# Transport characteristics of a $\mathcal{PT}$-symmetric non-Hermitian
system:
Effect of environmental interaction

Sudin Ganguly<EMAIL_ADDRESS>Department of Physics, School of Applied
Sciences, University of Science and Technology Meghalaya, Ri-Bhoi-793 101,
India  Souvik Roy  Physics and Applied Mathematics Unit, Indian Statistical
Institute, 203 Barrackpore Trunk Road, Kolkata-700 108, India  Santanu K.
Maiti<EMAIL_ADDRESS>Physics and Applied Mathematics Unit, Indian
Statistical Institute, 203 Barrackpore Trunk Road, Kolkata-700 108, India

###### Abstract

The environmental influence is inevitable but often ignored in the study of
electronic transport properties of small-scale systems. Such an environment-
mediated interaction can generally be described by a parity-time symmetric
non-Hermitian system with a balanced distribution of physical gain and loss.
It is quite known in the literature that along with the conventional junction
current, another current called bias-driven circular current can be
established in a loop geometry depending upon the junction configuration. This
current, further, induces a strong magnetic field that can even reach to few
Tesla. What will happen to these quantities when the system interacts with its
surrounding environment? Would it exhibit a detrimental response? We address
such issues considering a two-terminal ring geometry where the junction setup
is described within a tight-binding framework. All the transport quantities
are evaluated using the standard Green’s function formalism based on the
Landauer-Büttiker approach.

###### pacs:

72.80.Vp, 72.25.-b, 73.23.Ad,

##  I Introduction

Recently, the study of non-Hermitian (NH) systems protected by parity-time
$(\mathcal{PT})$ symmetry bender1  has become a subject of great interest
where the environmental influence can be taken into account. The non-
Hermiticity arises by introducing complex potentials to the Hamiltonian which
represent the physical gain and loss of the system. The
$\mathcal{PT}$-symmetry is protected in NH systems when the loss and gain have
a balanced distribution bender2 . The Hamiltonian of such
$\mathcal{PT}$-symmetric NH systems may have entirely real eigenvalue spectra
bender1 ; bender3 . Owing to this remarkable feature, various
$\mathcal{PT}$-symmetric NH systems have been explored in the field of optics
optics1 ; optics2 ; optics3 ; optics4 ; kato ; rogen ; ganga ; ruter , where
two hallmark features of $\mathcal{PT}$-symmetric systems, namely, the
existence of exceptional points kato ; rogen ; ganga  and non-orthogonal
eigenmodes ruter  have been observed. Several interesting phenomena have been
reported so far near the vicinity of the exceptional point, such as, diffusive
coherent transport diffu , topological states topo1 ; topo2 ; topo3 ; topo4 ,
chirality chiral1 ; chiral2 , possibility to stop light light-stop , etc. The
$\mathcal{PT}$-symmetric systems have been discussed in other fields also,
like in atomic physics atomic1 ; atomic2 ; atomic3 ; atomic4 , electronics
electronics1 ; electronics2 , magnetic metamaterials magmeta , etc. Overall,
significant theoretical and experimental efforts have been made in the above-
mentioned works and the study of physical gain and loss is still an active
research field.

Simultaneously, some works paid attention to the quantum transport properties
in different kinds of tight-binding lattices where localization effect
localiz1 ; localiz2 ; localiz3 , transmission characteristics and some related
issues have been investigated trans-char1 ; trans-char2 ; trans-char3 ; trans-
char4 . In all these studies, $\mathcal{PT}$-symmetric complex potentials were
found to have a significant effect in changing the transport properties of the
systems. Likewise, it is also important to examine the effect of environmental
influence through gain and loss on the current distribution in different arms
of a nano junction containing simple and/or complex loop substructures that
have not been explored so far to the best of our concern. The study of current
distribution and the generation of bias-induced circular current in loop
geometries cc1 ; cc2 ; cc3 ; cc4 ; cc5  are some of the key aspects in
mesoscopic physics which we intend to explore in the present work.

Circular current, flowing in a conducting ring, can induce a strong local
magnetic field ($\sim$ Tesla) at the center of the ring, depending on the
physical condition of the junction configuration. Furthermore, the orientation
of a local spin that is placed at or near the center of the loop can be
manipulated with atomic precision by means of the induced magnetic field. This
particular feature can be utilized as a basic unit for quantum computers qc1 ;
qc2 ; qc3 ; qc4 . Few efforts have been made towards this direction for the
generation and possible tuning of the circular current and induced local
magnetic field in different kinds of quantum loop geometries cho ; lidar ;
pershin-prb ; anda ; cc5 ; patra-scirep ; sudin-jpcm . In these works, the
reported values of the induced magnetic field vary from several mTesla to a
few Tesla. In the present work, we aim to probe further in this regard.
Specifically, we wish to explore the environmental influence in a ring
geometry that has a balanced distribution of physical gain and loss. How the
transport current, circular current, and induced magnetic field behave in the
presence of the environment-mediated interaction, are the key things that we
want to investigate in the present work. Moreover, since the quantum
interference effect in different arms of the ring strongly depends on the
lead-ring interface geometry and ring size, we also study the behavior of the
transport properties under these scenarios.

As it is known that non-Hermiticity occurs when the exchange of energy or
particles takes place between a system and its surrounding environment. The
non-Hermiticity can be introduced in a system by assuming imaginary on-site
potential at different lattice sites, where different signs of the complex
potential represent gain and loss in the system. Asymmetric hopping integrals
also make the systems non-Hermitian ezawa . In the optical framework,
introducing complex on-site potentials or asymmetric hopping integrals are
quite easy by employing topoelectrical circuits electronics1 ; electronics2 ;
ezawa ; lcr2 ; lcr3 . We can introduce gain and loss to our system by
attaching non-dispersive (energy independent) 1D leads through which
electrons/energy can enter into the system or coming out of it bae-2002 . Such
a non-dispersive 1D lead can be viewed as a quantum dot (QD) with complex site
potential.

In non-equilibrium Green’s function (NEGF) formalism, the effect of a lead is
expressed through the self-energy term, which is a complex quantity. The sign
of the imaginary part of the self-energy term decides whether the lead takes
out electrons from the system or injects them into it. If the chemical
potential of the lead is higher than that of a QD, electrons will go in and
vice-versa. A similar kind of complex potentials are generally used to take
into account the dephasing effect through Büttiker voltage probes bruno ;
dutta-prb2007 ; niko . To incorporate dephasing, Büttiker introduced an
elegant concept of voltage probes butti  attached to the active region where
no net current flows through them. Here, one electron enters into the probe,
and another electron comes out which is not coherent with the ingoing one.
This is equivalent to adding complex energy to the on-site potential in the
Hamiltonian of the system. Particularly, we consider the non-dispersive 1D
leads, where the self-energy terms are purely imaginary quantities. This is
possible by tuning the site energy of the leads with a controlled gate
potential such that the site energy is equal to the energy of the incoming
electrons. For a better understanding of this fact, a mathematical description
is given in Appendix A. Motivated by this, we have assumed complex on-site
potentials in the QDs, through which an exchange of energy/particles can occur
between the system and its immediate environment.

We describe our system within the tight-binding framework. The quantum ring
interacts with its surrounding environment through the quantum dots (Fig. 1).
The transport quantities, such as the transmission coefficient, transport and
circular currents, induced local magnetic field, etc. are evaluated using the
standard Green’s function formalism based on the Landauer-Büttiker approach.
The key findings of our work are: (i) real and complex eigenvalue spectra
exhibit several interesting features such as the broken degeneracy levels in
the presence of environment-mediated interaction, the appearance of
exceptional points (EPs), etc., (ii) transmission spectra are significantly
modified around the band center, (iii) transport and junction currents show
high values, (iv) induced magnetic field is also reasonably large ($\sim$
Tesla), and finally, (v) pronounced effects of lead-ring interface geometry
and the ring size are also observed. Our analysis may lead to some interesting
features in transport phenomena in $\mathcal{PT}$-symmetric non-Hermitian
quantum systems.

Here it is relevant to note that generally for non-Hermitian systems, the
transmission coefficient may not be confined within unity. However, for our
chosen setup, the transmission coefficient always resides within unity. This
particular feature depends on the unitarity of the $S$-matrix. In Appendices B
and C, we give detailed derivation for that.

The rest of the paper is organized as follows. In the following section (Sec.
II), we begin by describing our $\mathcal{PT}$-symmetric NH quantum ring and
provide the necessary theoretical formulae to calculate the junction and
circular currents and the induced magnetic field. Subsequently, in Sec. III,
we include an elaborate discussion of the results where we have discussed the
effect of environment-mediated interaction on the transport properties. We end
with a summary of our results in Sec. IV. Separate three appendices are
included to describe how to implement imaginary site potential, transmission,
and reflection coefficients for the single-site system with only gain term,
and both gain and loss terms.

