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††thanks: Corresponding author, Electronic address<EMAIL_ADDRESS>
# Degradation of non-maximal entanglement of scalar and Dirac fields in non-
inertial frames

Qiyuan Pan Jiliang Jing  Institute of Physics and Department of Physics, Hunan
Normal University, Changsha, Hunan 410081, P. R. China
and
Key Laboratory of Low Dimensional Quantum Structures and Quantum Control of
Ministry of Education, Hunan Normal University , Changsha, Hunan 410081, P.R.
China

###### Abstract

The entanglement between two modes of the free scalar and Dirac fields as seen
by two relatively accelerated observers has been investigated. It is found
that the same initial entanglement for an initial state parameter $\alpha$ and
its “normalized partner” $\sqrt{1-\alpha^{2}}$ will be degraded by the Unruh
effect along two different trajectories except for the maximally entangled
state, which just shows the inequivalence of the quantization for a free field
in the Minkowski and Rindler coordinates. In the infinite acceleration limit
the state doesn’t have the distillable entanglement for any $\alpha$ for the
scalar field but always remains entangled to a degree which is dependent of
$\alpha$ for the Dirac field. It is also interesting to note that in this
limit the mutual information equals to just half of the initially mutual
information, which is independent of $\alpha$ and the type of field.

###### pacs:

03.65.Ud, 03.67.Mn, 04.70.Dy, 97.60.Lf

The quantum information theory has made rapid progress in recent years and
more and more efforts have been expended on the study of quantum information
in the relativistic framework PT-B-P-E . Especially, the entanglement in a
relativistic setting has received considerable attention because it is
considered to be a major resource for quantum information tasks such as
quantum teleportation, quantum computation and so on DAA . Despite the
potential interest to quantum information, the study of entanglement can also
help us get a deeper understanding of the black hole thermodynamics Bombelli-
Callen  and the black hole information paradox Hawking-Terashima . Thus, many
authors have investigated the entanglement in the relativistic frames inertial
or not for various fields PST ; Mann ; Alsing .

More recently, Fuentes-Schuller _et al._ Mann  and Alsing _et al._ Alsing
explicitly demonstrated that the entanglement is a quantity depending on a
relative acceleration of one of the observers who, before being accelerated,
shared a maximally entangled bosonic or fermionic pair. Their results also
showed that the different type of field will have a qualitatively different
effect on the degradation of entanglement produced by the Unruh effect Davies-
Unruh . Choosing a generic state as the initial entangled state in this Brief
Report

$\displaystyle|\Psi_{sk}\rangle=\sqrt{1-\alpha^{2}}|0_{s}\rangle^{M}|1_{k}\rangle^{M}+\alpha|1_{s}\rangle^{M}|0_{k}\rangle^{M},$
(1)

where $\alpha$ is some real number which satisfies $|\alpha|\in(0,1)$,
$\alpha$ and $\sqrt{1-\alpha^{2}}$ are the so-called “normalized partners”, we
will try to see what effects this uncertainly initial entangled state will on
the degradation of entanglement for two relatively accelerated observers due
to the presence of an initial state parameter $\alpha$. Notice that the
Schwarzschild space-time very close to the event horizon resembles the Rindler
space in the infinite acceleration limit Wald ; Mann . Hence, as in Mann ;
Alsing  our results in this limit can be applied to discuss the entanglement
between two free bosonic or fermionic modes seen by observers when one
observer falls into a black hole and the other barely escapes through eternal
uniform acceleration.

Rindler coordinates are appropriate for describing the viewpoint of an
observer moving with uniform acceleration. The world lines of uniformly
accelerated observers in the Minkowski coordinates correspond to hyperbolae in
the left (region I) and right (region II) of the origin which are bounded by
light-like asymptotes constituting the Rindler horizon Mann ; Alsing , so two
Rindler regions are causally disconnected from each other Birrell . An
observer undergoing uniform acceleration remains constrained to either Rindler
region I or II and has no access to the other sector. The system in Eq. (1) is
bipartite from an inertial perspective, but in a non-inertial frame an extra
set of modes in region II becomes relevant. Thus, we will study the mixed-
state entanglement of the state as seen by an inertial observer Alice
detecting the mode $s$ and a uniformly accelerated observer Bob with proper
acceleration $a$ in region I detecting the second mode $k$.

Bosonic entanglement For a free scalar field, the Minkowski vacuum state can
be expressed as a two-mode squeezed state in the Rindler frame Davies-Unruh ;
Birrell

$\displaystyle|0_{k}\rangle^{M}=\frac{1}{\cosh
r}\sum_{n=0}^{\infty}\tanh^{n}r|n_{k}\rangle_{I}|n_{k}\rangle_{II},$ (2)

where $\cosh r=(1-e^{-2\pi|k|c/a})^{-1/2}$, $k$ is the wave vector and $r$ is
the acceleration parameter, $|n\rangle_{I}$ and $|n\rangle_{II}$ indicate the
Rindler-region-I-particle mode and -II-antiparticle mode respectively. Using
Eq. (2) and the first excited state Mann ; Birrell

$\displaystyle|1_{k}\rangle^{M}=\frac{1}{\cosh^{2}r}\sum_{n=0}^{\infty}\tanh^{n}r\sqrt{n+1}|(n+1)_{k}\rangle_{I}|n_{k}\rangle_{II},$

we can rewrite Eq. (1) in terms of Minkowski modes for Alice and Rindler modes
for Bob. Since Bob is causally disconnected from region II, we will trace over
the states in this region and obtain

