimensions. Therefore, once
the solution enters the Kasner epoch, the massive gravity contributions are
not important.

##  6 Dynamical epochs at the large $\kappa/\mu$ limit

In this section, let us focus our attention on the interior dynamics in the
presence of large amounts of the massive gravity parameter. As mentioned, the
quantity $c_{1}/c_{2}$ determines the intensity of the occurrence of the ER
collapse at the would-be inner horizon. As shown in Fig. 3, the ratio
$c_{1}/c_{2}$ is described numerically by the straight line
$a(1-T/T_{c})^{1/2}+b$ in which its slope $a$ is growing by increasing
$\kappa/\mu$. Therefore, we have depicted the behavior of the slope $a$ for
large amounts of the massive parameter $\kappa/\mu$ in Fig. 14. As illustrated
in Fig. 14, for massive configurations $\alpha=0$, $\beta\neq 0$ and
$\alpha,\beta\neq 0$ one observes the slope $a$ shifts to a constant at large
$\kappa/\mu$ values, whereas it continues to grow even for so large amounts of
$\kappa/\mu$ for the pure non-linear massive configuration $\alpha\neq 0$ and
$\beta=0$. Notice that the slope values presented here have been read for all
range of temperature as presented in Fig. 3. In other words, the slope $a$
depends only on the massive parameter, i.e. $a=a(\kappa/\mu)$.

This finding confirms that the amount of $c_{1}/c_{2}\propto
a(\kappa/\mu)(1-T/T_{c})^{1/2}$ is affected by varying both the massive
parameter and the temperature. In comparison, in the standard holographic
model Hartnoll:2020fhc , the slope $a$ will be constant and $c_{1}/c_{2}$
varies only by temperature.

As a consequence of this finding, for a sufficiently large massive parameter,
by fine tuning the temperature, one could get the largest amount for
$c_{1}/c_{2}$ in all massive configurations.

Figure 14:  The slope of $c_{1}/c_{2}$ with respect to $\kappa/\mu$.

Hence, in this sense, we expect that the time component of the metric
$g_{tt}$, which in general cases is exponentially vanishing after
$r_{\mathcal{I}}$ according to the discussion bellow Eq. (25), undergoes a
linear function,

$g_{tt}\propto
e^{-(c_{2}/c_{1})^{2}(r-c_{3})}\approx\Big{(}\frac{c_{2}}{c_{1}}\Big{)}^{2}(c_{3}-r)+\dots.$
(42)

both sides of the would-be inner horizon. Namely, the ER collapse disappears
from intrior dynamics epoches 555A similar result has been reported in
Sword:2021pfm  for the holographic superconductor model including an axion
field term and an Einstein-Maxwell-scalar (EMS) coupling term. At the large
EMS coupling parameter, the ER collapse does not occur near the would-be inner
horizon and is shifted outward to a large radial value. In fact, there is a
delay in the collapse of the ER and it is not completely removed from the
intrior dynamical epochs.. In fact, unlike the standard holographic
superconductor Hartnoll:2020rwq , $g_{tt}$ does not exponentially collapse
after the would-be inner horizon, while one sees a smooth crossing at this
horizon.

As a concrete example, in Fig. 15 we have depicted $g_{tt}$ behavior for large
$\kappa/\mu=\\{40,60,80\\}$ at $T/T_{c}=0.995$. As can be seen, in the pure
non-linear configuration, the ER bridge collapse is completely removed near
the would-be inner horizon (the vertical dashed line), whereas this collapse
still happens and is not sensitive to the change of the massive parameter in
the other configurations. In addition, by keeping the massive gravity
parameter fixed, we examine the impact of the temperature on the ER bridge
collapse. It can be seen in Fig. 16 that the ER bridge collapse can be
vanished at a given temperature in all massive configurations. For an example,
in pure non-linear massive configuration, the accuracy of Eq. (42) breaks down
for so small values of the scalar field at the horizon, i.e., as $T\to T_{c}$.

Figure 15:  The metric component $g_{tt}$ as a function of $r$ near the would-
be inner horizon (the vertical dashed lines) for large values of $\kappa/\mu$.
Here we take $T/T_{c}=0.995$.

Figure 16:  The metric component $g_{tt}$ as a function of $r$ near the would-
be inner horizon (the vertical dashed line) for different temperature values.
Here we take $\kappa/\mu=40$.

Figure 17:  Left: The evolution of $\phi$ for large amounts of the massive
parameter $\kappa/\mu$ in the pure non-linear massive configuration case.
Right: A comparison of the numerical solution (solid blue curve) of $\phi$ for
$\kappa/\mu=40$ in the pure non-linear massive configuration case and fits to
the relation (24) (green dashed curve) describing ER collapse before would-be
inner horizon and analytic relation (44) (red dashed curve) after the would-be
inner horizon. Here we have chosen $T/T_{c}=0.995$.

On the other hand, what happened in the ER collapse affects directly the
subsequent Josephson oscillations epoch. As illustrated in Fig. 17, the
Josephson oscillation epoch is completely disappeared for large amounts of
$\kappa/\mu$ in the pure non-linear massive configuration case. Now we attempt
to find the analytic relation describing the dynamics of $\phi$ around the
would-be inner horizon. From Eqs. (28) and (25), one immediately obtains the
below relation for the argument inside the Bessel functions in (29).

$\frac{|q\Phi_{0}|d_{1}}{2r^{2}}\approx\frac{\sqrt{2}r_{\mathcal{I}}^{5/2}}{2\phi_{0}r^{2}}\Big{(}\frac{c_{2}}{c_{1}}\Big{)}.$
(43)

It is obvious that this relation is proportional to the ratio $c_{2}/c_{1}$.
Therefore, it is expected to be very small for large values of the massive
parameter in the pure non-linear massive gravity. We have checked numerically
this function at the would-be inner horizon for large massive gravity
parameter in Fig. 18. Clearly, the amount of this expression is very small at
large $\kappa/\mu$ values. Note that after the would-be inner horizon
($r>r_{\mathcal{I}}$), its values becomes smaller.

Figure 18:  The behavior of the relation (43) with respect to $\kappa/\mu$.

