††thanks: Corresponding author

# On the two-dimensional time-dependent anisotropic harmonic oscillator in a
magnetic field

Pinaki Patra<EMAIL_ADDRESS>Department of Physics, Brahmananda Keshab
Chandra College, Kolkata, India-700108

###### Abstract

A Charged harmonic oscillator in a magnetic field, Landau problems, and an
oscillator in a noncommutative space, share the same mathematical structure in
their Hamiltonians. We have considered a two-dimensional anisotropic harmonic
oscillator (AHO) with arbitrarily time-dependent parameters (effective mass
and frequencies), placed in an arbitrarily time-dependent magnetic field. A
class of quadratic invariant operators (in the sense of Lewis and Riesenfeld)
have been constructed. The invariant operators ($\hat{\mathcal{I}}$) have been
reduced to a simplified representative form by a linear canonical
transformation (the group $Sp(4,\mathbb{R})$). An orthonormal basis of the
Hilbert space consisting of the eigenvectors of $\hat{\mathcal{I}}$ is
obtained. In order to obtain the solutions of the time-dependent Schrödinger
equation corresponding to the system, both the geometric and dynamical phase-
factors are constructed. Generalized Peres-Horodecki separability criterion
(Simon’s criterion) for the ground state corresponding to our system has been
demonstrated.

##  I Introduction

The long list of applications of a time-dependent harmonic oscillator (TDHO)
in modelling a variety of physical phenomena makes the TDHO in a subject of
active research tdho1 ; tdho2 ; tdho3 ; tdho4 ; tdho5 ; tdho6 . Dynamics of a
harmonic oscillator (HO) placed in a magnetic field is equivalent to the
dynamics of a HO in a deformed noncommutative space ncs1 ; ncs2 ; ncs3 ; ncs4
, which is supposedly compatible with quantum theory of gravity qg1 ; qg2 ;
qg3 ; qg4 ; qg5 ; qg6 ; qg7 ; qg8 ; qg9 . Moreover, Maxwell-Chern-Simons model
in long wavelength limit and a non-relativistic anyon in a magnetic field
resembles with the two dimensional HO in a magnetic field mcs1 ; mcs2 ; mcs3 ;
mcs4 ; mcs5 ; mcs6 ; mcs7 ; mcs8 . The systems with time-dependent parameters
(mass, frequency, external magnetic fields etc.) appear to be self-righteous
for the most physical systems, in particular for a system, which interacts
with its environment tdho1 ; tdho2 ; tdho3 ; tdho4 ; tdho5 ; tdho6 . For
example, the time-dependent effective mass (TDEM) appears for the energy
density functional approach to many-body problems tdem1 , the study of
electronic properties of condensed matter systems tdem2 , the Schrödinger
equation in curved space in the light of deformed algebras tdem3 , nonlinear
optical properties in quantum well, the cosmological models to quantum
information theory tdem5 , the asymmetric shape of crackling noise pulses
emitted by a diverse range of noisy systems can be modelled with the help of
time-dependent effective mass tdem6 , and many moretdem7 ; tdem8 .
The eigenstates of an annihilation operator (displaced ground state of HO),
namely the coherent states (CS) are particularly important in interferometry
cs1 ; cs2 ; cs3 ; cs4 . In particular, the maximum contrast in the
interference pattern is achieved for a CS cs5 . In the context of CS, a
bipartite Gaussian states are particularly important for its frequent
appearance in the systems with small oscillations cs6 ; cs7 . For example,
vibration of the test mass (corresponding to the masses on which the
interferometric mirrors are placed) under the effects of the tidal forces of
gravitational waves are demonstrated by the anisotropic oscillators cs8 ; cs9
; cs10 , which exhibits the bipartite Gaussian states upon consideration in a
two-dimensional noncommutative space csncs1 ; csncs2 ; csncs3 ; csncs4 .
Therefore, an anisotropic harmonic oscillator is an important problem on its
own right. In this article, we have considered a most general form of time-
dependent anisotropic harmonic oscillator placed in a time-dependent external
magnetic field. The time-dependent system is completely solved with the help
of Lewis-Riesenfeld (LR) invariant method Lewis1 ; Lewis2 ; Lewis3 ; Lewis4 ;
Lewis5 ; Lewis6 ; Lewis7 , which is based on the existence of an invariant
operator $\hat{\mathcal{I}}(t)$, corresponding to the Hamiltonian
$\hat{H}(t)$. If $\phi$ is an eigen-function of $\hat{\mathcal{I}}$ (i.e.,
$\hat{\mathcal{I}}\phi=\lambda\phi$), then $\psi=\phi e^{i\theta}$ will
satisfy the time-dependent Schrödinger equation (TDSE)
$i\hbar\partial\psi/\partial t=\hat{H}\psi$, for some time-dependent phase
factor $\theta(t)$ Lewis5 ; Lewis6 ; Lewis7 . LR-invariant (LRI) operators for
time-dependent (TD) isotropic harmonic oscillators (HO) are well known in the
literature ncs2 . On the other hand, time-independent anisotropic oscillators
are farely well known ncs1 ; anisotropic1 . However, for arbitrary time-
dependent anisotropic oscillator there is a gap in the literature. Our present
study aims to fulfil this gap. We have constructed a quadratic invariant
operator, which is transformed to a simpler form with the aid of a similarity
transformation (symplectic group $Sp(4,\mathbb{R})$ anisotropic1 ; sp1 ). The
eigen-states and eigenvalues of $\hat{\mathcal{I}}$ are constructed by
factorizing it with the annihilation and creation operators. The phase factors
(both the geometrical and dynamical phase1 ; phase2 ; phase3 ; phase4 ) are
obtained in order to solve the TDSE. Then the separability conditions for the
bipartite ground state (CS) have been illustrated with the generalized Peres-
Horodecki criterion (Simon’s criterion) sep1 ; sep2 ; sep3 ; sep4 ; sep5 ;
sep6 .
The organization of our paper is the followings. A brief discussion on our
system is followed by the Lewis-Riesenfeld invariant method. An unitary
transformation is constructed in order to transform the invariant operator in
a simpler one, keeping the intrinsic symplectic structure intact. At last, the
explicit form of the ground state is constructed and the explicit form of the
separability condition for the concerned state is mentioned.

##  II Systems under consideration

Hamiltonian of a charged particle inside a magnetic field can be expressed as
a combination of the kinetic part (quadratic in momentum), along with a
paramagnetic term (cross term of vector potential and momentum operator) and a
diamagnetic contribution (quadratic in vector potential). Using the Coulomb
gauge condition ($\vec{\nabla}.\vec{A}=0$) of the magnetic vector potential
$\vec{A}=(-\alpha_{01}(t)x_{2},\alpha_{02}(t)x_{1})$, the paramagnetic term of
the Hamiltonian for an anisotropic harmonic oscillator with charge $e$ and
effective mass $M=(\mu_{1}(t),\mu_{2}(t))$ is reduced to

$\hat{L}_{d}=\nu_{1}\left\\{\hat{x}_{2},\hat{p}_{1}\right\\}-\nu_{2}\left\\{\hat{x}_{1},\hat{p}_{2}\right\\},$
(1)

whereas, the diamagnetic contribution is reduced to

$V_{diamag}=\frac{1}{2}(k_{02}x_{1}^{2}+k_{01}x_{2}^{2}),$ (2)

where

$\nu_{j}=\frac{e}{2\mu_{j}}\alpha_{0j},\;\;k_{0j}=\frac{e^{2}\alpha_{0j}^{2}}{\mu_{j}},\;j=1,2.$
(3)

Here we have set the speed of light in vacuum to unity, which will be followed
throughout in this paper. The notation
$\\{\mathcal{O}_{1},\mathcal{O}_{2}\\}=\mathcal{O}_{1}\mathcal{O}_{2}+\mathcal{O}_{2}\mathcal{O}_{1}$
stands for the anticommutator of the operators $\mathcal{O}_{1}$ and
$\mathcal{O}_{2}$.
Thus the Hamiltonian ($\hat{H}$) for an anisotropic harmonic oscillator
(potential $V_{an}(x_{1},x_{2})=k_{1}(t)x_{1}^{2}/2+k_{2}(t)x_{2}^{2}/2$) with
charge $e$ and effective mass $M=(\mu_{1}(t),\mu_{2}(t))$, under the magnetic
vector potential ($-\alpha_{01}(t)x_{2},\alpha_{02}(t)x_{1}$) reads

$\hat{H}=\frac{1}{2\mu_{1}}\hat{p}_{1}^{2}+\frac{1}{2\mu_{2}}\hat{p}_{2}^{2}+\frac{1}{2}\alpha_{1}\hat{x}_{1}^{2}+\frac{1}{2}\alpha_{2}\hat{x}_{2}^{2}+\hat{L}_{d},$
(4)

Where

$\alpha_{1}=k_{1}+4\mu_{2}\nu_{2}^{2},\;\alpha_{2}=k_{2}+4\mu_{1}\nu_{1}^{2}.$
(5)

The system (4) is equivalent to an anisotropic oscillator (AHO) in a two
dimensional noncommutative space (NC)
($[\hat{\tilde{X}}_{1},\hat{\tilde{X}}_{2}]=i\theta,[\hat{\tilde{P}}_{1},\hat{\tilde{P}}_{2}]=i\eta,[\hat{\tilde{X}}_{i},\hat{\tilde{P}}_{j}]=i\hbar_{e}\delta_{ij};i,j=1,2.$),
with the spatial NC parameter $\theta$ and momentum NC parameter $\eta$. Here
$\hbar_{e}=\hbar(1+(\theta\eta)/(4\hbar^{2}))$, and $\delta_{ij}$ is Kronecker
delta, which equals to one for $i=j$ and zero otherwise. The usual Heisenberg
algebra for the commutative space
($[\hat{x}_{i},\hat{x}_{j}]=[\hat{p}_{i},\hat{p}_{j}]=0,\;[\hat{x}_{i},\hat{p}_{j}]=i\hbar\delta_{ij};i,j=1,2.$)
is connected with the NC-space algebra by the following Bopp’s shift.

$\displaystyle\hat{\tilde{X}}_{j}$ $\displaystyle=$
$\displaystyle\hat{x}_{j}+\frac{\theta}{2\hbar}\epsilon^{ij}\hat{p}_{i},\;\hat{\tilde{P}}_{j}=\hat{p}_{j}+\frac{\eta}{2\hbar}\epsilon^{ji}\hat{x}_{i};\;i,j=1,2.$
(6)

Here $\epsilon^{ij}$ is completely antisymmetric tensor with
$\epsilon^{12}=1$. With the transformations (6), it is evident that the NC-
space Hamiltonian

$\hat{H}_{nc}=\frac{1}{2m_{1}}\hat{\tilde{P}}_{1}^{2}+\frac{1}{2m_{2}}\hat{\tilde{P}}_{2}^{2}+\frac{1}{2}m_{1}\tilde{\omega}_{1}^{2}\hat{\tilde{X}}_{1}^{2}+\frac{1}{2}m_{2}\tilde{\omega}_{2}^{2}\hat{\tilde{X}}_{2}^{2},$
(7)

is equivalent to (4), with the following identification of the time-dependent
parameters.

$\displaystyle\frac{1}{\mu_{1}}$ $\displaystyle=$
$\displaystyle\frac{1}{m_{1}}+\frac{1}{4\hbar^{2}}m_{2}\tilde{\omega}_{2}^{2}\theta^{2},\;\frac{1}{\mu_{2}}=\frac{1}{m_{2}}+\frac{1}{4\hbar^{2}}m_{1}\tilde{\omega}_{1}^{2}\theta^{2},$
(8) $\displaystyle\alpha_{1}$ $\displaystyle=$ $\displaystyle
m_{1}\tilde{\omega}_{1}^{2}+\frac{\eta^{2}}{4\hbar^{2}m_{2}},\;\alpha_{2}=m_{2}\tilde{\omega}_{2}^{2}+\frac{\eta^{2}}{4\hbar^{2}m_{1}},$
(9) $\displaystyle\nu_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{4m_{1}\hbar}(\eta+m_{1}m_{2}\tilde{\omega}_{2}^{2}\theta),\;\nu_{2}=\frac{1}{4m_{2}\hbar}(\eta+m_{1}m_{2}\tilde{\omega}_{1}^{2}\theta).$
(10)

The aim of the present paper is to study the Hamiltonian (4), which has the
following quadratic form.