##  II Quantum system and theoretical framework

###  II.1 Tight-binding model

A $\mathcal{PT}$-symmetric NH quantum ring is schematically shown in Fig. 1,
where the sites of the ring are denoted with black balls. Each of these parent
lattice sites is again directly connected to two quantum dots (QDs),
representing the physical gain and loss of the system, and are denoted with
red and blue balls, respectively. The ring is coupled to two semi-infinite
one-dimensional (1D) perfect electrodes, namely,

Figure 1: (Color online) Schematic view of a $\mathcal{PT}$-symmetric non-
Hermitian ring attached to two electrodes (source ${\mathcal{S}}$ and drain
${\mathcal{D}}$). The sites of the ring (we can refer to them as parent
lattice sites) are represented by black balls. Each of these sites is attached
with two QDs representing the physical gain (red balls) and loss (blue balls)
of the system.

source (${\mathcal{S}}$) and drain (${\mathcal{D}}$), respectively. We couple
these electrodes asymmetrically into the ring, and this is the primary
condition to have a bias-driven circular current in the nano ring cc1 ; cc2 ;
patra-scirep ; sudin-jpcm ; patra-prb . We consider the quantum ring through a
tight-binding (TB) Hamiltonian, which is extremely suitable to describe a
physical system, especially in small-scale regions. The TB Hamiltonian of the
full setup can be written as a sum of the following sub-Hamiltonians

$H=H_{R}+H_{S}+H_{D}+H_{C}.$ (1)

The first term, $H_{R}$ represents the TB Hamiltonian for the quantum ring and
can be expressed as feng

$H_{R}=\sum\limits_{n}\epsilon_{n}c_{n}^{\dagger}c_{n}+t_{r}\sum\limits_{\langle
nm\rangle}\left(c_{n}^{\dagger}c_{m}+h.c.\right),$ (2)

where $\epsilon_{n}$ denotes the on-site energy at the $n$-th site. $t_{r}$
represents the nearest-neighbor hopping (NNH) strength between the parent
lattice sites and the hopping between the parent lattice sites and the QDs.
$c_{n}^{\dagger}\,(c_{n})$ is the creation (annihilation) operator of an
electron at the $n$-th site. The on-site energies for the parent lattice sites
of the ring are taken as zero, while that for the QDs are considered as
$\epsilon_{n}=\pm i\eta$. The positive (negative) complex potential denotes
the physical gain (loss) when the quantum ring interacts with the environment
through the QDs. The QDs are arranged in a manner such that the gain and loss
have a balanced distribution, and hence the Hamiltonian of the ring in the
given case becomes $\mathcal{PT}$-symmetric.

The second and third terms of Eq. 1 denote the Hamiltonians for the source and
drain, and the last term describes the Hamiltonian for the ring-to-electrode
coupling. They read as

$\displaystyle H_{S}$ $\displaystyle=$ $\displaystyle
H_{D}=\epsilon_{0}\sum\limits_{n}d_{n}^{\dagger}d_{n}+t_{0}\sum\limits_{\langle
nm\rangle}\left(d_{n}^{\dagger}d_{m}+h.c.\right),$ (3) $\displaystyle H_{C}$
$\displaystyle=$ $\displaystyle H_{S,ring}+H_{D,ring}$ (4) $\displaystyle=$
$\displaystyle\tau_{S}\left(c_{p}^{\dagger}d_{0}+h.c.\right)+\tau_{D}\left(c_{q}^{\dagger}d_{N+1}+h.c.\right).$

Here $\epsilon_{0}$ and $t_{0}$ are the on-site energy and NNH strength in the
electrodes. $d_{n}^{\dagger}\,(d_{n})$ is the creation (annihilation) operator
of an electron at the $n$-th site in the electrodes. The coupling strength
between the source and ring is $\tau_{S}$ and that between the drain and the
ring is $\tau_{D}$. The source and drain are connected to the ring at the
$p$-th and $q$-th sites ($p$ and $q$ are the variables), respectively.

###  II.2 Transmission probability and junction current

The transmission probability is calculated using the Green’s function
technique, which is obtained from the relation etms ; qtat

$T=\text{Tr}\left[\Gamma_{S}{\cal G}^{r}\Gamma_{D}{\cal G}^{a}\right],$ (5)

where $\Gamma_{S}$ and $\Gamma_{D}$ are the coupling matrices corresponding to
the source and drain electrodes respectively. ${\cal G}^{r}$ and ${\cal
G}^{a}\left(=\left({\cal G}^{r}\right)^{\dagger}\right)$ are the retarded and
advanced Green’s functions respectively. The retarded Green’s function is
defined as

${\cal G}^{r}=\left(E-H_{R}-\Sigma_{S}-\Sigma_{D}\right)^{-1},$ (6)

where $E$ is the electronic energy, $\Sigma_{S}$ and $\Sigma_{D}$ are the
self-energies due to the source and drain electrodes respectively.

Upon getting the transmission probability using Eq. 5, we compute the
transport current through the junction following the Landauer-Büttiker
formalism etms . Here the transmission function is integrated over a suitable
energy window associated with the applied bias voltage $V$. At absolute zero
temperature, the current flowing through the quantum ring takes the form

$I_{T}(V)=\frac{2e}{h}\int\limits_{E_{F}-\frac{eV}{2}}^{E_{F}+\frac{eV}{2}}T(E)\,dE,$
(7)

where $E_{F}$ is the equilibrium Fermi energy.

###  II.3 Circular current and induced magnetic field

To obtain the circular current in a loop geometry, the currents in the
individual bonds of the loop need to be calculated. The current for a
particular bond, connecting the sites $i$ and $j$, is given by cc1 ; cc2

$I_{ij}=\int\limits_{E_{F}-\frac{eV}{2}}^{E_{F}+\frac{eV}{2}}J_{ij}(E)\,dE$
(8)

where $J_{ij}(E)$ is the bond current density and it is defined as cc-greens

$J_{ij}(E)=\frac{4e}{h}\mbox{Im}\left[H_{ij}{\cal G}^{n}_{ij}\right],$ (9)

where $H_{ij}$ is the $(ij)$ element of the Hamiltonian $H_{R}$, ${\cal
G}^{n}$ is the correlation function and defined as ${\cal G}^{n}={\cal
G}^{r}\Gamma_{S}{\cal G}^{a}$. With the individual bond currents, the net
circular current in the ring can be evaluated using the relation cc1 ; cc2

$I_{C}=\frac{1}{N}\sum_{\langle ij\rangle}I_{ij}.$ (10)

Finally, we compute the circular current induced magnetic field at any
arbitrary point $\vec{r}$ inside the conducting ring using Biot-Savart’s law
bs

$\vec{B}(\vec{r})=\sum_{\langle ij\rangle}\frac{\mu_{0}}{4\pi}\int
I_{ij}\frac{d\vec{r}^{\prime}\times\left(\vec{r}-\vec{r}^{\prime}\right)}{\lvert\vec{r}-\vec{r}^{\prime}\rvert^{3}},$
(11)

where $\mu_{0}$ is the free space permeability.

##  III Results and Discussion

Now, we present and discuss our essential results. The common parameter values
throughout the work are considered as follows. We set the on-site energies for
the parent lattice sites and the leads as zero, while those for the QDs are
assumed as $\mathcal{PT}$-symmetric complex potentials, that is
$\epsilon_{n}=\pm i\eta\;(n\in\text{QDs})$. Here $\eta$ involves the
interaction of the physical system with the environment. Depending on its
sign, we have the gain or loss, as mentioned earlier. We follow the wideband
limit, where the hopping strength in the electrodes $(t_{0})$ is quite large
compared to the NNH strength in the ring system $(t_{r})$. Here we set
$t_{0}=2\,$eV and $t_{r}=1\,$eV. The coupling strengths of the quantum ring to
the source and drain are taken as $\tau_{S}=\tau_{D}=0.8\,$eV. Throughout the
analysis, the equilibrium Fermi energy is set at zero. All the energies are
measured in units of eV. The distance between two sites in the parent lattice
is considered about $1\,$Å. The area of a ring can then readily be calculated.
For example, a 20-site ring has the radius $\sim 3.2\,$Å. For a current-
carrying ring, the magnetic field at the center is calculated in unit of
Tesla.

###  III.1 Energy spectra for an isolated quantum ring

We begin with Fig 2, where the energy eigenvalues of an isolated
$\mathcal{PT}$-symmetric quantum ring (not coupled to the source and drain
electrodes) are shown. From the nature of the energy spectra, the
characteristic behavior of electronic transport can be clearly understood as
the latter one is directly involved with the energy eigenvalues of the ring
conductor. Here the number of parent lattice sites is $N=20$. Figures 2(a) and
(c) represent the variation of the real and imaginary parts of the eigenvalues
as a function of the complex potential $\eta$.

Figure 2: (Color online) Energy eigenvalues as a function of $\eta$ for a
$\mathcal{PT}$-symmetric isolated quantum ring (not attached to
${\mathcal{S}}$ and ${\mathcal{D}}$). (a) The real energy eigenvalues, (b)
real energy values within a selected small energy window for better viewing,
and (c) imaginary part of the energy values. Here we fix $N=20$.

In Fig. 2(b), we select a small energy window to show the dependence more
clearly. For $\eta=0$, when the system is Hermitian, the eigenvalues are real,
as expected, with several degenerate levels spreading within the energy window
-3 to 3 (Fig. 2(a)). As we turn on the complex potential, the spectrum of real
eigenvalues changes significantly with $\eta$. The most notable changes are
observed between the window $\pm 1$. At first, the degenerate zero-energy
levels start to spread out very slowly with $\eta$. Then, near $\eta=0.4$, a
closely packed band is formed by the real eigenvalues. A careful inspection
reveals that the degeneracy of the zero-energy levels completely diminishes by
the $\mathcal{PT}$-symmetric potential and several non-degenerate real
eigenvalues emerge at about $\eta=0.4$ as shown in Fig. 2(b). We also observe
that a few degenerate eigenvalues reappear beyond $\eta=0.4$. These degenerate
levels are nearly flat and are confined within the energy window $\pm 0.5$.
Outside the energy window $\pm 1$, the real eigenvalues tend to incline slowly
towards the zero energy level throughout the given range of $\eta$. We have
also varied $\eta$ over a wide range (not shown here) and we observe that the
real eigenvalues, outside the energy window $\pm 1$, become almost constant as
a function of $\eta$. Since the essential features are mostly confined within
the small range of $\eta$, here we plot the energy spectra by varying $\eta$
within this range. If we plot the energy values in a wide range, we may miss
some spectral features due to the large scale window.

In case of the spectrum involving imaginary eigenvalues (Fig. 2(c)), non-zero
values start to occur for $\eta\sim 0.4$. Therefore, even in the presence of
the $\mathcal{PT}$-symmetric complex potential $\eta$, a pure real eigenvalue
spectrum can be obtained up to a certain value of $\eta$ ($\sim 0.4$ in the
given case) optics1 ; mois-chap . There are a total of five EPs at which non-
zero imaginary eigenvalues start to appear. An interesting feature we note
here is that the degenerate real values for $\eta>0.4$ (Fig. 2(b)) are
accompanied by these EPs. Therefore, the real eigenvalue degeneracy and the
EPs occur exactly at the same values of $\eta$ in our quantum ring trans-char3
. The imaginary levels, originated from their respective EPs, vary
continuously with $\eta$ in a parabolic manner, and their values are spanned
in the energy window $\pm 1$. Beyond $\eta=1$, all the non-zero imaginary
eigenvalues merge in two values (not shown here), one with the positive and
the other with the negative signs having equal magnitudes.