As a result of the above discussion, the expression for the Josephson
oscillations in Eq. (29) for $\phi$ at the large massive parameter is written
approximately as

$\phi|_{r\sim
r_{\mathcal{I}}}\simeq\frac{2d_{3}}{\pi}\log\Big{(}\frac{c_{2}e^{\gamma_{E}}}{2\sqrt{2}c_{1}\phi_{0}}\frac{r_{\mathcal{I}}^{5/2}}{r^{2}}\Big{)}+d_{2}+...\simeq-\frac{4d_{3}}{\pi}\log(r)+s_{1}.$
(44)

The constant $d_{3}$ can be found through matching the above solution and its
derivative for the small $r$ with the ER bridge collapse solution (24) and its
derivative in the overlapping region at $r_{\mathcal{I}}$. Thus one finds

$d_{3}\approx\Big{(}\frac{c_{2}}{c_{1}}\Big{)}\sqrt{\frac{r_{\mathcal{I}}}{2}}\frac{\pi}{2}\frac{g^{\prime}_{tt}(r_{\mathcal{I}})}{g_{tt}(r_{\mathcal{I}})}\sin(\varphi_{0})=-\frac{\pi}{4}r_{\mathcal{I}}\phi^{\prime}(r_{\mathcal{I}}),$
(45)

where, we have used Eq. (24) in the last expression. As illustrated in the
right hand side of Fig. 17, the analytic relation (44) agrees well with the
numerics. More precisely, we find that the red dashed curve describes Eq. (44)
with $d_{3}=-1.51161$ and $s_{1}=-1.30245$ which is in excellent agreement
with the numerical data from Eq. (45). Interestingly, the logarithmic form of
$\phi$ implies that the system moves to Kasner cosmology epoch exactly after
the would-be inner horizon without experiencing neither the ER bridge collapse
nor the Josephson oscillation epoch as presented in Fig. 19.

Figure 19:  An example of an interior evolution with no ER bridge collapse and
Josephson oscillations epochs in the pure non-linear massive configuration.
Clearly, the system enters the Kasner geometry immediately after the would-be
inner horizon. Numerics are at $T/T_{c}=0.995$ and $\kappa/\mu=40,~{}60,~{}80$
from top to bottom.

In addition, using Eqs. (45) and (34), one can find

$c=-\frac{\sqrt{8}}{\pi}d_{3},\hskip 14.22636pt\text{or}\hskip
14.22636ptp_{t}=\frac{8d_{3}^{2}-\pi^{2}}{8d_{3}^{2}+3\pi^{2}}.$ (46)

It means that the Kasner exponent value is determined from the amount of the
derivative of the scalar field $\phi$ at the would-be inner horizon. For an
example, from Fig. 19, one finds $c=1.3603294$ which is in agreement with the
fitted graph in Fig. 17 with $d_{3}=-1.51161$ (or equivalently $c=1.36093$).
Notice that since the constant $c$ is larger than one, the Kasner inversion
does not happen in such cases. Note that at temperatures very close to the
critical temperature, all epochs can be observed in the pure non-linear
massive configuration (see Fig. 20 where we have shown a case with a Kasner
inversion).

Interestingly, as shown in Fig. 21, the relation (44) justifies for other
massive gravity configurations at small values of the temperature.

Figure 20:  Journey through the inside of the pure non-linear massive gravity
black hole.

Figure 21: Examples of the interior evolution. Black curves show the cases
with no Josephson oscillations epochs.

##  7 Conclusions

The purpose of the current study was to investigate the interior of the black
hole solution of the massive gravity in the presence of the charged scalar
hair. In this regard, we found a variety of dynamical epochs, with significant
features for the absence of the Cauchy (inner) horizon in such a solution.

In analogy with the intricate structure previously found inside a holographic
superconductor model Hartnoll:2020fhc , considering various configurations of
the massive gravity, we have seen that just below the critical temperature
where the charged massive black hole solution becomes unstable to form scalar
hair, the interior dynamics passes through three epochs. More precisely, one
can observe a collapse of the Einstein-Rosen (ER) bridge, the Josephson
oscillations of the scalar field, and then a Kasner epoch indicating that the
Cauchy horizon is not present.

While studying the ER collapse and Josephson oscillations epochs, we have
found that the instability of the inner horizon results in the collapse of the
ER bridge, for which $g_{tt}$ suddenly undergoes a very rapid collapse and
becomes exponentially small over a shot proper time. In addition, for a small
value of the dimensionless massive parameter $\kappa/\mu$, the ER bridge
collapse happens sooner while increasing this parameter causes the Josephson
oscillations epoch to be removed before entering the final epoch, i.e., Kasner
cosmology. Moreover, like the standard holographic superconductor case
Hartnoll:2020fhc , at temperatures which are far enough from $T_{c}$, the
number of Josephson oscillations tends to decrease and finally disappear.

Finally, we have shown that the interior evolution does not deviate from the
Kasner cosmology when the intermediate Kasner exponent is positive, i.e.,
$|c|>1$ ($p_{t}>0$) at the beginning of the Kasner cosmology epoch, while for
$|c|<1$ the Maxwell field growth (35) triggers a transition to a new Kasner
regime with the exponent $c\to c_{new}=1/c$, or equivalently $p_{t}\to
p_{t}^{new}=-p_{t}/(2p_{t}+1)$ at a certain radial position (called the Kasner
inversion). In addition, we have proved that the effect of the massive gravity
contribution is not significant in the final Kasner regime near the spacelike
singularity.

Interestingly, in the large massive gravity parameter regime for a given
temperature, the Einstein-Rosen bridge collapse and subsequent Josephson
oscillations epochs completely are removed from the interior dynamics. In
addition, the system moves to the final Kasner cosmology epoch exactly after
the would-be inner horizon and does not experience any Kasner inversion.

## Acknowledgements

We would like to thank Matteo Baggioli and Li Li for many useful comments and
discussions.