$\hat{H}=\frac{1}{2}X^{T}\hat{\mathcal{H}}X,$ (11)

with

$\displaystyle X=\left(\begin{array}[]{c}\hat{x}_{1}\\\ \hat{p}_{1}\\\
\hat{x}_{2}\\\
\hat{p}_{2}\end{array}\right),\;\mbox{and}\;\;\hat{\mathcal{H}}=\left(\begin{array}[]{cccc}\frac{\alpha_{1}}{2}&0&0&-\nu_{2}\\\
0&\frac{1}{2\mu_{1}}&\nu_{1}&0\\\ 0&\nu_{1}&\frac{\alpha_{2}}{2}&0\\\
-\nu_{2}&0&0&\frac{1}{2\mu_{2}}\end{array}\right).$ (20)

Here $T$ tenotes the matrix transposition.
One can see that the deformed angular momentum operator $\hat{L}_{d}$ will be
reduced to the usual angular momentum operator for an isotropic oscillator
(IHO). For an IHO, the usual angular momentum operator ($\hat{L}$) commutes
with the Hamiltonian. Thus the problem can be reduced to the two
noninteracting oscillators. However, for AHO ($\nu_{1}\neq\nu_{2}$), the
process of diagonalization is not starightforward. On the next section, we
have constructed a class of invariant operators (in the sense of Lewis-
Riesenfeld) corresponding to the time-dependent Hamiltonian (4).

##  III Invariant operators

At first we observe that, the set of operators

$\displaystyle\mathcal{A}=\left\\{\hat{x}_{1}^{2},\hat{x}_{2}^{2},\hat{p}_{1}^{2},\hat{p}_{2}^{2},\\{\hat{x}_{1},\hat{p}_{1}\\},\\{\hat{x}_{2},\hat{p}_{2}\\},\\{\hat{x}_{1},\hat{x}_{2}\\},\\{\hat{p}_{1},\hat{p}_{2}\\},\\{\hat{x}_{2},\hat{p}_{1}\\},\\{\hat{x}_{1},\hat{p}_{2}\\}\right\\}$
(21)

forms a closed quasi-algebra with respect to the Hamiltonian $\hat{H}$ as
follows.

$\displaystyle\left[\hat{H},\hat{x}_{1}^{2}\right]$ $\displaystyle=$
$\displaystyle-\frac{i\hbar}{\mu_{1}}\\{\hat{x}_{1},\hat{p}_{1}\\}-i\hbar\nu_{1}\\{\hat{x}_{1},\hat{x}_{2}\\}.$
(22) $\displaystyle\left[\hat{H},\hat{x}_{2}^{2}\right]$ $\displaystyle=$
$\displaystyle-\frac{i\hbar}{\mu_{2}}\\{\hat{x}_{2},\hat{p}_{2}\\}+i\hbar\nu_{2}\\{\hat{x}_{1},\hat{x}_{2}\\}.$
(23) $\displaystyle\left[\hat{H},\hat{p}_{1}^{2}\right]$ $\displaystyle=$
$\displaystyle
i\hbar\alpha_{1}\\{\hat{x}_{1},\hat{p}_{1}\\}-i\hbar\nu_{2}\\{\hat{p}_{1},\hat{p}_{2}\\}$
(24) $\displaystyle\left[\hat{H},\hat{p}_{2}^{2}\right]$ $\displaystyle=$
$\displaystyle
i\hbar\alpha_{2}\\{\hat{x}_{2},\hat{p}_{2}\\}+i\hbar\nu_{1}\\{\hat{p}_{1},\hat{p}_{2}\\}$
(25) $\displaystyle\left[\hat{H},\\{\hat{x}_{1},\hat{p}_{1}\\}\right]$
$\displaystyle=$
$\displaystyle-\frac{2i\hbar}{\mu_{1}}\hat{p}_{1}^{2}+2i\hbar\alpha_{1}\hat{x}_{1}^{2}-i\hbar(\nu_{1}\\{\hat{x}_{2},\hat{p}_{1}\\}+\nu_{2}\\{\hat{x}_{1},\hat{p}_{2}\\}).$
(26) $\displaystyle\left[\hat{H},\\{\hat{x}_{2},\hat{p}_{2}\\}\right]$
$\displaystyle=$
$\displaystyle-\frac{2i\hbar}{\mu_{2}}\hat{p}_{2}^{2}+2i\hbar\alpha_{2}\hat{x}_{2}^{2}+i\hbar(\nu_{1}\\{\hat{x}_{2},\hat{p}_{1}\\}+\nu_{2}\\{\hat{x}_{1},\hat{p}_{2}\\}).$
(27) $\displaystyle\left[\hat{H},\\{\hat{x}_{1},\hat{x}_{2}\\}\right]$
$\displaystyle=$
$\displaystyle-\frac{i\hbar}{\mu_{1}}\\{\hat{x}_{2},\hat{p}_{1}\\}-\frac{i\hbar}{\mu_{2}}\\{\hat{x}_{1},\hat{p}_{2}\\}+2i\hbar(\nu_{2}\hat{x}_{1}^{2}-\nu_{1}\hat{x}_{2}^{2}).$
(28) $\displaystyle\left[\hat{H},\\{\hat{p}_{1},\hat{p}_{2}\\}\right]$
$\displaystyle=$ $\displaystyle
i\hbar\alpha_{1}\\{\hat{x}_{1},\hat{p}_{2}\\}+i\hbar\alpha_{2}\\{\hat{x}_{2},\hat{p}_{1}\\}+2i\hbar(\nu_{1}\hat{p}_{1}^{2}-\nu_{2}\hat{p}_{2}^{2}).$
(29) $\displaystyle\left[\hat{H},\\{\hat{x}_{2},\hat{p}_{1}\\}\right]$
$\displaystyle=$
$\displaystyle-\frac{i\hbar}{\mu_{2}}\\{\hat{p}_{1},\hat{p}_{2}\\}+i\hbar\alpha_{1}\\{\hat{x}_{1},\hat{x}_{2}\\}+i\hbar\nu_{2}(\\{\hat{x}_{1},\hat{p}_{1}\\}-\\{\hat{x}_{2},\hat{p}_{2}\\}).$
(30) $\displaystyle\left[\hat{H},\\{\hat{x}_{1},\hat{p}_{2}\\}\right]$
$\displaystyle=$
$\displaystyle-\frac{i\hbar}{\mu_{1}}\\{\hat{p}_{1},\hat{p}_{2}\\}+i\hbar\alpha_{2}\\{\hat{x}_{1},\hat{x}_{2}\\}+i\hbar\nu_{1}(\\{\hat{x}_{1},\hat{p}_{1}\\}-\\{\hat{x}_{2},\hat{p}_{2}\\}).$
(31)

The closed quasi-algebra of $\mathcal{A}$ suggests an ansatz for the invariant
operator $\hat{\mathcal{I}}$ to be

$\displaystyle\hat{\mathcal{I}}=u_{11}\hat{x}_{1}^{2}+u_{22}\hat{x}_{2}^{2}+v_{11}\hat{p}_{1}^{2}+v_{22}\hat{p}_{2}^{2}+w_{11}\\{\hat{x}_{1},\hat{p}_{1}\\}+w_{22}\\{\hat{x}_{2},\hat{p}_{2}\\}$
$\displaystyle+u_{12}\\{\hat{x}_{1},\hat{x}_{2}\\}+v_{12}\\{\hat{p}_{1},\hat{p}_{2}\\}+w_{21}\\{\hat{x}_{2},\hat{p}_{1}\\}+w_{12}\\{\hat{x}_{1},\hat{p}_{2}\\}.$
(32)

The invariance of $\hat{\mathcal{I}}$ implies

$\frac{d\hat{\mathcal{I}}}{dt}=\frac{\partial\hat{\mathcal{I}}}{\partial
t}+\frac{1}{i\hbar}\left[\hat{\mathcal{I}},\hat{H}\right]=0.$ (33)

Using the ansatz (III) in (33) we get the following set of coupled equations
of the coefficients.

$\displaystyle\dot{w}=\nu v.$ (34) $\displaystyle\dot{u}=2\alpha v.$ (35)
$\displaystyle\dot{v}=\beta u+\mu w.$ (36)

The vectors $w,u,v$ are given by

$\displaystyle
w=(w_{12},w_{21})^{T},\;u=(u_{11},v_{11},u_{22},v_{22})^{T},\;v=(v_{12},w_{22},w_{11},u_{12})^{T}.$
(37)

The coefficient matrices are as follows.

$\displaystyle\mu=\left(\begin{array}[]{cc}-\frac{1}{\mu_{1}}&-\frac{1}{\mu_{2}}\\\
-\nu_{1}&-\nu_{2}\\\ \nu_{1}&\nu_{2}\\\
\alpha_{2}&\alpha_{1}\end{array}\right),\;\;\alpha=\left(\begin{array}[]{cccc}0&0&\alpha_{1}&\nu_{2}\\\
\nu_{1}&0&-\frac{1}{\mu_{1}}&0\\\ 0&\alpha_{2}&0&-\nu_{1}\\\
-\nu_{2}&-\frac{1}{\mu_{2}}&0&0\end{array}\right).$ (46)
$\displaystyle\nu=\sigma_{x}\mu^{T}\mathcal{S}_{x},\;\;\beta=\left(\begin{array}[]{cccc}0&-\nu_{2}&0&\nu_{1}\\\
0&0&-\frac{1}{\mu_{2}}&\alpha_{2}\\\ -\frac{1}{\mu_{1}}&\alpha_{1}&0&0\\\
-\nu_{1}&0&\nu_{2}&0\end{array}\right).$ (51)

With

$\displaystyle\mathcal{S}_{x}=\left(\begin{array}[]{cc}0&\sigma_{x}\\\
\sigma_{x}&0\end{array}\right),\;\mbox{and}\;\sigma_{x}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right).$ (56)

For a given set of time dependent parameters, one can solve the above set of
equations (34), (35) and (36), in principle.
The invariant operator (III) is transformed into a simple form on the next
section.

##  IV Diagonalization of the system

We can express the invariant operator (III) in the following quadratic form.

$\hat{\mathcal{I}}=\frac{1}{2}X^{T}\hat{\mathcal{H}}_{\mathcal{I}}X,$ (57)

with

$\displaystyle\hat{\mathcal{H}}_{\mathcal{I}}=\left(\begin{array}[]{cc}\hat{C}&\hat{A}^{T}\\\
\hat{A}&\hat{B}\end{array}\right).$ (60)

Where

$\displaystyle\hat{C}=\left(\begin{array}[]{cc}u_{11}&w_{11}\\\
w_{11}&v_{11}\end{array}\right),\hat{B}=\left(\begin{array}[]{cc}u_{22}&w_{22}\\\
w_{22}&v_{22}\end{array}\right),\hat{A}=\left(\begin{array}[]{cc}u_{12}&w_{21}\\\
w_{12}&v_{12}\end{array}\right).$ (67)

One can identify that

$\displaystyle\hat{\mathcal{S}}\hat{\mathcal{J}}_{4}+\hat{\mathcal{J}}_{4}\hat{\mathcal{S}}^{T}=0,$
(68)

where

$\displaystyle\hat{\mathcal{S}}=\hat{\mathcal{J}}_{4}\hat{\mathcal{H}}_{\mathcal{I}},\;\mbox{with}\;\;\hat{\mathcal{J}}_{2n}=\left(\begin{array}[]{cc}0&\hat{\mathbb{I}}_{n}\\\
-\hat{\mathbb{I}}_{n}&0\end{array}\right);$ (71)

$\hat{\mathbb{I}}_{n}$ being the $n\times n$ identity matrix. From the
equation (68), we can conclude that $\mathcal{S}\in sp(4,\mathbb{R})$. Since
our co-ordinates are the Cartesian type (the natural range (spectrum) for each
of them is the entire real line), the commutation relations

$[\hat{X}_{\alpha},\hat{X}_{\beta}]=-(\Sigma_{y})_{\alpha\beta};\;(\alpha,\beta=1,2,3,4),$
(72)

generates an intrinsic symplectic matrix

$\displaystyle\hat{\Sigma}_{y}=\mbox{diag}(\sigma_{y},\sigma_{y}).$ (73)

Here we have used the unit such that $\hbar=1$, which will be followed
throughout the present paper, unless otherwise specified. $\hat{\sigma}_{y}$
being the second Pauli matrix

$\displaystyle\sigma_{y}=\left(\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right).$ (76)

The aim of the present section is to diagonalize
$\hat{\mathcal{H}}_{\mathcal{I}}$ keeping the symplectic structure (72)
intact.
The structure constants of the closed quasi-algebra of the co-ordinates
($\hat{X}_{i},i=1,..,4.$) with respect to the Hamiltonian ($\hat{H}$), induce
the following commutation relation.