We also notice several interesting features of the EPs and the energy spectrum
at the EPs. For $\eta\sim 0.4$, At the first EP, depending on the ring size, a
few pairs of eigenvalues become complex, where each pair is complex conjugate
to each other mois-chap . This feature prevails beyond the first EP. At the
first EP, there are four degenerate complex eigenvalues when the number of
sites ($N$) in the parent lattice is even, whereas this number becomes two for
odd $N$. After the first EP, whenever we reach the next EP, the number of
degenerate complex eigenvalues increases in a specific manner. Between two
successive EPs, the number of complex eigenvalues remains the same and that
number increases only at the occurrence of the next EP. For the ring with even
$N$, that number increases by eight and it is four for odd $N$. Interestingly,
we find that rings with odd $N$ can produce more EPs compared to the rings
with even $N$. For example, we get ten EPs when $N=19$, while the number of
EPs becomes six for $N=20$. However, the total number of degenerate energy
levels due to EP is equal to the number of QDs in a ring. This is true for any
ring with even or odd $N$. Another notable feature is that for a given
physical parameter of the system, the first EP is robust and it occurs at a
universal value of $\eta$ irrespective of the system size, which we confirm
through our exhaustive analysis.

Zero mode: The number of zero modes for $\eta=0$ is equal to the number of
sites in the parent lattice. This is true for any even or odd $N$ ring. We
find that an infinitesimally small $\eta$ breaks the zero-mode degeneracy,
where the number of zero-modes drops from $N$ to 2 for even-$N$. Beyond a
certain $\eta$-value, that number increases to 6 and remains the same.
Surprisingly, we find that the zero-mode energy levels do not appear for any
finite $\eta$ in the case of odd-$N$ rings.

Now, we concentrate on the central part of our analysis which includes the
characteristics of transmission co-efficient, junction current, circular
current, and the current induced magnetic field at the ring center. To
investigate all these features we clamp the ring to the source and drain
electrodes (Fig. 1).

###  III.2 Transmission coefficient

Figure 3 shows the behavior of transmission coefficient in the absence and
presence of the $\mathcal{PT}$-symmetric complex potential. Here we choose the
identical lattice sites in the ring that is considered in Fig. 2 to establish
a direct link between the energy spectra and the transmission function. The
drain is connected to the ring at an angle $3\pi/2$ with respect to the
source. When the system is Hermitian, the transmission coefficient exhibits a
few sharp resonant peaks as shown in Fig. 3(a).

Figure 3: (Color online) $T$ as a function of $E$ for $N=20$ with (a) $\eta=0$
(black), (b) $\eta=0.25$ (red), and (c) $\eta=0.5$ (green). Drain position is
fixed at an angle $3\pi/2$ with respect to the source.

Within the energy window $\pm 1$, the transmission coefficient is completely
zero even though a degenerate eigenlevel at $E=0$ is present as we have seen
in the real eigenvalues spectrum (Fig. 2(a)). This is a clear indication of
the presence of localized states, which we further confirm by investigating
the inverse participation ratio (IPR) characteristics. The IPR is a reliable
measure of the localization phenomena. The IPR is close to zero for an
extended state while it is of the order of unity for a localized state. We
find that the IPR for $\eta=0$ is about one order of magnitude higher than the
$\eta\neq 0$ cases and therefore we are certain that there exists localized
states at and around $E=0$ in the absence of the environmental influence. The
localization phenomenon is completely governed by the geometry of the system.
Such localization phenomena are also observed in different geometries in the
absence of disorder. For instance, in the absence of disorder, a diamond-
shaped periodic network exhibits a localized state maiti-prb , an armchair
graphene nanoribbon with particular widths shows insulating behavior fujita .

For $\eta=0.25$, a new broad peak appears in the transmission spectrum around
$E=0$ (Fig. 3(b)), while the other sharp resonant peaks are almost similar to
the $\eta=0$ case. The broad peak around the zero energy consists of closely
packed several peaks since there are several real eigenvalues close to zero
energy, and these peaks suggest the appearance of conducting states.
Therefore, by introducing $\eta$, a localization to delocalization (LTD)
transition can be achieved. Most importantly, such an LTD transition is robust
irrespective of the ring size or ring-lead interface geometry. Several peaks
are observed for $\eta=0.5$ (Fig. 3(c)) inside the energy window $\pm 1$ since
the zero level degeneracy is broken and several distinct non-degenerate levels
are formed at the given value of $\eta$ (see Figs. 2(a) and (b)). The rest of
the resonant transmission peaks are also similar to the previous two cases.
This is again simply because of the behavior of the real eigenvalues spectrum,
where the real values are almost constant with $\eta$, outside the energy
window $\pm 1$. It should be noted here that the localized state at $E=0$
remains localized even in the presence of the complex potential. From the
transmission spectra, it can be manifested that the complex potential $\eta$
has a significant role mois-pra-21 , and its direct consequence will
definitely be reflected in the current spectra. These features can be
substantiated from our next analysis.

###  III.3 Junction current, circular current, and induced magnetic field

With the knowledge of the transmission function, we can now analyze the
behavior of the junction current. Subsequently, we also discuss one by one the
nature of bias-driven circular current and the induced magnetic field in the
nano loop. All these features are thoroughly discussed as follows. The
behavior of junction current $I_{T}$ as a function of voltage is shown in Fig.
4(a). Here the nature of the transmission spectrum is directly reflected in
the current-voltage characteristics since the current is determined by
integrating the transmission probability over a specific energy zone
associated with the voltage bias. For $\eta=0$, the current is zero below a
certain threshold voltage $\sim 1.5\,$V. Once the voltage is increased beyond
the threshold value, a non-zero current is obtained and $I_{T}$ increases with
the voltage in a step-like fashion.

Figure 4: (Color online) (a) Transport current $I_{T}$ and (b) circular
current $I_{C}$ and induced local magnetic field B as a function of voltage.
All the physical parameters and color convention are identical with Fig. 3.

This step-like feature is also observed for the other two non-zero $\eta$
values. Interestingly, the threshold voltages for the later two cases are
infinitesimally small. Thus, we can selectively regulate the threshold bias
voltage by regulating the complex potential. It seems that the system behaves
like a variable bandgap semiconductor. Another key feature is that in the
presence of the complex potential, the junction current is always higher than
the $\eta=0$ case at any particular voltage. The maximum current for
$\eta=0.25$ is about $75\,\mu$A, while it becomes $\sim 90\,\mu$A for
$\eta=0.5$. The step-like feature in the current-voltage characteristics for
all the three cases can be explained from the transmission spectra (Fig. 3) as
follows. Whenever a transmitting channel appears in the allowed energy window,
we get a finite junction current. Now, increasing the bias means more allowed
energy window, and thus when a next transmission peak appears into this
window, a step in the current takes place. The reason for the reduced
threshold voltage for non-zero $\eta$ is quite obvious. For $\eta=0$, up to
the threshold voltage, no finite transmission peak is observed within the
allowed energy window, and hence the net junction current is zero. On the
other hand, for non-zero $\eta$, transmission peaks are observed very close to
the energy $E=0$, thus finite current is obtained at an infinitesimally small
bias.

The complex potential is also found to have a significant effect in modulating
the circular current and the induced magnetic field at the ring center as
clearly shown in Fig. 4(b). However, the generation of a bias-driven circular
current completely differs from that of a junction current. The junction
current always increases with increasing the bias window as more resonant
peaks are captured, provided there is no negative differential effect (NDR)
ndr . On the other hand, circular current can be both positive or negative
depending upon the current distribution in the two arms of the ring and their
resultant response. We find that the circular current is significantly large
($\sim\,$mA) than the junction current ($\sim\,\mu$A). Here $I_{C}$ shows
almost similar behavior that we observed for the junction current, but in the
present case, it acquires a negative phase. For $\eta=0.5$, the maximum
$I_{C}$ is about $0.5\,$mA. Due to such a large current, the induced magnetic
field at the center of the ring is also reasonably strong. The maximum value
for the induced magnetic field for $\eta=0.5$ is about 0.3 Tesla. Overall, the
quantum ring exhibits favorable responses when it interacts with the
environment.

Now, we show the explicit dependence of the complex potential on the current
properties and the induced magnetic field. The results are shown in Fig. 5.
Here we plot the junction current, the absolute maximum of the circular
current $(I_{C}^{max})$, and the induced

Figure 5: (Color online) (a) Transport current $I_{T}$ and (b) maximum
magnitude of circular current $I_{C}^{max}$ and induced magnetic field
$B^{max}$ as a function of $\eta$. All the physical parameters are similar as
mentioned in Fig. 3.

magnetic field $(B^{max})$ as a function of $\eta$. The maximum value of
$I_{C}$, and thus, $B$ is computed by taking the maximum over the voltage
window between 0 to 6 V. All the physical parameters and the system
configuration are identical as mentioned in Fig. 3. In case of junction
current, $I_{T}$ increases monotonically with $\eta$ up to $\eta\sim 0.4$,
acquiring a value around 110$\,\mu$A (see Fig. 5(a)). Then $I_{T}$ suddenly
gets reduced providing a dip where the first EP occurs. Within the given
$\eta$-range, we find a total of three dips in the transport current profile,
where each dip denotes the existence of an EP. However, it is quite hard to
detect all the EPs from the current-voltage plot, since there are also other
factors playing on, such as the positioning of the leads. The maximum
transport current is noted about 120$\,\mu$A at $\eta\sim 0.8$. All these
features are in accordance with the eigenvalue spectrum as shown in Fig. 2.
For example, the gradual increase of the current is because, with the
enhancement of $\eta$, more and more non-degenerate energy levels come into
the allowed energy window, capturing more resonant transmission peaks, which
finally contribute to the current. The sudden dip in $I_{T}$ is due to the
presence of the first EP mois-prl-20 , at which degenerate levels appear (Fig.
2(b)). At this EP, fewer transmission peaks are available within the allowed
energy window, making the current smaller than the other values of $\eta$.
Like the junction current, the bias-driven circular current shows almost
similar behavior as a function of $\eta$ as given in Fig. 5(b). The maximum
value is about 0.65 mA at $\eta\sim 0.4$. At this value of the complex
potential, the maximum strength of the induced magnetic field is $\sim
0.42\,$Tesla, which is certainly a reasonably strong field. Most importantly,
all of them exhibit favorable responses throughout the given $\eta$ range.