## References

  * (1) H. Reissner, _Über die eigengravitation des elektrischen feldes nach der einsteinschen theorie_ , _Annalen der Physik_ 355 (1916) 106–120.
  * (2) G. Nordström, _On the energy of the gravitation field in einstein’s theory_ , _Kon. Ned. Akad. Wetensch. Proc. Series B Physical Sciences_ 20 (1918) 1238–1245.
  * (3) R. P. Kerr, _Gravitational field of a spinning mass as an example of algebraically special metrics_ , _Physical review letters_ 11 (1963) 237.
  * (4) R. Geroch, _Domain of dependence_ , _Journal of Mathematical Physics_ 11 (1970) 437–449.
  * (5) R. Penrose, _Gravitational collapse: The role of general relativity_ , _Riv. Nuovo Cim._ 1 (1969) 252–276.
  * (6) K. Papadodimas, S. Raju and P. Shrivastava, _A simple quantum test for smooth horizons_ , _JHEP_ 12 (2020) 003, [1910.02992].
  * (7) V. Balasubramanian, A. Kar and G. Sárosi, _Holographic Probes of Inner Horizons_ , _JHEP_ 06 (2020) 054, [1911.12413].
  * (8) A. Frenkel, S. A. Hartnoll, J. Kruthoff and Z. D. Shi, _Holographic flows from CFT to the Kasner universe_ , _JHEP_ 08 (2020) 003, [2004.01192].
  * (9) 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, [2006.10056].
  * (10) 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_ , 2009.06277.
  * (11) S. A. H. Mansoori, L. Li, M. Rafiee and M. Baggioli, _What’s inside a hairy black hole in massive gravity?_ , _JHEP_ 10 (2021) 098, [2108.01471].
  * (12) S. A. Hartnoll, G. T. Horowitz, J. Kruthoff and J. E. Santos, _Diving into a holographic superconductor_ , _SciPost Phys._ 10 (2021) 009, [2008.12786].
  * (13) R.-G. Cai, L. Li and R.-Q. Yang, _No Inner-Horizon Theorem for Black Holes with Charged Scalar Hairs_ , _JHEP_ 03 (2021) 263, [2009.05520].
  * (14) M. Henneaux, _The final Kasner regime inside black holes with scalar or vector hair_ , _JHEP_ 03 (2022) 062, [2202.04155].
  * (15) R.-G. Cai, C. Ge, L. Li and R.-Q. Yang, _Inside anisotropic black hole with vector hair_ , _JHEP_ 02 (2022) 139, [2112.04206].
  * (16) L. Sword and D. Vegh, _Kasner geometries inside holographic superconductors_ , 2112.14177.
  * (17) Y. Liu, H.-D. Lyu and A. Raju, _Black hole singularities across phase transitions_ , _JHEP_ 10 (2021) 140, [2108.04554].
  * (18) D. O. Devecioglu and M.-I. Park, _No Scalar-Haired Cauchy Horizon Theorem in Einstein-Maxwell-Horndeski Theories_ , 2101.10116.
  * (19) M. Van de Moortel, _Violent nonlinear collapse in the interior of charged hairy black holes_ , 2109.10932.
  * (20) R. Auzzi, S. Bolognesi, E. Rabinovici, F. I. Schaposnik Massolo and G. Tallarita, _On the time dependence of holographic complexity for charged AdS black holes with scalar hair_ , 2205.03365.
  * (21) Y.-S. An, L. Li, F.-G. Yang and R.-Q. Yang, _Interior Structure and Complexity Growth Rate of Holographic Superconductor from M-Theory_ , 2205.02442.
  * (22) R.-Q. Yang, R.-G. Cai and L. Li, _Constraining the number of horizons with energy conditions_ , _Class. Quant. Grav._ 39 (2022) 035005, [2104.03012].
  * (23) Y.-S. An, L. Li and F.-G. Yang, _No Cauchy horizon theorem for nonlinear electrodynamics black holes with charged scalar hairs_ , _Phys. Rev. D_ 104 (2021) 024040, [2106.01069].
  * (24) E. Caceres, A. Kundu, A. K. Patra and S. Shashi, _Trans-IR Flows to Black Hole Singularities_ , 2201.06579.
  * (25) O. J. C. Dias, G. T. Horowitz and J. E. Santos, _Inside an asymptotically flat hairy black hole_ , _JHEP_ 12 (2021) 179, [2110.06225].
  * (26) D. Vegh, _Holography without translational symmetry_ , 1301.0537.
  * (27) R. A. Davison, _Momentum relaxation in holographic massive gravity_ , _Phys. Rev. D_ 88 (2013) 086003, [1306.5792].
  * (28) S. A. Hartnoll, C. P. Herzog and G. T. Horowitz, _Building a Holographic Superconductor_ , _Phys. Rev. Lett._ 101 (2008) 031601, [0803.3295].
  * (29) H. B. Zeng and J.-P. Wu, _Holographic superconductors from the massive gravity_ , _Phys. Rev. D_ 90 (2014) 046001, [1404.5321].
  * (30) R.-G. Cai, Y.-P. Hu, Q.-Y. Pan and Y.-L. Zhang, _Thermodynamics of black holes in massive gravity_ , _Physical Review D_ 91 (2015) 024032.


# Scaling down Deep Learning

Sam Greydanus

###### Abstract

Though deep learning models have taken on commercial and political relevance,
many aspects of their training and operation remain poorly understood. This
has sparked interest in “science of deep learning” projects, many of which are
run at scale and require enormous amounts of time, money, and electricity. But
how much of this research really needs to occur at scale? In this paper, we
introduce MNIST-1D: a minimalist, low-memory, and low-compute alternative to
classic deep learning benchmarks111Code at github.com/greydanus/mnist1d. The
training examples are 20 times smaller than MNIST examples yet they
differentiate more clearly between linear, nonlinear, and convolutional models
which attain 32, 68, and 94% accuracy respectively (these models obtain 94,
99+, and 99+% on MNIST). Then we present example use cases which include
measuring the spatial inductive biases of lottery tickets, observing deep
double descent, and metalearning an activation function.

MNIST, dataset, benchmark, deep learning, convolution

##  1 Introduction

By any scientific standard, the Human Genome Project was enormous: it involved
billions of dollars of funding, dozens of institutions, and over a decade of
accelerated research (Lander et al., 2001). But that was only the tip of the
iceberg. Long before the project began, scientists were hard at work
assembling the intricate science of human genetics. And most of the time, they
were not studying humans. The foundational discoveries in genetics centered on
far simpler organisms such as peas, molds, fruit flies, and mice. To this day,
biologists use these simpler organisms as genetic “minimal working examples”
in order to save time, energy, and money. A well-designed experiment with
Drosophilia, such as Feany and Bender (2000), can teach us an astonishing
amount about humans.

Figure 1: Constructing the MNIST-1D dataset. Like MNIST, the classifier’s
objective is to determine which digit is present in the input. Unlike MNIST,
each example is a one-dimensional sequence of points. To generate an example,
we begin with a digit template and then randomly pad, translate, and transform
it to produce sequences like the ones shown.

The deep learning analogue of Drosophilia is the MNIST dataset. A large number
of deep learning innovations including dropout, Adam, convolutional networks,
generative adversarial networks, and variational autoencoders began life as
MNIST experiments (Srivastava et al., 2014; Kingma and Ba, 2014; LeCun et al.,
1989; Goodfellow et al., 2014; Kingma and Welling, 2013). Once these
innovations proved themselves on small-scale experiments, scientists found
ways to scale them to larger and more impactful applications.

