# Stability of black holes with non-minimally coupled scalar hair

Nikos Chatzifotis<EMAIL_ADDRESS>Physics Division, National Technical
University of Athens, 15780 Zografou Campus, Athens, Greece  Christoforos
Vlachos<EMAIL_ADDRESS>Physics Division, National Technical University
of Athens, 15780 Zografou Campus, Athens, Greece  Kyriakos Destounis
<EMAIL_ADDRESS>Theoretical Astrophysics, IAAT, University
of T$\ddot{u}$bingen, 72076 T$\ddot{u}$bingen, Germany  Eleftherios
Papantonopoulos<EMAIL_ADDRESS>Physics Division, National Technical
University of Athens, 15780 Zografou Campus, Athens, Greece

###### Abstract

General relativity admits a plethora of exact compact object solutions. The
augmentation of Einstein’s action with non-minimal coupling terms leads to
modified theories with rich structure, which, in turn, provide non-trivial
solutions with intriguing phenomenology. Thus, assessing their viability under
external generic fluctuations is of utmost importance for gravity theories. We
consider static and spherically-symmetric solutions of a Horndeski subclass
which includes a massless scalar field non-minimally coupled to the Einstein
tensor. Such theory possesses second-order field equations and admits an exact
black hole solution with scalar hair. Here, we study the stability of such
solution under axial perturbations and find that it is gravitationally stable
at the linear level. The qualitative features of the ringdown waveform depend
solely on ratio of the two available parameters of spacetime, namely the black
hole mass $m$ and the non-minimal coupling strength $\ell_{\eta}$. Finally, we
demonstrate that the gravitational-wave ringdown transitions from one which
exhibits echoes to one with a typical quasinormal ringing phase followed by a
late-time tail as the ratio $m/\ell_{\eta}$ increases.

##  I Introduction

Compact objects play a decisive role in contemporary astrophysics, as their
relativistic collisions may provide crucial information concerning
astrophysical processes in extreme-gravity conditions. The latest
gravitational-wave (GW) detections by ground-based interferometers [1, 2, 4,
3, 5] have provided intelligence regarding the strong-field regime. The early
stage of the gravitational ringdown of black hole (BH) mergers, described by
quasinormal modes (QNMs) [6, 7, 8, 9], further contributes to the
understanding of their relaxation properties, as well as the governing theory
of gravity. Nevertheless, a conclusive interpretation of the underlying
gravitational theory has not yet been met. Therefore, it is expected that
future ground and space-borne detectors will improve our perception of
gravitational interactions, and in particular will shed light into the
existence of exotic compact objects [10, 11, 12, 13] which may possess
unexpected near-horizon structures that differ significantly from those of BH
event horizons.

Even though general relativity (GR) has withstood many experimental tests,
remaining consistent with plenty of observations so far, modified theories of
gravity attempt to describe phenomena where GR seems to fail, such as the
construction of viable cosmological models for inflation and dark energy [14].
The most general scalar-tensor theory of gravity in four dimensions whose
Lagrangian is constructed by the metric tensor and a non-minimally coupled
scalar field is Horndeski’s theory [15]. This gravity theory preserves a
classical Galilean symmetry [16, 17], leads to second-order field equations
and is free of ghost instabilities [18, 19, 20, 21]. The most extensively
studied subclass of Horndeski theory is represented by a Lagrangian with a
scalar field non-minimally coupled to the Einstein tensor.

On large scales, the non-minimal coupling term of the aforementioned Horndeski
subclass possesses very intriguing effects on inflationary dynamics. From an
inflationary model-building point of view, it allows for a very effective
implementation of a slow-roll phase, due to the fact that it acts as a
friction mechanism [22, 23], allowing potentials such as the Standard-Model
Higgs [24], thus making it a very attractive term in Horndeski theory. A
generalization of the non-minimal kinetic term and its application to
inflation was recently analyzed in [25, 26, 27]. During the inflationary phase
of the Universe in GR, scalar and tensor perturbations result in spectra which
are red-shifted [28]. One then expects that the presence of a non-minimal
kinetic term will magnify the red-shift behavior of the perturbation spectra
because of the friction effect, which is subsequently related to the decrease
of the Hubble parameter $H$ during inflation (for a review of the effects of
the non-minimal kinetic term in inflation see [29]). On the contrary, if the
scalar fields are phantom fields with negative kinetic energy non-minimally
coupled to the Einstein tensor, then the spectra of scalar and tensor
perturbations produced during an inflationary phase are typically blue-shifted
[30]. Their dynamics in cosmological setups, as well as the instabilities at
which tachyons or ghosts appear in the infrared region around the present
Hubble scale were discussed in [31]. Beyond inflation, the non-minimal kinetic
coupling has also been utilized to construct cosmological models [32].

Apart from cosmological applications, the particular subclass of Horndeski
theory allows the construction of BH solutions with scalar hair [33, 34, 35,
37, 36]. Consequently, an important aspect of these compact objects is their
stability properties. Regarding the formation of stable hairy BHs, the ‘no-
hair’ theorem should be evaded [38, 39], which translates to the existence of
a balance mechanism to outweigh the gravitational force outside the BH event
horizon. A typical example is offered by holography. A charged scalar field
theory embedded into an anti-de Sitter (AdS) Lagrangian leads to the formation
of horizon hair, as a result of the counterbalance between the attractive
gravitational and the repulsive electromagnetic force [40, 41]. Then,
according to the gauge/gravity duality, such mechanism allows a holographic
phase transition which results to a conformal field theory describing a
holographic superconductor on the AdS boundary [42, 43, 44].

Although Horndeski theory admits a plethora of compact object solutions when
the scalar field is coupled to the Einstein tensor, there exists an exact
spacetime solution describing a BH with scalar hair [34]. Therefore, by
assessing the stability properties of such object we can determine if it poses
a viable candidate for astrophysical compact objects. The aforementioned BH’s
stability, against test scalar perturbations, was recently assessed in [45].
At the linearized level, the non-minimal coupling constant sources an
asymptotic AdS-like boundary in the effective potential of the wave equation
that governs the propagation of scalar perturbations. Such boundary serves as
a perfect reflector for incident scalar waves and generates a trapping region
outside the photon sphere. As a result, the ringdown signal of the BH exhibit
successively damped echoes. Thus, scalarized compact objects in the particular
Horndeski class can serve as alternatives to the standard echo sources which
possess trapping regions beyond the photon sphere due to near-horizon
structures [46, 47, 48, 49, 50, 51, 52, 53, 54, 55].

Inspired by the behavior of tensor perturbations induced by real or phantom
matter during the cosmological phase of inflation, and motivated by recent
findings [45, 56], we study gravitational perturbations on the aforementioned
compact object solutions of scalar-tensor Hordenski theory. The work is
organized as follows. In Section II we review the BH solution of the Horndeski
theory with a scalar field kinetically coupled to the Einstein tensor. In
Section III we discuss the general framework of axial gravitational
perturbations and we derive the effective potential of the considered BH
solution. In Section IV we demonstrate the numerical scheme of time-domain
integration. In Section V we study the evolution of the axial gravitational
perturbations and finally in Section VI we conclude this work.

##  II Black hole solution with a scalar field kinetically coupled to Einstein
tensor

In what follows, we will consider static solutions of a scalar-tensor theory
in which the scalar field is kinetically coupled to the Einstein tensor. This
is part of the most general scalar-tensor theory which yields second-order
field equations, the Horndeski theory. The full Lagrangian is given by

$\displaystyle\mathcal{L}=\sum_{i=2}^{i=5}\mathcal{L}_{i}\leavevmode\nobreak\
,$ (1) $\displaystyle\mathcal{L}_{2}$
$\displaystyle=K(\Phi,X)\leavevmode\nobreak\ ,$ $\displaystyle\mathcal{L}_{3}$
$\displaystyle=-G_{3}(\Phi,X)\square\Phi\leavevmode\nobreak\ ,$
$\displaystyle\mathcal{L}_{4}$
$\displaystyle=G_{4}(\Phi,X)R+G_{4,X}\left[(\square\Phi)^{2}-(\nabla_{\mu}\nabla_{\nu}\Phi)^{2}\right]\leavevmode\nobreak\
.$ $\displaystyle\mathcal{L}_{5}$
$\displaystyle=G_{5}(\Phi,X)G_{\mu\nu}\nabla^{\mu}\nabla^{\nu}\Phi-\frac{1}{6}G_{5,X}\left[(\square\Phi)^{3}-3\square\Phi(\nabla_{\mu}\nabla_{\nu}\Phi)^{2}+2(\nabla_{\mu}\nabla_{\nu}\Phi)^{3}\right]\leavevmode\nobreak\
,$

where $X=-\frac{1}{2}\nabla_{\mu}\Phi\nabla^{\mu}\Phi$. We consider a
particular subset of Horndeski theory with non-trivial
$\mathcal{L}_{2}=K(\Phi,X)=2\varepsilon X$ and $G_{4}(\Phi,X)=(8\pi)^{-1}-\eta
X$ terms. The theory is described by the following action,

$S=\int d^{4}x\sqrt{-g}\left[\frac{R}{8\pi}-(\varepsilon g_{\mu\nu}+\eta
G_{\mu\nu})\partial^{\mu}\Phi\partial^{\nu}\Phi\right],$ (2)

where $g_{\mu\nu}$ is the metric tensor, $g=det(g_{\mu\nu})$, $R$ is the
scalar curvature, $G_{\mu\nu}$ is the Einstein tensor, $\Phi$ is a real
massless scalar field and $\eta$ is a parameter of non-minimal kinetic
coupling with the dimension of length-squared. The $\varepsilon$ parameter
equals $\pm 1$. In the case $\varepsilon=+1$, we have a canonical scalar field
with positive kinetic term, while the case $\varepsilon=-1$ corresponds to a
phantom scalar field with negative kinetic term. In the original Hordenski
theory the kinetic energy of the scalar field is positive. In this work we
will also considered the case where the scalar field has negative kinetic
energy.