$\displaystyle\rho_{AB}=\frac{1}{\cosh^{2}r}\sum_{n=0}^{\infty}\tanh^{2n}r\rho_{n},$
$\displaystyle\rho_{n}=\alpha^{2}|1n\rangle\langle
1n|+\frac{\alpha\sqrt{(1-\alpha^{2})(n+1)}}{\cosh r}|1n\rangle\langle 0(n+1)|$
$\displaystyle\qquad+\frac{\alpha\sqrt{(1-\alpha^{2})(n+1)}}{\cosh
r}|0(n+1)\rangle\langle 1n|$
$\displaystyle\qquad+\frac{(1-\alpha^{2})(n+1)}{\cosh^{2}r}|0(n+1)\rangle\langle
0(n+1)|,$ (3)

where $|nm\rangle=|n_{s}\rangle^{M}|m_{k}\rangle_{I}$. The partial transpose
criterion provides a sufficient condition for the existence of entanglement in
this case peres : if at least one eigenvalue of the partial transpose is
negative, the density matrix is entangled; but a state with positive partial
transpose can still be entangled. It is well-known bound or nondistillable
entanglement Vidal-Plenio . Interchanging Alice’s qubits, we get the
eigenvalues of the partial transpose $\rho_{AB}^{T_{A}}$ in the ($n$,$n+1$)
block

$\displaystyle\lambda_{\pm}^{n}=\frac{\tanh^{2n}r}{2\cosh^{2}r}\left[\xi_{n}\pm\sqrt{\xi_{n}^{2}+\frac{4\alpha^{2}(1-\alpha^{2})}{\cosh^{2}r}}\right],$

where $\xi_{n}=\alpha^{2}\tanh^{2}r+(1-\alpha^{2})n/\sinh^{2}r$. Obviously the
eigenvalue $\lambda_{-}^{n}$ is always negative for finite acceleration
($r<\infty$). Hence, this mixed state is always entangled for any finite
acceleration of Bob. In the limit $r\rightarrow\infty$, the negative
eigenvalue will go to zero. To discuss this further, we will use the
logarithmic negativity which serves as an upper bound on the entanglement of
distillation Vidal-Plenio . This entanglement monotone is defined as
$N(\rho)=\log_{2}||\rho^{T}||$, where $||\rho^{T}||$ is the trace norm of the
partial transpose $\rho^{T}$. We therefore find

$\displaystyle
N(\rho_{AB})=\log_{2}\left\\{\frac{\alpha^{2}}{\cosh^{2}{r}}+\sum_{n=0}^{\infty}\frac{\tanh^{2n}r}{\cosh^{2}r}\right.$
$\displaystyle\times\left.\sqrt{\left[\alpha^{2}\tanh^{2}r+\frac{(1-\alpha^{2})n}{\sinh^{2}r}\right]^{2}+\frac{4\alpha^{2}(1-\alpha^{2})}{\cosh^{2}r}}\right\\}.$

For vanishing acceleration ($r=0$),
$N(\rho_{AB})=\log_{2}(1+2|\alpha|\sqrt{1-\alpha^{2}})$. In the range
$0<|\alpha|\leq 1/\sqrt{2}$ the larger $\alpha$, the stronger the initial
entanglement; but in the range $1/\sqrt{2}\leq|\alpha|<1$ the larger $\alpha$,
the weaker the initial entanglement. For finite acceleration, the monotonous
decrease of $N(\rho_{AB})$ with increasing $r$ for different $\alpha$ means
that the entanglement of the initial state is lost to the thermal fields
generated by the Unruh effect. From Fig. 1 it is surprisingly found that the
same initial entanglement for $\alpha$ and its “normalized partner”
$\sqrt{1-\alpha^{2}}$ will be degraded along two different trajectories except
for the maximally entangled state, i.e., $|\alpha|=1/\sqrt{2}$. This
phenomenon, due to the coupling of $\alpha$ and the hyperbolic functions
related to $r$, just shows the inequivalence of the quantization for a scalar
field in the Minkowski and Rindler coordinates. The logarithmic negativity is
exactly zero for any $\alpha$ in the limit $r\rightarrow\infty$, which
indicates that the state doesn’t have the distillable entanglement.

Figure 1:  Logarithmic negativity of the bosonic modes versus $r$ for
different $\alpha$.

The mutual information, which can be used to estimate the total amount of
correlations between any two subsystem of the overall system, is defined as
RAM

$\displaystyle I(\rho_{AB})=S(\rho_{A})+S(\rho_{B})-S(\rho_{AB}),$ (4)

where $S(\rho)=-\text{Tr}(\rho\log_{2}\rho)$ is the entropy of the density
matrix $\rho$. From Eq. (Degradation of non-maximal entanglement of scalar and
Dirac fields in non-inertial frames), we can obtain the entropy of this joint
state

$\displaystyle
S(\rho_{AB})=-\sum_{n=0}^{\infty}\frac{\tanh^{2n}r}{\cosh^{2}r}\left[\alpha^{2}+\frac{(1-\alpha^{2})(n+1)}{\cosh^{2}r}\right]$
$\displaystyle\qquad\times\log_{2}\frac{{\tanh^{2n}r}}{\cosh^{2}r}\left[\alpha^{2}+\frac{(1-\alpha^{2})(n+1)}{\cosh^{2}r}\right].$
(5)