They key advantage of Drosophilia and MNIST is that they dramatically
accelerate the iteration cycle of exploratory research. In the case of
Drosophilia, the fly’s life cycle is just a few days long and its nutritional
needs are negligible. This makes it much easier to work with than mammals,
especially humans. In the case of MNIST, training a strong classifier takes a
few dozen lines of code, less than a minute of walltime, and negligible
amounts of electricity. This is a stark contrast to state-of-the-art vision,
text, and game-playing models which can take months and hundreds of thousands
of dollars of electricity to train (Sharir et al., 2020).

Yet in spite of its historical significance, MNIST has three notable
shortcomings. First, it does a poor job of differentiating between linear,
nonlinear, and translation-invariant models. For example, logistic, MLP, and
CNN benchmarks obtain 94, 99+, and 99+% accuracy on it. This makes it hard to
measure the contribution of a CNN’s spatial priors or to judge the relative
effectiveness of different regularization schemes. Second, it is somewhat
large for a toy dataset. Each input example is a 784-dimensional vector and
thus it takes a non-trivial amount of computation to perform hyperparameter
searches or debug a metalearning loop. Third, MNIST is hard to hack. The ideal
toy dataset should be procedurally generated so that researchers can smoothly
vary parameters such as background noise, translation, and resolution.

In order to address these shortcomings, we propose the MNIST-1D dataset. It is
a minimalist, low-memory, and low-compute alternative to MNIST, designed for
exploratory deep learning research where rapid iteration is a priority.
Training examples are 20 times smaller but they are still better at measuring
the difference between 1) linear and nonlinear classifiers and 2) models with
and without spatial inductive biases (eg. translation invariance). The dataset
is procedurally generated but still permits analogies to real-world digit
classification.

Figure 2: Visualizing the performance of common models on the MNIST-1D
dataset. Whereas most ML models perform within a few percent accuracy on
MNIST, this dataset separates them cleanly according to their characteristics.
Logistic regression models fare worse than MLPs because they cannot use
nonlinearities. MLPs, meanwhile, fare worse than CNNs because they cannot use
translation invariance and local connectivity to bias optimization towards
solutions that generalize well. These results suggest MNIST-1D is a good
dataset for studying the inductive biases of ML models.

##  2 Context

Table 1: Test accuracies of common classifiers on the MNIST and MNIST-1D datasets. Most classifiers achieve similar test accuracy on MNIST. The MNIST-1D dataset is much smaller than MNIST and does a better job of separating models with different inductive biases. The drop in CNN and GRU performance when using shuffled data indicates that spatial priors are important on this dataset. Dataset | Logistic regression | Fully connected model | Convolutional model | GRU model | Human expert
---|---|---|---|---|---
MNIST | $94\pm 0.5$ | $>99$ | $>99$ | $>99$ | $>99$
MNIST-1D | $32\pm 1$ | $68\pm 2$ | $94\pm 2$ | $91\pm 2$ | $96\pm 1$
MNIST-1D (shuffled) | $32\pm 1$ | $68\pm 2$ | $56\pm 2$ | $57\pm 2$ | $\approx 30\pm 10$

The machine learning community has grown rapidly in recent years. This growth
has accelerated the rate of scientific innovation, but it has also produced
multiple competing narratives about the field’s ultimate direction and
objectives. In this section, we will explore three such narratives in order to
place MNIST-1D in its proper context.

Scaling trends. One of the defining features of machine learning in the 2010’s
was a massive increase in the scale of datasets, models, and compute
infrastructure (Amodei and Hernandez, 2018). This scaling pattern allowed
neural networks to achieve breakthrough results on a wide range of benchmarks
(Krizhevsky et al., 2012; Szegedy et al., 2015; Radford et al., 2019). Yet
while this scaling effect has helped neural networks take on commercial and
political relevance, opinions differ about how much more intelligence it can
generate. One one hand, many researchers and organizations argue that scaling
is a crucial path to making neural networks behave more intelligently (Amodei
and Hernandez, 2018). On the other hand, there is a healthy but marginal
population of researchers who are not primarily motivated by scale. They are
united by a common desire to change research methodologies, advocating a shift
away from human-engineered datasets and architectures (Clune, 2019), an
emphasis on human-like learning patterns (Chollet, 2019), and better
integration with traditional symbolic AI approaches (Marcus, 2018).

Once again, the genetics analogy is useful. In genetics, scale has been most
effective when small-scale experiments have helped to guide the direction and
vision of large-scale experiments. For example, the organizers of the Human
Genome Project regularly used yeast and fly genomes to guide analysis of the
human genome (Lander et al., 2001). Thus one should be suspicious of research
agendas that place disproportionate emphasis on large-scale experiments, since
a healthy research ecosystem needs both. The fast, small scale projects permit
creativity and deep understanding, whereas the large-scale projects expose
fertile new research territory.

Understanding vs. performance. Researchers are also divided over the value of
understanding versus performance. Some contend that a high-performing
algorithm need not be interpretable so long as it saves lives or produces
economic value. Others argue that hard-to-interpret deep learning models
should not be deployed in sensitive real-world contexts. Both arguments have
merit, but the best path forward seems to be to focus on understanding high-
performing algorithms better so that this tradeoff becomes less severe. One
way to do this is by identifying things we don’t understand about neural
networks, reproducing these things on a toy problem like MNIST-1D, and then
performing ablation studies to isolate the causal mechanisms.

Ecological impacts. A growing number of researchers and organizations claim
that deep learning will have positive environmental applications (Loehle,
1987; Rolnick et al., 2019). This may be true in the long run, but so far
artificial intelligence has done little to solve environmental problems. In
the meantime, deep learning models are consuming massive amounts of
electricity to train and deploy (Strubell et al., 2019). Our hope is that
benchmarks like MNIST-1D will encourage researchers to spend more time
iterating on small datasets and toy models before scaling, making more
efficient use of electricity in the process.

##  3 Methods

Given modern machine learning’s emphasis on scale and performance, we see a
hidden demand for alternative projects – projects that prioritize creativity
over scale and understanding over performance. We designed MNIST-1D for that
sort of research. In particular, we wanted a dataset that was

  * •

Extremely small: smaller than MNIST

  * •

Able to identify models with (spatial) inductive biases

  * •

Easy to hack, extend, or modify

  * •

Analogous to large-scale problems

Dimensionality. Our first choice was to make the data one-dimensional (eg time
series) rather than two-dimensional (eg images). Our rationale was that there
were already many good image datasets so the value in adding another was
small. Meanwhile, one-dimensional signal processing requires less computation
but has many of the same scientific properties.

Constructing the dataset. We began with ten one-dimensional template patterns
which resemble the digits 0-9 when plotted as in Figure 1. Each of these
platonic forms consists of 12 hand-selected $x$ coordinates. Next, we padded
each sequence with 36-60 additional points, translated the digit at random,
scaled it, added Gaussian noise, and added a constant linear signal analogous
to shear in a 2D image. We used a Gaussian filter with $\sigma=2$ to induce
correlations that would be easy to confuse with the templates. Last of all, we
downsampled the pattern to 40 data points. Figure 1 shows class-wise examples
before and after these transformations.