Variation of the action (2) with respect to the metric $g_{\mu\nu}$ and scalar
field $\Phi$ provides the following field equations

$\displaystyle G_{\mu\nu}=8\pi\big{[}\varepsilon
T_{\mu\nu}+\eta\Theta_{\mu\nu}\big{]}\leavevmode\nobreak\ ,$ (3a)
$\displaystyle[\varepsilon g^{\mu\nu}+\eta
G^{\mu\nu}]\nabla_{\mu}\nabla_{\nu}\Phi=0\leavevmode\nobreak\ ,$ (3b)

where

$\displaystyle T_{\mu\nu}$
$\displaystyle=\nabla_{\mu}\phi\nabla_{\nu}\Phi-\frac{1}{2}g_{\mu\nu}(\nabla\Phi)^{2}\leavevmode\nobreak\
,$ (4) $\displaystyle\Theta_{\mu\nu}$
$\displaystyle=-\frac{1}{2}\nabla_{\mu}\Phi\,\nabla_{\nu}\Phi\,R+2\nabla_{\alpha}\Phi\,\nabla_{(\mu}\phi
R^{\alpha}_{\nu)}$
$\displaystyle+\nabla^{\alpha}\Phi\,\nabla^{\beta}\Phi\,R_{\mu\alpha\nu\beta}+\nabla_{\mu}\nabla^{\alpha}\Phi\,\nabla_{\nu}\nabla_{\alpha}\Phi$
$\displaystyle-\nabla_{\mu}\nabla_{\nu}\Phi\,\square\Phi-\frac{1}{2}(\nabla\Phi)^{2}G_{\mu\nu}$
(5)
$\displaystyle+g_{\mu\nu}\big{[}-\frac{1}{2}\nabla^{\alpha}\nabla^{\beta}\Phi\,\nabla_{\alpha}\nabla_{\beta}\Phi+\frac{1}{2}(\square\Phi)^{2}-\nabla_{\alpha}\Phi\,\nabla_{\beta}\Phi\,R^{\alpha\beta}\big{]}\leavevmode\nobreak\
.$

A static and spherically-symmetric BH solution to the aforementioned theory
has been found in [34], where the scalar field of the theory depends only on
the radial coordinate. The solution yields the constraint $\varepsilon\eta<0$,
which leads to the definition of the following coupling parameter

$\ell_{\eta}=\left|\varepsilon\eta\right|^{1/2}\leavevmode\nobreak\ .$ (6)

In terms of the line element

$ds^{2}=-g_{tt}(r)dt^{2}+g_{rr}(r)dr^{2}+g_{\theta\theta}(r)d\Omega^{2}\leavevmode\nobreak\
,$ (7)

the BH solution corresponds to $g_{\theta\theta}(r)=r^{2}$ with
$r\in(0,+\infty)$ and yields the following metric components

$\displaystyle g_{tt}(r)$ $\displaystyle=-\frac{1}{4}F(r)\leavevmode\nobreak\
,$ (8a) $\displaystyle g_{rr}(r)$
$\displaystyle=\frac{(r^{2}+2\ell_{\eta}^{2})^{2}}{(r^{2}+\ell_{\eta}^{2})^{2}F(r)}\leavevmode\nobreak\
,$ (8b) $\displaystyle F(r)$
$\displaystyle=\left[3+\frac{r^{2}}{3\ell^{2}_{\eta}}-\frac{8m}{r}+\frac{\ell_{\eta}}{r}\arctan\left(\frac{r}{\ell_{\eta}}\right)\right],$
(8c)

while the scalar hair of the theory reads

$(\partial_{r}\Phi)^{2}=\Psi^{2}=-\frac{\varepsilon}{8\pi\ell_{\eta}^{2}}\frac{r^{2}g_{rr}}{r^{2}+\ell_{\eta}^{2}}\leavevmode\nobreak\
,$ (9)

which implies that

$4\pi\eta\Psi^{2}=\frac{1}{2}\frac{r^{2}}{r^{2}+\ell_{\eta}^{2}}g_{rr}>0\leavevmode\nobreak\
,\qquad\forall r>r_{h}$ (10)

where $r=r_{h}$ is the event horizon radius. An important note is that the
equations of motion of the scalar field, (3b), can be expressed as the
conservation of the Noether current that corresponds to the shift symmetry of
the Galileon, i.e. $\Phi\rightarrow\Phi+\delta\Phi$. It can straightforwardly
be found that the current is defined as

$J^{\mu}=(\varepsilon g^{\mu\nu}+\eta
G^{\mu\nu})\nabla_{\nu}\Phi\leavevmode\nobreak\ .$ (11)

The BH solution satisfies the physical requirement that the norm of this
current does not diverge at the horizon, by virtue of (6). The scalar hair,
however, diverges at the horizon as one can readily see from (9). One can also
deduce from (9) that the metric components can be expressed in terms of the
scalar hair. As such, the scalar hair of the theory can be understood as an
intrinsic part of the geometry. Finally, we note that due to the Bianchi
identity, $\nabla^{\mu}G_{\mu\nu}=0$, the equation (3a) leads to a
differential consequence

$\nabla^{\mu}\big{[}\varepsilon
T_{\mu\nu}+\eta\Theta_{\mu\nu}\big{]}=0\leavevmode\nobreak\ .$ (12)

The substitution of expressions (4) and (5) into the Bianchi identity yields
Eq. (3b). In other words, the equation of motion of scalar field (3b) is the
differential consequence of the system (3a), which verifies the fact that the
scalar hair is indeed an intrinsic part of the geometry. Let us also note that
the solution reproduces the Schwarzschild BH in the limit of
$l_{\eta}\rightarrow+\infty$, therefore the metric can be understood as a
hairy BH generalization of the Schwarzschild spacetime with effective AdS-
asymptotics, when the spin-0 degree of freedom also acquires dynamics from the
kinetic mixing with the graviton, i.e. the
$G^{\mu\nu}\partial_{\mu}\Phi\partial_{\nu}\Phi$ term.

As one can observe, the metric functions (8) of the BH solution [34] depend on
the parameter $\ell_{\eta}$, therefore they do not understand if the produced
compact object is made of normal or phantom matter as long as
$\epsilon\eta<0$. Even though in [34] the scalar field is assumed to be
canonical ($\epsilon>0$) with the non-minimal coupling being negative
($\eta<0$), we have checked that the same solution is obtain if instead the
scalar field is phantom ($\epsilon<0$) and the non-minimal coupling is
positive ($\eta>0$).

In [23] it was argued that the subclass of the Horndeski theory under
consideration yields wormhole solutions as well. We wish to note here that the
wormhole solution derived there is just a coordinate artifact of the BH and
does not correspond to a traversable wormhole. Let us consider the BH metric
we previously described

$ds^{2}=-\frac{1}{4}F(r)dt^{2}+\frac{(r^{2}+2l_{\eta}^{2})^{2}}{(r^{2}+l_{\eta}^{2})^{2}F(r)}dr^{2}+r^{2}d\Omega^{2}.$
(13)

We perform the following coordinate transformation

$u^{2}=r^{2}-a^{2},\qquad u\in(-\infty,+\infty)$ (14)

and the coordinate redefinition

$dt^{2}=\frac{4}{\left(3-\frac{8m}{a}+\frac{a^{2}}{3l_{\eta}^{2}}+\frac{l_{\eta}}{a}\arctan\left(\frac{a}{l_{\eta}}\right)\right)}dT^{2}=CdT^{2}\,.$
(15)

Note that this coordinate transformation covers the BH region for $r>a$ twice.
Fixing $a>r_{h}$, the corresponding geometry will cover only the region
$r>r_{h}$ twice. Performing the coordinate transformation (14), one finds:

$\displaystyle ds^{2}$ $\displaystyle=-CF(\sqrt{u^{2}+a^{2}})dT^{2}$
$\displaystyle+\frac{u^{2}(u^{2}+a^{2}+2l_{\eta}^{2})^{2}}{(u^{2}+a^{2})(u^{2}+a^{2}+l_{\eta}^{2})^{2}F(\sqrt{u^{2}+a^{2})}}du^{2}$
$\displaystyle+(u^{2}+a^{2})d\Omega^{2},$ (16) $\displaystyle
F(\sqrt{u^{2}+a^{2})}$
$\displaystyle=\left(3-\frac{8m}{\sqrt{u^{2}+a^{2}}}+\frac{u^{2}+a^{2}}{3l_{\eta}^{2}}+\frac{l_{\eta}}{\sqrt{u^{2}+a^{2}}}\arctan\left(\frac{\sqrt{u^{2}+a^{2}}}{l_{\eta}}\right)\right).$
(17)

The corresponding metric is the solution derived in [57], modulo the form of
the $g_{tt}$ component, which was left as an indefinite integral. Obviously, a
compact object cannot change nature due to a coordinate transformation, thus,
the metric (16) is just the BH solution (8) written in a “bad” coordinate
system and was falsely identified as a wormhole. This result is in accordance
with the findings in [58] regarding the absence of static and spherically-
symmetric wormhole solutions in Horndeski theory.

##  III Axial gravitational perturbations: general analysis

In this section we will undergo an analysis of axial perturbations of the BH
solution using Chandrasekhar’s method [59]. The most general metric for an
axisymmetric non-stationary spacetime is given by

$ds^{2}=-e^{2f_{0}}dt^{2}+e^{2f_{3}}(d\phi-q_{0}dt-q_{1}dr-
q_{2}d\theta)^{2}+e^{2f_{1}}dr^{2}+e^{2f_{2}}d\theta^{2}\leavevmode\nobreak\
.$ (18)

This result is found by use of the Cotton-Darboux theorem, which states that
any 3-dimensional metric, $g=g^{ij}\partial_{i}\partial_{j}$, can always be
brought to a diagonal form by a local coordinate transformation. It is clear
that the background metric of our solutions can be described by $q_{i}=0$. In
this gauge, axial perturbations are described by the non-vanishing values of
$q_{i}$, while polar perturbations are described by $f_{i}\rightarrow
f_{i}+\delta f_{i}$ and $q_{i}=0$.