###  III.4 Interface sensitivity

The quantum interference among the electronic waves flowing through different
branches of the ring geometry significantly affects the transport phenomena,
especially when the ring size is quite small. It is thus important to study
this effect to check whether we can get a much favorable response for any
other ring-electrode junction configuration. So far, we have considered a ring
with $N=20$, and the angular separation between the source and drain was fixed
at $\theta=3\pi/2$. Now, we wish to see how the different transport quantities
behave if the position of the drain is varied with respect to the source. In
order to have more possible lead-ring interface geometries, we take $N=40$ and
compute the maximum magnitudes of the transport quantities as described above.
Figure 6(a) shows the variation of the transport current $I_{T}$ as a function
of $\theta$ where each $\theta$ gives rise to a distinct lead-ring interface
geometry. The interface sensitivity is reflected in the $I_{T}$-$\theta$
spectrum. For the asymmetric lead-ring interface geometries, $I_{T}$ varies
between 60 to 110$\,\mu$A and oscillates with $\theta$. The oscillation is
solely due to the effect of quantum interference in this interferometric
geometry. For the symmetric configuration,

Figure 6: (Color online) (a) Transport current $I_{T}$ and (b) the maximum of
circular current $I_{C}^{max}$ and induced magnetic field $B^{max}$ as a
function of $\theta$, where $\theta$ is the angle between the source and the
drain electrodes. Here we set $\eta=0.5$ and $N=40$.

that is, when $\theta=\pi$, the current is $\sim 160\,\mu$A and is the largest
among all the given interface geometries. Such a feature is quite common in
ring geometries when the source and drain electrodes are symmetrically
connected to the ring. For the symmetric configuration, the two arms of the
ring are identical in every aspect (e.g., length, hopping, etc.) and a
constructive interference takes place. This further aids to produce the
highest current compared to any other configuration, where a partial
destructive interference is always present due to the difference in the two
arm lengths of the ring. $I_{T}$ provides a symmetric behavior around
$\theta=\pi$. This is because of the mirror-symmetric scattering matrix about
the $y$-axis ($y\rightarrow-y$) kislev .

The interface sensitivity is also found to have a prominent effect on the
circular current as depicted in Fig. 6(b). Here $I_{C}^{max}$ is antisymmetric
about $\theta=\pi$ due to the same reason as mention earlier in Fig. 6(a). The
circular current is negative for the large angles while the current is
positive for the small angles. The maximum circular current we observe when
the angular separation between the source and drain electrodes is very small
or very large and $I_{C}^{max}\sim\pm 1\,$mA. The associated magnetic field,
in this case, is about $\pm 0.3\,$Tesla. For $\theta=\pi$, that is for the
symmetric configuration, the circular current is zero, due to the mutual
cancellation of the currents flowing through the two arms of the ring, which
are equal in magnitudes but opposite in phases. The key observation is that
the circular current can be controlled between $\pm 1\,$mA by changing the
position of the drain. This feature is also applicable for the induced
magnetic field which can be varied over a wide range between $\pm 0.3\,$Tesla.

###  III.5 Effect of ring size

Finally, we explore the size dependence on the above-mentioned transport
quantities which sometimes brings several interesting new features as it is
directly involved with the quantum interference effect. To do so, we vary the
system size from $N=8$ to 100 and plot $I_{T}$, $I_{C}^{max}$, and $B^{max}$
in Figs. 7(a), (b), and (c) respectively. The drain electrode is connected to
the ring at an angle $\theta=\pi/2$ with respect to the source. We choose the
number of sites in the ring in a restricted manner, that is $N=4m$

Figure 7: (Color online) Dependence of the system size on the (a) transport
current $I_{T}$ and the maximum magnitude of (b) circular current
$I_{C}^{max}$, and (c) induced magnetic field $B^{max}$ for $\eta=0.5$. Drain
position is fixed at an angle $\pi/2$ with respect to the source.

($m=2,3,4,...$) such that the particular choice of the lead-ring interface
geometry is always preserved irrespective of the system size. The strength of
the complex potential is fixed at $\eta=0.5$. Both $I_{T}$, $I_{C}^{max}$ show
oscillatory behavior as a function of $N$ due to the quantum interference.
Both the junction and circular currents can have moderate values even when the
system size is quite large. For example, the noted junction current is about
80$\,\mu$A and the circular one is $\sim 0.55\,$mA for $N=100$. The circular
current-induced magnetic field gets reduced with increasing the ring size, as
expected. But the fact is that even for much bigger rings, we get a reasonably
large magnetic field as reflected from Fig. 7(c) (more clearly it can be
viewed from the inset of this figure). We find that $B^{max}$ is about 60 mT
for $N=100$.

General Remarks: Before we end our discussion, it is worth noting for the
benefit of readers that in the non-Hermitian system, the transmission
probability may not always be confined within unity. It can also be higher
than unity. Such a feature has already been established in the literature. For
instance, it has been shown that in a non-Hermitian Aharonov-Bohm ring q-li ,
the transmission probability exceeds the value 1. A similar feature has also
been reported by Shobe et al. in a non-Hermitian ${\mathcal{P}T}$-symmetric 1D
chain shobe . $T>1$ implies that the probability current is not conserved in
the non-Hermitian systems. We find that in some cases the unitarity of the
$S$-matrix gets broken, and the transmission probability becomes greater than
unity. However, in our case, the unitarity of the $S$-matrix remains preserved
and $T$ is always confined within unity. To have a better understanding about
that we analytically show the expressions of the reflection and transmission
probabilities for a single site with the only gain term in Appendix B and a
single site with ${\mathcal{P}T}$-symmetric gain and loss terms in Appendix C.
For such a toy model, we can easily find analytical forms of reflection and
transmission coefficients in terms of the gain/loss, and analytical
expressions always help us to understand the physical phenomena from the
fundamental level. This mathematical prescription can easily be extended to a
multi-site system.

##  IV Summary

In the present work, we have analyzed the transport properties of a two-
terminal $\mathcal{PT}$-symmetric non-Hermitian quantum ring, implemented via
a balanced distribution of physical gain and loss. The gain and loss have been
incorporated by assuming that each site of the quantum ring is attached with
two QDs possessing complex on-site potentials. The system has been described
within a tight-binding framework. All the transport quantities have been
evaluated using the standard Green’s function formalism based on the Landauer-
Büttiker approach. The present analysis has been carried out with a thorough
discussion of the spectrum of energy eigenvalues, the transmission
coefficients, junction and bias-driven circular currents, and finally the
induced magnetic field. Our essential findings are summarized as follows.

$\bullet$ The degeneracy of the zero-energy level is broken in the presence of
the $\mathcal{PT}$-symmetric complex potential and gives rise to several
distinct levels.

$\bullet$ Even though the system is non-Hermitian, a real energy spectrum can
be obtained up to a certain limit of the strength of the complex potential.

$\bullet$ Several EPs are observed accompanied by degenerate real eigenvalues.

$\bullet$ The number of EPs is higher for the rings with odd $N$ compared to
the rings with even $N$. However, the number of degenerate complex levels is
always greater for the latter case. The $\eta$ value at which the first EP
occurs is robust and is independent of the ring size.

$\bullet$ The transmission spectra are significantly modified in the presence
of environment-mediated interaction.

$\bullet$ The junction and circular currents exhibit favorable responses. The
circular current is very high ($\sim\,$mA) compared to the junction current
($\sim 100\,\mu$A).

$\bullet$ The induced magnetic field is also reasonably strong, $\sim
0.4\,$Tesla for a particular strength of the complex potential.

$\bullet$ A uniform oscillation in both the junction and circular currents has
been observed with ring size $N$. It is associated with the quantum
interference effect. The induced magnetic field decreases monotonically with
$N$, still yielding moderate strengths for bigger system sizes. For example,
we observe $B\sim 50\,$mT for $N=96$.

At the end, we would like to point out that here we have considered a simple
ring-like geometry to encapsulate the essence of the environmental influence
on transport phenomena. However, our approach can easily be extended to any
other simple or complex ring-like structures or multi-connected geometries
expecting several non-trivial signatures.

###### Acknowledgements.

SKM thankfully acknowledges the financial support of the Science and
Engineering Research Board, Department of Science and Technology, Government
of India (Project File Number: EMR/2017/000504). The authors of this work
would like to thank both the reviewers for their valuable comments and
suggestions to enhance the quality of the work. SG and SKM thanks B. K.
Nikolić, R. Thomale, S. Sil, and S. Z. Bin for fruitful discussions.

## Data Availability Statements

The datasets generated during and/or analysed during the current study are
available from the corresponding author on reasonable request.

##  Appendix A Non-dispersive lead – implementation of imaginary site energy

Let us consider a semi-infinite 1D lead that is coupled to a conductor. In
NEGF formalism, the effect of a lead is described by a self-energy function
$\Sigma$, which has the following expression etms ; qtat ,

$\Sigma=\frac{t_{c}^{2}}{2t_{0}^{2}}\left[E-\epsilon_{0}\pm
i\sqrt{4t_{0}^{2}-\left(E-\epsilon_{0}\right)^{2}}\right]$ (12)

where $t_{c}$ is the coupling strength between the lead and the conductor,
$t_{0}$ is the NNH strength of the lead and $\epsilon_{0}$ is the site energy.
$E$ is the energy of the incoming electrons.