Tracing over Alice’s states for $\rho_{AB}$, we get Bob’s density matrix in
region I; its entropy is

$\displaystyle
S(\rho_{BI})=-\sum_{n=0}^{\infty}\frac{\tanh^{2n}r}{\cosh^{2}r}\left[\alpha^{2}+\frac{(1-\alpha^{2})n}{\sinh^{2}r}\right]$
$\displaystyle\qquad\times\log_{2}\frac{\tanh^{2n}r}{\cosh^{2}r}\left[\alpha^{2}+\frac{(1-\alpha^{2})n}{\sinh^{2}r}\right].$
(6)

In the same way, we have Alice’s density matrix by tracing over Bob’s states;
its entropy is given by

$\displaystyle
S(\rho_{A})=-[\alpha^{2}\log_{2}\alpha^{2}+(1-\alpha^{2})\log_{2}(1-\alpha^{2})].$
(7)

We draw the behaviors of the mutual information $I(\rho_{AB})$ versus $r$ for
different $\alpha$ in Fig. 2. For vanishing acceleration, the initially mutual
information is
$I_{bi}=-2[\alpha^{2}\log_{2}\alpha^{2}+(1-\alpha^{2})\log_{2}(1-\alpha^{2})]$.
In the range $0<|\alpha|\leq 1/\sqrt{2}$ the larger $\alpha$, the stronger
$I_{bi}$; but in the range $1/\sqrt{2}\leq|\alpha|<1$ the larger $\alpha$, the
weaker $I_{bi}$. As the acceleration increases, the mutual information becomes
smaller. It is interesting to note that except for the maximally entangled
state, the same initially mutual information for $\alpha$ and
$\sqrt{1-\alpha^{2}}$ will be degraded along two different trajectories.
However, in the infinite acceleration limit, the mutual information converges
to the same value again, i.e.,
$I_{bf}=-[\alpha^{2}\log_{2}\alpha^{2}+(1-\alpha^{2})\log_{2}(1-\alpha^{2})]$,
which equals to just half of $I_{bi}$. Obviously if $I_{bi}$ is higher, it is
degraded to a higher degree in this limit. Since the distillable entanglement
in the infinite acceleration limit is zero, we are safe to say that the total
correlations consist of classical correlations plus bound entanglement in this
limit.

Figure 2:  Mutual information of the bosonic modes versus $r$ for different
$\alpha$.

Fermionic entanglement With the single-mode approximation used by Alsing _et
al._ , the fermionic Minkowski vacuum can be written as Alsing

$\displaystyle|0\rangle^{M}=\cos r|0\rangle_{I}|0\rangle_{II}+\sin
r|1\rangle_{I}|1\rangle_{II},$ (8)

and the only excited state is given by

$\displaystyle|1\rangle^{M}=|1\rangle_{I}|0\rangle_{II},$ (9)

where $\cos r=(1+e^{-2\pi\omega c/a})^{-1/2}$ and the acceleration parameter
$r$ is in the range $0\leq r\leq\pi/4$ for $0\leq a\leq\infty$ in this case.
Using Eq. (8) and (9) for the Minkowski particle states $|0_{k}\rangle^{M}$
and $|1_{k}\rangle^{M}$ and tracing over the modes in the region II, we get

$\displaystyle\rho_{AB}=(1-\alpha^{2})|01\rangle\langle 01|$
$\displaystyle\qquad~{}+\alpha\sqrt{(1-\alpha^{2})}\cos r(|01\rangle\langle
10|+|10\rangle\langle 01|)$
$\displaystyle\qquad~{}+\alpha^{2}(\cos^{2}r|10\rangle\langle
10|+\sin^{2}r|11\rangle\langle 11|),$ (10)

with $|mn\rangle=|m\rangle_{A}^{M}|n\rangle_{BI}$. The partial transpose
criterion provides a necessary and sufficient condition for entanglement in a
mixed state of two qubits peres : if at least one eigenvalue of the partial
transpose is negative, the density matrix is entangled. Interchanging Alice’s
qubits, we obtain an eigenvalue of the partial transpose $\rho_{AB}^{T_{A}}$

$\displaystyle\lambda_{-}=\frac{1}{2}\left[\alpha^{2}\sin^{2}{r}-\sqrt{\alpha^{4}\sin^{4}{r}+4\alpha^{2}(1-\alpha^{2})\cos^{2}{r}}\right],$

which is always negative for $0\leq r\leq\pi/4$. Thus, the state is always
entangled for any uniform acceleration of Bob. The logarithmic negativity is
expressed as

$\displaystyle N(\rho_{AB})=\log_{2}[1-\alpha^{2}\sin^{2}{r}$
$\displaystyle\quad\quad+\sqrt{\alpha^{4}\sin^{4}{r}+4\alpha^{2}(1-\alpha^{2})\cos^{2}{r}}~{}].$

Figure 3:  Logarithmic negativity of the fermionic modes versus $r$ for
different $\alpha$ (notice that $\alpha_{N_{m}}=\sqrt{(4-\sqrt{2})/7}$ ).