Implementation. Our goals during implementation were to make the code as
simple, modular, and hackable as possible. The code for generating the dataset
occupies two Python files and a total of 150 lines. The get_dataset method has
a simple API for changing dataset features such as maximum digit translation,
correlated noise scale, shear scale, final sequence length, and more. The
default train/test split is 4000/1000.

Benchmarking the dataset. We used PyTorch to implement and train simple
logistic, MLP, CNN, and GRU baselines. All models used the Adam optimizer and
early stopping for model selection. We also trained the same models on a
version of the dataset which was permuted along the spatial dimension. We
refer to this as the “shuffled” version since it measures each model’s
performance in the absence of local spatial structure. A number of related
works use the same shuffling process to remove spatial priors (Zhang et al.,
2016; Li et al., 2018). Table 1 shows that the test accuracy of CNNs and GRUs
decreases by about 38% on the shuffled data whereas the MLP and linear models
perform about the same. This is a good sanity check since the former two
models have spatial and temporal locality priors whereas the latter two do
not. As with the dataset itself, the code for implementing and training the
benchmark models occupies roughly 150 lines and is clean and modular. You can
reproduce the benchmarks in your browser in a few minutes222bit.ly/3fghqVu.

##  4 Example use cases

Figure 3: Finding and analyzing lottery tickets. In a-b), we isolate a minimum
viable example of the effect. Recent work by Morcos et al. (2019) shows that
lottery tickets can transfer between datasets. We wanted to determine whether
spatial inductive biases played a role. So we performed a series of
experiments: in c) we plot the asymptotic performance of a 92% sparse ticket.
In d) we reverse all the 1D signals in the dataset, effectively preserving
spatial structure but changing the location of individual datapoints. This is
analogous to flipping an image upside down. Under this ablation, the lottery
ticket continues to win. Next, in e) we permute the indices of the 1D signal,
effectively removing spatial structure from the dataset. This ablation hurts
lottery ticket performance significantly more, suggesting that part of the
lottery ticket’s performance can be attributed to a spatial inductive bias.
Finally, in f) we keep the lottery ticket sparsity structure but initialize
its weights with a different random seed. Contrary to results reported in
(Frankle and Carbin, 2019), we see that our lottery ticket continues to
outperform a dense baseline, aligning well with our hypothesis that the
sparsity pattern represents a spatial inductive bias. In g), we verify our
hypothesis by measuring how often unmasked weights are adjacent to one another
in the first layer of our model. The lottery ticket has many more adjacent
weights than chance would predict, implying a local connectivity structure
which helps gives rise to spatial biases. Figure 9 in the Appendix visualizes
the actual masks, revealing how this local connectivity is manifested.

In this section we will explore several examples of how MNIST-1D can be used
to study core “science of deep learning” phenomena.

Finding lottery tickets. It is not unusual for deep learning models to have
ten or even a hundred times more parameters than necessary. This
overparameterization helps training but increases computational overhead. One
solution is to progressively prune weights from a model during training so
that the final network is just a fraction of its original size. Although this
approach works, conventional wisdom holds that sparse networks do not train
well from scratch. Recent work by Frankle and Carbin (2019) challenges this
conventional wisdom. The authors report finding sparse subnetworks inside of
larger networks that train to equivalent or even higher accuracies. These
“lottery ticket” subnetworks can be found through a simple iterative
procedure: train a network, prune the smallest weights, and then rewind the
remaining weights to their original initializations and retrain.

Since the original paper was published, a multitude of works have sought to
explain this phenomenon and then harness it on larger datasets and models.
However, very few works have attempted to isolate a “minimal working example”
of this effect so as to investigate it more carefully. Figure 3 shows that the
MNIST-1D dataset not only makes this possible, but also enables us to
elucidate, via carefully-controlled experiments, some of the reasons for a
lottery ticket’s success. Unlike many follow-up experiments on the lottery
ticket, this one took just two days of researcher time to produce. The curious
reader can also reproduce these results in their browser in a few
minutes333bit.ly/3nCEIaL.

Observing deep double descent. Another intriguing property of neural networks
is the “double descent” phenomenon. This phrase refers to a training regime
where more data, model parameters, or gradient steps can actually reduce a
model’s test accuracy (Trunk, 1979; Belkin et al., 2019; Geiger et al., 2019;
Nakkiran et al., 2020). The intuition is that during supervised learning there
is an interpolation threshold where the learning procedure, consisting of a
model and an optimization algorithm, is just barely able to fit the entire
training set. At this threshold there is effectively just one model that can
fit the data and this model is very sensitive to label noise and model mis-
specification.

Figure 4: Observing deep double descent. MNIST-1D is a good environment for
determining how to locate the interpolation threshold of deep models. This
threshold is fairly easy to predict in fully-connected models, less easy to
predict for other models like CNNs, RNNs, and Transformers. Here we see that a
CNN has a double descent peak at the same interpolation threshold, although
the effect is much less pronounced.

Several properties of this effect, such as what factors affect its width and
location, are not well understood in the context of deep models. We see the
MNIST-1D dataset as a good tool for exploring these properties. In fact, we
were able to reproduce the double descent pattern after a few hours of
researcher effort. Figure 4 shows our results for a fully-connected network
and a convolutional model444Run in browser: bit.ly/2UBWWNu. We also observed a
nuance that we had not seen mentioned in previous works: when using a mean
square error loss, the interpolation threshold lies at $n*K$ model parameters
where $n$ is the number of training examples and $K$ is the number of model
outputs. But when using a negative log likelihood loss, the interpolation
threshold lies at $n$ model parameters – it does not depend on the number of
outputs. This is an interesting empirical observation that may explain some of
the advantage in using a log likelihood loss over a MSE loss on this type of
task.

Gradient-based metalearning. The goal of metalearning is to “learn how to
learn.” A model does this by having two levels of optimization: the first is a
fast inner loop which corresponds to a traditional learning objective and the
second is a slow outer loop which updates the “meta” properties of the
learning process. One of the simplest examples of metalearning is gradient-
based hyperparameter optimization. The concept was was proposed by Bengio
(2000) and then scaled to deep learning models by Maclaurin et al. (2015). The
basic idea is to implement a fully-differentiable neural network training loop
and then backpropagate through the entire process in order to optimize
hyperparameters like learning rate and weight decay.