By setting the site energy of the lead equal to the energy of the incoming
electrons, that is $E=\epsilon_{0}$, the self-energy function becomes

$\Sigma=\frac{t_{c}^{2}}{2t_{0}^{2}}\left(\pm i2t_{0}\right)=\pm
i\frac{t_{c}^{2}}{t_{0}}\equiv\pm i\eta$ (13)

From Eq. 13, we see that the self-energy function is independent of energy
(non-dispersive) and purely imaginary. Hence, a non-dispersive lead can always
be viewed as a QD with complex site energy. By tuning the coupling strength
$t_{c}$ (that can be easily done experimentally), the strength of the complex
potential can be modulated. The site energy of the lead can be varied with the
help of a controlled gate potential fano .

##  Appendix B Single site system with gain

First, we show the analytical expressions of transmission and reflection
probabilities for a single site system with only the gain term, using the
wave-guide theory approach.

Figure 8: Schematic diagram of a 1D chain with a single site having gain term
only.

In Fig. 8, a single site (red circle) is shown which is attached with two
leads (formed by green circles). The parent lattice site is placed at $x=0$,
which is subjected to a potential $i\eta$. The lattice sites in the incoming
and outgoing leads are placed at $x=\pm 1,\pm 2,...$, etc. The negative and
positive signs are corresponding to the incoming and outgoing leads,
respectively.

The wave function is assumed to have the form,

$\psi_{x}=\begin{cases}Ae^{ikx}+Be^{-ikx}&\text{for }x\leq 0\\\
Ce^{ikx}&\text{for }x\geq 0\end{cases}$ (14)

where $\psi_{x}\equiv\langle x|\psi\rangle$ and $k$ is the wave vector.

From the continuity condition, at $x=0$, we have,

$\psi_{0}=A+B=C.$ (15)

With the boundary condition (Eq. 14), we solve the Schrödinger equation,

$H|\psi\rangle=E|\psi\rangle,$ (16)

where

$|\psi\rangle=\left(\cdot\cdot\cdot,\psi_{-1},\psi_{0},\psi_{1},\cdot\cdot\cdot\right)^{T}.$
(17)

For the leads, we have the dispersion relation

$\displaystyle E(k)=2t_{0}\cos{k}$ (18) $\displaystyle\Rightarrow\quad$
$\displaystyle k=\cos^{-1}{\left(\frac{E}{2t_{0}}\right)}.$ (19)

At $x=0$, we get

$\displaystyle(E-i\eta)\psi_{0}=t_{0}\left(\psi_{-1}+\psi_{1}\right),$ (20)

where $t_{0}$ is the nearest-neighbor hopping (NNH) strength.

The wave function at $x=0$ is given in Eq. 15. From Eq. 14, the wave functions
$\psi_{-1}$ and $\psi_{1}$ are

$\displaystyle\psi_{-1}$ $\displaystyle=$
$\displaystyle\left(Ae^{-ik}+Be^{ik}\right),$ (21) $\displaystyle\psi_{1}$
$\displaystyle=$ $\displaystyle Ce^{ik}.$ (22)

Plugging the explicit forms of $\psi_{0}$, $\psi_{1}$, and $\psi_{-1}$ in Eq.
20 and with some algebra, we get the transmission probability $T$ as,

$T=\left|\frac{C}{A}\right|^{2}=\frac{4t_{0}^{2}-E^{2}}{\left(\sqrt{4t_{0}^{2}-E^{2}}+\eta\right)^{2}}\,,$
(23)

and, the reflection probability $R$,

$R=\left|\frac{B}{A}\right|^{2}=\frac{\eta^{2}}{\left(\sqrt{4t_{0}^{2}-E^{2}}+\eta\right)^{2}}\,.$
(24)

From Eqs. 23 and 24, it is clearly reflected that for $\eta=0$, $T=1$ and
$R=0$. Therefore, for the Hermitian case $R+T=1$, and, the unitarity of the
$S$-matrix is preserved. However, the unitarity of the $S$-matrix is always
broken when $\eta\neq 0$ and thus $R+T\neq 1$.

We assume that $t_{0}<0$, which is the usual case for normal crystals.
However, it is also possible to prepare a specific type of crystal, either
electronic or photonic, in which $t_{0}>0$, and then the meaning of gain and
lossy potential is reversed shobe . For simplicity, we fix $t_{0}<0$, in which
case, the scatterer acts as a source for $\eta>0$, while for $\eta<0$, it acts
as a sink.

Now, from Eq. 23, we see that the transmission probability diverges at

$E=\pm\sqrt{4t_{0}^{2}-\eta^{2}}$ (25)

At $E=0$, the transmission probability becomes (from Eq. 23),

$T=\frac{4t_{0}^{2}}{\left(2t_{0}+\eta\right)^{2}}.$ (26)

With $t_{0}=-1$ and $\eta=2.1$, we get $T=400$, that is the transmission
probability is much higher than unity. On the other hand, for $\eta=-2.1$,
$T=0.98$, which means the transmission probability is restricted within unity.
It is now evident that whether the system is electronic or photonic, the
transmission probability is greater than unity for the non-Hermitian case
shobe  having a gain term, but that is not the case for the lossy scatterer.

##  Appendix C Single site system with gain and loss
(${\mathcal{P}T}$-symmetric case)

Figure 9: Schematic diagram of a 1D chain with single site having a balanced
distribution of gain and loss.

In Fig. 9, a single site system (black circle) is shown attached with two
leads (green circles). The parent lattice site is placed at $x=0$, which is
attached with two QDs. One QD (red circle) is having a potential $i\eta$
(gain) and the other one (blue circle) having a potential $-i\eta$ (loss). The
lattice sites in the incoming and outgoing leads are placed at $x=\pm 1,\pm
2,...$, etc. The negative and positive signs are corresponding to the incoming
and outgoing leads, respectively.

The wave function and the boundary condition are the same as given in Eqs. 14
and 15. The only thing changes here is the wave function $\psi$ in the
Schrödinger equation (Eq. 16), where

$|\psi\rangle=\left(\cdot\cdot\cdot,\psi_{-1},\psi_{G},\psi_{0},\psi_{L},\psi_{1},\cdot\cdot\cdot\right)^{T},$
(27)

$\psi_{G}$ and $\psi_{L}$ correspond to the gain and loss QDs respectively.

Away from the scattering region, for the leads, we have the same dispersion
relation as given in Eqs. 18 and 19.

At $x=0$, instead of one, now, we get three equations. They are

$\displaystyle(E-\epsilon)\psi_{0}$ $\displaystyle=$ $\displaystyle
t_{0}\left(\psi_{-1}+\psi_{1}\right),$ (28) $\displaystyle(E-i\eta)\psi_{G}$
$\displaystyle=$ $\displaystyle t_{0}\psi_{0},$ (29)
$\displaystyle(E+i\eta)\psi_{L}$ $\displaystyle=$ $\displaystyle
t_{0}\psi_{0},$ (30)

where $\epsilon$ is the on-site potential at the parent site and is fixed at
zero for simplicity. Expressing $\psi_{G}$ and $\psi_{L}$ in terms of
$\psi_{0}$, Eq. 28 becomes

$E\left(1-\frac{2t_{0}}{E^{2}+\eta^{2}}\right)\psi_{0}=t_{0}\left(\psi_{-1}+\psi_{1}\right).$
(31)

With $\psi_{0}=A+B=C$, $\psi_{1}=Ce^{ik}$, and $\psi_{-1}=Ae^{-ik}+Be^{ik}$,
the transmission and reflection probabilities finally take the forms,

$\displaystyle T=\left|\frac{C}{A}\right|^{2}$ $\displaystyle=$
$\displaystyle\frac{4E^{2}t_{0}^{4}/\left(E^{2}+\eta^{2}\right)^{2}}{4t_{0}^{2}-E^{2}+4E^{2}t_{0}^{4}/\left(E^{2}+\eta^{2}\right)^{2}}$
(32) $\displaystyle R=\left|\frac{B}{A}\right|^{2}$ $\displaystyle=$
$\displaystyle\frac{4t_{0}^{2}-E^{2}}{4t_{0}^{2}-E^{2}+4E^{2}t_{0}^{4}/\left(E^{2}+\eta^{2}\right)^{2}}\,.$
(33)

Interestingly, from Eqs. 32 and 33, we get $T+R=1$ even for $\eta\neq 0$. This
indicates that though the system is non-Hermitian, the unitarity of the
$S$-matrix remains preserved and that is why the transmission probability
always stays within unity. For this particular reason, we consider such a
system in our work.

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# A survey on deep supervised hashing methods for image retrieval

Xiaopeng Zhang
Department of Mathematics
Nankai University
<EMAIL_ADDRESS>

###### Abstract

Hashing has been widely used in approximate nearest search for large-scale
database retrieval for its computation and storage efficiency.Deep hashing,
which devises convolutional neural network architecture to exploit and extract
the semantic information or feature of images, has received increasing
attention recently. In this survey, several deep supervised hashing methods
for image retrieval are evaluated and I conclude three main different
directions for deep supervised hashing methods. Several comments are made at
the end. Moreover, to break through the bottleneck of the existing hashing
methods, I propose a $\displaystyle Shadow$ $\displaystyle Recurrent$
$\displaystyle Hashing$(SRH) method as a try. Specifically, I devise a CNN
architecture to extract the semantic features of images and design a loss
function to encourage similar images projected close. To this end, I propose a
concept: ”shadow” of the CNN output. During optimization process, the CNN
output and its shadow are guiding each other so as to achieve the optimal
solution as much as possible. Several experiments on dataset CIFAR-10 show the
satisfying performance of SRH.