For the purposes of writing down the explicit form of the equations (3a) for
the most general form of the metric of (18), we shall obtain the components of
the curvature tensors via Cartan’s structure equations. We choose the
following tetrads to work with

$\displaystyle\epsilon^{0}_{\mu}=(e^{f_{0}},0,0,0)\leavevmode\nobreak\ ,$
(19a) $\displaystyle\epsilon^{1}_{\mu}=(0,e^{f_{1}},0,0)\leavevmode\nobreak\
,$ (19b)
$\displaystyle\epsilon^{2}_{\mu}=(0,0,e^{f_{2}},0)\leavevmode\nobreak\ ,$
(19c)
$\displaystyle\epsilon^{3}_{\mu}=(-e^{f_{3}}q_{0},-e^{f_{3}}q_{1},-e^{f_{3}}q_{2},e^{f_{3}})\leavevmode\nobreak\
.$ (19d)

Under these tetrads, the basis is found to be

$\displaystyle\epsilon^{0}=e^{f_{0}}dt$ $\displaystyle\rightarrow
dt=e^{-f_{0}}\epsilon^{0}\leavevmode\nobreak\ ,$ (20a)
$\displaystyle\epsilon^{1}=e^{f_{1}}dt$ $\displaystyle\rightarrow
dr=e^{-f_{1}}\epsilon^{1}\leavevmode\nobreak\ ,$ (20b)
$\displaystyle\epsilon^{2}=e^{f_{2}}d\theta$ $\displaystyle\rightarrow
d\theta=e^{-f_{2}}\epsilon^{2}\leavevmode\nobreak\ ,$ (20c)
$\displaystyle\epsilon^{3}=-e^{f_{3}}q_{0}dt-e^{f_{3}}q_{1}dr-e^{f_{3}}q_{2}d\theta+e^{f_{3}}d\phi$
$\displaystyle\rightarrow
d\phi=e^{-f_{3}}\epsilon^{3}+q_{0}e^{-f_{0}}\epsilon^{0}+q_{1}e^{-f_{1}}\epsilon^{1}+q_{2}e^{-f_{2}}\epsilon^{2}\leavevmode\nobreak\
.$ (20d)

The reasoning behind these tetrads is that they associate the perturbations to
a single tetrad and thus allow for a decoupling of the equations of motion at
first order. The spin connections can be derived from the tetrad postulate,
where we associate the zero torsion condition with the Levi-Civita connection
as

$\omega^{a}_{\mu
b}=\epsilon^{a}_{\nu}\epsilon^{\lambda}_{b}\Gamma^{\nu}_{\mu\lambda}-\epsilon^{\lambda}_{b}\partial_{\mu}\epsilon^{a}_{\lambda}\leavevmode\nobreak\
.$ (21)

Using Cartan’s second structure equation, we can derive the Riemann tensor,

$R^{a}_{b\mu\nu}=2(\partial_{[\mu}\omega^{a}_{\nu]b}+\omega^{a}_{c[\mu}\omega^{c}_{\nu]b})\leavevmode\nobreak\
,$ (22)

and from (22) all the necessary tensors for the equations of motion of the
underlying gravity theory. From (3a), we know that

$G_{ab}=\hat{T}_{ab}\leavevmode\nobreak\ ,$ (23)

where $\hat{T}_{ab}=8\pi\big{[}\varepsilon T_{ab}+\eta\Theta_{ab}\big{]}$
(note that indices $a,b$ are Lorentz and not spacetime indices). However, the
Einstein and stress-energy tensors acquire contributions from the
perturbations. From the linearization of the equations of motion we find that
only the $G_{03},G_{13},G_{23}$ terms are important at first order. In fact,
equation $\delta G_{03}=\delta\hat{T}_{03}$ is degenerate, i.e. it is
automatically satisfied by the other two equations. In particular, we find the
following results

$\displaystyle\delta
G_{13}=\frac{1}{2r^{3}\sin^{2}\theta}\frac{1}{\sqrt{g_{tt}}}\frac{d}{d\theta}Q+\frac{r\sin\theta}{2g_{tt}\sqrt{grr}}\frac{d}{dt}\left[\frac{d}{dr}q_{a}-\frac{d}{dt}q_{b}\right]\leavevmode\nobreak\
,$ (24a)
$\displaystyle\delta\hat{T}_{13}=\frac{4\pi\eta\Psi^{2}}{g_{rr}}\delta
G_{13}\leavevmode\nobreak\ ,$ (24b) $\displaystyle\delta
G_{23}=-\frac{1}{2r^{2}\sin^{2}\theta}\frac{1}{\sqrt{g_{tt}g_{rr}}}\frac{d}{dr}Q+\frac{\sin\theta}{2g_{tt}}\frac{d}{dt}\left[\frac{d}{d\theta}q_{a}-\frac{d}{dt}q_{c}\right]\leavevmode\nobreak\
,$ (24c)
$\displaystyle\delta\hat{T}_{23}=\frac{4\pi\eta\Psi^{2}}{g_{rr}}\left(\frac{1}{2r^{2}\sin^{2}\theta}\frac{-1}{\sqrt{g_{tt}g_{rr}}}\frac{d}{dr}Q-\frac{\sin\theta}{2g_{tt}}\frac{d}{dt}\left[\frac{d}{d\theta}q_{a}-\frac{d}{dt}q_{c}\right]\right)$
(24d)
$\displaystyle\qquad-4\pi\eta\frac{d}{dr}\left(\frac{\Psi^{2}}{2g_{rr}}\right)\frac{1}{r^{2}\sin^{2}\theta\sqrt{g_{tt}}\sqrt{g_{rr}}}Q\leavevmode\nobreak\
,$

where
$\displaystyle{Q=\left[r^{2}\sin^{3}\theta\frac{\sqrt{g_{tt}}}{\sqrt{g_{rr}}}\frac{d}{d\theta}q_{b}-r^{2}\sin^{3}\theta\frac{\sqrt{g_{tt}}}{\sqrt{g_{rr}}}\frac{d}{dr}q_{c}\right]}$.
Using the redefinition

$F=\left(1-\frac{4\pi\eta\Psi^{2}}{grr}\right)Q\leavevmode\nobreak\ ,$ (25)

the differential equations we get from $\delta G_{ab}=\delta\hat{T}_{ab}$,
using (24a-24d), read

$\displaystyle\frac{1}{r^{4}\sin^{3}\theta}\sqrt{g_{tt}}\sqrt{g_{rr}}\left(\frac{g_{rr}}{grr-4\pi\eta\Psi^{2}}\right)\frac{d}{d\theta}F$
$\displaystyle=\frac{d}{dt}\left[\frac{d}{dt}q_{b}-\frac{d}{dr}q_{a}\right]\leavevmode\nobreak\
,$ (26a)
$\displaystyle\frac{1}{r^{2}\sin^{3}\theta}\frac{\sqrt{g_{tt}}}{\sqrt{g_{rr}}}\left(\frac{g_{rr}}{grr+4\pi\eta\Psi^{2}}\right)\frac{d}{dr}F$
$\displaystyle=\frac{d}{dt}\left[\frac{d}{d\theta}q_{a}-\frac{d}{dt}q_{c}\right]\leavevmode\nobreak\
.$ (26b)

Differentiating (26a) and (26b) by $\theta$ and $r$ respectively, and adding
them together yields the differential equation that governs axial
perturbations

$\frac{r^{4}}{\sqrt{g_{tt}}\sqrt{g_{rr}}}\left(\frac{g_{rr}-4\pi\eta\Psi^{2}}{g_{rr}}\right)\frac{d}{dr}\left[\frac{1}{r^{2}}\frac{\sqrt{g_{tt}}}{\sqrt{g_{rr}}}\left(\frac{g_{rr}}{grr+4\pi\eta\Psi^{2}}\right)\frac{d}{dr}F\right]-\frac{r^{2}}{g_{tt}}\frac{d^{2}F}{dt^{2}}+\sin^{3}\theta\frac{d}{d\theta}\left[\frac{1}{\sin^{3}\theta}\frac{d}{d\theta}F\right]=0\leavevmode\nobreak\
.$ (27)

Using a separation of variables, the angular component of equation (27) can be
understood as the known ultraspherical differential equation with solutions
the Gegenbauer polynomials, i.e. the angular component of the perturbation is
the same as in the Schwarzschild case. This is to be expected, since both
spacetimes are spherically symmetric. Thus, setting $F=R(r,t)S(\theta)$, (27)
can be rewritten as

$\frac{r^{4}}{\sqrt{g_{tt}}\sqrt{g_{rr}}}\left(\frac{g_{rr}-4\pi\eta\Psi^{2}}{g_{rr}}\right)\frac{d}{dr}\left[\frac{1}{r^{2}}\frac{\sqrt{g_{tt}}}{\sqrt{g_{rr}}}\left(\frac{g_{rr}}{g_{rr}+4\pi\eta\Psi^{2}}\right)\frac{d}{dr}R\right]-\frac{r^{2}}{g_{tt}}\frac{d^{2}R}{dt^{2}}-(l+2)(l-1)R=0\leavevmode\nobreak\
.$ (28)

In order to continue, it will prove useful to set

$h=\frac{\sqrt{g_{tt}}}{\sqrt{g_{rr}}}\,,\qquad
A=\left(\frac{g_{rr}}{g_{rr}+4\pi\eta\Psi^{2}}\right)\,,\qquad
B=\left(\frac{g_{rr}-4\pi\eta\Psi^{2}}{g_{rr}}\right)\leavevmode\nobreak\
,\qquad$ (29)

thus, simplifying equation (28) to

$Br^{2}h\frac{d}{dr}\left[\frac{1}{r^{2}}Ah\frac{d}{dr}R\right]-\frac{d^{2}R}{dt^{2}}-\frac{(l+2)(l-1)}{r^{2}}g_{tt}R=0\leavevmode\nobreak\
.$ (30)

By further defining a scalar field

$u=\frac{R}{r}\sqrt{A}\leavevmode\nobreak\ ,$ (31)

and using the tortoise coordinate transformation $h=dr/dr_{*}$, we find that
the equation that governs the axial perturbations reads