For vanishing acceleration ($r=0$),
$N(\rho_{AB})=\log_{2}(1+2|\alpha|\sqrt{1-\alpha^{2}})$. For finite
acceleration, the entanglement is degraded by the Unruh effect just as shown
in Fig. 3. We find that in the range $0<|\alpha|\leq 1/\sqrt{2}$ for the
larger $\alpha$, the initial entanglement is higher, but it isn’t always
degraded to a higher degree. It should be noted that for
$1/2<|\alpha|<1/\sqrt{2}$ the final entanglement of the initial state is
higher than that of the maximally entangled state, i.e., $\log_{2}(3/2)\simeq
0.585$, and for $|\alpha|=\sqrt{(4-\sqrt{2})/7}$ the maximally final
entanglement is $\log_{2}[(5+4\sqrt{2})/7]\simeq 0.606$. In the range
$1/\sqrt{2}\leq|\alpha|<1$ the larger $\alpha$, the weaker the initial
entanglement and the lower the final entanglement. Unlike the behaviors of the
bosonic case, except for the maximally entangled state, the same initial
entanglement of the fermionic modes for $\alpha$ and $\sqrt{1-\alpha^{2}}$
will be degraded along two different trajectories and asymptotically reach two
differently nonvanishing minimum values in the infinite acceleration limit
($r=\pi/4$) due to the coupling of $\alpha$ and the trigonometric functions
related to $r$. In the infinite acceleration limit
$N(\rho_{AB})=\log_{2}(1-\alpha^{2}/2+|\alpha|\sqrt{2-7\alpha^{2}/4})\neq 0$,
which means that the state is always entangled. This is in strong contrast to
the bosonic case and shows that the fermionic system can be used as a resource
for performing certain quantum information processing tasks.

Similar to the bosonic case, we go through the same process again and get the
mutual information for these fermionic modes

$\displaystyle
I(\rho_{AB})=(1-\alpha^{2}\sin^{2}{r})\log_{2}(1-\alpha^{2}\sin^{2}{r})$
$\displaystyle\quad\quad+\alpha^{2}\sin^{2}{r}\log_{2}\alpha^{2}\sin^{2}{r}$
$\displaystyle\quad\quad-(1-\alpha^{2}\cos^{2}{r})\log_{2}(1-\alpha^{2}\cos^{2}{r})$
$\displaystyle\quad\quad-\alpha^{2}\cos^{2}{r}\log_{2}\alpha^{2}\cos^{2}{r}$
$\displaystyle\quad\quad-\alpha^{2}\log_{2}\alpha^{2}-(1-\alpha^{2})\log_{2}(1-\alpha^{2}),$
(11)

whose trajectories versus $r$ for different $\alpha$ are shown by Fig. 4.

Figure 4:  Mutual information of the fermionic modes versus $r$ for different
$\alpha$.

For vanishing acceleration, the initially mutual information is
$I_{fi}=-2[\alpha^{2}\log_{2}\alpha^{2}+(1-\alpha^{2})\log_{2}(1-\alpha^{2})]$,
whose behaviors are the same to $I_{bi}$ of the bosonic modes. The mutual
information becomes smaller as the acceleration increases, and again we
surprisingly find that the same initially mutual information for $\alpha$ and
$\sqrt{1-\alpha^{2}}$ will be degraded along two different trajectories except
for the maximally entangled state. In the infinite acceleration limit the
mutual information converges to
$I_{ff}=-[\alpha^{2}\log_{2}\alpha^{2}+(1-\alpha^{2})\log_{2}(1-\alpha^{2})]$,
which is just half of $I_{fi}$. This behavior is reminiscent of that seen for
the bosonic case, so we conclude that

$\displaystyle I_{f}=\frac{1}{2}I_{i},$ (12)

which is independent of $\alpha$ and the type of field.

It should be noted that if we set the initial entangled state as

$\displaystyle|\Psi_{sk}\rangle=\alpha|0_{s}\rangle^{M}|0_{k}\rangle^{M}+\sqrt{1-\alpha^{2}}|1_{s}\rangle^{M}|1_{k}\rangle^{M},$
(13)

we will have the same behavior of the entanglement degradation for the same
$\alpha$ just as shown in Figs. 1-4.

Summarizing, the entanglement of the scalar and Dirac fields in non-inertial
frames is degraded by the Unruh effect as the Bob’s rate of acceleration
increases, but their behaviors of the degradation of entanglement are
different for the same initial state parameter $\alpha$. It is surprisingly
found that the same initial entanglement for $\alpha$ and
$\sqrt{1-\alpha^{2}}$ will be degraded along two different trajectories except
for the maximally entangled state, which just shows the inequivalence of the
quantization for a free field in the Minkowski and Rindler coordinates. In the
infinite acceleration limit, which can be applied to the case Alice falling
into a black hole while Bob barely escapes, the state doesn’t have the
distillable entanglement for any $\alpha$ for the scalar field but always
remains entangled to a degree which is dependent of $\alpha$ for the Dirac
field. It should be noted that for $|\alpha|=\sqrt{(4-\sqrt{2})/7}$, we will
have the maximally final entanglement for the fermionic state in this limit.
Further analysis shows that the mutual information is degraded to a
nonvanishing minimum value which is dependent of $\alpha$ for these two fields
with increasing acceleration parameter $r$. However, it is interesting to note
that the mutual information in the infinite acceleration limit equals to just
half of the initially mutual information, which is independent of $\alpha$ and
the type of field.