$\frac{1}{2\hbar}\left[i\hat{\mathcal{I}},X\right]=\hat{\Omega}X.$ (77)

Where

$\hat{\Omega}=\left(\begin{array}[]{cccc}w_{11}&v_{11}&w_{21}&v_{12}\\\
-u_{11}&-w_{11}&-u_{12}&-w_{12}\\\ w_{12}&v_{12}&w_{22}&v_{22}\\\
-u_{12}&-w_{21}&-u_{22}&-w_{22}\end{array}\right).$ (78)

We observe the followings.

$\displaystyle\hat{\Omega}=\left(\begin{array}[]{cc}A_{1}&B_{1}\\\
C_{1}&D_{1}\end{array}\right),$ (81)

where

$\displaystyle A_{1}=\left(\begin{array}[]{cc}w_{11}&v_{11}\\\
-u_{11}&-w_{11}\end{array}\right),\;B_{1}=\left(\begin{array}[]{cc}w_{21}&v_{12}\\\
-u_{12}&-w_{12}\end{array}\right),$ (86) $\displaystyle
C_{1}=\left(\begin{array}[]{cc}w_{12}&v_{12}\\\
-u_{12}&-w_{21}\end{array}\right),\;D_{1}=\left(\begin{array}[]{cc}w_{22}&v_{22}\\\
-u_{22}&-w_{22}\end{array}\right).$ (91)

Since

$\displaystyle A_{1}=i\sigma_{y}C,\;C_{1}=i\sigma_{y}A,\;D_{1}=i\sigma_{y}B,$
(92)

we have

$\displaystyle\Delta_{A_{1}}=\Delta_{C},\;\Delta_{B_{1}}=\Delta_{A},$ (93)
$\displaystyle\Delta_{C_{1}}=\Delta_{A},\;\Delta_{D_{1}}=\Delta_{B}.$ (94)

Where the notation $\Delta_{Q}$ denotes the determinant of the matrix $Q$.
We wish to diagonalize $\hat{\Omega}$, and construct an equivalent co-ordinate
system for our purpose to be served. The construction of $\hat{Q}^{-1}$ and
$\hat{Q}$ can be done by arranging the normalized eigenvectors ( left and
right) column-wise. Although, $\hat{\Omega}$ is a normal matrix
$([\hat{\Omega}^{\dagger},\hat{\Omega}]=0$, it is not a symmetric one
$(\hat{\Omega}^{T}\neq\hat{\Omega})$). Therefore, the left and right eigen-
vectors of $\hat{\Omega}$ are not same (although, the left and right eigen-
values are the same). The characteristic polynomial
($p(\lambda)=\det(\lambda\mathbb{\hat{I}}-\hat{\Omega}),$) of $\hat{\Omega}$
is given by

$p(\lambda)=\lambda^{4}+\Delta\lambda^{2}+\Delta_{\Omega},\;\mbox{with}\;\;\Delta=\Delta_{C}+\Delta_{B}+2\Delta_{A}.$
(95)

One can see that $p(\lambda)$ has four purely imaginary roots for the possible
time-dependent real parameters in our problem.

$\lambda\in\\{\pm
i\sigma_{j},\;j=1,2|\sigma_{j}^{2}=\frac{1}{2}(\Delta\pm\sqrt{\Delta^{2}-4\Delta_{\Omega}})\\}.$
(96)

Let the eigenvalue equations for $\hat{\Omega}$ are given as follows.

$\displaystyle\chi_{lj}\hat{\Omega}$ $\displaystyle=$
$\displaystyle-i\sigma_{j}\chi_{lj},\;j=1,2.$ (97)
$\displaystyle\chi_{lj}^{*}\hat{\Omega}$ $\displaystyle=$ $\displaystyle
i\sigma_{j}\chi_{lj}^{*}\;,j=1,2.$ (98) $\displaystyle\hat{\Omega}\chi_{rj}$
$\displaystyle=$ $\displaystyle-i\sigma_{j}\chi_{rj},\;j=1,2.$ (99)
$\displaystyle\hat{\Omega}\chi_{rj}^{*}$ $\displaystyle=$ $\displaystyle
i\sigma_{j}\chi_{rj}^{*},\;j=1,2.$ (100)

Where the suffix $l$ and $r$ stands for the left and right eigen-vectors
respectively. The following identities hold quite generally

$\displaystyle\chi_{lj}^{*}\chi_{rj}$ $\displaystyle=$
$\displaystyle\chi_{lj}\chi_{rj}^{*},\forall i,j=1,2.$ (101)
$\displaystyle\chi_{lj}\chi_{rj}$ $\displaystyle=$
$\displaystyle\chi_{lj}^{*}\chi_{rj}^{*}=\delta_{ij}.$ (102)

This suggests the relationship between the left and right eigenvalues as

$\chi_{rj}=-\Sigma_{y}\chi_{lj}^{\dagger},\;(j=1,2),$ (103)

which can be used as the normalization conditions. In our case, the explicit
form of the left eigenvectors $\chi_{lj},\;(j=1,2)$ are given by

$\displaystyle\chi_{lj}=\frac{1}{k_{j}}\left(s_{j1}+iq_{j1},s_{j2}+iq_{j2},s_{j3},s_{j4}+iq_{j4}\right).$
(104)

Where

$\displaystyle s_{j1}$ $\displaystyle=$ $\displaystyle
u_{11}^{2}(v_{12}u_{22}-w_{21}w_{22})+u_{11}u_{12}(\sigma_{j}^{2}+w_{12}w_{21}$
(105) $\displaystyle+w_{11}w_{22}-v_{12}u_{12})-u_{11}u_{22}w_{11}w_{12}.$
$\displaystyle q_{j1}$ $\displaystyle=$
$\displaystyle\sigma_{j}u_{11}(w_{12}u_{22}-u_{12}w_{22}-u_{11}w_{21}+u_{12}w_{11}).$
(106) $\displaystyle s_{j2}$ $\displaystyle=$
$\displaystyle\sigma_{j}^{2}(u_{11}w_{21}+u_{12}w_{22}-u_{22}w_{12})+w_{11}^{2}(u_{12}w_{22}-u_{22}w_{12})$
(107)
$\displaystyle+w_{21}w_{11}(u_{12}w_{12}-u_{11}w_{22})+w_{11}v_{12}(u_{11}u_{22}-u_{12}^{2}).$
$\displaystyle q_{j2}$ $\displaystyle=$
$\displaystyle\sigma_{j}^{3}u_{12}+\sigma_{j}[u_{12}(w_{12}w_{21}-v_{12}u_{12}+w_{11}^{2})+u_{11}(v_{12}u_{22}-w_{21}w_{22}-w_{11}w_{21})].$
(108) $\displaystyle s_{j3}$ $\displaystyle=$ $\displaystyle
u_{11}u_{22}(\sigma_{j}^{2}+w_{11}^{2}-u_{11}v_{11})+u_{11}u_{12}(v_{11}u_{12}-w_{11}w_{21})$
(109) $\displaystyle+u_{11}w_{21}(u_{11}w_{21}-u_{12}w_{11}).$ $\displaystyle
s_{j4}$ $\displaystyle=$ $\displaystyle
u_{11}w_{22}\sigma^{2}+u_{11}^{2}(v_{12}w_{21}-v_{11}w_{22})+u_{12}u_{11}(v_{11}w_{12}-v_{12}w_{11})$
(110) $\displaystyle+u_{11}w_{11}(w_{11}w_{22}-w_{12}w_{21}).$ $\displaystyle
q_{j4}$ $\displaystyle=$ $\displaystyle
u_{11}\sigma_{1}^{3}+u_{11}\sigma_{1}(w_{11}^{2}-u_{11}v_{11}+w_{21}w_{12}-v_{12}u_{12}).$
(111)

$k_{j}$ is the normalization factor.
For the time-independent Hamiltonian for an anisotropic harmonic oscillator we
have

$w_{11}=w_{22}=u_{12}=v_{12}=0.$ (112)

In that case the components of the left eigenvectors simplifies to

$\displaystyle s_{j1}=q_{j2}=s_{j4}=0.$ (113) $\displaystyle
q_{j1}=\sigma_{j}u_{11}(w_{12}u_{22}-u_{11}w_{21}).$ (114) $\displaystyle
s_{j2}=\sigma_{j}^{2}(u_{11}w_{21}-u_{22}w_{12}).$ (115) $\displaystyle
s_{j3}=u_{11}u_{22}\sigma_{j}^{2}+u_{11}^{2}(w_{21}^{2}-v_{11}u_{22}).$ (116)
$\displaystyle
q_{j4}=u_{11}\sigma_{1}^{3}+u_{11}\sigma_{1}(w_{21}w_{12}-u_{11}v_{11}).$
(117)

That means, for time-independent case, the components are either real or
purely imaginary, which simplifies the calculations.
However, for general time-dependent systems, we have to consider the
components (105) to (111). The right eigenvectors $(\chi_{rj},\;j=1,2)$ can be
constructed from (103) in a straighforward manner.
Now the similarity transformation $\hat{Q}$ (as well as $\hat{Q}^{-1}$) can be
constructed by arranging the eigen-vectors columnwise. In particular,

$\displaystyle\hat{Q}$ $\displaystyle=$
$\displaystyle(\chi_{r1},\chi_{r1}^{*},\chi_{r2},\chi_{r2}^{*}),$ (118)
$\displaystyle\hat{Q}^{-1}$ $\displaystyle=$
$\displaystyle(\chi_{l1}^{T},(\chi_{l1}^{*})^{T},\chi_{l2}^{T},(\chi_{l2}^{*})^{T}).$
(119)

One can verify that

$\hat{Q}^{\dagger}=-\hat{\Sigma}_{z}\hat{Q}^{-1}\hat{\Sigma}_{y},$ (120)

with

$\hat{\Sigma}_{z}=\mbox{diag}(\sigma_{z},\sigma_{z}).$ (121)

Where $\sigma_{z}$ is the third Pauli matrix

$\displaystyle\sigma_{z}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right).$ (124)

Now, the diagonalization gives

$\hat{Q}_{D}=\hat{Q}^{-1}\hat{\Omega}\hat{Q}=\mbox{diag}(-i\sigma_{1},i\sigma_{1},-i\sigma_{2},i\sigma_{2}).$
(125)

Whether this diagonalization helps our purpose, or not, can be seen by the
followings.
Let the new co-ordinate vector $\hat{a}$ is given by

$\hat{a}=\hat{Q}^{-1}\hat{\zeta}.$ (126)

Then, using (120), we can write

$\displaystyle\therefore\zeta^{\dagger}$ $\displaystyle=$
$\displaystyle\hat{a}^{\dagger}(-\hat{\Sigma}_{z}\hat{Q}^{-1}\hat{\Sigma}_{y}).$
(127)

Then, the time-independent eigen-value equation for the invariant operator
$\hat{\mathcal{I}}$

$\hat{\mathcal{I}}\psi=\lambda\hat{\mathcal{I}},$ (128)

is transformed as

$\displaystyle\hat{Q}^{-1}\frac{1}{2}\zeta^{T}\hat{\mathcal{I}}\zeta\psi\hat{Q}$
$\displaystyle=$ $\displaystyle\lambda\hat{Q}^{-1}\hat{\mathcal{I}}\hat{Q}$
(129) $\displaystyle\therefore\frac{1}{2}\hat{a}^{\dagger}\Sigma\hat{a}\psi$
$\displaystyle=$ $\displaystyle\lambda\hat{\Sigma}\psi.$ (130)

Where

$\hat{\mathcal{I}}_{D}=\hat{\Sigma}=\mbox{diag}(\sigma_{1},\sigma_{1},\sigma_{2},\sigma_{2}).$
(131)

That means, the new equivalent diagonal Hamiltonian $\hat{\Sigma}$
corresponding to the invariant operator $\hat{\mathcal{I}}$ obeys the same
dynamics as that of our original one ($\hat{\mathcal{I}}$).