$h\frac{d}{dr}\left[h\frac{d}{dr}u\right]-\frac{1}{AB}\frac{d^{2}u}{dt^{2}}-\left[\frac{(l+2)(l-1)}{r^{2}}\frac{g_{tt}}{AB}+\frac{r}{\sqrt{A}}h\frac{d}{dr}\left(h\frac{d}{dr}\left(\frac{\sqrt{A}}{r}\right)\right)\right]u=0\leavevmode\nobreak\
.$ (32)

If we consider a time dependence of the form $\exp(i\omega t)$, then (32)
yields a non-trivial wave equation of the following form

$h\frac{d}{dr}\left[h\frac{d}{dr}u\right]+\frac{\omega^{2}}{AB}u-\left[\frac{(l+2)(l-1)}{r^{2}}\frac{g_{tt}}{AB}+\frac{r}{\sqrt{A}}h\frac{d}{dr}\left(h\frac{d}{dr}\left(\frac{\sqrt{A}}{r}\right)\right)\right]u=0\leavevmode\nobreak\
.$ (33)

As such, the corresponding Regge-Wheeler-like potential reads

$V(r)=\frac{(l+2)(l-1)}{r^{2}}\frac{g_{tt}}{AB}+\frac{r}{\sqrt{A}}h\frac{d}{dr}\left(h\frac{d}{dr}\left(\frac{\sqrt{A}}{r}\right)\right)\leavevmode\nobreak\
.$ (34)

It is important to note that by fixing the metric components to those of the
BH solution, Eq. (34) asymptotes to the standard Schwarzschild Regge-Wheeler
effective potential in the limit where $\ell_{\eta}\rightarrow+\infty$.
Another crucial aspect of Eq. (33) is the presence of a multiplication factor
$1/AB$ to the gravitational perturbation frequency $\omega$, which signifies
the existence of a modified speed of GW propagation [60].

Eq. (33) demonstrates that one can reduce the problem of axial gravitational
perturbations around the compact object under consideration into a single one-
dimensional scattering problem with an effective potential. By applying the
procedure outlined above, on the BH solution (8)-(9) we find the corresponding
effective potentials a gravitational perturbation induces. An illustration of
these potentials for various choices of the angular index of the perturbation
$\ell$ is given in Fig. 1. The effective potential possesses a peak for
sufficiently large $\ell_{\eta}$, corresponding to the location of the photon
sphere [61] which is responsible for the prompt ringdown and QNMs found in
various spacetimes [62, 63, 64, 65, 66, 67, 68]. Asymptotically, the potential
approaches a constant positive value, a behavior very different from the case
of scalar perturbations [45], but still, one which encodes the non-
asymptotically-flat nature of spacetime. A similar behavior was also observed
in [69].

At spatial infinity, the BH potential at zeroth order yields

$V(r\rightarrow\infty)\sim-
C_{1}\,\frac{r^{2}}{\ell_{\eta}^{4}}+\frac{C_{2}+C_{3}\ell+c_{4}\ell^{2}}{\ell_{\eta}^{2}}+\mathcal{O}\left(\frac{1}{r}\right)$
(35)

which is a rather unusual result since the first term implies that
$V\rightarrow-\infty$ at large distances, a behavior that we do not observe in
Fig. 1. The reason is that the constant $C_{1}$ is of order $10^{-99}$ while
the constants $C_{2},\,C_{3},\,C_{4}$ are of order $10^{-1}$, hence, the
effect of the first term is wildly suppressed whereas the second term results
in dominating the asymptotic behavior of the potential as is clearly
illustrated in Fig. 1. A similar asymptotic behavior was also observed in [70]
for the case of vector perturbations in the scalarized BH we consider here.

Figure 1: Left: BH effective potential with $\ell_{\eta}=100\,,\,m=0.1$ and
different values of angular numbers $\ell$. Right: BH effective potential with
$m=0.5\,,\,\ell=2$ and various coupling constants $\ell_{\eta}$.

##  IV Time-domain integration scheme

Here, we briefly demonstrate the numerical scheme of time-domain integration,
first proposed in [71], which yields the temporal response of a metric
perturbation as it propagates on a fixed background spacetime. By defining
$u(r_{\ast},t)=u(i\Delta r_{\ast},j\Delta t)=u_{i,j}$,
$V(r(r_{*}))=V(r_{\ast})=V(i\Delta r_{\ast})=V_{i}$, $A(r_{\ast})=A(i\Delta
r_{\ast})=A_{i}$ and $B(r_{\ast})=B(i\Delta r_{\ast})=B_{i}$ equation (33)
takes the form

$\displaystyle\frac{u_{i+1,j}-2u_{i,j}+u_{i-1,j}}{\Delta
r^{2}_{\ast}}-\frac{1}{A_{i}B_{i}}\frac{u_{i,j+1}-2u_{i,j}+u_{i,j-1}}{\Delta
t^{2}}-V_{i}\,u_{i,j}=0\;.$ (36)

Then, by using as initial condition a Gaussian wave-packet of the form
$\psi(r_{\ast},t)=\exp\left[-\frac{(r_{\ast}-c)^{2}}{2\sigma^{2}}\right]$ and
$\psi(\rho_{\ast},t<0)=0$, where $c$ and $\sigma$ correspond to the median and
width of the wave-packet, we can derive the time evolution of $u$

$\displaystyle u_{i,j+1}=A_{i}B_{i}\,\Delta
t^{2}\left(\frac{u_{i+1,j}-2u_{i,j}+u_{i-1,j}}{\Delta
r^{2}_{\ast}}-V_{i}u_{i,j}\right)+2u_{i,j}-u_{i,j-1}\leavevmode\nobreak\ ,$
(37)

where the Courant-Friedrichs-Lewy condition requires $\Delta t/\Delta
r_{\ast}<1/(A_{i}B_{i})$. To calculate the precise values of the potential
$V_{i}$, we integrate numerically the equation for the tortoise coordinate and
then solve with respect to the corresponding radial coordinate. Moreover we
require the vanishing of perturbations at radial infinity by imposing
reflective boundary boundary conditions $u_{i_{max},j}=0$, since our solution
is asymptotically AdS (for further details regarding boundary conditions on
asymptotically AdS compact objects, see [72, 8]).

##  V Evolution of axial gravitational perturbations

As stated in [73], a Schwarzschild-AdS BH is determined by two dimensionful
parameters, namely the AdS radius $r=R_{AdS}$ and the event horizon radius
$r=r_{h}$. The BH solution in the present study depends also on two
parameters: $m$ and $\ell_{\eta}$. The value of mass controls the position of
the event horizon $r_{h}$ (and consequently of the photon sphere) and
$\ell_{\eta}$ dictates the value of the effective AdS radius $R_{AdS}$.
However, one key difference between the two solutions is that the first is a
“bald” BH embedded in AdS spacetime, whereas the scalarized solution is
dressed with scalar hair whose existence creates the AdS asymptotics of the
solution. In this sense, the parameter $\ell_{\eta}$ controls the strength of
the scalar hair and as a consequence the value of $R_{AdS}$. In our case, the
effective AdS radius at spatial infinity is given by
$R_{AdS}=\sqrt{3}\,\ell_{\eta}$. One may categorize AdS BHs as (i) small size
BHs with $r_{h}<<R_{AdS}$, (ii) intermediate size BHs with $r_{h}\sim R_{AdS}$
and (iii) large BHs with $r_{h}>>R_{AdS}$ (see Fig. 2). In what follows, we
apply the numerical procedure outlined above, to calculate the temporal
response of axial gravitational perturbations on the BHs of the above
categories. Note that in the following figures we obtain the perturbation
response at a position arbitrarily close to the event horizon, though the same
results are obtained if we calculate the response at any position outside the
event horizon. Furthermore, we have performed some typical tests to ensure the
validity of the integration method. Specifically, we have calculated the
response of gravitational perturbations on the BH considered here, in the
limit where $\ell_{\eta}\rightarrow\infty$, where the effect of the scalar
hair is suppressed, (we have chosen $\ell_{\eta}=1000$, with $V(r)$ rapidly
converging to the Regger-Wheeler potential). By using the Prony method [74] we
have extracted the QNMs from the temporal response at this limit and have
found that they agree very well with the standard gravitational QNMs of
Schwarzschild BHs [59].

Figure 2: The difference between the AdS and event horizon radius as a
function of $m$ and $\ell_{\eta}$, where $R_{AdS}=\sqrt{3}\,\ell_{\eta}$. For
$m<<\ell_{\eta}$ we have $r_{h}<<R_{AdS}$, for $m\simeq\ell_{\eta}$ we have
$r_{h}\simeq R_{AdS}$, while for $m>>\ell_{\eta}$ we have $r_{h}>>R_{AdS}$.
The minima of the curves occur at approximately $m/\ell_{\eta}\simeq 3.1$ and
indicate the points in the parametric space $(m,\ell_{\eta})$ for which
$r_{h}>>R_{AdS}$.

Figure 3 presents the temporal response of the BH against gravitational
perturbations for masses in the region $m<<\ell_{\eta}$
($m/\ell_{\eta}\sim\mathcal{O}(10^{-3})$) . It is worth noting that the
initial quasinormal ringdown is very similar to that of Schwarzschild BHs.
Such behavior is expected and can be attributed to the high value of
$\ell_{\eta}$ relative to the mass. Perturbations with higher multipole number
$\ell$ decay faster and with higher frequency since more energy is carried
away from the photon sphere. The late time behavior however, shows that,
instead of a power-law cutoff, the field settles down to constant value which
directly relates to the asymptotic value that the effective potential acquires
(see Fig. 1).

In Fig. 4 the evolution of perturbations for $m<\ell_{\eta}$
($m/\ell_{\eta}\sim\mathcal{O}(10^{-1})$) is displayed. The most obvious
effect one observes is the emergence of echoes following the initial
quasinormal ringdown. In this parametric region the relation between the mass
of the BH and the coupling $\ell_{\eta}$ becomes more transparent. By keeping
$\ell_{\eta}$ fixed and increasing the mass, perturbations will have to travel
a smaller distance between the photon sphere and the reflective AdS boundary
leading to repetitions in the signal which appear in smaller timescales.
Analogously, the same behavior is obtained when one keeps the mass fixed and
decreases the coupling. This pattern was also observed in [45] for the case of
scalar perturbations indicating that scalar fields can serve as a good probe
for gravitational fluctuations.