Figure 5: Metalearning a learning rate. Unlike many gradient-based
metalearning implementations, ours takes seconds to run and occupies a few
dozen lines of code. This allows researchers to iterate on novel ideas before
scaling. Best viewed with zoom.

Metalearning is a promising topic but it is very difficult to scale. First of
all, metalearning algorithms consume enormous amounts of time and compute.
Second of all, implementations tend to grow complex since there are twice as
many hyperparameters (one set for each level of optimization) and most deep
learning frameworks are not set up well for metalearning. This places an
especially high incentive on debugging and iterating metalearning algorithms
on small-scale datasets such as MNIST-1D. For example, it took just a few
hours to implement and debug the gradient-based hyperparameter optimization
shown in Figure 5555Run in browser: bit.ly/38OSyTu.

Metalearning an activation function. Having implemented a “minimal working
example” of gradient-based metalearning, we realized that it permitted a
simple and novel extension: metalearning an activation function. With a few
more hours of researcher time, we were able to parameterize our classifier’s
activation function with a second neural network and then learn the weights
using meta-gradients. As Figure 6 shows, our learned activation function
substantially outperforms baseline nonlinearities such as ReLU, Elu, and
Swish.666Run in browser: bit.ly/38V4GlQ We note that previous works (Clevert
et al., 2016; Ramachandran et al., 2018; Vercellino and Wang, ) have tried to
optimize activation functions, but none have done so with analytical gradients
computed via bilevel optimization.

Figure 6: Metalearning an activation function. Starting from an ELU shape, we
use gradient-based metalearning to find the optimal activation function for a
neural network trained on the MNIST-1D dataset. The activation function itself
is parameterized by a second (meta) neural network. Note that the ELU baseline
(red) is obscured by the tanh baseline (blue) in the figure above.

We transferred this activation function to convolutional models trained on
MNIST and CIFAR10 images and found that it achieves middle-of-the-pack
performance. It is especially good at producing low training loss early in
optimization, which is the objective that it was trained on in MNIST-1D. When
we rank nonlinearities by final test loss, though, it achieves middle-of-the-
pack performance. We suspect that running the same metalearning algorithm on
larger models and datasets would further refine our activation function,
allowing it to at least match the best hand-designed activation function. We
leave line of inquiry to future work.

Figure 7: Benchmarking common pooling methods. We observe that pooling helped
performance in low-data regimes and hindered it in high-data regimes. While we
do not entirely understand this effect, we hypothesize that pooling is a
mediocre architectural prior that is better than nothing in low-data regimes
but becomes overly restrictive in high-data regimes.

Measuring the spatial priors of deep networks. A large part of deep learning’s
success is rooted in “deep priors” which include hard-coded translation
invariances (e.g., convolutional filters), clever architectural choices (e.g.,
self-attention layers), and well-conditioned optimization landscapes (e.g.,
batch normalization). Principle among these priors is the translation
invariance of convolution. A primary motivation for this dataset was to
construct a toy problem that could effectively quantify a model’s spatial
priors. Figure 2 illustrates that this is indeed possible with MNIST-1D. One
could imagine that other models with more moderate spatial priors would sit
somewhere along the continuum between the MLP and CNN benchmarks.

Benchmarking pooling methods. Our final case study begins with a specific
question: What is the relationship between pooling and sample efficiency? We
had not seen evidence that pooling makes models more or less sample efficient,
but this seemed an important relationship to understand. With this in mind, we
trained models with different pooling methods and training set sizes and found
that, while pooling tended to be effective in low-data regimes, it did not
make much of a difference in high-data regimes (see Figure 7). We do not fully
understand this effect, but hypothesize that pooling is a mediocre
architectural prior which is better than nothing in low-data regimes and then
ends up restricting model expression in high-data regimes. By the same token,
max-pooling may also be a good architectural prior in the low-data regime, but
start to delete information – and thus perform worse compared to L2 pooling –
in the high-data regime. As with the other examples, you can reproduce these
results in your browser in a few minutes777bit.ly/3lGmTqY.

##  5 When to scale

We should emphasize that this paper is not an argument against large-scale
machine learning research. That sort of research has proven its worth time and
again and has come to represent one of the most exciting aspects of the ML
research ecosystem. Rather, we are arguing in favor of small-scale machine
learning research. Neural networks do not have problems with scaling or
performance – but they do have problems with interpretability,
reproducibility, and iteration speed. We see carefully-controlled, small-scale
experiments as a great way to address these problems.

In fact, small-scale research is complimentary to large-scale research. As in
biology, where fruit fly genetics helped guide the Human Genome Project, we
believe that small-scale research should always have an eye on how to
successfully scale. For example, several of the findings reported in this
paper are at the point where they should be investigated at scale. We would
like to show that large scale lottery tickets also learn spatial inductive
biases, and show evidence that they develop local connectivity. We would also
like to try metalearning an activation function on a larger model in the hopes
of finding an activation that will outperform ReLU and Swish in generality.

##  6 Related work

The core inspiration for this work stems from an admiration for all that the
MNIST dataset (LeCun et al., 1998) has done for deep learning. While it has
some notable flaws – some of which we have addressed – it also has
underappreciated strengths: it is simple, intuitive, and provides the perfect
sandbox for exploring creative new ideas.

Our work also bears philosophical similarities to the Synthetic Petri Dish by
Rawal et al. (2020). It was published concurrently to this work and the
authors make similar references to biology in order to motivate the use of
small synthetic datasets for exploratory research. Their work differs from
ours in that they use metalearning to obtain their datasets whereas we
construct ours by hand. In doing so, we are able to control various causal
factors, such as amount of noise, translation, and padding independently.
Also, our dataset is more intuitive to humans: a human can outperform a strong
CNN on the MNIST-1D task. These traits make MNIST-1D a better dataset for
investigating “science of deep learning” questions on a small scale. The
Synthetic Petri Dish, meanwhile, is not designed to answer the same questions;
its objective is specifically to accelerate neural architecture search.

There are a number of other small-scale datasets that are commonly used to
investigate “science of deep learning” questions. The CIFAR-10 dataset by
Krizhevsky et al. (2009) is larger than the MNIST dataset in that individual
examples are four times larger, but the number of images is the same. It
generally does a better job of discriminating between MLP and CNN
architectures, and between various CNN architectures such as vanilla CNNs
versus ResNets (He et al., 2015). The FashionMNIST dataset by Xiao et al.
(2017) is the same size as MNIST but somewhat more difficult; it aims to
rectify some of the most serious problems with MNIST, in particular, that it
is too easy and does not discriminate properly between different machine
learning models.