##  V Eigen-functions and eigen-values of the invariant operator

If we compute the commutators of the components of $\hat{a}$, then we can
identify that they forms a set of independent annihilation and creation
operators. In particular, we can identify that

$\displaystyle\hat{\zeta}^{T}=(\hat{a}_{1},\hat{a}_{1}^{\dagger},\hat{a}_{2},\hat{a}_{2}^{\dagger}),$
(132)

with the following commutation relations.

$\displaystyle[\hat{a}_{k},\hat{a}_{l}^{\dagger}]=\delta_{kl},\;[\hat{a}_{k},\hat{a}_{l}]=[\hat{a}_{k}^{\dagger},\hat{a}_{l}^{\dagger}]=0;\;k,l=1,2.$
(133)

The eigenstates of $\hat{\mathcal{I}}$ can be expressed as

$\displaystyle|n_{1},n_{2}\rangle=\frac{1}{\sqrt{n_{1}!n_{2}!}}(\hat{a}_{1}^{\dagger})^{n_{1}}(\hat{a}_{2}^{\dagger})^{n_{2}}|0,0\rangle;n_{1},n_{2}=0,1,2,3,.....$
(134)

The corresponding energy eigenvalues are

$E_{n_{1},n_{2}}=\left(n_{1}+\frac{1}{2}\right)\sigma_{1}+\left(n_{2}+\frac{1}{2}\right)\sigma_{2}.$
(135)

##  VI Time-dependent Phase factor and the solutions of Schrödinger equation

If $|n_{1},n_{2}\rangle$ is an eigenstate of the invariant operator
$\mathcal{\hat{I}}$, then the time-dependent Schrödinger equation will be
satisfied by the same function multiplied with some time-dependent phase
factor ($\theta_{n_{1},n_{2}}(t)$). In particular,

$\hat{H}\left(|n_{1},n_{2}\rangle
e^{i\theta_{n_{1},n_{2}}(t)}\right)=i\frac{\partial}{\partial
t}\left(|n_{1},n_{2}\rangle e^{i\theta_{n_{1},n_{2}}(t)}\right).$ (136)

That means

$\dot{\theta}_{n_{1},n_{2}}=\langle
n_{1},n_{2}|\left(i\frac{\partial}{\partial
t}-\hat{H}\right)|n_{1},n_{2}\rangle.$ (137)

We can split the phase-factor into geometric phase $\theta^{g}$ (including the
Berry phase phenomenon for slowly varying parameters) and dynamical phase
$\theta^{d}$ as follows.

$\displaystyle\dot{\theta}_{n_{1},n_{2}}^{g}=i\langle
n_{1},n_{2}|\frac{\partial}{\partial t}|n_{1},n_{2}\rangle.$ (138)
$\displaystyle\dot{\theta}_{n_{1},n_{2}}^{d}=-\langle
n_{1},n_{2}|\hat{H}|n_{1},n_{2}\rangle.$ (139)

For the ground state of time-dependent anisotropic oscillator

$\dot{\theta}_{0,0}^{g}=\frac{\dot{N}_{0}}{N_{0}}-\dot{a}\langle\hat{x}_{1}^{2}\rangle-\dot{b}\langle\hat{x}_{2}^{2}\rangle-\dot{c}\langle\hat{x}_{1}\hat{x}_{2}\rangle.$
(140)

Using the equations (126) and (127), the quadratic form (11) can be rewritten
as

$\hat{H}=\hat{a}^{\dagger}\left(-\Sigma_{z}\hat{Q}^{-1}\Sigma_{y}\right)\hat{\mathcal{H}}\hat{Q}\hat{a}.$
(141)

Since

$\tilde{\Omega}=\Sigma_{y}\mathcal{H},$ (142)

is a subclass of the matrix $\Omega$, it will be diagonalized by $Q$. The
characteristic polynomial ($P_{\tilde{\Omega}}(\tilde{\lambda})$) of
$\tilde{\Omega}$ is

$P_{\tilde{\Omega}}(\tilde{\lambda})=\tilde{\lambda}^{4}+\tilde{\Delta}\tilde{\lambda}^{2}+\Delta_{\tilde{\Omega}},$
(143)

with

$\displaystyle\tilde{\Delta}$ $\displaystyle=$
$\displaystyle-\frac{\alpha_{1}}{4\mu_{1}}-\frac{\alpha_{2}}{4\mu_{2}}-2\nu_{1}\nu_{2},$
(144) $\displaystyle\Delta_{\tilde{\Omega}}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{1}\alpha_{2}}{16\mu_{1}\mu_{2}}-\frac{\nu_{2}^{2}\alpha_{2}}{4\mu_{1}}-\frac{\nu_{1}^{2}\alpha_{1}}{4\mu_{2}}+\nu_{1}^{2}\nu_{2}^{2}.$
(145)

$P_{\tilde{\Omega}}(\tilde{\lambda})$ has positive discriminant

$D_{\tilde{\Omega}}=\left(\frac{\alpha_{1}}{4\mu_{1}}-\frac{\alpha_{2}}{4\mu_{2}}\right)^{2}+(\alpha_{1}\nu_{1}+\alpha_{2}\nu_{2})\left(\frac{\nu_{1}}{\mu_{2}}+\frac{\nu_{2}}{\mu_{1}}\right)\geq
0.$ (146)

In other words, $\tilde{\lambda}^{2}$ is real. Moreover,

$\tilde{\lambda}^{2}=\frac{1}{2}\left(-\tilde{\Delta}+\sqrt{D_{\tilde{\Omega}}}\right)\geq
0,$ (147)

gives two real roots for $\tilde{\lambda}$, which reads

$\tilde{\sigma}_{\pm}=\pm\frac{1}{\sqrt{2}}\sqrt{-\tilde{\Delta}+\sqrt{D_{\tilde{\Omega}}}}=\pm\tilde{\sigma}_{1}.$
(148)

Other two roots will be either real or purely imaginary depending on the
parameter values. In particular,

$\displaystyle(\alpha_{2}-4\mu_{1}\nu_{1}^{2})(\alpha_{1}-4\mu_{2}\nu_{2}^{2})\geq
0\implies\tilde{\lambda}^{2}\geq 0,$ (149)
$\displaystyle(\alpha_{2}-4\mu_{1}\nu_{1}^{2})(\alpha_{1}-4\mu_{2}\nu_{2}^{2})\leq
0\implies\tilde{\lambda}^{2}\leq 0.$ (150)

It is worth noting that, for an isotropic oscillator $\tilde{\lambda}^{2}$
will be positive for all parameter values. For our present case (anisotropic),
the roots

$\pm\tilde{\sigma}_{2}=\pm\frac{1}{\sqrt{2}}\sqrt{-\tilde{\Delta}-\sqrt{D_{\tilde{\Omega}}}},$
(151)

are either real or purely imaginary, depending on the parameter values (149).
The (141) is reduced to the following.

$\hat{H}=-\tilde{\sigma}_{1}(2\hat{a}_{1}^{\dagger}\hat{a}_{1}+1)-\tilde{\sigma}_{2}(2\hat{a}_{2}^{\dagger}\hat{a}_{2}+1)$
(152)

Then the dynamical phase (139) can be expressed as

$\dot{\theta}_{n_{1},n_{2}}^{d}=(2n_{1}+1)\tilde{\sigma}_{1}+(2n_{2}+1)\tilde{\sigma}_{2}.$
(153)

For the ground state

$\dot{\theta}_{0,0}^{d}=\sqrt{-\tilde{\Delta}+2\sqrt{\Delta_{\tilde{\Omega}}}}.$
(154)

Given the time dependent parameters, the first order ordinary differential
equations for the dynamical and geometrical phases can be integrated in a
straightforward manner.

##  VII Separability of bipartite ground states

Given a bipartite density operator $\hat{\rho}$, we can define

$\Delta\hat{X}=\hat{X}-\langle\hat{X}\rangle,\;\;\mbox{where}\;\langle\hat{X}_{\alpha}\rangle=Tr(\hat{X}_{\alpha}\hat{\rho}),\;\alpha=1,2,3,4.$
(155)

The components of $\Delta\hat{X}$ satisfy the commutation relation (72) as of
$\hat{X}$. Uncertainties (variances) may be defined as expectation velues of
the Hermitian operators

$\hat{\mathcal{V}}_{\alpha\beta}=\frac{1}{2}\\{\Delta\hat{X}_{\alpha},\Delta\hat{X}_{\beta}\\},\;\alpha,\beta=1,2,3,4,$
(156)

which enables us to define the variance matrix $\hat{V}$ through the matrix
elements

$\mathcal{V}_{\alpha\beta}=\langle\hat{\mathcal{V}}_{\alpha\beta}\rangle=\frac{1}{2}\langle\\{\hat{X}_{\alpha},\hat{X}_{\beta}\\}\rangle-\langle\hat{X}_{\alpha}\rangle\langle\hat{X}_{\beta}\rangle,\;\alpha,\beta=1,2,3,4.$
(157)

The statement of the uncertainty principle is then given by

$\hat{V}-\frac{1}{2}\hat{\Sigma}_{y}\geq 0,$ (158)

which has $Sp(4,\mathbb{R})$ invariant form. Under the Peres-Horodecki partial
transpose ($\hat{\mathcal{T}}$), the Wigner distribution
$(W(x_{1},p1,x_{2},p_{2})=\pi^{-2}\int dx_{1}^{\prime}dx_{2}^{\prime}\langle
x-x^{\prime}|\hat{\rho}|x+x^{\prime}\rangle\exp(2i\vec{x}^{\prime}.\vec{p}))$
undergoes mirror reflection ($\hat{\mathcal{T}}:W(x_{1},p1,x_{2},p_{2})\to
W(x_{1},p1,x_{2},-p_{2})$, with $\hat{\mathcal{T}}=diag(1,1,1,-1)$). Peres-
Horodecki criterion states that if the bipartite state is separable then
$W(\hat{\mathcal{T}}X)$ has to be Wigner distribution. In other words, for a
separable state, after the partial transpose operation, the transformed
variance matrix $\hat{\tilde{V}}$ should satisfy same uncertainty relation
i.e., $\hat{\tilde{V}}-\frac{1}{2}\hat{\Sigma}_{y}$ should be a nonnegative
matrix

$\hat{\tilde{V}}-\frac{1}{2}\hat{\Sigma}_{y}\geq
0,\;\mbox{with}\;\hat{\tilde{V}}=\hat{\mathcal{T}}\hat{V}\hat{\mathcal{T}}.$
(159)

(158) and (159) is preserved only under $Sp(2,\mathbb{R})\bigotimes
Sp(2,\mathbb{R})$ subgroup of $Sp(4,\mathbb{R})$ corresponding to independent
local linear canonical transformations $\hat{S}_{1}$ and $\hat{S}_{2}$ on the
subsystems of two independent observers respectively. In other words,
$\hat{S}_{local}=diag(\hat{S}_{1},\hat{S}_{2})$ with
$\hat{S}_{1}\mathcal{J}_{2}\hat{S}_{1}^{T}=\hat{S}_{2}\mathcal{J}_{2}\hat{S}_{2}^{T}=\mathcal{J}_{2}$,
where $\mathcal{J}_{2}$ is given in (71). On the other hand, since
$[\hat{x}_{j},\hat{p}_{k}]=i\delta_{jk}$, from (157), we can see that
$\hat{V}$ can be written as the following block form

$\displaystyle\hat{V}=\left(\begin{array}[]{cc}\hat{V}_{11}&\hat{V}_{12}\\\
\hat{V}_{12}^{T}&\hat{V}_{22}\end{array}\right).$ (162)

The local groups transform $2\times 2$ blocks as follows.