Figure 3: Effective potential (left) and time evolution (right) of
gravitational perturbations, with varying $\ell$, of the scalarized BH with
$\ell_{\eta}=100$, $m=0.1$. The parametric region considered here corresponds
to small BHs in the sense that $r_{h}<<R_{AdS}$
($m/\ell_{\eta}\sim\mathcal{O}(10^{-3})$).

Figure 4: Effective potential (left) and time evolution (right) of an $\ell=2$
gravitational perturbation of the scalarized BH with $\ell_{\eta}=5$ for
varying mass $m$. The cases we consider here correspond to intermediate size
BHs with $r_{h}\sim R_{AdS}$ ($m/\ell_{\eta}\sim\mathcal{O}(10^{-1})$).

To obtain a more complete picture about the effect of the ratio
$m/\ell_{\eta}$ on the BH’s response to fluctuations we have plotted in Fig. 5
the time evolution of perturbations for a wide range of masses keeping
$\ell_{\eta}$ fixed. As $m$ grows the echoes are replaced by quasinormal
oscillations, while further increment of the mass leads to a single
quasinormal ringdown followed by a late-time tail. We conclude that this
behavior stems from the shape of the effective potential which decreases in
amplitude as $m$ increases. This leads to an increasingly smaller region where
trapped modes, which lead to echoes, can occur, and thus the quasinormal
ringing of the BH dominates over the echoes which are suppressed.

When the mass becomes proportional ($m/\ell_{\eta}\sim\mathcal{O}(10^{0})$) or
significantly larger than $\ell_{\eta}$
($m/\ell_{\eta}\sim\mathcal{O}(10^{2})$), negative wells develop in the
effective potential in the vicinity of the event horizon (see Figs. 6, 7).
Despite the negative well formation, the time-domain profiles show an
exponential decay of the signal without any indication of an instability. On
the contrary, more massive objects lead to signals with shorter quasinormal
ringing stages, due to the absence of a photon sphere peak, and with faster
decay rates even though the corresponding effective potentials develop even
deeper negative wells.

Figure 5: Effective potential (left) and time evolution (right) of an $\ell=2$
gravitational perturbation of the scalarized BH with $\ell_{\eta}=0.1$ for
varying mass $m$.

Figure 6: Effective potential (left) and time evolution (right) of an $\ell=2$
gravitational perturbation of the scalarized BH with $\ell_{\eta}=1$ for
varying mass $m$. The cases we consider here correspond to intermediate size
BHs with $r_{h}\sim R_{AdS}$ ($m/\ell_{\eta}\sim\mathcal{O}(10^{0})$). The
responses were shifted in time for illustration purposes.

Figure 7: Effective potential (left) and time evolution (right) of an $\ell=1$
gravitational perturbation of the scalarized BH with $\ell_{\eta}=5$ for
varying mass $m$. The cases we consider here correspond to large BHs with
$r_{h}>>R_{AdS}$ ($m/\ell_{\eta}\sim\mathcal{O}(10^{2})$). A shift in time and
amplitude was applied on the responses for illustration purposes.

Figure 8: Time evolution of an $\ell=2$ gravitational perturbation of the
scalarized BH with fixed ratio $m/\ell_{\eta}\simeq 0.1$ (left),
$m/\ell_{\eta}\simeq 0.3$ (middle) and $m/\ell_{\eta}=1$ (right).

From the above, we conclude that the scalarized BH spacetime is modally stable
under axial gravitational perturbations, where the qualitative features of the
response depend solely on the ratio $m/\ell_{\eta}$. In Fig. 8 we demonstrate
the above statement for the case of intermediate size BHs. Our numerics show
that a similar analogy occurs irregardless of the BH’s size. Even though the
source of echoes in our BH is related to the asymptotics of spacetime, and not
to the nature of the near-horizon regime, our results are in accordance with
perturbations in wormholes with decreasing throat radii [55] and black-bounce
models, which transition from regular BHs to wormholes [75].

##  VI Discussion

In this work we studied static and spherically symmetric solutions of a
Horndeski subclass which includes a massless scalar field non-minimally
coupled to the Einstein tensor. Such theory admits an exact BH solution
dressed with scalar hair. We have studied the stability of such solution under
axial perturbations and report that it is gravitationally stable at the linear
level. The qualitative features of the ringdown waveform depend solely on
ratio of the two available parameters of spacetime, namely the black hole mass
$m$ and non-minimal coupling strength $\ell_{\eta}$. We have further
demonstrated that as $m/\ell_{\eta}$ increases, the gravitational-wave
ringdown transitions from one with echoes to one with a typical quasinormal
ringing followed by a late-time tail.

In a recent analysis [76], a class of mechanical models were studied, where a
canonical degree of freedom interacts with another one with a negative kinetic
term, i.e. with a ghost. Surprisingly, it was shown that the classical motion
of the system is completely stable for all initial conditions, even though one
would expected that such system to be unstable due to the presence of a ghost
field. In our case, we have dealt with a conceptually analog system,
consisting of a scalarized BH for which the kinetic energy of the scalar hair
can be positive or negative (first degree of freedom) provided that the
strength of the non-minimal coupling to the Einstein tensor has the opposite
sign (second degree of freedom), being attractive of repulsive respectively.
Regardless of the case, we find that the BH is stable under axial
perturbations, thus providing an illustration that the purely classical
analysis in [76] can potentially apply to perturbed gravitating systems.
Therefore, the interplay of the kinetic energy of the scalar field and the
strength of its coupling to the Einstein tensor deserves further study which
we leave for the future.

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# Elastic-Link for Binarized Neural Network

Jie Hu1, Ziheng Wu3, Vince Tan4, Zhilin Lu2, Mengze Zeng4, Enhua Wu1,5
corresponding author

###### Abstract

Recent work has shown that Binarized Neural Networks (BNNs) are able to
greatly reduce computational costs and memory footprints, facilitating model
deployment on resource-constrained devices. However, in comparison to their
full-precision counterparts, BNNs suffer from severe accuracy degradation.
Research aiming to reduce this accuracy gap has thus far largely focused on
specific network architectures with few or no $1\times 1$ convolutional
layers, for which standard binarization methods do not work well. Because
$1\times 1$ convolutions are common in the design of modern architectures
(e.g. GoogleNet, ResNet, DenseNet), it is crucial to develop a method to
binarize them effectively for BNNs to be more widely adopted. In this work, we
propose an “Elastic-Link” (EL) module to enrich information flow within a BNN
by adaptively adding real-valued input features to the subsequent
convolutional output features. The proposed EL module is easily implemented
and can be used in conjunction with other methods for BNNs. We demonstrate
that adding EL to BNNs produces a significant improvement on the challenging
large-scale ImageNet dataset. For example, we raise the top-1 accuracy of
binarized ResNet26 from 57.9% to 64.0%. EL also aids convergence in the
training of binarized MobileNet, for which a top-1 accuracy of 56.4% is
achieved. Finally, with the integration of ReActNet, it yields a new state-of-
the-art result of 71.9% top-1 accuracy.

## Introduction

Convolutional Neural Networks (CNNs) have led to a series of breakthroughs for
a variety of visual tasks (Krizhevsky, Sutskever, and Hinton 2012; Long,
Shelhamer, and Darrell 2014; Ren et al. 2015; Toshev and Szegedy 2014; Zhu et
al. 2016b). However, the challenge of resource constraints in terms of latency
and memory storage is often faced when deploying CNNs on mobile or embedded
devices. Previous work (Cai et al. 2017; Jacob et al. 2018; McKinstry et al.
2019; Jain et al. 2020; Shuang Wu and Shi 2016; Sung, Shin, and Hwang 2015;
Yang et al. 2019; Zhou et al. 2016) has demonstrated that quantizing the real-
valued weights and activations of CNNs into low-precision representations can
reduce memory footprint while still achieving good performances. This class of
methods allows fixed-point arithmetic to be applied, which substantially
accelerates inference and reduces energy costs. Taken to an extreme, both the
weights and the activations can be represented with binary tensors {-1, +1}.
Such networks are termed Binarized Neural Networks (BNNs) (Courbariaux et al.
2016). In BNNs, arithmetic operations for convolutions can be replaced by the
more efficient xnor and bitcount operations.

Figure 1: Example images illustrating the same features on full-precision
ResNet26 (2nd column), Bi-Real-ResNet26 (3rd column) and our proposed EL-
ResNet26 (4th column). Figure 2: Diagram of the Elastic-Link module. $\oplus$
denotes element-wise summation. The process of applying the scaling factor to
activations is depicted within the dashed box, indicating that it is omitted
in some cases for better performance. $GetScale$ and $ApplyScale$ operations
refer to XNOR-Net (Rastegari et al. 2016).

However, BNNs suffer from significant accuracy degradation as a result of
information loss at each binarized layer. To conduct binarization, a real-
valued signal is passed through the $Sign$ activation, which eliminates the
signal’s amplitude and retains only its sign information. Because this process
is irreversible, information loss increases with each layer of the BNN.
Therefore, a central challenge to improving BNN accuracy is the reduction of
this information loss.

One approach seeks to minimize the quantization error between the real-valued
and binary forms of the weights and activations. Rastegari et al. (2016)
utilizes scaling factors to reduce the euclidean distances between the two
forms. More recently, Liu et al. (2018) employs a sophisticated fine-tuning
strategy from a full-precision network with customized gradient approximation
methods. Liu et al. (2018) additionally proposes a shortcut connection to
forward the real-valued activation, drastically reducing the extent of
information loss. However, these methods apply best to networks which consist
primarily of $3\times 3$ or $5\times 5$ convolutions, such as AlexNet
(Krizhevsky, Sutskever, and Hinton 2012), VGGNet (Simonyan and Zisserman 2015)
and Basic-Block ResNet (He et al. 2016).