##  1 Introduction

With the explosive growing of data in real applications, approximate nearest
neighbor (ANN) search has attracted much attention in machine learning, with a
lot of applications in information retrieval, computer vision and so on. Given
a query, our goal is to retrieve images that are similar to the query, where
“similar” could refer to visually similar or semantically similar. In fact, it
is not that hard if we do not make any restriction. The simplest way of
looking for similar images is to find every feature of images (output of the
AlexNet fc7, for example) and rank them according to the distances to the
query in the feature space. However, with millions of images on the Internet,
we can hardly do so for the high storage consuming and expensive searching
time. For a database with millions of images, which is quite common in today’s
world, even a linear scan through the database would cost a great deal of time
and memory. To address the inefficiency of the linear searching, recently a
widely used technique for ANN search, hashing, has attracted lots of
intention. Hashing methods represents images by binary codes instead of real-
valued features, thus the time costs from searching and retrieval can be
largely reduced.

The image retrieval approaches can be categorized into two main streams,
namely, classical retrieval approaches using local features and deep methods
based on CNNs producing feature extractor and hash functions. Most existing
classical methods basically extract features from hand-crafted extractors,
thus the performance of such methods could heavily depends on the feature they
use or the approaches to extract feature they design. So for classical methods
for fast image retrieval, they are more suitable for dealing with visual
similarity rather than semantically similarity. Moreover, most classical
hashing methods only perform well on specific problems for the corresponding
feature extractors are designed temporarily. While traditional hashing methods
have showed their successful performances on some real world datasets, CNNs
based methods demonstrate more impressive learning power recently, especially
in cases where the local features are not good enough. These deep methods have
shown the state-of-the-art performance. In this survey, we mainly focus on
deep hashing methods for image retrieval and find out those pros and cons for
them. Moreover, inspired by those deep hashing methods, I propose a new method
on large scale image retrieval and test the performance on CIFAR-10 dataset.

##  2 Extensive evaluation

While the very first hashing methods based on deep learning should be dated
back to semantic hashing Hinton et.al.[1], this method imposes difficult
optimization and its performance is surpassed by lots of classical hashing
methods. After a decade, CNN began to show much power in several applications
and soon a CNN based deep hashing method for image retrieval was proposed.

Figure 1: model architecture of CNNH

###  2.1 Convolutional Neural Network Hashing

Convolutional Neural Network Hashing(CNNH) [2] is the first approach to make
full use of the CNN architecture. It first proposes a scalable coordinate
descent method to decompose the pairwise similarity matrix $\displaystyle S$
into $\displaystyle HH^{T}$, where $\displaystyle H$ is a matrix with each of
its row being the approximate hash code associated to a training image. Then
feature representations of training images are learnt through a CNN
architecture as well as a set of hash functions. Moreover, the author further
improve the stage of hash functions learning by incorporating the image class
labels as a part in the output layer.

Figure1 shows the overview of CNNH. It consists of two main stages. In stage1,
approximate hash codes are calculated by minimizing the following
reconstruction errors:

$\displaystyle\min_{H}\sum_{i=1}^{n}\sum_{j=1}^{n}(S_{i,j}-\frac{1}{q}H_{i}\cdot
H_{j}^{T})^{2}=\min_{H}\left\|S-\frac{1}{q}HH^{T}\right\|_{F}^{2}$

where $\displaystyle H$ is $\displaystyle n\times q$ matrix, with each row
being the target q-bit hash code of image $\displaystyle I_{k}$.
$\displaystyle S_{i,j}=1$ when $\displaystyle I_{i},I_{j}$ are semantically
similar, $\displaystyle S_{i,j}=-1$ respectively. In stage2, CNNH
simultaneously learn a feature extractor for query images as well as a set of
hash functions.Herein, CNNH leverages the convolutional neural network to take
advantage of the 2D structure of images and to learn the underlying features,
and then uses these features to control the outcomes of the corresponding hash
codes.

CNNH devises a CNN architecture in which the CNN can be viewed as a feature
extractor and capture the underlying semantic structure of images in spite of
significant appearance variations. From this aspect, it surpasses most
classical hashing methods. However, since it is the first try with the CNN
architecture, it inevitably has some drawbacks. In this two-stage method, the
learned approximate hash codes are used to guide the learning of the image
representation, but the learned image representation cannot give feedback for
learning better approximate hash codes. It seems like these two stages are in
a bit of reverse order.

One should note that in supervised learning, similarity set $\displaystyle
S=\\{s_{ij}\\}$ can be constructed from semantic labels of data points or
relevance feedback from click-through data in real retrieval systems. Although
CNNH is not an end-to-end method, the requirement of the similarity matrix
should not be viewed as a main drawback. In fact, such matrix is involved as
the supervised information in most of the deep supervised methods.

###  2.2 Deep neural network hashing

In CNNH, the approximate hash codes are determined at the very beginning of
the whole process, which seems to be contradictory to our common logic. The
approximate hash codes guide the learning process of hash function, which
fails to give any feedback to the hash codes learning process. In other words,
a pair of semantically similar or dissimilar images may not have feature
vectors with relatively small or large Euclidean distance. This one-way
interaction is limited and needs to be changed.

Figure 2: (a)A divide-and-encode approach (b)A fully-connected layer and a
sigmoid layer

Deep Neural Network Hashing(DNNH) [3] generates hash codes of images through a
CNN-based architecture. Throughout the deep network, input images are
converted into unified image representations via a shared subnetwork of
stacked convolution layers. Then, as shown in figure2, these intermediate
image representations are encoded into hash codes by divide-and-encode
procedure. Specifically, the input intermediate features are divided into
$\displaystyle q$ slices, with each of the slices mapped to one dimension by a
fully connected layer. Furthermore, a triple ranking loss is designed to
preserve relative similarities as well as pulling dissimilar pairs far away.

This work makes a few designs on the network architecture. First, at the end
of the full-connected layer, each hash bit is generated from a separated slice
of features in order to reduce the redundancy among the hash bits. Second, in
order to encourage the output of a divide-and-encode module to be binary
codes, a sigmoid activation function is used followed by a piece-wise
threshold function. Third, a triplet loss function is designed to characterize
that two images are similar and other two are dissimilar. Unlike CNNH, this
one-way method plans as a whole for the two learning stage and make sure the
two stages are instructing each other so that the coding process is compatible
with the feature-extraction process.

###  2.3 Deep supervised hashing

Deep Supervised Hashing(DSH) [4] constructs a CNN architecture and elaborately
designs a loss function to maximize the discriminability of the Hamming space
by encoding the supervised information from the input image, then imposes
regularization on the real-valued outputs to approximate the desired discrete
values. As shown in figure3, DSH devises a CNN model which processes pairs of
images to determine whether two images are similar and then generates the
corresponding hash code with relatively small distance. The loss function is
designed to make sure similar query points are projected to close outputs in
Hamming space

Figure 3: model architecture of DSH

and push the outputs of the dissimilar ones far away. The loss function is
defined as:

$\displaystyle
L(b_{1},b_{2})=\frac{1}{2}(1-y)D_{h}(b_{1},b_{2})+\frac{1}{2}y\max(m-D_{h}(b_{1},b_{2}),0)$
$\displaystyle s.t.\quad b_{j}\in\\{+1,-1\\}^{k},j\in\\{1,2\\}$

where $\displaystyle m>0$ is a margin threshold parameter. The first term
punishes similar pairs when projected far away. The second term punishes
dissimilar pairs projected close when their distance fall in the threshold
$\displaystyle m$. Suppose N pairs are trained, the loss function is defined
as :

$\displaystyle L=\sum_{i=1}^{N}L(b_{i,1},b_{i,2},y_{i})$ $\displaystyle
s.t.\quad b_{i,j}\in\\{+1,-1\\}^{k},i\in\\{1,..,N\\},j\in\\{1,2\\}$

.

Herein, we cannot directly optimize this loss function due to the discrete
constraints. A common relaxation method is to use $\displaystyle tanh$ or
$\displaystyle sigmoid$ function to approximate the discrete constraints.
However, working with such non-linear function would slow down the network.
DSH designs the loss function by adding a regularizer as follows.

$\displaystyle
L(b_{1},b_{2})=\frac{1}{2}(1-y)\left\|b_{1}-b_{2}\right\|_{2}^{2}+\frac{1}{2}y\max(m-\left\|b_{1}-b_{2}\right\|_{2}^{2},0)+\alpha(\left\|\left|b_{1}\right|-1\right\|_{1}+\left\|\left|b_{2}\right|-1\right\|_{1})$

In this work, a novel idea is to impose an additional regularizer to replace
the binary constraints and substitute the sigmoid function or tanh function,
non-linear functions restraining and slowing the convergence of the network,
especially when the output is near the boundary. The proposed regularizer
overcome the problem because its gradient is always 1. More importantly, DSH
is an end-to-end architecture. The learned approximate hashing codes are used
to guide the learning of the image representation and the learned
representation gives feedback to the learning of hash codes.

###  2.4 Deep Anchor Graph Hashing

Deep Anchor Graph Hashing(DAGH) [5] mainly addresses the mini-batch issue. Due
to the high computation cost and limited hardware’s memory, lots of deep
supervised hashing methods will first select a subset from the training set,
and then form a mini-batch to update the network in each iteration. Therefore,
the remaining labeled data cannot be fully utilized and the model cannot
directly obtain the binary codes of the entire training set for retrieval.
This method regards the samples in subsets as anchors and design a regression
term to establish the connections between the anchors and all the binary
codes. A training subset is selected in every iteration and is used to update
the neural network by back-propagation.

The DAGH gives a try to tackle the mini-batch issue by using the anchor graph
to characterize the similarities between the deep features and all the binary
codes. But we should note that this issue is caused by CNNs, whether or not
the time-consuming is less by this work which actually do not improve the CNNs
architecture should be tested further. However, we should admit that this work
provides us with a new perspective towards the mini-batch issue.

###  2.5 Deep Incremental Hashing Network

Although the mini-batch issue is difficult to solve, Deep Incremental Hashing
Network(DIHN) [6] open a new path to somehow tackle the issue. DIHN learns the
hash codes for the new coming images directly, while keeping the old ones
unchanged. We should remind that the incremental hashing codes are not simply
quantified through the original deep hashing network. Instead, DIHN first
devises a incremental hashing loss to preserve the semantic similarity between
training points, and then learns the new incremental binary codes by
integrating the CNN learning model into the incremental hashing loss. DIHN is
the first deep incremental hashing approach, It simultaneously generate hash
codes for incremental database images and learn a CNN model for producing hash
codes for query images. Predictably, DIHN performs well on large scale
retrieval system and decrease the training time with no accuracy decline.