This work was supported by the National Natural Science Foundation of China
under Grant No. 10675045; the FANEDD under Grant No. 200317; the Hunan
Provincial Natural Science Foundation of China under Grant No. 07A0128; and
the construct program of the key discipline in hunan province.

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morphism $\mathrm{Spec}(K)\rightarrow\mathfrak{M}_{\mathrm{tot}}$ admits a
unique extension over $\mathfrak{M}_{0}$. Therefore by properness of
$\mathfrak{M}$ we have that $\mathcal{Z}_{\omega_{P}}\subseteq\mathfrak{M}$,
hence $\omega_{P}\in\Psi$. Again since $\mathcal{Z}_{\omega_{P}}$ is contained
inside the polygon space contained in $\mathfrak{M}$, we have that
$\omega_{P}\in\Phi_{\Delta}$, hence $\omega_{P,K}\in\Psi_{\Delta}$. Therefore
$\Psi=\Psi_{\Delta}$.

_Step 6_. Given a maximally-biconnected complex $\Delta$, let
$\mathfrak{M}_{\Delta}\vcentcolon=(\Psi_{\Delta})\operatorname{\\!/\\!\\!/\\!}T$.
To finish the proof it suffices to show that
$\mathfrak{M}_{\Delta}\subset\mathfrak{M}_{\mathrm{tot}}$ together with its
natural morphism to $\mathfrak{M}_{0}$ is a crepant resolution of
$\mathfrak{M}_{0}$. Then the map in the statement of the Theorem is well-
defined, and by the previous steps it is surjective.

First note that $U(\Psi_{\Delta})\operatorname{\\!/\\!\\!/\\!}T$ is an open
subscheme of $\mathfrak{M}_{\mathrm{tot}}$, which is covered by
$\mathfrak{M}_{\theta}$, for $\theta\in F$ generic. Hence each point of
$\mathfrak{M}_{\Delta}$ is smooth, and locally around each point the natural
morphism to $\mathfrak{M}_{0}$ is crepant. Moreover since
$\mathfrak{M}_{\Delta}$ is separated, it suffices to show that it satisfies
the existence part of the valuative criterion for properness over
$\mathfrak{M}_{0}$.

We have seen that if $\\{i\\}^{c}\in\Delta$ then
$\mathfrak{M}_{\Delta}=\mathfrak{M}_{\theta}$, for $\theta\in C_{i}^{\circ}$.
Thus we may assume $\Delta$ is full. ∎

We have already seen that the number of maximally-biconnected complexes on
$[n]$ equals $\lambda(n)$.

######  Corollary 5.15.

There are $\lambda(n)$ crepant resolutions of $\mathfrak{M}_{0}$ up to
isomorphism over $\mathfrak{M}_{0}$.

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# DMKD: Improving Feature-based Knowledge Distillation for Object Detection
Via Dual Masking Augmentation

###### Abstract

Recent mainstream masked distillation methods function by reconstructing
selectively masked areas of a student network from the feature map of its
teacher counterpart. In these methods, the masked regions need to be properly
selected, such that reconstructed features encode sufficient discrimination
and representation capability like the teacher feature. However, previous
masked distillation methods only focus on spatial masking, making the
resulting masked areas biased towards spatial importance without encoding
informative channel clues. In this study, we devise a Dual Masked Knowledge
Distillation (DMKD) framework which can capture both spatially important and
channel-wise informative clues for comprehensive masked feature
reconstruction. More specifically, we employ dual attention mechanism for
guiding the respective masking branches, leading to reconstructed feature
encoding dual significance. Furthermore, fusing the reconstructed features is
achieved by self-adjustable weighting strategy for effective feature
distillation. Our experiments on object detection task demonstrate that the
student networks achieve performance gains of 4.1% and 4.3% with the help of
our method when RetinaNet and Cascade Mask R-CNN are respectively used as the
teacher networks, while outperforming the other state-of-the-art distillation
methods.

Index Terms—  Masked Knowledge Distillation, Dual Attention, Masked Feature
Reconstruction, Object Detection

##  1 INTRODUCTION

It is well known that knowledge distillation is capable of transferring the
dark knowledge learned in a complex teacher network to a lightweight student
network. The earlier knowledge distillation algorithms primarily focus on the
output head of the networks, and the representative schemes include logit-
based distillation for classification [1, 2] and head-based distillation for
detection [3, 4, 5]. More recently, feature-based distillation strategies have
received increasing attention due to its desirable task-agnostic flexibility
[6, 7, 8, 9]. In particular, feature distillation helps the student network
imitate the teacher feature with enhanced representational capability. In
object detection, a variety of advanced feature distillation methods have been
developed. To be specific, the classic FitNet [10] performed distillation at
the global feature level. FGD [11] was developed to distinguish foreground and
background distillation within a unified framework. CWD [12] simply minimized
the Kullback-Leibler divergence [13] between the channel probability maps of
the two networks, and the distillation process is inclined to the most
significant regions of each channel.

Fig. 1: Illustration of our dual masking strategies. Obviously, our method can
simultaneously capture the spatially salient and object-specific channel-wise
clues which are highlighted on the spatial and channel attention map for dual
masking. $\tau_{s}$ and $\tau_{c}$ are introduced as two hyperparameters which
are used for adjusting the respective masking ratio.