Performing binarization with the above methods on networks in which $1\times
1$ convolutions play a crucial role - for example, GoogleNet (Szegedy et al.
2015), Bottleneck ResNet or efficient networks with separated convolutions
(Howard et al. 2017) \- causes substantially greater accuracy degradations.
$1\times 1$ convolutions fuse information across channels and are already used
to reduce computational cost via dimensionality reduction. We hypothesize that
the marginal information loss from binarization is the proverbial last straw.
(Howard et al. 2017) observed that the training or fine-tuning of binarized
MobileNet fails to even converge, giving credence to this hypothesis.

In order to make binarization more widely applicable, we introduce an
effective and universal module named “Elastic-Link” (EL). In order to
compensate for the loss incurred by binarization, we adaptively add the real-
valued input features (i.e. the features before feeding into the binarization
function) to the output features of the subsequent convolution to retain the
original real-valued signal. Liu et al. (2018) demonstrated that adding extra
shortcut connections, implemented by an element-wise summation, on the Basic-
Block ResNet (He et al. 2016) produces considerable improvement in accuracy.
This can be viewed as a special case of our proposed EL in which the input and
output have the same shape. To generalize this finding, we develop a method to
enable feature addition even if the feature size is changed by the
convolution. EL uses a Squeeze or Expand operation to align the feature sizes
between the input and the output. Furthermore, we do not simply perform a
direct summation after the Squeeze or Expand operation, but rather learn a
scaling factor to balance the relative extents of preserving the real-valued
information and convolutional transformation, unifying these mechanisms and
fusing them in a learnable, light-weight manner. EL is applicable to any
architecture without structural limitations. To better visualize the effects
of information preservation, we illustrate the feature-maps of a full-
precision model, a Bi-Real model and our model with EL separately in Fig. 1.
The feature maps of our EL model show clear object contours and the retention
of important information for recognition, with less noise. By contrast, the
foreground and background in the Bi-Real model are not as easily
discriminated. We believe that the EL module benefits information flow in the
binarized neural networks.

Moreover, as shown in Fig. 2, the design of the EL module is simple and can be
directly applied to existing modern architectures. To assess the effectiveness
of EL, we conduct extensive experiments on the ImageNet dataset. We outperform
the current state-of-the-art result with a top-1 accuracy of 68.9%. We also
contribute comprehensive ablation studies and discussions to further
understanding of the intrinsic characteristics of BNNs.

## Related Work

Quantized Weights with Real-Valued Activations. Restricting weights to be
either +1 or -1 allows the time-consuming multiply-accumulate (MAC) operations
to be replaced with simple addition operations. Recent studies (Courbariaux,
Bengio, and David 2015; Rastegari et al. 2016) rely on this insight and employ
the straight-through estimator (STE) (Bengio, LšŠonard, and Courville 2013) to
tackle non-differentiability in the back-propagation of gradients during
training. Courbariaux et al. proposed BinaryConnect (Courbariaux, Bengio, and
David 2015) which drastically decreases computational complexity and storage
requirements while achieving good results on the CIFAR-10 and SVHN datasets,
but lacks experiments on large-scale datasets like ImageNet (Russakovsky et
al. 2015). In Binarized Weight Networks (BWN) (Rastegari et al. 2016), the
full-precision weights are binarized during each forward and backward pass on
the fly, updating only the full-precision weights. BWN achieves a notable
accuracy increase, especially on large-scale classification tasks. Finally,
TWN (Li, Zhang, and Liu 2016) and TTQ (Zhu et al. 2016a) use ternary instead
of binary weights {$-\alpha^{-}$, 0, $\alpha^{+}$} to pass more information.

Quantized Weights and Activations. Another approach which has recently gained
popularity restricts both the weights and the activations to {-1, +1}. This
allows the convolutional operations to be completely replaced by the efficient
xnor and bitcount operations (Courbariaux et al. 2016; Rastegari et al. 2016),
thus gaining extreme efficiency. XNOR-Net (Rastegari et al. 2016) is one of
the most representative works of this approach and achieved remarkable
accuracy with various networks. Bi-Real net (Liu et al. 2018) introduced
additional shortcuts to retain the information of the real-valued activations,
so as to alleviate information loss from binarization. The paper additionally
introduced a custom gradient approximation method with a sophisticated fine-
tuning strategy to further increase accuracy. Recently, Bethge et al. (2019)
demonstrated that the said fine-tuning strategy and gradient approximation
methods are not necessary for training BNNs - even without these, the authors
achieved superior accuracy training from scratch with the simple straight-
through estimator. Another method proposed to counteract the information
degradation phenomenon is the linear combination of multiple binary weights to
approximate full-precision weights, as introduced by ABC-net (Lin, Zhao, and
Pan 2017). Finally, TBN (Wan et al. 2018) takes ternary inputs{-1, 0, +1} and
binary weights with scale factors {$-\alpha$, $\alpha$}, demonstrating the
method on both image classification and object detection tasks.

## Methodology

In this section, we first revisit the standard process for training BNNs, then
subsequently introduce a novel module, “Elastic-Link” (EL), to reduce the
information loss in BNNs. Lastly, we demonstrate the EL module on Bottleneck
ResNet (He et al. 2016) and MobileNet (Howard et al. 2017).

### Gradient Approximation

It is standard to use the $Sign$ function to binarize a CNN. Real values are
converted to the binary set of {-1, +1} by the following equation:

$Sign(x)=\left\\{\begin{aligned} &+1&if\quad x\geq 0\\\ &-1&otherwise\\\
\end{aligned}\right.$ $None$

where $x$ refers to a real-valued weight or input/activation. To facilitate
training, binarization is typically executed on the fly and only the real-
valued weights are updated by the gradients, as described in (Courbariaux et
al. 2016; Rastegari et al. 2016). During inference, the real-valued weights
are unused and binary weights are used as a drop-in replacement.

In the backward pass, since the $Sign$ function is non-differentiable
everywhere, an approximation is used. In this work, we follow the conventional
“straight through estimator” (STE) (Bengio, LšŠonard, and Courville 2013)
unless otherwise stated. The approximated gradient in STE is formulated as:

$\frac{\partial Sign(x)}{\partial x}=\left\\{\begin{aligned} &1&if-1\leq x\leq
1\\\ &0&otherwise\\\ \end{aligned}\right.$ $None$

### Scaling Factor for Weights and Activations

As proposed in XNOR-Net (Rastegari et al. 2016), a binary convolutional
operation can be given as follows:

$\displaystyle
BinConv\left(\mathbf{A},\mathbf{W}\right)\approx\left(Sign\left(\mathbf{A}\right)\otimes
Sign\left(\mathbf{W}\right)\right)\odot\mathbf{K}\alpha$ $None$

where $\mathbf{A}\in\mathbb{R}^{c\times h\times w}$ is the real-valued input
activation and $\mathbf{W}\in\mathbb{R}^{c\times k_{h}\times k_{w}}$ is the
real-valued convolutional kernel. Here ($c$, $h$, $w$, $k_{h}$, $k_{w}$) refer
to number of input channels, input height, input width, kernel height, and
kernel width respectively. $\otimes$ denotes the efficient XNOR-Bitcounting
operation (Rastegari et al. 2016) that replaces the time-consuming arithmetic
operations.

$\alpha$ is a scaling factor given by a vector L1-Normalization
$\alpha=\frac{1}{n}\|\mathbf{W}\|_{\ell 1}$, which helps minimize the L2 error
between the real-valued weights and the binary weights with scalar coefficient
$\alpha$. $K$ is a two-dimensional scaling matrix for the input activation,
whose shape corresponds to the convolutional output. It is given by setting
each element with the same principle as $\alpha$. XNOR-Net concluded that
$\alpha$ is more effective than $K$, which can even be entirely ignored for
simplicity due to the relatively small improvement realized. Similarly, Liu et
al. (2018) and Lin, Zhao, and Pan (2017) validated the effectiveness of
$\alpha$ across various networks and datasets. Recently, Bethge et al. (2019)
found that these scaling factors did not result in accuracy gains when
BatchNorm (Ioffe and Szegedy 2015) is applied after each convolutional layer.
In our experiments, we did observe the same experimental phenomenon with the
Basic-Block ResNet, which is constructed with only $3\times 3$ convolutions.
However, we find that this principle holds only for $3\times 3$ convolutions.
When binarizing $1\times 1$ convolutions, the scaling factor is still
influential, which will be elaborated on later section.

### Elastic-Link

Inspired by the shortcut connection mechanism, we use an element-wise
summation operation to add real-valued input features to the output features
generated by a binary convolution. In our Elastic-Link module, instead of an
identity shortcut, we apply either a Squeeze or an Expand operation when a
convolution alters the feature shape. A diagram of our proposed Elastic-Link
module is shown in Fig. 2. Formally, let $X_{r}$ denote the real-valued input
feature where $\mathbf{X_{r}}\in\mathbb{R}^{H_{i}\times W_{i}\times C_{i}}$.
We binarize $X_{r}$ through a Sign activation function and obtain the binary
$X_{b}$. Next, a binary convolution and standard BatchNorm are applied to
obtain the convolutional output feature $Y_{r}^{n}\in\mathbb{R}^{H_{o}\times
W_{o}\times C_{o}}$.