###  2.6 Asymmetric Deep Supervised Hashing

Most deep supervised hashing methods adopt a symmetric strategy to learn one
deep hash function for both query points and database points. The training of
these symmetric deep supervised hashing methods is typically time-consuming.
The storage and computational cost for these hashing methods with pairwise
supervised information is $\displaystyle\mathcal{O}(n^{2})$ where n is the
number of database points. Besides the requirement of mini-batch for CNNs,
most deep supervised hashing methods subject to the time-consuming hashing
learning procedure so that only a small subset from the whole database are
sampled. Asymmetric Deep Supervised Hashing(ADSH) [7] treats the query points
and database points in an asymmetric way. ADSH learns a deep hashing function
only for query points, while the hash codes for database points are directly
learned. The computational cost of ADSH is $\displaystyle\mathcal{O}(n)$.

Figure 4: model architecture of ADSH

As shown in Figure4, the model architecture of ADSH consists of two parts,
namely feature learning part and loss function part. Unlike previous hashing
methods, ADSH assumes that the whole database contains $\displaystyle m$ query
points $\displaystyle X=\\{x_{i}\\}_{i=1}^{m}$ and $\displaystyle n$ database
points $\displaystyle Y=\\{u_{j}\\}_{j=1}^{n}$. The corresponding binary hash
code are denoted as $\displaystyle
U=\\{u_{i}\\}_{i=1}^{m}\in\\{-1,+1\\}^{m\times k}$ and $\displaystyle
V=\\{v_{j}\\}_{j=1}^{n}\in\\{-1,+1\\}^{n\times c}$ . In the feature learning
part, ADSH adopt the CNN architecture to extract semantic feature from the
query images and leverages the outputs to construct the following loss
function.

$\displaystyle\min_{U,V}L(U,V)=\sum_{i=1}^{m}\sum_{j=1}^{n}(u_{i}^{T}v_{j}-cS_{i,j})^{2}$
$\displaystyle s.t.\quad U\in\\{-1,+1\\}^{m\times c},V\in\\{-1,+1\\}^{n\times
c},u_{i}=h(x_{i}),i=1,2..,m$

However, it is difficult to optimize $\displaystyle h(x_{i})$ due to the
discrete constraints, so ADSH sets $\displaystyle
h(x_{i})=sign(F(x_{i},\mathcal{O}))$, where $\displaystyle\mathcal{O}$ is the
parameter of the CNN model. Moreover, ADSH adds a regularizer to keep
$\displaystyle v_{i}$ and $\displaystyle tanh(F(y_{i},\mathcal{O}))$ as close
as possible. So the final loss function is designed as:

$\displaystyle\min_{\mathcal{O},V}=\sum_{i=1}^{m}\sum_{j=1}^{n}[tanh(F(y_{i};\mathcal{O}))^{T}v_{j}-cS_{i,j}]^{2}+\gamma\sum_{i=1}^{m}[v_{i}-tanh(F(y_{i};\mathcal{O}))]^{2}$
$\displaystyle s.t.\quad V\in\\{-1,+1\\}^{n\times c}$

In the loss function stage, the learning algorithm consists of two stages. In
the first steps, we learn $\displaystyle\mathcal{O}$ with $\displaystyle V$
fixed. By back propagation algorithm, the neural network parameters
$\displaystyle\mathcal{O}$ is updated. In the second stage, we learn
$\displaystyle V$ with $\displaystyle\mathcal{O}$ fixed. So the problem can be
rewritten as

$\displaystyle\min_{V}L(V)=\left\|\tilde{U}V^{T}-cS\right\|^{2}_{F}+\gamma\left\|V-\tilde{U}\right\|^{2}_{F}=\left\|\tilde{U}V^{T}\right\|^{2}_{F}-2ctr(V^{T}S^{T}\tilde{U})-2\gamma
tr(V\tilde{U^{T}})$ $\displaystyle s.t.\quad V\in\\{-1,+1\\}^{n\times c}$

. We can get the optimal solution as:

$\displaystyle
V_{*k}=-sign(2\tilde{V_{k}}\tilde{U^{T}_{k}}\tilde{U_{*k}}+Q_{*k})$

where, $\displaystyle U_{*k}$ denotes the $\displaystyle k$th column of
$\displaystyle U$ and $\displaystyle\tilde{U_{k}}$ denotes the matrix of
$\displaystyle U$ excluding $\displaystyle U_{*k}$

Although it seems incredible to reduce the computational complexity so much,
we have to notice that the feature of the database points are not taken into
account throughout the whole algorithm since the hash codes for database
points are directly learned without the CNNs extracting the features. As a
result, the binary codes for database points are not compact with respect to
those query points. However, this work gives us a new perspective. And a
possible improvement for this work can be adding an additional procedure in
which the database points and the query points reverse their positions. A
whole new loss function is probably able to be given. What’s more, the
computational cost should still be O(n). We can go on looking for this trying
then.

###  2.7 HashNet

HashNet [8] mainly focuses on two problems in deep supervised hashing, ill-
posed problem and data imbalanced problem.

First, most deep learning hash methods have to give a continuous
representation through CNNs and generate binary hash codes in separated step,
which would result in suboptimal binary codes. HashNet addresses this problem
by gradually reducing the amount of smoothing during the training, which
results in a sequence of optimization problems converging to the original
optimization problem. Second, the similarity information is usually very
sparse in real retrieval systems. The number of similar pairs is much smaller
than the number of dissimilar pairs. This will result in the data imbalanced
problem, making similarity-preserving ineffective. HashNet tackles the problem
by weighting the training pairs according to the importance of misclassifying
the pair.

HashNet addresses deep learning to hash from imbalanced similarity data by a
continuation method. Admittedly, this continuation method cannot be applied to
large-scale dataset since it will definitely cost too much running time. On
the other hand, it is better to adopt this method to check out those updates
during every iteration in most deep supervised hashing methods.

###  2.8 Deep Cauchy Hashing

Based on HashNet, Deep Cauchy Hashing(DCH) [9] improves the Hamming space
retrieval by designing a pairwise cross-entropy loss based on Cauchy
distribution, which penalizes significantly on similar image pairs with
Hamming distance larger than the given Hamming radius threshold. Most state-
of-the-art deep hashing methods usually adopt generalize sigmoid function as
the probability function. However, we can observe that the probability of
generalized sigmoid function stays high when the Hamming distance between hash
codes is much larger than 2 and only starts to decrease obviously when the
Hamming distance becomes close to $\displaystyle K/2$. DCH solves the
misspecification problem of sigmoid function by constructing the probability
function based on the Cauchy distribution instead of the sigmoid function. As
we know, for large-scale dataset, to quantize the distances among those
compact and concentrated hash codes is significant. The DCH approach enables
more efficient and effective Hamming space retrieval based on a Bayesian
learning framework. However, the drawbacks of DCH is also apparent. To
compensate the efficient retrieval among concentrated hash codes, most of the
distances in Hamming spaces are probability stay quite large, thus obscuring
the original image retrieval.

DCH mainly focuses on concentrating relevant images to be within a small
Hamming ball, thus enables the most efficient constant-time search, which is
Hamming space retrieval, instead of using linear scan. DCH first proposed a
Bayesian learning framework to perform deep hashing on query images by
quantizing the similarity of pairwise images. The goal is converted to
maximize the likelihood function:

$\displaystyle logP(\textbf{H}|\mathcal{S})\propto
logP(\mathcal{S}|\textbf{H})P(\textbf{H})=\sum_{s_{i,j}\in\mathcal{S}}\omega_{i,j}logP(s_{i,j}|h_{i},h_{j})+\sum_{i=1}^{N}logP(h_{i})$

where $\displaystyle
P(\mathcal{S}|\textbf{H})=\prod_{s_{i,j}\in\mathcal{S}}[P(s_{i,j}|h_{i},h_{j})]^{\omega_{ij}}$,$\displaystyle
P(s_{i,j}|h_{i},h_{j})=\sigma(d(h_{i},h_{j}))^{s_{ij}}(1-\sigma(d(h_{j},h_{j})))^{1-s_{ij}}$,$\displaystyle\sigma$
is probability function.

Most deep hashing methods adopt sigmoid function
$\displaystyle\sigma=\frac{1}{1+e^{-\alpha x}}$ as probability function. DCH
proposed to use a novel probability function based on Cauchy distribution so
as to concentrate relevant images to be within a small Hamming ball and make
efficient Hamming space retrieval possible.

$\displaystyle\sigma(d(h_{i},h_{j}))=\frac{\gamma}{\gamma+d(h_{i},h_{j})}$

##  3 Comments

The whole framework of deep supervised hashing methods for image retrieval can
be categorized into three main streams. First, address the ill-posed gradient
problems caused by the sign function. Second, design various of loss functions
to improve the hash code learning and to make full use of the supervised
feature information. Third, design different structures of networks to reduce
the redundancy and to recover loss of features during CNN learning.

Comments:

1\. It seems that most deep supervised hashing methods focus too much on the
semantic features to generate hash codes but ignore texture or spatial
features. We have to admit that to achieve better retrieval performance,
spatial details should be highly taken into account. For example, if we have a
query point which is a picture of a people wearing Nike shoes, we probably
retrieve results of people wearing shoes with various of brands. As we know,
throughout the CNN architecture, the early convolutional layer tends to retain
spatial details asymptotically while the last convolutional layer can capture
more semantic and less spatial details. If we extensively exploit features
from different convolutional layers, we will probably achieve better
performance.

2\. Multimedia data on the Internet exists as a range of different types and
are represented by different modalities. Since the features that CNNs produce
are sometimes too general, cross-modal retrieval should be more suitable for
practical application.[10]

3\. When comparing with other deep supervised hashing methods, some authors
employ more complex structures on their own model while leaving those simple
structures on other models in order to get their own state-of-the-art
performances. It’s quite unfair. We’d better compare the performance on a
common benchmark.