$\hat{V}_{11}\to\hat{S}_{1}\hat{V}_{11}\hat{S}_{1}^{T},\;\hat{V}_{22}\to\hat{S}_{2}\hat{V}_{22}\hat{S}_{2}^{T},\;\hat{V}_{12}\to\hat{S}_{1}\hat{V}_{12}\hat{S}_{2}^{T}.$
(163)

Therefore, the $Sp(2,\mathbb{R})\bigotimes Sp(2,\mathbb{R})$ invariants
corresponding to $\hat{V}$ are

$\Delta_{1}=Det(\hat{V}_{11}),\;\Delta_{2}=Det(\hat{V}_{22}),\;\Delta_{12}=Det(\hat{V}_{12}),\;\tau_{V}=\mbox{Trace}(\hat{V}_{11}\hat{\tilde{V}}_{12}\hat{V}_{22}\hat{\tilde{V}}_{21}).$
(164)

Where

$\displaystyle\hat{\tilde{V}}_{jk}=\mathcal{J}_{2}V_{jk}\mathcal{J}_{2},\;\mbox{with}\;\hat{V}_{21}=\hat{V}_{12}^{T}.$
(165)

It was shown by Simon sep1  that the necessary and sufficient condition for
separability condition ( generalized Peres-Horodecki criterion) is equivalent
to the following inequality

$\displaystyle\Delta_{1}\Delta_{2}+\left(\frac{1}{4}-|\Delta_{12}|\right)^{2}-\mbox{Trace}(V_{11}\tilde{V}_{12}V_{22}\tilde{V}_{21})\geq(\Delta_{1}+\Delta_{2}).$
(166)

To construct the exact form of the inequality corresponding to our system we
need to construct the state of our system under consideration. We shall
restrict ourselves only to the ground state, which is given by

$\hat{a}_{1}|0,0\rangle=\hat{a}_{2}|0,0\rangle=0.$ (167)

In the position ($\left\\{|x\rangle\right\\}$) representation, equation (167)
reads

$\displaystyle\left(\Lambda_{1}^{(l)}x-i\hbar\Lambda_{2}^{(l)}\partial_{x}\right)\psi_{00}=0.$
(168)

Where

$\displaystyle\Lambda_{1}^{(l)}=\left(\begin{array}[]{cc}\chi_{l11}&\chi_{l13}\\\
\chi_{l21}&\chi_{l23}\end{array}\right),\;\;\Lambda_{2}^{(l)}=\left(\begin{array}[]{cc}\chi_{l12}&\chi_{l14}\\\
\chi_{l22}&\chi_{l24}\end{array}\right),$ (173)

with

$\chi_{lj}=(\chi_{lj1},\chi_{lj2},\chi_{lj3},\chi_{lj4}),\;j=1,2.$ (174)

The notations $x=(x_{1},x_{2})^{T},\;\partial_{x}=(\frac{\partial}{\partial
x_{1}},\frac{\partial}{\partial x_{2}})^{T}$ have been used.

Let us consider the following ansatz for the solution of (168).

$\psi_{00}=N_{0}e^{-S(x_{1},x_{2})},$ (175)

with

$S(x_{1},x_{2})=\frac{1}{2}x^{T}\Lambda x.$ (176)

Using (175) in (168), we get

$\Lambda=i(\Lambda_{2}^{(l)})^{-1}\Lambda_{1}^{(l)}.$ (177)

The explicit form in the $\\{|x\rangle\\}$ representation
($\psi(x_{1},x_{2})=\langle x_{1},x_{2}|0,0\rangle$) reads

$\psi(x_{1},x_{2})=N_{0}e^{-\frac{1}{2}(\Lambda_{11}x_{1}^{2}+\Lambda_{22}x_{2}^{2}+(\Lambda_{12}+\Lambda_{21})x_{1}x_{2})},$
(178)

$N_{0}$ being the normalization constant. The matrix elements of $\Lambda$ are
as follows.

$\displaystyle\Lambda_{11}$ $\displaystyle=$
$\displaystyle\frac{i}{\Delta_{\Lambda_{2}^{(l)}}}\left(\chi_{l24}\chi_{l11}-\chi_{l14}\chi_{l21}\right).$
(179) $\displaystyle\Lambda_{12}$ $\displaystyle=$
$\displaystyle\frac{i}{\Delta_{\Lambda_{2}^{(l)}}}\left(\chi_{l24}\chi_{l13}-\chi_{l14}\chi_{l23}\right).$
(180) $\displaystyle\Lambda_{21}$ $\displaystyle=$
$\displaystyle\frac{i}{\Delta_{\Lambda_{2}^{(l)}}}\left(\chi_{l12}\chi_{l21}-\chi_{l22}\chi_{l11}\right).$
(181) $\displaystyle\Lambda_{22}$ $\displaystyle=$
$\displaystyle\frac{i}{\Delta_{\Lambda_{2}^{(l)}}}\left(\chi_{l12}\chi_{l23}-\chi_{l22}\chi_{l13}\right).$
(182)

The expectation values of the observables are given by the followings.

$\displaystyle\langle\hat{x}_{1}\rangle$ $\displaystyle=$
$\displaystyle\langle\hat{x}_{2}\rangle=\langle\hat{p}_{1}\rangle=\langle\hat{p}_{2}\rangle=0.$
(183) $\displaystyle\langle\hat{x}_{1}^{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{\Lambda_{22,r}}{2\Delta_{r}},\;\langle\hat{x}_{2}^{2}\rangle=\frac{\Lambda_{11,r}}{2\Delta_{r}},\;\langle\hat{x}_{1}\hat{x}_{2}\rangle=-\frac{\Lambda_{12,r}}{2\Delta_{r}}.$
(184) $\displaystyle\langle\hat{p}_{1}^{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2\Delta_{r}}\left(\Lambda_{22,r}|\Lambda_{11}|^{2}+\Lambda_{11,r}\Lambda_{12}^{2}-2i\Lambda_{12,i}\Lambda_{12,r}\Lambda_{11}^{*}\right).$
(185) $\displaystyle\langle\hat{p}_{2}^{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2\Delta_{r}}\left(\Lambda_{11,r}|\Lambda_{22}|^{2}+\Lambda_{22,r}\Lambda_{12}^{2}-2i\Lambda_{12,i}\Lambda_{12,r}\Lambda_{22}^{*}\right).$
(186) $\displaystyle\langle\hat{p}_{1}\hat{p}_{2}\rangle$ $\displaystyle=$
$\displaystyle\Lambda_{12}-\Lambda_{11}\Lambda_{12}\langle\hat{x}_{1}^{2}\rangle-\Lambda_{22}\Lambda_{12}\langle\hat{x}_{2}^{2}\rangle-(\Lambda_{11}\Lambda_{22}+\Lambda_{12}^{2})\langle\hat{x}_{1}\hat{x}_{2}\rangle.$
(187) $\displaystyle\langle\left\\{\hat{x}_{1},\hat{p}_{1}\right\\}\rangle$
$\displaystyle=$ $\displaystyle
2i\Lambda_{11}\langle\hat{x}_{1}^{2}\rangle+2\Lambda_{12}\langle\hat{x}_{1}\hat{x}_{2}\rangle-1.$
(188) $\displaystyle\langle\left\\{\hat{x}_{2},\hat{p}_{2}\right\\}\rangle$
$\displaystyle=$ $\displaystyle
2i\Lambda_{22}\langle\hat{x}_{2}^{2}\rangle+2\Lambda_{12}\langle\hat{x}_{2}\hat{x}_{1}\rangle-1.$
(189) $\displaystyle\langle\hat{x}_{1}\hat{p}_{2}\rangle$ $\displaystyle=$
$\displaystyle
i\Lambda_{12}\langle\hat{x}_{1}^{2}\rangle+\Lambda_{22}\langle\hat{x}_{1}\hat{x}_{2}\rangle,\;\langle\hat{x}_{2}\hat{p}_{1}\rangle=i\Lambda_{12}\langle\hat{x}_{2}^{2}\rangle+\Lambda_{11}\langle\hat{x}_{2}\hat{x}_{1}\rangle.$
(190) $\displaystyle\mbox{Where}\;\Delta_{r}$ $\displaystyle=$
$\displaystyle\Lambda_{11,r}\Lambda_{22,r}-\Lambda_{12,r}^{2}.$ (191)

Here the suffix $r$ and $i$ denotes the real and imaginary part of the time-
dependent parameters. For example
$\Lambda_{11}=\Lambda_{11,r}+i\Lambda_{11,i}$.
Now from (183), we find that $\mathcal{V}_{\alpha\beta}$ in (157) simplifies
to
$\mathcal{V}_{\alpha\beta}=\frac{1}{2}\langle\\{\hat{X}_{\alpha},\hat{X}_{\beta}\\}\rangle$,
i.e.,

$\displaystyle\hat{V}_{\alpha\alpha}=\left(\begin{array}[]{cc}\langle\hat{x}_{\alpha}^{2}\rangle&\langle\frac{1}{2}\\{\hat{x}_{\alpha},\hat{p}_{\alpha}\\}\rangle\\\
\langle\frac{1}{2}\\{\hat{x}_{\alpha},\hat{p}_{\alpha}\\}\rangle\rangle&\langle\hat{p}_{\alpha}^{2}\rangle\end{array}\right),\;\hat{V}_{12}=\left(\begin{array}[]{cc}\langle\hat{x}_{1}\hat{x}_{2}\rangle&\langle\hat{x}_{1}\hat{p}_{2}\rangle\\\
\langle\hat{p}_{1}\hat{x}_{2}\rangle&\langle\hat{p}_{1}\hat{p}_{2}\rangle\end{array}\right),\;\alpha=1,2.$
(196)

Using the explicit forms of expectation values in (196), the Simon’s
separability criterion (166) corresponding to the ground state of our system
reads

$\displaystyle|\Lambda_{12}|^{2}\Delta_{r}^{2}-(\Lambda_{12,r}^{4}+2\Lambda_{12,i}^{2}\Lambda_{11,r}\Lambda_{22,r})\Delta_{r}+2\Lambda_{12,i}^{2}(2\Lambda_{12,r}^{4}-\Lambda_{11,r}\Lambda_{22,r}\Lambda_{12,i}^{2})$
$\displaystyle+3\Lambda_{11,r}\Lambda_{22,r}(\Lambda_{11,r}^{2}\Lambda_{22,r}^{2}-2\Lambda_{12,r}^{4})+3\Lambda_{11,r}^{2}\Lambda_{22,r}^{2}\Lambda_{12,r}^{2}\geq
0.$ (197)

##  VIII Conclusions

The exact solutions of the time-dependent Schrödinger equation (SE) for an
anisotropic harmonic oscillator are constructed with the help of Lewis-
Riesenfeld invariant method. We had considered an invariant operator, which is
quadratic in co-ordinates and momentum. The coupled first order differential
equations of the coefficients in the linear combination of the constituent
operators in the invariant operator are expressed in a closed matrix form.
These coupled matrix equations can be solved in principle for a given set of
time-dependent parameters (mass, field strength, frequencies, NC parameter -
whichever is relevant). The quadratic form is expressed in a simpler form by
diagonalizing the system keeping the intrinsic symplectic structure intact.
The annihilation and creation operators are introduced. The eigenstates and
eigenvalues are demonstrated in detail. Both the geometrical and dynamical
phases are constructed, and thus Schrödinder equation is completely solved for
our system. The ground state for the bipartite system is demonstrated along
with its separability criterion.
The results discussed in this paper can be directly used for various types of
systems. In particular, for an anisotropic oscillator in noncommutative space
(for both the spatial and momentum noncommutativity), charged oscillator in a
anisotropic magnetic field, for Chern-Simon model (in long wavelength limit)
and of the similar types.

##  IX Author Declarations

The authors have no conflicts to disclose.

##  X Availability of data

Data sharing is not applicable to this article as no new data were created or
analyzed in this study.

##  XI Acknowledgement

This research was supported in part by the International Centre for
Theoretical Sciences (ICTS) for the online program - Non-Hermitian Physics
(code: ICTS/nhp2021/3).

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# Streaming LifeLong Learning With Any-Time Inference

Soumya Banerjee1, Vinay Kumar Verma2, and Vinay P. Namboodiri3 1 IIT Kanpur,
India<EMAIL_ADDRESS>Duke University, USA,
<EMAIL_ADDRESS>University Of Bath, UK<EMAIL_ADDRESS>
###### Abstract

Despite rapid advancements in lifelong learning (LLL) research, a large body
of research mainly focuses on improving the performance in the existing static
continual learning (CL) setups. These methods lack the ability to succeed in a
rapidly changing dynamic environment, where an AI agent needs to quickly learn
new instances in a ‘single pass’ from the non-i.i.d (also possibly temporally
contiguous/coherent) data streams without suffering from catastrophic
forgetting. For practical applicability, we propose a novel lifelong learning
approach, which is streaming, i.e., a single input sample arrives in each time
step, single pass, class-incremental, and subject to be evaluated at any
moment. To address this challenging setup and various evaluation protocols, we
propose a Bayesian framework, that enables fast parameter update, given a
single training example, and enables any-time inference. We additionally
propose an implicit regularizer in the form of snap-shot self-distillation,
which effectively minimizes the forgetting further. We further propose an
effective method that efficiently selects a subset of samples for online
memory rehearsal and employs a new replay buffer management scheme that
significantly boosts the overall performance. Our empirical evaluations and
ablations demonstrate that the proposed method outperforms the prior works by
large margins.