Scikit-learn by Pedregosa et al. (2011) provides dozens of toy datasets – some
synthetic and others real – that are meant for evaluating machine learning
models on a small scale. These datasets are appropriate for some “science of
machine learning” questions but not for others. For example, the two_moons
dataset is useful for exploring the role of nonlinearity in classification.
However, none of these synthetic tasks is good for exploring the role of
spatial inductive biases in deep learning architectures. Making real world
analogies to, say, digit classification, is not possible with these datasets,
and one can often do very well on them using simple linear or kernel-based
methods.

##  7 Discussion

There is a counterintuitive possibility that in order to explore the limits of
how large we can scale neural networks, we may need to explore the limits of
how small we can scale them first. Scaling models and datasets downward in a
way that preserves the nuances of their behaviors at scale will allow
researchers to iterate quickly on fundamental and creative ideas. This fast
iteration cycle is the best way of obtaining insights about how to incorporate
progressively more complex inductive biases into our models. We can then
transfer these inductive biases across spatial scales in order to dramatically
improve the sample efficiency and generalization properties of large-scale
models. We see the humble MNIST-1D dataset as a first step in that direction.

## Acknowledgements

Thanks to Luke Metz for all the fun conversations. Thanks to Tony Zador for
encouraging me to finally publish this. Thanks to Dmitry Kobak for helping
create the tSNE plots comparing MNIST and MNIST-1D shown in Figure 10.

## References

  * Amodei and Hernandez [2018] D. Amodei and D. Hernandez.  Ai and compute, May 2018.
  * Belkin et al. [2019] M. Belkin, D. Hsu, S. Ma, and S. Mandal.  Reconciling modern machine-learning practice and the classical bias–variance trade-off.  _Proceedings of the National Academy of Sciences_ , 116(32):15849–15854, 2019.  ISSN 0027-8424.  doi: 10.1073/pnas.1903070116.  URL https://www.pnas.org/content/116/32/15849.
  * Bengio [2000] Y. Bengio.  Gradient-based optimization of hyperparameters.  _Neural computation_ , 12(8):1889–1900, 2000\.
  * Chollet [2019] F. Chollet.  The measure of intelligence.  _arXiv preprint arXiv:1911.01547_ , 2019.
  * Clevert et al. [2016] D.-A. Clevert, T. Unterthiner, and S. Hochreiter.  Fast and accurate deep network learning by exponential linear units (elus).  _International Conference on Learning Representations_ , 2016.
  * Clune [2019] J. Clune.  Ai-gas: Ai-generating algorithms, an alternate paradigm for producing general artificial intelligence.  _arXiv preprint arXiv:1905.10985_ , 2019.
  * Feany and Bender [2000] M. B. Feany and W. W. Bender.  A drosophila model of parkinson’s disease.  _Nature_ , 404(6776):394–398, 2000.
  * Frankle and Carbin [2019] J. Frankle and M. Carbin.  The lottery ticket hypothesis: Finding sparse, trainable neural networks.  In _International Conference on Learning Representations (ICLR)_ , volume abs/1803.03635, 2019.  URL http://arxiv.org/abs/1803.03635.
  * Geiger et al. [2019] M. Geiger, S. Spigler, S. d’Ascoli, L. Sagun, M. Baity-Jesi, G. Biroli, and M. Wyart.  Jamming transition as a paradigm to understand the loss landscape of deep neural networks.  _Physical Review E_ , 100(1):012115, 2019.
  * Goodfellow et al. [2014] I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio.  Generative adversarial nets.  In _Advances in neural information processing systems_ , pages 2672–2680, 2014.
  * He et al. [2015] K. He, X. Zhang, S. Ren, and J. Sun.  Deep residual learning for image recognition.  _CoRR_ , abs/1512.03385, 2015.  URL http://arxiv.org/abs/1512.03385.
  * Kingma and Ba [2014] D. P. Kingma and J. Ba.  Adam: A method for stochastic optimization.  _International Conference on Learning Representations_ , 2014.
  * Kingma and Welling [2013] D. P. Kingma and M. Welling.  Auto-encoding variational bayes.  _arXiv preprint arXiv:1312.6114_ , 2013.
  * Krizhevsky et al. [2009] A. Krizhevsky, G. Hinton, et al.  Learning multiple layers of features from tiny images.  2009\.
  * Krizhevsky et al. [2012] A. Krizhevsky, I. Sutskever, and G. Hinton.  Imagenet classification with deep convolutional neural networks.  In _Advances in Neural Information Processing Systems 25_ , pages 1106–1114, 2012.
  * Lander et al. [2001] E. S. Lander, L. M. Linton, B. Birren, C. Nusbaum, M. C. Zody, J. Baldwin, K. Devon, K. Dewar, M. Doyle, W. FitzHugh, et al.  Initial sequencing and analysis of the human genome.  _Nature_ , 2001.
  * LeCun et al. [1989] Y. LeCun, B. Boser, J. S. Denker, D. Henderson, R. E. Howard, W. Hubbard, and L. D. Jackel.  Backpropagation applied to handwritten zip code recognition.  _Neural computation_ , 1(4):541–551, 1989.
  * LeCun et al. [1998] Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner.  Gradient-based learning applied to document recognition.  _Proceedings of the IEEE_ , 86(11):2278–2324, 1998.
  * Li et al. [2018] C. Li, H. Farkhoor, R. Liu, and J. Yosinski.  Measuring the Intrinsic Dimension of Objective Landscapes.  In _International Conference on Learning Representations_ , Apr. 2018\.
  * Loehle [1987] C. Loehle.  Applying artificial intelligence techniques to ecological modeling.  _Ecological modelling_ , 38(3-4):191–212, 1987\.
  * Maclaurin et al. [2015] D. Maclaurin, D. Duvenaud, and R. Adams.  Gradient-based hyperparameter optimization through reversible learning.  In _International Conference on Machine Learning_ , pages 2113–2122, 2015.
  * Marcus [2018] G. Marcus.  Deep learning: A critical appraisal.  _arXiv preprint arXiv:1801.00631_ , 2018.
  * Morcos et al. [2019] A. Morcos, H. Yu, M. Paganini, and Y. Tian.  One ticket to win them all: generalizing lottery ticket initializations across datasets and optimizers.  In _Advances in Neural Information Processing Systems_ , pages 4933–4943, 2019.
  * Nakkiran et al. [2020] P. Nakkiran, G. Kaplun, Y. Bansal, T. Yang, B. Barak, and I. Sutskever.  Deep double descent: Where bigger models and more data hurt.  _International Conference on Learning Representations_ , 2020.
  * Pedregosa et al. [2011] F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay.  Scikit-learn: Machine learning in Python.  _Journal of Machine Learning Research_ , 12:2825–2830, 2011\.
  * Radford et al. [2019] A. Radford, J. Wu, R. Child, D. Luan, D. Amodei, and I. Sutskever.  Language models are unsupervised multitask learners.  2019\.
  * Ramachandran et al. [2018] P. Ramachandran, B. Zoph, and Q. V. Le.  Searching for activation functions.  _International Conference on Learning Representations (workshop tract)_ , 2018.
  * Rawal et al. [2020] A. Rawal, J. Lehman, F. P. Such, J. Clune, and K. O. Stanley.  Synthetic petri dish: A novel surrogate model for rapid architecture search.  _arXiv preprint arXiv:2005.13092_ , 2020.
  * Rolnick et al. [2019] D. Rolnick, P. L. Donti, L. H. Kaack, K. Kochanski, A. Lacoste, K. Sankaran, A. S. Ross, N. Milojevic-Dupont, N. Jaques, A. Waldman-Brown, et al.  Tackling climate change with machine learning.  _Neural Information Processing Systems Workshop on Climate Change AI_ , 2019.
  * Sharir et al. [2020] O. Sharir, B. Peleg, and Y. Shoham.  The cost of training nlp models: A concise overview.  _arXiv preprint arXiv:2004.08900_ , 2020.
  * Srivastava et al. [2014] N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, and R. Salakhutdinov.  Dropout: a simple way to prevent neural networks from overfitting.  _The journal of machine learning research_ , 15(1):1929–1958, 2014.
  * Strubell et al. [2019] E. Strubell, A. Ganesh, and A. McCallum.  Energy and policy considerations for deep learning in nlp.  _57th Annual Meeting of the Association for Computational Linguistics (ACL_ , 2019.
  * Szegedy et al. [2015] C. Szegedy, W. Liu, Y. Jia, P. Sermanet, S. Reed, D. Anguelov, D. Erhan, V. Vanhoucke, and A. Rabinovich.  Going deeper with convolutions.  _Computer Vision and Pattern Recognition_ , Sept. 2015.
  * Trunk [1979] G. V. Trunk.  A problem of dimensionality: A simple example.  _IEEE Transactions on pattern analysis and machine intelligence_ , (3):306–307, 1979.
  * [35] C. J. Vercellino and W. Y. Wang.  Hyperactivations for activation function exploration.
  * Xiao et al. [2017] H. Xiao, K. Rasul, and R. Vollgraf.  Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms, 2017.
  * Zhang et al. [2016] C. Zhang, S. Bengio, M. Hardt, B. Recht, and O. Vinyals.  Understanding deep learning requires rethinking generalization.  _CoRR_ , abs/1611.03530, 2016.  URL http://arxiv.org/abs/1611.03530.