In feature distillation methods, recent research suggests it is preferable for
the student model to reconstruct important features from the teacher model in
the first place instead of following the teacher for generating competitive
representations. For instance, MGD [14] was proposed to randomly mask pixels
in the feature map of the student network, leading to reconstructed features
of the teacher model via a simple block. AMD [15] performed attention-guided
feature masking on the feature map of the student network, such that region-
specific important features can be identified via spatially adaptive feature
masking instead of random masking in the previous method.

Fig. 2: Visualisation of the feature maps obtained by different distillation
methods.

Although current masked feature distillation methods contribute to improving
the performance of the student model, the selectively masked areas in the
aforementioned approaches are generated only considering spatial importance.
For instance, MGD [14] performs random feature masking that is position-
agnostic, resulting in non-informative and semantically irrelevant regions to
be reconstructed. This limitation is somewhat alleviated in AMD [15] which
utilizes spatial-wise masking for identifying the spatially salient regions,
whereas it still fails to identify the informative channels which are critical
for dense prediction task as indicated in [12, 16]. In this study,
differently, we attempt to capture spatially important regions and object-
aware channel-wise clues both of which can be obtained by dual attention
mechanisms. In particular, the dual masking strategy illustrated in Fig. 1 is
used for masked feature reconstruction to comprehensively characterize the
significant object-aware features. As shown in Fig. 2, in contrast to MGD and
AMD which only capture discriminative local features (head and body of a
bird), our DMKD provides a more comprehensive object-specific encoding for
improved representation capability.

##  2 Methodology

Fig. 3: Overall pipeline of the proposed DMKD. After generating the spatial
and channel-wise attention maps from the teacher feature, we perform dual
masking guided by the attention maps for reconstructing important object-aware
features. Finally, the reconstructed features are fused with two learnable
parameters $\alpha$ and $\beta$ for self-adjustable weighting. Table 1:
Comparison of our method with other distillation methods for object detection
on COCO.

Teacher | Student | mAP | $AP_{S}$ | $AP_{M}$ | $AP_{L}$ | mAR | $AR_{S}$ | $AR_{M}$ | $AR_{L}$
---|---|---|---|---|---|---|---|---|---
| RetinaNet-ResNet50 (baseline) | 37.4 | 20.6 | 40.7 | 49.7 | 53.9 | 33.1 | 57.7 | 70.2
| FKD [17] | 39.6 (+2.2) | 22.7 | 43.4 | 52.5 | 56.1 (+2.2) | 36.8 | 60.0 | 72.1
| FGD [11] | 40.7 (+3.3) | 22.9 | 45.0 | 54.7 | 56.8 (+2.9) | 36.5 | 61.4 | 72.8
| MGD [14] | 41.0 (+3.6) | 23.4 | 45.3 | 55.7 | 57.0 (+3.1) | 37.2 | 61.7 | 72.8
| AMD [15] | 41.3 (+3.9) | 23.9 | 45.4 | 55.7 | 57.4 (+3.5) | 38.2 | 61.7 | 73.5
RetinaNet [18] ResNeXt101 (41.0) | DMKD (Ours) | 41.5 (+4.1) | 24.0 | 45.8 | 55.8 | 57.8 (+3.9) | 38.5 | 62.3 | 73.2
| RetinaNet-ResNet50 (baseline) | 38.6 | 22.5 | 42.2 | 50.4 | 55.1 | 34.9 | 59.4 | 70.3
| FKD [17] | 40.6 (+2.0) | 23.4 | 44.6 | 53.0 | 56.9 (+1.8) | 37.3 | 60.9 | 71.4
| FGD [11] | 42.0 (+3.4) | 24.0 | 45.7 | 55.6 | 58.2 (+3.1) | 37.8 | 62.2 | 73.3
| MGD [14] | 42.3 (+3.7) | 24.4 | 46.2 | 55.9 | 58.4 (+3.3) | 40/4 | 62.3 | 73.9
| AMD [15] | 42.7 (+4.1) | 24.8 | 46.5 | 56.3 | 58.8 (+3.7) | 40.6 | 62.4 | 74.1
RepPoints [19] ResNeXt101 (44.2) | DMKD (Ours) | 42.9 (+4.3) | 25.1 | 46.9 | 56.4 | 60.1 (+4.0) | 40.9 | 62.9 | 74.4
| RetinaNet-ResNet50 (baseline) | 38.4 | 21.5 | 42.1 | 50.3 | 52.0 | 32.6 | 55.8 | 66.1
| FKD [17] | 41.5 (+3.1) | 23.5 | 45.0 | 55.3 | 54.4 (+2.4) | 34.0 | 58.2 | 69.9
| FGD [11] | 42.0 (+3.6) | 23.8 | 46.4 | 55.5 | 55.4 (+3.4) | 35.5 | 60.0 | 70.0
| MGD [14] | 42.1 (+3.7) | 23.7 | 46.4 | 56.1 | 55.5 (+3.5) | 35.4 | 60.0 | 70.5
| AMD [15] | 42.4 (+4.0) | 24.1 | 46.5 | 56.2 | 55.8 (+3.8) | 35.3 | 60.0 | 70.8
Cascade Mask RCNN [20] ResNeXt101 (47.3) | DMKD (Ours) | 42.7 (+4.3) | 24.4 | 46.9 | 56.5 | 56.0 (+4.0) | 35.5 | 60.4 | 71.0