##  I INTRODUCTION

In this paper, we aim to achieve ‘any-time-inference’ ability in _streaming
lifelong learning_ (SL). This is an appealing ability as it allows practical
applicability of AI agents/robots, which can be continually updated and
deployed without catastrophic forgetting [1]. While deep learning models can
achieve state-of-the-art performance throughout various tasks ranging across
vision, language, speech, etc., these models suffer from catastrophic
forgetting [2], if trained incrementally with sequentially arriving data.
LifeLong Learning (LLL) or continual learning (CL) [3] studies the problem of
training a deep neural network (DNN) continuously with sequentially coming
data without forgetting. This paper introduces a novel approach, which enables
DNN for class-incremental streaming classification. _Class-incremental
streaming classification/learning_ (CISL) is a challenging variant of LLL [4,
5], with the following essential and desirable properties:

  1. i)

Learner sees each labeled instance only once (desirable),

  2. ii)

Learner needs to adapt the newly available data immediately (desirable),

  3. iii)

The input data stream can be non-i.i.d and temporally contiguous/coherent
(essential),

  4. iv)

The AI agent can be evaluated at any moment, i.e., ‘any-time-inference’
(essential),

  5. v)

The AI agent should be able to predict a class label among all the observed
classes so far during inference without having the task-id [6] (essential),

  6. vi)

For practical applicability, learner must limit their memory usage
(desirable).

A wide variety of CL approaches [3, 7, 8, 9, 10, 11, 12] have been proposed to
alleviate catastrophic forgetting in DNN. However, most of the existing
approaches mainly focus on ‘incremental-batch-learning’ (IBL) [3], where the
input data in each incremental step arrives in batches, and the agent visits
them multiple times in order to enable CL. While these methods are applicable
in a static environment, these methods are ill-suited for a rapidly changing
dynamic environment, where an AI agent requires to learn from the newly
available knowledge immediately with no forgetting. Different from this,
‘streaming learning’ (SL) imposes a unique set of restrictions/challenges (as
mentioned earlier) and has not received widespread attention like the other
existing CL approaches. However, its utility is apparent, as it enables the
practical deployment of autonomous AI agents. Hayes et al. [5] argued that SL
is closer to biological learning than other existing CL scenarios. Fig. 1
demonstrates an example of streaming classification scenario. In Table I, we
classify the existing CL approaches in accordance to the underlying assumption
that they impose. It can be observed that ExStream [4] and REMIND [5] are the
only two CL approaches, that focus on streaming classification. However, it is
worth mentioning that ExStream [4] uses full buffer replay, which violates the
subset buffer replay constraint of CISL. Furthermore, while REMIND [5]
satisfies all the CISL constraints, it stores a comparably large number of
past samples in memory, which limits its applicability.

Figure 1: SL addresses the problem of learning continuously from a stream of
temporally correlated (non-i.i.d) labeled data, for e.g., learning to
recognize an object from multiple viewpoints with no forgetting. In this fig.,
3 temporally correlated frames of bottle are taken from iCubWorld 1.0 [13].
TABLE I: Categorization of the baseline approaches depending on the underlying
simplifying assumptions they impose. In $\zeta(n)$, $n$ represents the number
of gradient steps required to train the corresponding model.
$\zeta(n)\gg\zeta(1)$. ‘-’ indicates, that we are unable to find the exact
value.

|  |  |  | ‘Class-Incremental Streaming Learning’ (CISL) Crucial Properties |  |  |  |
---|---|---|---|---|---|---|---|---
Methods |  Type |  Bayesian Framework |  Batch-Size $(N_{t})$ |  Fine-tunes |  Single Pass Learning |  CIL |  Subset Buffer Replay |  Training Time |  Inference Time |  Violates Any _CISL_ Constraint |  Memory Capacity |  Regularization Based |  Memory Based
EWC [3] | Batch | ✗ | $N_{t}\gg 1$ | ✗ | ✗ | ✗ | n/a | $\zeta(n)$ | $\zeta(1)$ | ✓ | n/a | ✓ | ✗
MAS [7] | Batch | ✗ | $N_{t}\gg 1$ | ✗ | ✗ | ✗ | n/a | $\zeta(n)$ | $\zeta(1)$ | ✓ | n/a | ✓ | ✗
VCL [8] | Batch | ✓ | $N_{t}\gg 1$ | ✗ | ✗ | ✗ | n/a | $\zeta(n)$ | $\zeta(1)$ | ✓ | n/a | ✓ | ✗
Coreset VCL [8] | Batch | ✓ | $N_{t}\gg 1$ | ✓ | ✗ | ✗ | ✗ | $\zeta(n)$ | $\zeta(n)$ | ✓ | - | ✓ | ✓
GDumb [14] | Online | ✗ | $N_{t}\gg 1$ | ✓ | ✗ | ✓ | ✗ | $\zeta(1)$ | $\zeta(n)$ | ✓ | - | ✗ | ✓
TinyER [15] | Online | ✗ | $N_{t}\gg 1$ | ✗ | ✓ | ✓ | ✓ | $\zeta(1)$ | $\zeta(1)$ | ✗ | $\leq 5\%$ | ✗ | ✓
DER [16] | Online | ✗ | $N_{t}\gg 1$ | ✗ | ✓ | ✓ | ✓ | $\zeta(1)$ | $\zeta(1)$ | ✗ | $\leq 5\%$ | ✓ | ✗
DER++ [16] | Online | ✗ | $N_{t}\gg 1$ | ✗ | ✓ | ✓ | ✓ | $\zeta(1)$ | $\zeta(1)$ | ✗ | $\leq 5\%$ | ✓ | ✓
ExStream [4] | Streaming | ✗ | $N_{t}=1$ | ✗ | ✓ | ✓ | ✗ | $\zeta(1)$ | $\zeta(1)$ | ✓ | $\leq 5\%$ | ✗ | ✓
REMIND [5] | Streaming | ✗ | $N_{t}=1$ | ✗ | ✓ | ✓ | ✓ | $\zeta(1)$ | $\zeta(1)$ | ✗ | $\gg 10\%$ | ✗ | ✓
Ours | Streaming | ✓ | $N_{t}=1$ | ✗ | ✓ | ✓ | ✓ | $\zeta(1)$ | $\zeta(1)$ | ✗ | $\leq 5\%$ | ✓ | ✓
CISL Constraints |  |  | $N_{t}=1$ | ✗ | ✓ | ✓ | ✓ | $\zeta(1)$ | $\zeta(1)$ |  |  |  |

In this work, we overcome the aforementioned limitations in a principled
manner and introduce a novel CISL method, dubbed as Bayesian Class Incremental
Streaming Learning (BaSiL), that seeks to incorporate the ability of rehearsal
with a functional regularization into a simple yet effective framework of
streaming variational Bayes. We leverage both the replay samples and the
sequential data to compute the joint likelihood in order to approximate the
posterior in each incremental step. As a consequence, the proposed approach
does not require explicit finetuning unlike approaches like [14], and can be
evaluated ‘on-the-fly’ at any moment. We also propose a novel online ‘loss-
aware’ buffer management policy and various sample selection strategies that
include the most informative sample(s) in memory or select for rehearsal,
which significantly boosts the model’s performance. Our experimental results
on three benchmark datasets demonstrate the effectiveness of our proposed
method to circumvent forgetting in highly challenging streaming scenarios.

Our contributions can be summarized as follows:

  * •

We introduce BaSiL, a novel rehearsal-based dual regularization framework,
which includes a streaming Bayesian framework and a functional regularizer, to
alleviate forgetting in challenging CISL setup,

  * •

We propose a novel online ‘loss-aware’ replay buffer management policy and
various sampling strategies which significantly boosts the model’s
performance,

  * •

We empirically show that the proposed method can be evaluated at any time
without requiring to finetune unlike [14, 8] with no significant drop in final
accuracy,

  * •

Empirical evaluations and ablations on three benchmark datasets validate the
superiority of BaSiL over the existing baselines.

##  II RELATED WORK

Parameter-isolation-based approaches train different subsets of model
parameters on sequential tasks. PNN [17], DEN [18] expand the network to
accommodate the new task. PathNet [19], PackNet [12], Piggyback [20], and HAT
[21] train different subsets of network parameters on each task.

Regularization-based approaches use an extra regularization term in the loss
function to enable continual learning. LWF [22] uses knowledge distillation
[23] loss to prevent forgetting. EWC [3], IMM [24], SI [11] and MAS [7]
regularize by penalizing changes to the important weights of the network.

Rehearsal-based approaches replay a subset of past training data during
sequential learning. iCaRL [10], SER [25], and TinyER [15] use memory replay
when training on a new task. DER/DER++ [16] uses knowledge distillation [23]
and memory replay while learning a new task. DGR [26], MeRGAN [27], and CloGAN
[9] uses generative models to replay past samples during continual learning.

VCL [8] leverages Bayesian inference to mitigate forgetting [2]. However, the
approach, when naïvely adapted, performs poorly in SL. Additionally, it
requires task-id during inference. Furthermore, explicit finetuning with the
coreset samples restricts it from performing any time inference.

REMIND [5] is a recently proposed rehearsal-based SL approach, which follows a
setting close to the one proposed. However, it stores a large number of past
examples compared to the other baselines; for e.g., iCaRL [10] stores 10K past
examples for ImageNet experiment, whereas REMIND stores 1M past examples.
Furthermore, it actually uses a lossy compression to store past samples, which
is merely an engineering technique, not an algorithmic improvement, and can be
used by any CL approach.

##  III PROPOSED APPROACH

###  III-A Streaming Learning With A Single Example

In the following, we introduce BaSiL, which trains a convolutional neural
network (CNN) in CISL setup. Formally, we separate the CNN into two networks
(Fig. 2): $(i)$ non-plastic feature extractor $G(\cdot)$, containing the
initial $U$ layers of CNN, and $(ii)$ plastic neural network $F(\cdot)$,
containing the final $V$ layers. For a given input image $\bm{x}$, the
predicted class label is computed as: $y=F(G(\bm{x}))$. We initialize the
parameters of feature extractor $G(\cdot)$ with the pre-trained weights and
keep it frozen throughout SL. We model the plastic network $F(\cdot)$ as a
_Bayesian-neural-network_ (BNN) [28, 29] and optimize its parameters with the
sequentially coming data in CISL setup without forgetting.

In the below, we describe how the plastic network $F(\cdot)$ (BNN) is
optimized with a single posterior approximation in each incremental step, $t$,
with a single data point:
$\bm{D}_{t}=\left\\{{d}_{t}\right\\}=\left\\{\left(\bm{x}_{t},y_{t}\right)\right\\}$,
arriving sequentially in a streaming manner with no forgetting [2].

Figure 2: Schematic representation of the proposed model. $\bm{\theta}_{G}$
denotes the parameters of the frozen feature extractor $G(\cdot)$, whereas
$\bm{\theta}_{F}$ denotes the parameters of BNN $F(\cdot)$.