In order to compensate for the information loss, we add the real-valued input
$X_{r}$ to the normalized convolutional output $Y_{r}^{n}$. If the input size
is equal to the convolutional output size, an identity shortcut connection
with element-wise summation is applied as proposed in Bi-Real net (Liu et al.
2018). However, this condition is rarely true. In practice, a convolution
operation usually changes the number of channels, and occasionally changes the
height and width as well. In the channel-reduction case, we design a Squeeze
operation in which the real-valued input $X_{r}$ is split into multiple groups
along the channel axis without overlap. The number of channels for each group
is $\lceil\frac{C_{i}}{C_{o}}\rceil$. We additionally zero-pad the features on
input $X_{r}$ to ensure that $C_{i}$ can be exactly divided by $C_{o}$. Next,
we sum these feature groups together to yield the squeezed features which will
be of the same shape as $Y_{r}^{n}$. To reduce the effect of amplitude
increase from the summation, and to offer a self-balancing tradeoff between
information preservation and transformation, we divide the squeezed feature by
a learnable scalar $\gamma$ initialized as the number of groups. We take a
similar approach for the channel expansion case. In an Expand operation, the
real-valued input feature is repeated several times and then concatenated to
match the feature size of the convolutional output. The expanded feature is
correspondingly divided by same learnable factor of $\gamma$. Finally, the
output of the Squeeze or Expand operation is added to the convolutional output
feature $Y_{r}^{n}$, giving the overall output of the binarized convolution
module. If spatial downsampling is required, a $2\times 2$ max-pooling with
stride 2 is applied before Squeeze or Expand to ensure spatial compatibility.
The Elastic-Link module is formulated as follows:

$\displaystyle
EL\left(\mathbf{X_{r}},\mathbf{W},\gamma\right)=BN\left(BinConv\left(\mathbf{X_{r}},\mathbf{W}\right)\right)$
$None$ $\displaystyle+SEI\left(\mathbf{X_{r}},\gamma\right)$

Where $\mathbf{X_{r}}$ refers to the real-valued input activation and
$\mathbf{W}$ refers to the convolutional weight. $BN$ and $Sign$ refer to the
BatchNorm and Sign function respectively. $SEI$ refers to $Squeeze$, $Expand$
or $Identity$ operation, depending on the ratio of input and output channels.
$\gamma$ is the aforementioned learnable parameter that balances information
preservation and transformation. We initialize $\gamma$ by the following
equation and optimize it through back-propagation:

$\gamma=\left\\{\begin{aligned}
&\lceil\frac{C_{i}}{C_{o}}\rceil,&C_{i}>=C_{o}\\\
&\lceil\frac{C_{o}}{C_{i}}\rceil,&C_{i}<C_{o}\\\ \end{aligned}\right.$ $None$

Where $C_{i}$ and $C_{o}$ refer to the number of channels for the input and
output of a convolution respectively. $\lceil\;\rceil$ denote the ceiling or
round-up operation. The max-pooling operation in the downsample case as well
as the additional ReLU for efficient networks are omitted here for clarity.

Instantiations. The Elastic-Link module easily plugs into many modern
architectures. Taking Bottleneck ResNet as an example, we integrate the
Elastic-Link module into ResNet26 which consists of 8 bottleneck blocks. All
bottleneck blocks are replaced by EL-Bottlenecks, as depicted in Fig. 3. The
first convolution (of kernel size $7\times 7$) and the classification layers
remain full-precision to keep essential information at the input and output of
the whole network. The downsampling shortcut in the first block of each stage,
originally a $1\times 1$ convolution of stride 2, is replaced by a $2\times 2$
average pooling with stride 2 followed by a $1\times 1$ convolution in full-
precision. By integrating an EL module into the first $1\times 1$ convolution
of each Bottleneck block, more full-precision information flows to the middle
$3\times 3$ convolution, which is essential for capturing features at larger
receptive fields.

Next we apply the EL module to efficient networks composed of separable
convolutions. To the best of our knowledge, efficient networks have so far
been considered incompatible with binarization. MobileNet (Howard et al. 2017)
is one of the most representative efficient architectures. By adding Elastic-
Link to each pointwise convolution and depthwise convolution (see Fig. 3), we
are able to overcome the non-convergence problem typically encountered in
training binarized MobileNet. We additionally find that keeping the ReLU
activation achieves better performance. Similar to the ResNet case, we keep
the first convolution, classifier and downsample components at full-precision.

Figure 3: Schema of EL-Bottleneck module (Left) and EL-MobileNet module
(Right).

## Computational Complexity

For the Elastic-Link module to be considered practical, it must offer a good
tradeoff between improved performance and additional computational burden. To
illustrate the increased complexity associated with the Elastic-Link module,
we compare Bi-ResNet50 with EL-ResNet50. The additional computational cost
incurred by the EL module originates from the $\gamma$ scaling after the
Squeeze, Expand or Identity operation as well as the element-wise summation of
each $1\times 1$ convolution, because the Squeeze and Expand can be
implemented with address mapping without any overhead. In total, EL-ResNet50
requires an extra $\sim$8M FLOPs over Bi-ResNet50’s $\sim$300M FLOPs for a
single forward pass with an input image of $224\times 224$, corresponding to a
2.6% increase. The number of FLOPs is computed as described in (Liu et al.
2018). For a practical comparison, we use the BMXNet library (Yang et al.
2017) on an Intel Core i7-9700K CPU to measure the actual time taken. Bi-Real-
ResNet50 takes on average 22.2 ms for a single forward pass (over 10 runs),
compared to 22.9 ms for our proposed EL-ResNet50. We believe that this small
additional cost is justified by the increase in model performance.

## Experiments

In this section, we conduct extensive evaluations to demonstrate the
effectiveness of Elastic-Link for binarized neural networks on the large scale
image classification task ImageNet (Russakovsky et al. 2015). ImageNet is
challenging and often used to validate the performance of proposed methods in
BNNs. The dataset consists of about 1.28 million training images and 50
thousand validation images, annotated for 1000 classes.

### Training Details

During training, we perform binarization on-the-fly in the forward pass and
follow the STE gradient approximation strategy. Input images are resized such
that the shorter edge is 256 pixels, then randomly cropped to $224\times 224$
pixels, followed by a random horizontal flip. Mean channel subtraction is used
to normalize the input. All networks are trained from scratch using the Adam
optimizer without weight decay normalization. The entire training process
consists of 100 epochs with a mini-batch size of 256. The initial learning
rate is set to 1e-3 and decreases after the 50th and 80th epochs by a factor
of 10 each.

During inference, we center-crop patches of size $224\times 224$ from each
image on the validation set and report the top-1 accuracy for comparison.

### Effect of Scaling Activation

We have previously discussed the importance of the scaling weights and
activations in binarized neural networks. Scaling is intended to mitigate the
differences between the distributions of full-precision tensors and their
binary counterparts. However, we obverse that scaling only the weights results
in a minimal increase of accuracy when training from scratch, as previously
observed in (Bethge et al. 2019). We expect that scaling the activations will
greatly improve performance.

To investigate the effects of the scaling factor $K$ on activations, we
conduct experiments on Bi-Real nets - binarized ResNet26 with an additional
shortcut at the middle $3\times 3$ convolution of each block - here named
S-ResNet26 for convenience. ResNet26 is constructed from a set of homogenous
Bottleneck blocks which each comprises three convolutions: a first $1\times 1$
channel-reduction convolution to reduce computation burden, a middle $3\times
3$ convolution to capture spatial information, and a final channel-expansion
$1\times 1$ convolution to align the channel count with the input features for
residual connection.

The experimental results are shown in Table 1. Applying the scaling factor to
only the first channel-reduction convolution of each bottleneck block results
in a top-1 accuracy of 59.8%, a significant increase over the original 58.7%
wherein no scaling factors are applied. When scaling is also applied to the
other two convolutions of each block, a slight accuracy drop is observed.
Based on this phenomenon, we speculate that the scaling factor for activations
facilitates optimization of the channel-reduction convolution. Noting that
improper application of the scaling factor can cause adverse effects, we apply
scaling on only the first (channel-reduction) convolution of each block in our
subsequent experiments on Bottleneck ResNet.

### Integration with Basic-Block ResNet

We next apply the EL module to the Basic-Block ResNet to validate its
benefits. A Basic-Block is constructed by two successive isomorphic
convolutions, and thus a feature passing through the block maintains its
shape, except where downsampling is applied. In a Basic-Block, the EL module
resolves to the identity shortcut connection as proposed in Bi-Real nets (Liu
et al. 2018). The binarized ResNet model constructed using Basic-Blocks with
the EL module is almost the same as the Bi-Real net in terms of model
architecture. However, the Bi-Real net utilizes a multi-stage fine-tuning
strategy and the custom differentiable approximation ApproxSign, whereas the
EL models are trained from scratch with simple STE. As can be observed in the
results listed in table 3, both EL-ResNet18 and EL-ResNet34 are superior to
their Bi-Real variants by a margin of 2.2% and 1.0% in top-1 accuracy
respectively, demonstrating that a simple training schedule suffices to
achieve a well-trained binarized model.

### Integration with Bottleneck ResNet

We next investigate the effectiveness of the proposed EL module on Bottleneck
ResNet. Using the activation scaling strategy reported above, we obtain a
strong baseline on S-ResNet26. Subsequently, we apply the EL module to each
bottleneck block (see Fig. 3) and conduct extensive ablation studies. The
results are reported in Table 2 and we can obverse the following phenomena:

  * -

From the result of model $EL_{1}$, $EL_{2}$ and $EL_{3}$, we can see that
applying Elastic-Link on more convolutions within a network can monotonically
bring benefits ($59.8\%\rightarrow 61.8\%\rightarrow 62.1\%\rightarrow
63.2\%$), demonstrating the effectiveness of the EL module.

  * -

Comparing model $EL_{3}$ with $EL_{4}$, when the original residual shortcuts
are removed throughout the network, there is a significant drop of top-1
accuracy from 63.2% to 59.2%. This proves that although EL provides more
information via the additional connection for each convolution, residual
connections are still essential to forward primitive uncompressed information.

  * -

Comparing model $EL_{6}$ with $EL_{3}$, allowing $\gamma$ to be learnable
further increases the top-1 accuracy from 63.2% to 64.0%.

  * -

From the results of models $EL_{5}$ and $EL_{6}$, it can be observed that EL
is compatible with the scaling factor strategy and achieves better accuracy
when used with other techniques.

We term $EL_{6}$ in Table 2 as EL-ResNet26, and this is constructed using EL-
Bottleneck (see Fig. 3). Including EL with learnable $\gamma$ in the
convolutional layers results in dramatic gains, increasing top-1 accuracy by
an absolute value of 4.2%.