###  2.1 Preliminaries

Before elaborating on our proposed method, we briefly review the basic
feature-based distillation framework formulated as:

$\mathcal{L}_{base}=\sum_{c=1}^{C}\sum_{h=1}^{H}\sum_{w=1}^{W}\textit{MSE}\left(\Theta_{align}(\mathrm{F}_{c;\,h;\,w}^{S}),\mathrm{F}_{c;\,h;\,w}^{T}\right)$
(1)

where $\mathrm{F}_{c;\,h;\,w}^{S/T}\in\mathbb{R}^{C\times H\times W}$ is the
feature map generated from the student or teacher network. $MSE(\cdot)$
denotes the fitting error between the student feature aligned by a learnable
function $\Theta_{align}$ and the teacher feature. It can be observed from the
above formulation that the vanilla feature-based distillation model forces the
student raw feature to be as close as possible to the teacher counterpart. In
this sense, the teacher acts as a template to be mimicked directly and feature
distillation can be identified as a straightforward and no-frills algorithm
because the features of student and teacher networks are directly used for
representational matching.

###  2.2 Our DMKD framework

To comprehensively encode object-aware semantics into the student network, we
develop a dual masked knowledge distillation (DMKD) framework in which both
the spatially salient regions and informative channels are uncovered for
preferable masked feature reconstruction. Notably, our model can be treated as
a plug-and-play augmentation module that is applicable to the vanilla feature-
based distillation methods. As shown in Fig. 3, our DMKD consists of three key
steps: dual attention map generation, attention-guided masking, and self-
adjustable weighted fusion. To begin with, the spatial and channel-wise
attention maps are produced from the teacher feature formulated as follows:

$\displaystyle
A^{s}=\phi_{align}\left(Sigmoid\left(\frac{1}{C\mathcal{T}}\left\langle\left\|\mathrm{F}^{T}_{1}\right\|_{2}^{2},...,\left\|\mathrm{F}^{T}_{n}\right\|_{2}^{2}\right\rangle\right)\right)$
(2) $\displaystyle
A^{c}=Sigmoid\left(\frac{1}{HW\mathcal{T}}\sum_{h=1}^{H}\sum_{w=1}^{W}\left\langle\mathrm{F}_{h,w,1}^{T},...,\mathrm{F}_{h,w,C}^{T}\right\rangle\right)$
(3)

where $A^{s}\in\mathbb{R}^{1\times H\times W}$ and
$A^{c}\in\mathbb{R}^{C\times 1\times 1}$ are spatial and channel-wise
attention map, respectively. $\mathrm{F}^{T}_{n}$ in Eq. (2) is the $n$-th
vector of teacher’s feature map, while $\mathrm{F}^{T}_{h,w,C}$ in Eq. (3) is
the feature map of $C$-th channel. $\mathcal{T}$ is introduced for adjusting
the distribution [1].

Next, the resulting dual attention maps $A^{s}$ and $A^{c}$ are used for
deriving the corresponding masking maps $M^{s}$ and $M^{c}$ with the
thresholds $\tau_{s}$ and $\tau_{c}$:

$\displaystyle M^{s}_{i\subseteq[0,H),j\subseteq[0,W)}$
$\displaystyle=\left\\{\begin{aligned} &0,\quad A_{i,j}^{s}\geq\tau_{s}\\\
&1,\quad Otherwise\end{aligned}\right.$ (4) $\displaystyle
M^{c}_{k\subseteq[0,C)}$ $\displaystyle=\left\\{\begin{aligned} &0,\quad
A_{k}^{c}\geq\tau_{c}\\\ &1,\quad Otherwise\end{aligned}\right.$ (5)

where $M^{s}$ and $M^{c}$ share the same shape with $A^{s}$ and $A^{c}$,
respectively. Then, we utilize them to perform spatial and channel masking on
the student feature $\mathrm{F}^{S}$, generating two masked features
$\mathrm{F}^{S}_{s}$ and $\mathrm{F}^{S}_{c}$:

$\displaystyle\mathrm{F}^{S}_{s}=\Theta_{align}\left(\mathrm{F}^{S}\right)\otimes
M^{s}$ (6)
$\displaystyle\mathrm{F}^{S}_{c}=\Theta_{align}\left(\mathrm{F}^{S}\right)\otimes
M^{c}$ (7)

where $\otimes$ denotes the element-wise multiplication.

Finally, two different generation blocks based on convolution and Multilayer
Perceptrons (MLPs) are respectively employed to reconstruct the spatial and
channel-wise masked features. Notably, the channel reconstruction is achieved
by MLPs, which differs from the spatial one as in MGD and AMD because the
channel features are spatially independent. In other words, this generative
layer should only focus on the interaction of inter-channels which will be
demonstrated in the ablation study. In addition, pairwise self-adjustable
weighting factors $\alpha$ and $\beta$ are introduced to fuse the dual
reconstructed features, leading to the final student feature formulated as:

$\mathrm{F}^{rec}=\alpha\cdot\Theta_{Conv}(\mathrm{F}^{S}_{s})+\beta\cdot\Theta_{MLP}(\mathrm{F}^{S}_{c})$
(8)

Thus, the distillation loss can be built on the reconstructed student feature
$F^{rec}$ and the teacher counterpart $F^{T}$ as follows:

$\mathcal{L}_{DMKD}=\sum_{c=1}^{C}\sum_{h=1}^{H}\sum_{w=1}^{W}\textit{MSE}\left(\mathrm{F}_{c;\,h;\,w}^{rec},\mathrm{F}_{c;\,h;\,w}^{T}\right)$
(9)

##  3 Experiments

###  3.1 Experimental Setup

We have evaluated our DMKD on the public MS COCO2017 [21] benchmarking dataset
for the object detection task. Besides, Average Precision (AP) and Average
Recall (AR) are used for performance measures. The hyperparameters are
empirically set to $\tau_{s}=0.55$, $\tau_{c}=0.65$,
$\alpha=\beta=0.5(initial)$, $\mathcal{T}=0.5$. Besides, $\gamma$ is set as
$5\times 10^{-7}$ and $5\times 10^{-6}$ for the one-stage and the two-stage
models respectively. During the training process, we employ the SGD optimizer
to train all detectors over the course of 24 epochs. Besides, ResNeXt101 [22]
and ResNet50 [23] are used as the respective backbones for the teacher and the
student networks.

###  3.2 Results

As shown in Table 1, in the first group of experiments, the results indicate
that our DMKD not only significantly boosts the baseline student model by 4.1%
mAP but also outperforms the teacher network by 0.5% mAP. The advantages of
DMKD against the state-of-the-art MGD and AMD are manifested by respective
further performance gains of 0.5% and 0.2%. Similar performance superiority of
our method can also be observed regarding mAR metric. When the RetinaNet
network is replaced with the other detectors in the remaining experiments,
analogous results are also reported.

###  3.3 Ablation studies

In this section, we conduct different ablation studies to gain further insight
into the interior design of our DMKD framework with RetinaNet used as the
detector.

Table 2: Ablation study of different masking strategies. $C$ and $S$ denote
single Channel and Spatial-wise masking respectively.

| Masking strategy
---
Dual (Ours) | w/o C | w/o S | w/o masking
mAP (%) | 41.5 | 41.3 (-0.2) | 41.2 (-0.3) | 39.6 (-1.9)

####  3.3.1 Masking strategy

We first explore the effect of feature masking on the distillation performance
and compare different masking strategies. As shown in Table 2, a significant
performance drop of exceeding 1.5% can be observed when any masking strategy
is not involved, suggesting that masking strategy plays a beneficial role in
feature distillation for enhanced feature reconstruction. Furthermore, when
comparing our dual masking scheme with both of the single masking strategies,
consistent performance gains are reported, demonstrating the advantage of dual
masking in capturing both spatially salient regions and informative channel-
wise clues for encoding comprehensive object-aware representations.

Table 3: Ablation study of spatial and channel masking thresholds $\tau_{s}$
and $\tau_{c}$.

$\tau_{s}$ ($\tau_{c}=0.65$) | 0.55 | 0.45 (mask ratio $\uparrow$) | 0.65 (mask ratio $\downarrow$)
---|---|---|---
mAP (%) | 41.5 | 41.5 (+0.0) | 41.4 (-0.1)
$\tau_{c}$ ($\tau_{s}=0.55$) | 0.65 | 0.55 (mask ratio $\uparrow$) | 0.75 (mask ratio $\downarrow$)
mAP (%) | 41.5 | 41.3 (-0.2) | 41.4 (-0.1)

####  3.3.2 Masking ratios

Table 3 illustrates the performance of our method with varying masking ratios.
It can be clearly observed that the model performance slightly deteriorates
when decreasing the spatially masking ratio with increased $\tau_{s}$, which
is consistent with the finding in MAE [24]. This implies some spatially
important regions may be excluded from spatial masking. On the other hand,
when increasing the channel-wise masking ratio, a similar performance decline
can also be observed, suggesting that the model excessively focuses on
redundant and non-informative channels. In our experiments, the dual
thresholds $\tau_{s}$ and $\tau_{c}$ are empirically set as 0.55 and 0.65.

####  3.3.3 Feature generative blocks

Table 4: Ablation study of two different generative blocks for channel
reconstruction.

Generative block |  Conv-based $\Theta_{Conv}$ |  MLP-based $\Theta_{MLP}$
---|---|---
mAP (%) | 41.0 (-0.5) | 41.5

We also discuss the effect of feature generation block design on channel-wise
feature reconstruction. As shown in Table 4, the MLP-based module within our
DMKD achieves better performance than the Conv-based one. This can be
explained by the properties of the MLP operator, i.e., in contrast with the
spatial interaction as in ConvNets, it acts independently on the spatial
dimension and only models the inter-channel relations. Specifically, we use
two successive linear projections
$\theta_{MLP}^{{}^{\prime}}\in\mathbb{R}^{C\times 2C}$ and
$\theta_{MLP}^{{}^{\prime\prime}}\in\mathbb{R}^{2C\times C}$ followed by GELU
[25] and Layer Normalization [26] in practice.

##  4 Conclusion

In this paper, we explore the masked-based distillation algorithms and propose
a simple and effective feature augmentation framework termed dual masking
knowledge distillation (DMKD). In contrast with previous masked-based
algorithms, we simultaneously focus on spatial and channel dimensions which
respectively reflect the important spatial regions and object-aware
information. Moreover, DMKD is a dynamic algorithm that adaptively captures
the most semantically relevant spatial areas and informative channels,
improving flexibility over the previous methods. Extensive experiments
demonstrate the superiority of our design.

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