Variational Posterior Estimation. Streaming learning naturally emerges from
the Bayes’ rule [30]; given the posterior $p(\bm{\theta}|\bm{D}_{1:{t-1}})$,
whenever a new data: $\bm{D}_{t}=\\{{d}_{t}\\}=\\{(\bm{x}_{t},y_{t})\\}$
arrives, the new posterior $p(\bm{\theta}|\bm{D}_{1:t})$ can be computed by
combining the previous posterior and the new data likelihood, i.e.,
$p(\bm{\theta}|\bm{D}_{1:t})\propto
p(\bm{D}_{t}|\bm{\theta})\;p(\bm{\theta}|\bm{D}_{1:{t-1}})$, where the old
posterior is treated as the prior. However, for any complex model, the exact
Bayesian inference is not tractable, and an approximation is needed. A
Bayesian neural network (BNN) [28, 31] commonly approximates the posterior
with a variational posterior $q(\bm{\theta})$ by minimizing the KL divergence
(in Eq. 1), or equivalently by maximizing the evidence lower bound (ELBO) (in
Eq. LABEL:eq:2):

$\displaystyle\bm{L}_{t}(\bm{\theta})$
$\displaystyle=\operatorname*{arg\,min}_{q\in{Q}}\text{KL}\left(q_{t}(\bm{\theta})\left|\right|\frac{1}{Z_{t}}q_{t-1}(\bm{\theta})p(\bm{D}_{t}|\bm{\theta})\right)$
(1)
$\displaystyle\begin{split}&\backsimeq\operatorname*{arg\,max}_{q\in{Q}}\mathbb{E}_{\bm{\theta}\sim
q_{t}(\bm{\theta})}\left[\log
p(\bm{D}_{t}|\bm{\theta})\right]-\text{KL}\left(q_{t}(\bm{\theta})\left|\right|q_{t-1}(\bm{\theta})\right)\end{split}$
(2)

KL-divergence in Eq. LABEL:eq:2 works as an implicit parameter regularizer, by
keeping the prior and the posterior distribution close to each other, which
can minimize forgetting in sequential learning. However, there are a few
concerns. Firstly, optimizing the ELBO (in Eq. LABEL:eq:2) to approximate the
posterior $p(\bm{\theta}|\bm{D}_{1:t})$ with only a single training example,
i.e.,
$\bm{D}_{t}=\left\\{{d}_{t}\right\\}=\left\\{\left(\bm{x}_{t},y_{t}\right)\right\\}$,
in each incremental step, $t$, can fail in SL [32]. Furthermore, likelihood
estimation from a single example:
$\bm{D}_{t}=\left\\{{d}_{t}\right\\}=\left\\{\left(\bm{x}_{t},y_{t}\right)\right\\}$
can bias the model towards the new data disproportionately and can maximize
the confusion during inference in class incremental learning (CIL) setup [6].

We, therefore, propose to estimate the likelihood from both the new example:
$\bm{D}_{t}$ and the previously observed examples in order to approximate the
new posterior: $p(\bm{\theta}|\bm{D}_{1:t})$. For this purpose, a small
fraction $(\leq 5\%)$ of past observed examples are stored as representatives
in a _fixed-sized_ tiny episodic memory buffer $\bm{M}$. However, instead of
storing raw pixels (images) $\bm{x}$, we store the embedding
$\bm{z}=G(\bm{x})$, where $\bm{z}\in\mathbb{R}^{d}$. This allows us to avoid
the cost of recomputing the image-embeddings and expedites the learning even
further. It also saves a significant amount of space and allows us to keep
more examples in a small budget.

During training on the new example:
$\bm{D}_{t}=\left\\{{d}_{t}\right\\}=\left\\{\left(\bm{x}_{t},y_{t}\right)\right\\}$,
we select a subset of samples: $\bm{D}_{\bm{M},t}$ from the memory $\bm{M}$,
where:
$(i)\bm{D}_{\bm{M},t}\subset\bm{M},\quad(ii)|\bm{D}_{\bm{M},t}|={N}^{\prime}_{1}\ll|\bm{M}|$,
_instead of replaying the whole buffer_ , and compute the likelihood jointly
with the new example $\bm{D}_{t}$ to estimate the new posterior. We,
therefore, compute the new posterior as follows:
$p(\bm{\theta}|\bm{D}_{1:t})\propto
p(\bm{D}_{t}|\bm{\theta})\;p(\bm{D}_{\bm{M},t}|\bm{\theta})\;p(\bm{\theta}|\bm{D}_{1:{t-1}})$.
Since the exact Bayesian inference is intractable, we approximate it with a
variational posterior $q_{t}(\bm{\theta})$ as follows:

$\displaystyle\bm{L}^{1}_{t}(\bm{\theta})=\operatorname*{arg\,min}_{q\epsilon{Q}}\text{KL}\left(q_{t}(\bm{\theta})\left|\right|\frac{1}{Z_{t}}q_{t-1}(\bm{\theta})p(\bm{D}_{t}|\bm{\theta})p(\bm{D}_{\bm{M},t}|\bm{\theta})\right)$
(3)

_Note that Eq. 3 consists of two likelihood terms, that is, one for the new
data and other for the subset buffer replay. It minimizes the confusion in the
network parameters while predicting a class label over all the observed
classes in CIL setup. Further, it allows the model to be evaluated ‘on-the-
fly’ at any moment._

The above minimization (in Eq. 3) can be equivalently written as the
maximization of the evidence lower bound (ELBO) as follow:

$\displaystyle\bm{L}^{1}_{t}(\bm{\theta})$
$\displaystyle=\begin{multlined}\mathbb{E}_{\bm{\theta}\sim
q_{t}(\bm{\theta})}\left[\log{p(y_{t}|\bm{\theta},G(\bm{x}_{t}))}\right]\\\
+\sum_{n=1}^{N_{1}^{\prime}}\mathbb{E}_{\bm{\theta}\sim
q_{t}(\bm{\theta})}\left[\log{p(y_{\bm{M},t}^{(n)}|\bm{\theta},\bm{z}_{\bm{M},t}^{(n)})}\right]\\\
-\lambda_{1}\cdot\text{KL}\left(q_{t}(\bm{\theta})\left|\right|q_{t-1}(\bm{\theta})\right)\end{multlined}\mathbb{E}_{\bm{\theta}\sim
q_{t}(\bm{\theta})}\left[\log{p(y_{t}|\bm{\theta},G(\bm{x}_{t}))}\right]\\\
+\sum_{n=1}^{N_{1}^{\prime}}\mathbb{E}_{\bm{\theta}\sim
q_{t}(\bm{\theta})}\left[\log{p(y_{\bm{M},t}^{(n)}|\bm{\theta},\bm{z}_{\bm{M},t}^{(n)})}\right]\\\
-\lambda_{1}\cdot\text{KL}\left(q_{t}(\bm{\theta})\left|\right|q_{t-1}(\bm{\theta})\right)$
(7)

where: $(i)$ $\bm{D}_{t}=\\{{d}_{t}\\}=\\{(\bm{x}_{t},y_{t})\\}$, $(ii)$
$\bm{D}_{\bm{M},t}=\\{{d}^{(n)}_{\bm{M},t}\\}^{N_{1}^{\prime}}_{n=1}=\\{(\bm{z}_{\bm{M},t}^{(n)},y_{\bm{M},t}^{(n)})\\}^{N_{1}^{\prime}}_{n=1}$,
$(iii)$ $\bm{D}_{\bm{M},t}\subset\bm{M}$, $(iv)$
$|\bm{D}_{\bm{M},t}|=N_{1}^{\prime}\ll|\bm{M}|$, and $(v)$ $\lambda_{1}$ is a
hyper-parameter.

Snap-Shot Self Distillation. It is worth noting that KL divergence
minimization (in Eq. 7) between prior and posterior distribution works as an
inherent parameter regularizer, and minimizes the changes in the network
parameters in SL. However, its effect may weaken over time due to the presence
of distribution shift and temporal coherence in the input data-stream.
Furthermore, the initialization of the prior with the old posterior at each
incremental step can introduce information loss in the network for a longer
sequence of SL. On these grounds, we propose a functional regularizer, which
encourages the network to mimic the output responses as produced in the past
for the previously observed samples. Specifically, we propose to minimize KL
divergence between class-probability scores obtained in past and current
incremental step $t$. In particular, we uniformly select $N^{\prime}_{2}$
samples along with their logits and minimize the objective in Eq. 8, or
equivalently under mild assumptions of knowledge-distillation [23], we
minimize the mean-squared-error between the corresponding logits as in Eq.
LABEL:eq:6:

$\displaystyle\bm{L}^{2}_{t}(\bm{\theta})$
$\displaystyle=\lambda_{2}\cdot\sum^{N_{2}^{\prime}}_{j=1}\mathbb{E}_{\bm{\theta}\sim
q_{t}(\bm{\theta})}\left[\text{KL}\left(\text{softmax}(\bm{h}_{j})\;\left|\right|\;\text{softmax}(f_{\bm{\theta}}(\bm{z}_{j}))\right)\right]$
(8)
$\displaystyle\begin{split}&=\lambda_{2}\cdot\sum^{N_{2}^{\prime}}_{j=1}\mathbb{E}_{\bm{\theta}\sim
q_{t}(\bm{\theta})}\left[\left|\left|\bm{h}_{j}-f_{\bm{\theta}}(\bm{z}_{j})\right|\right|^{2}_{2}\right]\end{split}$
(9)

where: $(i)$ $f(\cdot)$ denotes plastic network without softmax activation,
$(ii)$ $\lambda_{2}$: hyper-parameter, $(iii)$ $\bm{h}_{j}$ denotes logits.

One important highlight of Eq. LABEL:eq:6 is that the stored logits are
updated in an online manner. That is, we replace the stored logits with the
newly predicted logits whenever a sample is used for rehearsal. Therefore, it
is called _snap-shot self-distillation_ [33]. It essentially prevents the
model from being constrained to match the sub-optimal initial logits and
minimizes the information loss in the network further.

###  III-B Informative Past Sample Selection For Replay

We consider the following strategies for selecting past informative samples
for memory replay:

Uniform Sampling (Uni). In this approach, samples are selected uniformly
random from memory.

Uncertainty-Aware Positive-Negative Sampling (UAPN). UAPN selects
${N_{1}^{\prime}}/{2}$ samples with the highest uncertainty scores (negative
samples) and ${N_{1}^{\prime}}/{2}$ samples with the lowest uncertainty scores
(positive samples).

Loss-Aware Positive-Negative Sampling (LAPN). LAPN selects
${N_{1}^{\prime}}/{2}$ samples with the highest loss-values (negative-
samples), and ${N_{1}^{\prime}}/{2}$ samples with the lowest loss-values
(positive-samples).

###  III-C Memory Buffer Replacement Policy

TABLE II: Memory buffer capacity used for various datasets. Dataset |  ImageNet100 |  iCubWorld 1.0 |  CORe50
---|---|---|---
Buffer Capacity | 1000 | 180 | 1000
Training-Set Size | 127778 | 6002 | 119894

In this work, we consider the following strategies to replace a stored example
with a newly arriving sample.

Loss-Aware Weighted Class Balancing Replacement (LAWCBR). In this approach,
whenever the buffer is full, we remove a sample from the majority class, i.e.,
$y_{r}=\operatorname*{arg\,max}\text{ClassCount}(\bm{M})$. We weigh each
sample of the majority class inversely w.r.t their loss, i.e.,
$w^{y_{r}}_{i}\propto\frac{1}{l^{y_{r}}_{i}}$ and use these weights as the
replacement probability; the lesser the loss, the more likely to be removed.

Loss-Aware Weighted Random Replacement With A Reservoir (LAWRRR). In this
approach, we propose a novel variant of reservoir sampling [34] to replace an
existing sample whenever the buffer is full. We weigh each stored sample
inversely w.r.t the loss, i.e., $w_{i}\propto\frac{1}{l_{i}}$, and
proportionally to the number of examples representative of that particular
sample’s class, i.e., $w_{i}\propto\text{ClassCount}(\bm{M},y_{i})$. We
combine these two scores and use that as the replacement probability; the
higher the weight, the more likely to be replaced.

###  III-D Efficient Buffer Update

Loss/uncertainty-aware sampling strategies require computing these quantities
in each incremental step for all the stored examples. However, it becomes
computationally expensive with the larger replay buffer size. We, therefore,
store the corresponding loss-values and uncertainty-scores along with the
examples in the replay buffer. Since these quantities are just scalar values,
the additional storage cost is negligible, but it makes the learning pipeline
computationally efficient. Every time a sample is selected for memory replay,
its loss and uncertainty are replaced by the new values. Additionally, we also
replace the stored logits with the new logits. Empirically, we observe that
the model’s accuracy degrades if we don’t update the logits with the new
logits.

###  III-E Training

Training the plastic network (BNN) $F(\cdot)$ requires specification of
$q(\bm{\theta})$ and, in this work, we model $\bm{\theta}$ by stacking up the
parameters (weights & biases) of the network $F(\cdot)$. We use a Gaussian
mean-field posterior $q_{t}(\bm{\theta})$ for the network parameters, and
choose the prior distribution, i.e., $q_{0}(\bm{\theta})=p(\bm{\theta})$, as
multivariate Gaussian distribution. We train the network $F(\cdot)$ by
maximizing the ELBO in Eq. 7 and minimizing the mean-squared-error in Eq.
LABEL:eq:6. For memory replay in Eq. 7, we select past informative samples
using the strategies mentioned in Section III-B, and we use _uniform sampling_
to select samples from memory to be used in Eq. LABEL:eq:6. Fig. 2 shows the
schematic diagram of the proposed model as well as the learning process.