##  4 One possible attempt

###  4.1 Notation

We are given a training set of $\displaystyle N$ images: $\displaystyle
I={I_{1},I_{2},..,I_{N}},I_{i}\in R^{d_{1}\times d_{2}\times 3}$.
$\displaystyle Y={y_{i,j}}$ is the corresponding supervised information, which
is the similarity matrix in this case. For image $\displaystyle I_{i}$ and
$\displaystyle I_{j}$,$\displaystyle Y_{i,j}=0$ if they are similar, and
$\displaystyle Y_{i,j}=1$ otherwise. We denote the binary network output of
$\displaystyle I_{i}$ by $\displaystyle b_{i}$,where $\displaystyle
b_{i}\in\\{\pm 1\\}^{k}$, $\displaystyle k$ is the length of hashing codes.

Our goal is to give a map from $\displaystyle I$ to $\displaystyle k$-bit
binary codes: $\displaystyle F:I_{i}\rightarrow b_{i}$ , such that the
distances of hashing codes from similar pairs are small, while large for
dissimilar pairs.

###  4.2 Motivation

The purpose of supervised hashing is to preserve the small Hamming distances
between hashing codes from similar images. As a result, we should minimize the
following loss function:

$\displaystyle
L(Y,B)=\sum_{i,j}\frac{1}{2}(1-y_{i,j})H(b_{i},b{j})+\frac{1}{2}max(m-H(b_{1},b_{2}),0)$
$\displaystyle s.t.\quad b_{i}\in\\{\pm 1\\}^{k},\quad 1\leqslant i\leqslant
N\quad$

However, we cannot directly optimize (1) through back propagation due to the
discrete constraints. To fulfill the discrete constrains, lots of supervised
methods leverage a common relaxation scheme, which is to replace
$\displaystyle b_{i}$ by $\displaystyle sigmoid(b_{i})$ or $\displaystyle
tanh(b_{i})$ and get the final hashing codes by $\displaystyle signum$
function. Nevertheless, such nonlinear functions will definitely slow down the
learning process[222] and the $\displaystyle signum$ function may lead to
suboptimal results. To tackle this problem, a good idea is to transfer the
discrete constraints into the regularizer terms. Specifically, a loss function
is designed as:

$\begin{split}L(Y,B)=&\sum_{i,j}\frac{1}{2}(1-y_{i,j})\left\|b_{i}-b{j}\right\|_{2}^{2}+\frac{1}{2}y_{i,j}max(m-\left\|b_{i}-b_{j}\right\|_{2}^{2},0)\\\
&+\alpha(\left\|\left|b_{i}\right|-\mathbf{1}\right\|_{1}+\left\|\left|b_{j}\right|-\mathbf{1}\right\|_{1})\end{split}$
(4.1) $\displaystyle s.t.\quad b_{i}\in R^{k}$

Herein, L2-norm is used to measure the distances between pairs instead of
Hamming distance because the subgradient of Hamming distance will treat pairs
with different distances equally. Although this model achieves relatively good
performance, the disadvantage is obvious. We should notice that the absolute
value of subgradient of the regularizer always stays at
$\displaystyle\alpha\cdot\mathbf{1}$. During training, when the first term is
becoming relatively small compared to the regularizer, the subgradient causes
the loss function create jump movements. As a result, the loss function will
be basically caught at a value and fails to be optimized further.

###  4.3 Model description

Given that we have gave up utilizing nonlinear relaxation in loss function and
regularizer with subgradient like (1), how can we express the discrete
restriction in the loss function?

We propose the concept: the ”shadow” $\displaystyle u_{i}$ of the network
output $\displaystyle b{i}$. During training, we use the ”shadow” to instruct
the learning of $\displaystyle b_{i}$. In this process, ”shadow” stays fixed
until this epoch of training ends. Then in the next step, the ”shadow” is
determined according to the output of the neural network. In other word,
$\displaystyle b_{i}$ and its shadow $\displaystyle u_{i}$ are guiding each
other in the whole process. Specifically, our loss function is defined as:

$\begin{split}L(Y,B)=&\sum_{i,j}\frac{1}{2}(1-y_{i,j})\left\|b_{i}-b{j}\right\|_{2}^{2}+\frac{1}{2}y_{i,j}max(m-\left\|b_{i}-b_{j}\right\|_{2}^{2},0)\\\
&+\frac{\alpha}{2}\sum_{i}\left\|b_{i}-u_{i}\right\|_{2}^{2}+\frac{\beta}{2}\sum_{i}(b_{i}b_{i}^{T}-k)^{2}\end{split}$
(4.2) $\displaystyle s.t.\quad b_{i}\in R^{k},\quad u_{i}\in\\{\pm 1\\}^{k}$

During optimization, we sequentially update the parameters of the deep neural
network and binary matrix $\displaystyle U$ in the following two steps: First,
update weights with U fixed: we use back propagation to update the neural
network parameter. In each iteration, we randomly sample a minibatch of images
and use BP algorithm to update the weights. The gradient of the loss with
respect to $\displaystyle b_{i}$ follows:

$\displaystyle\frac{\partial L}{\partial
b_{i}}=\sum_{j}(1-y_{i,j})(b_{i}-b_{j})+y_{i,j}(b_{i}-b_{j})+\alpha(b_{i}-u_{i})+\beta(b_{i}b_{i}^{T}-k)b_{i}\quad
if\left\|b_{i}-u_{i}\right\|_{2}^{2}\leq m$ $\displaystyle\frac{\partial
L}{\partial
b_{i}}=\sum_{j}(1-y_{i,j})(b_{i}-b_{j})+\alpha(b_{i}-u_{i})+\beta(b_{i}b_{i}^{T}-k)b_{i}\quad
if\left\|b_{i}-u_{i}\right\|_{2}^{2}\geq m\quad(3)$

In the second step, we update U according to instant B. When B is fixed ,(2)
becomes:

$\displaystyle
min\tilde{L}=\left\|U-B\right\|_{F}^{2}=\mathop{tr}\limits_{U}(U-B)(U-B)^{T}=\mathop{tr}\limits_{U}(U^{T}B)\quad
s.t.\quad u_{i}\in\\{\pm 1\\}^{k}$

So we have,$\displaystyle U=sign(B)\quad(4)$.

We iteratively solve the above two steps until the loss function reaches the
optimal solution. The algorithm can be summarized as Algorithm1.

Algorithm 1 Framework of Shadow Recurrent Hashing

0: $\displaystyle N$ datapoints $\displaystyle I$; supervised similarity
matrix$\displaystyle S$; hashing code length $\displaystyle k$; parameters
$\displaystyle\alpha,\beta$; number of iteration $\displaystyle T$; batch size
$\displaystyle M$

0: hashing codes B

1: Initialize weights,$\displaystyle U$

2: for all $\displaystyle i=1:T$ do

3: for all $\displaystyle n=1:\frac{N}{m}$ do

4: Randomly sample a mini-batch from $\displaystyle I$

5: Calculate $\displaystyle b_{i}$ by forward propagation

6: Calculate the gradient according to (3)

7: Update weights by BP

8: end for

9: Update U according to (4)

10: end for

##  5 Experiment

Network parameters:

I tested the SRH method on CIFAR10 with GPU Geforce 920MX. The SRH method is
implemented with Pytorch framework. The network structure consists of three
convolution-pooling layers and two fully connected layers. The convolution
layers use $\displaystyle 32,32,645\times 5$ filers with stride 1, and the
pooling layers use $\displaystyle 3\times 3$ windows with stride 2. The first
fully connected layer has 500 nodes.The second one has $\displaystyle k$
nodes, where $\displaystyle k$ denotes the length of the hash codes. All
activation function are Relu. m is set to be $\displaystyle 2\times k$.

I initialize the weight layers with the ”Xavier” initialization. During the
whole training process, the mini-batch size is set to 160, momentum is set to
0.9, weight decay rate is set to 0.004. The initial learning rate is
$\displaystyle 10^{-3}$. The margin threshold $\displaystyle m$ is set to
$\displaystyle m=2k$ to make sure that more than $\displaystyle\frac{k}{2}$
bits in hash codes are different for dissimilar pairs.

Training methodology:

If we directly train the network with $\displaystyle n$ query points and
$\displaystyle\frac{n(n-1)}{2}$ pairs, storing the image pairs could be very
consuming and wasted and a common CPU cannot hold such huge batch size. So to
make better use of the computational resources and storage place, I propose to
generate hash codes of image pairs by exploiting all the image pairs in each
mini-batch. As for the image pairs across different batches, we cover the
tradeoff by randomly selecting the mini-batches in each iteration and run the
whole algorithm plenty of times. As a result, we avoid the inconvenience of
dealing with all pairs as a whole, thus my method can be scalable to large-
scale datasets.

Dataset CIFAR-10

This dataset is made up of 60000 images with $\displaystyle 32\times 32$
pixels each, which belong to 10 different categories. These images are
directly vectorized and used as the input of the CNN architecture. Moreover,
we can get the semantic information matrix $\displaystyle S$ from the direct
category indicator. Images from the same category are considered to be
semantically similar, and vice versa.

Models(with $\displaystyle epoch=150,\beta=0.01$) | CIFAR10
---|---
$\displaystyle\alpha/\beta=0.1$ | 0.5531
$\displaystyle\alpha/\beta=1$ | 0.6166
$\displaystyle\alpha/\beta=10$ | 0.5666
$\displaystyle\alpha/\beta=100$ | 0.1935
Table 1: mAP of models, all hashing codes are 12 bits.

##  6 Conclusion

In this paper, I evaluate several deep supervised hashing methods according to
the timeline. Different methods give solution to different existing problems
in deep hashing. Some of them has achieved the state-of-the-art performance.
Moreover, to break through the bottleneck of existing deep hash method, I
propose a new idea ”shadow” to accelerate the optimization process. SRH
elaborately convert the discrete constraints in loss function to regularizer
and design a new type of regularizer to make sure the regularizer term stay
calm during training process. Experiments on CIFAR-10 shows the satisfying
performance of the proposed method.

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