$K_{s}$ | $K_{i}$ | $K_{e}$ | Top-1
---|---|---|---
|  |  | 58.7
|  | ✓ | 58.5
✓ |  | ✓ | 59.3
✓ | ✓ |  | 59.8
✓ | ✓ | ✓ | 59.4
✓ |  |  | 59.8
Table 1: Top-1 accuracy (%) of S-ResNet26 variants on ImageNet. $K_{s}$, $K_{i}$ and $K_{e}$ mean applying activation scaling to the first convolution (channel-reduction), middle convolution (spatial) and last convolution (channel-expansion) of each bottleneck block respectively. | $K_{s}$ | $EL_{s}$ | $EL_{i}$ | $EL_{e}$ | $Id.$ | $\gamma_{l}$ | Top-1
---|---|---|---|---|---|---|---
Baseline | ✓ |  |  |  | ✓ |  | 59.8
$EL_{1}$ | ✓ | ✓ |  |  | ✓ |  | 61.8
$EL_{2}$ | ✓ | ✓ |  | ✓ | ✓ |  | 62.1
$EL_{3}$ | ✓ | ✓ | ✓ | ✓ | ✓ |  | 63.2
$EL_{4}$ | ✓ | ✓ | ✓ | ✓ |  |  | 59.2
$EL_{5}$ |  | ✓ | ✓ | ✓ | ✓ | ✓ | 63.6
$EL_{6}$ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | 64.0
Table 2: Top-1 accuracy (%) of binarized ResNet26 with different
configurations of EL on the ImageNet validation set. $EL_{s}$, $EL_{i}$ and
$EL_{e}$ denotes applying Elastic-Link to the first convolution (channel-
reduction), middle convolution (spatial) and the last convolution (channel-
expansion) of each bottleneck block respectively. $Id.$ means keeping residual
connections, and $\gamma_{l}$ means setting $\gamma$ to be learnable in the EL
module.

### Results on Deeper and Efficient Networks

In order to validate the generalizability of the EL module, we apply it to
deeper networks, such as ResNet50 (He et al. 2016), which are rarely reported
in existing literature. The results in Table 3 shows that EL-ResNet50 (65.6%
top-1 accuracy) is superior to the Bi-Real-ResNet50 (62.7% top-1 accuracy) by
a significant margin, proving that the EL module maintains strong performance
even as the network grows deeper.

We next demonstrate the efficacy of the EL module on MobileNet. This is much
more challenging as separable convolutions are weak at capturing spatial
features. More concretely, the depthwise convolutions lack inter-channel
information, while the pointwise convolutions which are expected to perform
the inter-channel aggregation are particularly sensitive to information loss.
As such, binarizing a MobileNet causes serious degradation in the network’s
ability to extract strong features. We add additional shortcut connection on
depthwise convolutions and also perform the multi-stage fine-tuning strategy
as discussed in Bi-Real net (Liu et al. 2018). However, this binarized
MobileNet still failed to converge in its training loss. Subsequently, we
included the EL module into MobileNet (see Fig. 3), and obtained the results
shown in Table 3. The EL module not only enabled convergence, but also
achieved an excellent performance of 56.4% top-1 accuracy, a strong baseline
for future work on binarized efficient networks.

| Bottleneck Block | Efficient Block | Basic Block
---|---|---|---
| RN26 | RN50 | MobileNet | RN18 | RN34
Full-Precision | 72.5 | 75.9 | 70.6 | 69.3 | 71.5
XNOR (Rastegari et al. 2016) | 52.1 | 54.2 | Not Converge | 51.2 | 53.2
ABCNet (Lin, Zhao, and Pan 2017) | 45.2 | 52.9 | Not Converge | 42.7 | 52.4
TBN (Wan et al. 2018) | - | - | - | 55.6 | 58.2
Bi-Real (Liu et al. 2018) | 57.8 | 62.7 | Not Converge | 56.4 | 62.2
BinaryE (Bethge et al. 2019) | 57.9 | 61.2 | Not Converge | 56.7 | 59.5
CI-Net (Wang et al. 2019) | - | - | - | 56.7 | 62.4
XNOR++ (Bulat and Tzimiropoulos 2019) | - | - | - | 57.1 | -
GBCN (Liu et al. 2019) | - | - | - | 57.8 | -
MoBiNet (Phan et al. 2020) | - | - | 54.4 | - | -
EL (Ours) | 64.0 | 65.6 | 56.4 | 60.1 | 63.2
Real-to-Bin (Brais, Bulat, and Tzimiropoulos 2020) | 64.8 | 65.9 | 54.8 | 65.4 | 66.1
EL† (Ours) | 67.1 | 68.9 | 61.2 | 65.7 | 66.5

Table 3: Comparison of top-1 accuracy (%) on the ImageNet validation set. EL
is specifically designed to improve $1\times 1$ convolution, and its
superiority compared to other works is correspondingly obvious across networks
constructed with Bottleneck or Efficient Block. For reference, we also include
networks built with Basic Block (which lack $1\times 1$ convolutions). “RN” is
short for ResNet. EL† refers to including Real-to-Bin (Brais, Bulat, and
Tzimiropoulos 2020) in our proposed EL networks.

### Comparison with the State-of-the-Art

Finally, in order to demonstrate the superiority of our proposed EL in BNNs,
we compare our results with other work on binary weights and activations. The
main results on ImageNet are listed in Table 3. We demonstrate that binarized
networks integrating with our EL module obtain considerable gains and also
achieve the best performance against previous methods. At the same time, our
approach integrates well with the Real-to-Binary (Brais, Bulat, and
Tzimiropoulos 2020) approach, which is important for practical applications.
Remarkably, we also integrated EL with the state-of-the-art result ReActNet-C
(Liu et al. 2020) with top-1 accuracy of 71.4% on the Reduction block by
replacing the duplicate activation parts with our EL modules. We obtained a
new state-of-the-art result with a top-1 accuracy of 71.9%.

## Discussion

### Effect of Binary $1\times 1$ Convolution

An EL module propagates the real-valued input signal without a change of
receptive field regardless of the change in channel depth. Similarly, a
$1\times 1$ convolution linearly fuses inter-channel information also without
a change of receptive field. Our experiments have comprehensively demonstrated
the effectiveness of the EL module as a solution to the information loss
incurred by binary $1\times 1$ convolutions. We next investigate whether the
EL module is effective outside of $1\times 1$ convolutions. We set up a
comparison on S-ResNet26 by replacing the first convolution (channel-
reduction) of each block with the following alternatives: 1) a full-precision
convolution, 2) an EL module and 3) an EL module without the inner $1\times 1$
convolution. The result in Table 4 shows that the model with the complete EL
module reaches almost the accuracy of the model with the full-precision
convolution. Removing the $1\times 1$ convolution from the EL module still
results in an absolute 1.2% improvement in top-1 accuracy. We therefore
conclude that between the two mechanisms of fusing inter-channel information
by binary $1\times 1$ convolution and forwarding real-valued information, the
latter is more crucial to improve accuracy in BNNs.

### The Role of the $\gamma$ Factor

The learnable factor $\gamma$ for the EL module controls the proportion of
information fusion between the real-valued input features and the
convolutional output features. In order to investigate the behavior of the
$\gamma$ factor, we study the distribution of $\gamma$ from EL-ResNet26 with
respect to the depth within the model. Fig. 4 illustrates the value changes of
all 16 $\gamma$ across 8 EL-Bottleneck blocks after training. Relative to
their initial values, all the $\gamma$ in the EL module that were applied to
the channel-reduction convolutions with kernel size $1\times 1$ increased
significantly. This indicates that less original information is retained as
input into the subsequent spatial convolution. In contrast, the $\gamma$
values in the EL module that were applied to the channel-expansion
convolutions slightly decreased, indicating that more information from
previous spatial convolution is forwarded to the next block. This phenomenon
is within expectations because if all the original information before a
convolution is forwarded, the said convolution would be of relatively little
utility as a feature transformer. It is also worth noting that all $\gamma$
applied to the channel-reduction convolutions in which downsampling occurred
increased substantially during training (i.e. little original information is
retained), as the convolutions in these shortcuts are full-precision, and thus
are able to output more accurate information.

FP $1\times 1$ | Binary $1\times 1$ | $EL^{-}$ | Top-1
---|---|---|---
| ✓ |  | 59.4
✓ |  |  | 62.0
|  | ✓ | 60.6
| ✓ | ✓ | 61.8
Table 4: Top-1 accuracy (%) of various S-ResNet26 with different forms for the
first convolutions of each block (the $1\times 1$, channel-reduction ones). FP
$1\times 1$ refers to full-precision $1\times 1$ convolution. Binary $1\times
1$ refers to binarized $1\times 1$ convolution. $EL^{-}$ refers to removing
the $1\times 1$ convolution from the original EL module, retaining only the
Squeeze operation. Figure 4: Learnable factor $\gamma$ in all the EL module of
EL-ResNet26 on ImageNet. The red histogram indicates an increase and the green
histogram indicates a decrease compared to the initial value after training.

## Conclusion

In this work, we proposed a novel Elastic-Link module for binarized neural
networks. The Elastic-Link module introduces a connectivity mechanism to
adaptively fuse real-valued input features and convolutional output features
regardless of whether or not the feature-size is altered in the convolution. A
learnable scaling factor enables an optimized tradeoff between information
preservation and feature transformation, significantly reducing information
degradation in the binarized form. The Elastic-Link module can be easily
embedded into any architecture. It greatly enhances the representational
ability of BNNs, especially for networks in which $1\times 1$ convolutions are
indispensable, such as Bottleneck ResNet and MobileNet. Extensive experiments
demonstrate that binarized networks with Elastic-Link achieve considerable
performance gains with negligible computational overhead. Moreover, the module
is compatible with other techniques that have been shown to improve accuracy,
e.g. the application of a scaling factor to activations. Combining Elastic-
Link with such techniques achieves a new state-of-the-art result. A key
challenge remaining is the observed increase in the degree of information
degradation with network depth. We plan to explore more effective approaches
to counteract this phenomenon in future work.

## Acknowledgments

The work is supported in part by NSFC Grants (62072449, 61972271), Macao FDCT
Grant (0018/2019/AKP) and Guangdong-Hongkong-Macao Joint Research Grant
(2020B1515130004).

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