##  Appendix A Supplementary figures

More on the human vs. CNN benchmarks. An experienced human can classify
MNIST-1D examples at almost 96% accuracy. The CNN can do so at 94% accuracy.
Both the human and the CNN struggle primarily with classifying 2’s and 7’s,
and to a lesser degree 4’s (see Figure 8). The human had a harder time
classifying 9’s whereas the CNN had a harder time classifying 1’s. Both had
zero errors classifying 3’s and 6’s.

Figure 8: Classwise errors on the MNIST-1D dataset. Some classes contribute
more than others, but most represent an appreciable fraction. This is good,
because if only one or two classes were responsible for most of the mistakes,
that would indicate that those classes are too difficult compared to the
others. It’s also interesting to note that humans and CNNs struggle with the
same classes, such as 7’s and 2’s.

Classification errors were fairly evenly balanced across classes, which is a
good sign. If only one or two classes were responsible for most of the
mistakes, that would have been a sign that those classes were too difficult
compared to the others.

It’s interesting that a human can outperform a CNN on this simple task. Part
of the issue is that the CNN is only given 4000 training examples – with more
examples it can match and eventually exceed the human baseline. Even though
the data is low-dimensional, the classification objective is quite difficult
and spatial/relational priors matter a lot. It may be that the architecture of
the CNN prevents it from learning all of the tricks that humans are capable of
using (eg, using relational reasoning about two signals to determine how they
work together to form the digit signal).

It’s worth noting that CNNs outperform human experts on most large-scale image
classification tasks like ImageNet and CIFAR-100. But here is a tiny benchmark
where humans are still competitive – this is a nice quality, as it suggests
that the key to performing well on the dataset does not rest on shortcut
learning such as memorizing specific numbers or patterns to machine precision.
A high-performing ML model, we can hope, would have to solve the problem using
strategies that would be intuitive to a human.

Further analysis of lottery tickets. In addition to the analysis of lottery
tickets shown in Figure 3, we include some further visualizations in this
appendix. In particular, Figure 9 shows the actual masks of the first layer
weights of random and lottery ticket masks. This qualitative comparison helps
highlight the spatial inductive bias of the lottery ticket we obtained.

Figure 9: Visualizing first layer weight masks of random tickets and lottery
tickets. For interpretabilty, we have sorted the mask along the hidden layer
axis according to the number of adjacent unmasked parameters. This helps
reveal a bias towards local connectivity in the lottery ticket masks. Notice
how there are many more vertically-adjacent unmasked parameters in the lottery
ticket masks. These vertically-adjacent parameters correspond to local
connectivity along the input dimension, which in turn biases the sparse model
towards data with spatial structure.

Dimensionality reduction. We used tSNE to reduce the dimensionality of MNIST
and MNIST-1D so as to plot the datasets in two dimensions. We show the results
in Figure 10 with each example colored according to its class label. We
observe ten well-defined clusters in the MNIST dataset, suggesting that most
examples in most classes are linearly separable from one another. By contrast,
we observe few well-defined clusters in the MNIST-1D dataset, suggesting that
a classifier must learn a nonlinear representation of the data in order to
separate the classes properly. Thanks to Dmitry Kobak for helping with this
visualization.

Figure 10: Visualizing the MNIST and MNIST-1D datasets with tSNE. The well-
defined clusters in the MNIST plot indicate that the majority of the examples
are linearly separable according to class. The MNIST-1D plot, meanwhile,
reveals a lack of well-defined clusters which suggests that nonlinear features
are much more important for successful classification.

##  Appendix B Hyperparameters

Table 2: Default hyperparameters of the MNIST-1D dataset. Hyperparameter | Value
---|---
Train/test split | 4k/1k
Template length | 12
Padding points | 36-60
Max translation | 48
Correlated noise scale | 0.25
Iid noise scale | $2\times 10^{-2}$
Shear scale | 0.75
Shuffle sequence | False
Final seq. length | 40
Seed | 42