##  IV Experiments

###  IV-A Datasets And Data Orderings

TABLE III: $\bm{\Omega}_{\text{all}}$ results with their associated standard
deviations. For each experiment, the method with best performance in
_‘streaming-learning-setup’_ is highlighted in Bold. The reported results are
averaged over $10$ runs with different permutations of the data. Offline model
is trained only once.
${\widehat{\text{Offline}}=\frac{1}{T}{\sum_{t=1}^{T}{\bm{\alpha}_{\text{offline},t}}}}$,
where $T$ is the total number of testing events. ‘-’ indicates experiments we
are unable to run, because of compatibility issues.

Method | iid | Class-iid | instance | Class-instance
---|---|---|---|---
iCubWorld 1.0 |  CORe50 |  ImageNet100 |  iCubWorld 1.0 |  CORe50 |  ImageNet100 |  iCubWorld 1.0 |  CORe50 |  iCubWorld 1.0 |  CORe50
Fine-Tune | 0.1369 | 0.1145 | 0.0127 | 0.3893 | 0.3485 | 0.1233 | 0.1307 | 0.1145 | 0.3485 | 0.3430
EWC | - | - | - | 0.3790 | 0.3508 | 0.1225 | - | - | 0.3487 | 0.3427
MAS | - | - | - | 0.3912 | 0.3432 | 0.1234 | - | - | 0.3486 | 0.3429
VCL | - | - | - | 0.3806 | 0.3462 | 0.1205 | - | - | 0.3473 | 0.3420
Coreset VCL | - | - | - | 0.3948 | 0.3424 | 0.1259 | - | - | 0.4705 | 0.4715
GDumb | 0.8993 | 0.9345 | 0.8361 | 0.9660 | 0.9742 | 0.9197 | 0.6715 | 0.7433 | 0.7908 | 0.6548
DER | 0.1437 | 0.1145 | 0.0126 | 0.4057 | 0.3432 | 0.1217 | 0.3759 | 0.1168 | 0.4082 | 0.3308
DER++ | 0.1428 | 0.1145 | 0.0130 | 0.4467 | 0.3431 | 0.1230 | 0.4518 | 0.1145 | 0.4499 | 0.3429
TinyER | 0.9590 | 1.0007 | 0.9415 | 0.9069 | 0.9573 | 0.8995 | 0.8726 | 0.8432 | 0.8215 | 0.8461
ExStream | 0.9235 | 0.9844 | 0.9293 | 0.8820 | 0.8760 | 0.8757 | 0.8954 | 0.8257 | 0.8727 | 0.8837
REMIND | 0.9260 | 0.9933 | 0.9088 | 0.8553 | 0.9448 | 0.8803 | 0.8157 | 0.8544 | 0.7615 | 0.7826
Ours | 0.9716 | 1.0069 | 0.9640 | 0.9480 | 0.9686 | 0.9171 | 0.9580 | 0.9824 | 0.9585 | 0.9384
Offline | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000
$\widehat{\text{Offline}}$ | 0.7626 | 0.8733 | 0.8520 | 0.8849 | 0.9070 | 0.8953 | 0.7646 | 0.8733 | 0.8840 | 0.9079

Datasets. To evaluate the efficacy of the proposed model we perform extensive
experiments on three benchmark datasets: ImageNet100, iCubWorld 1.0 [13], and
CORe50 [35]. ImageNet100 is a subset of ImageNet-1000 (ILSVRC-2012) [36]
containing randomly chosen 100 classes, with each class containing 700-1300
training samples and 50 validation samples. Since the test data for
ImageNet-1000 (ILSVRC-2012) [36] is not provided with labels, we use the
validation data for evaluating the model’s performance, similar to [5].
iCubWorld 1.0 is an object recognition dataset which contains the sequences of
video frames, with each containing a single object. There are 10 classes, each
containing 3 different object instances with each class containing 600-602
samples for training and 200-201 samples for testing. CORe50 is similar to
iCubWorld 1.0, containing images from temporally coherent sessions. There are
10 classes, each containing 5 different object instances with each class
containing 11983-12000 samples for training and 4495-4500 samples for testing.
_Technically, iCubWorld, and CORe50 are the ideal dataset for streaming
learning, as it requires learning from temporally coherent image sequences,
which are naturally non-i.i.d images_.

Evaluation Over Different Data Orderings. The proposed approach is robust to
the various _SL_ settings; we evaluate the model’s SL ability with the
following four [4, 5] challenging data ordering schemes: $(i)$ _‘streaming
iid’_ : where the data-stream is organized by the randomly shuffled samples
from the dataset, $(ii)$ _‘streaming class iid‘_ : where the data-stream is
organized by the samples from one or more classes, these samples are shuffled
randomly, $(iii)$ _‘streaming instance’_ : where the data-stream is organized
by temporally ordered samples from different object instances, and $(iv)$
_‘streaming class instance’_ : where the data-stream is organized by the
samples from different classes, the samples within a class are temporally
ordered based on different object instances. _Only iCubWorld 1.0 and CORe50
dataset contains the temporal ordering, therefore ‘streaming instance’, and
‘streaming class instance’ settings are evaluated only on these two datasets._

###  IV-B Metrics

For evaluating the performance of the streaming learner, we use
$\bm{\Omega_{\text{all}}}$ metric, similar to [37, 4, 5], where
$\bm{\Omega}_{\text{all}}$ represents normalized incremental learning
performance with respect to an offline learner:
$\bm{\Omega}_{\text{all}}=\frac{1}{T}\sum_{t=1}^{T}\frac{\bm{\alpha}_{t}}{\bm{\alpha}_{\text{offline},t}}$,
where $T$ is the total number of testing events, $\bm{\alpha}_{t}$ is the
performance of the incremental learner at time $t$, and
$\bm{\alpha}_{\text{offline},t}$ is the performance of a traditional offline
model at time $t$.

###  IV-C Baselines And Compared Methods

The proposed approach follows the _‘streaming learning setup’_ ; to the best
of our knowledge, recent works ExStream [4] and REMIND [5] are the only
methods that follow the same setting. We compare our approach against these
strong baselines. We also compare our model with $(i)$ a network trained with
one sample at a time (Fine-tuning/lower-bound) and $(ii)$ a network trained
offline, assuming all the data is available (Offline/upper-bound). Finally, we
choose recent popular ‘batch’ (IBL) and ‘online’ learning methods, such as EWC
[3], MAS [7], VCL [8], Coreset VCL [8], TinyER [15], GDumb [14], and DER/DER++
[16] as baselines and rigorously evaluate our model against these approaches.
For a fair comparison, we train all the methods in a SL setup, i.e., one
sample at a time, and no finetuning is used. Only Coreset VCL and GDumb
finetunes before each inference, therefore, these two methods utilize an extra
advantage over the other baselines. However, BaSiL still outperforms these
approaches by a significant margin.

Figure 3: Plots of $\bm{\Omega}_{\text{all}}$ as a function of streaming
learning model and data-ordering on $(a)$ iCubWorld 1.0, and $(b)$ CORe50.
Figure 4: $\bm{\Omega}_{\text{all}}$ results for different versions of
baselines on iCubWorld 1.0 on $(a)$ Class-iid, and $(b)$ Class-instance
ordering.

###  IV-D Results

The detailed results of BaSiL over various experimental settings along with
the strong baseline methods are shown in Table III. We can clearly observe
that BaSiL consistently outperforms all the baseline by a significant margin.
The proposed model is also robust to the different streaming learning
scenarios compared to the baselines. We repeat our experiment ten times and
report the average accuracy. We observe that ‘batch-learning’ methods severely
suffer from catastrophic forgetting. Moreover, replay-based ‘online-learning’
methods such as DER/DER++ also suffer from information loss badly.

While GDumb achieves higher final accuracy on class-i.i.d ordering, it
finetunes before each inference. Therefore, it enjoys an extra advantage over
other SL baselines and it is not considered as the best-performing model, even
when it achieves higher accuracy.

We believe it is important to highlight that iCubWorld 1.0 and CORe50 are two
challenging datasets, which evaluate the models in more realistic scenarios or
data-orderings. Particularly, class-instance and instance ordering require the
learner to learn from temporally ordered video frames one at a time. From
Table III, we observe that BaSiL obtain up to ${8.58}\%$ & ${6.26}\%$
improvement on iCubWorld 1.0, and ${5.47}\%$ & ${12.8}\%$ improvement on
CORe50, over the state-of-the-art streaming learning approaches. Fig. 3 shows
the impact of temporal orderings on the streaming learning model’s
performance. It is evident that class-instance and instance ordering are more
difficult, and the baselines continue to suffer from severe forgetting.

Finally, for completeness, we train BaSiL in ‘batch’ as well as ‘online’
learning setting to determine its effectiveness and compatibility in these
settings. In Fig. 4, we compare BaSiL in various settings. We can observe that
BaSiL outperforms the baselines by a significant margin on both class-i.i.d
and class-instance ordering on iCubWorld, _implying even though BaSiL designed
to work in SL, it can be thought of as a robust method for various LLL
scenarios with the widest possible applicability_.

###  IV-E Implementation Details

In all the experiments, models are trained with one sample at a time. Only
Coreset VCL and GDumb finetunes before each inference. For a fair comparison,
the same network structure is used throughout all the models. For all methods,
we use fully connected single-head networks with two hidden layers as the
plastic network $F(\cdot)$, where each layer contains $256$ nodes with ReLU
activations; for ‘VCL’, ‘Coreset VCL’, and ‘BaSiL’, $F(\cdot)$ is a BNN,
whereas for all other methods $F(\cdot)$ is a deterministic network. For a
fair comparison, we store the same number of past examples for all replay-
based approaches, as mentioned in Table II. We use Mobilenet-V2 [38] pre-
trained on ImageNet-1000 (ILSVRC-2012) [36] as the visual feature extractor.
For memory-replay, we use _‘uncertainty-aware positive-negative’_ sampling
strategy throughout all data-orderings, except for ‘streaming-i.i.d’ ordering,
we use _‘uniform’_ sampling. We use _‘loss-aware weighted random replacement
with a reservoir’_ sampling strategy as memory replacement policy for all the
experiments. We set $N_{1}^{\prime}=N_{2}^{\prime}=16$ throughout all
experiments. We set $\lambda_{1}=1$ and $\lambda_{2}=0.3$ across all
experiments; however, for online/batch learning experiments, we use
$\lambda_{2}=0.2$ and use _uniform sampling_ for memory replay. We repeat our
experiment $10$ times and report the average-accuracy.

##  V Ablation Study

TABLE IV: $\bm{\Omega}_{\text{all}}$ Results as a function of different sampling and buffer management policies. Memory Replacement |  Sample Selection | iCubWrold | ImageNet100
---|---|---|---
instance |  Class instance |  iid |  Class-iid
LAWCBR | Uni | 0.8975 | 0.8506 | 0.9582 | 0.9014
UAPN | 0.9346 | 0.8500 | 0.9327 | 0.9135
LAPN | 0.9172 | 0.8536 | 0.9253 | 0.9122
LAWRRR | Uni | 0.9269 | 0.9346 | 0.9640 | 0.8643
UAPN | 0.9580 | 0.9585 | 0.9578 | 0.9171
LAPN | 0.9558 | 0.9497 | 0.9575 | 0.9112

We perform extensive ablation to validate the significance of the proposed
components.

Significance Of Different Sampling Strategies. In Table IV, we compare the
performance of BaSiL while using various sampling strategies and memory
replacement policies. We observe that for the buffer replacement, LAWRRR
performs better compared to LAWCBR. Furthermore, for the sample replay, UAPN,
along with LAWRRR memory buffer policy, outperforms other sampling strategies,
except _uniform sampling_ (Uni) performs better on i.i.d ordering.

Choice Of Hyperparameter ( $\bm{\lambda}_{2}$). Fig. 5 shows the effect of
changing the knowledge-distillation loss weight $\bm{\lambda}_{2}$ on the
final $\bm{\Omega}_{\text{all}}$ accuracy for iCubWorld 1.0 on instance and
class-instance ordering, while using different sampling strategies and buffer
replacement policies. We observe the best model performance for
$\bm{\lambda}_{2}=0.3$, and use this value for all our experiments.

Significance Of Knowledge-Distillation Loss. Fig. 5 with
$\bm{\lambda}_{2}=0.0$ represents the model without kd-loss. We can observe
that the model performance significantly degrades without knowledge
distillation. Therefore, kd-loss is a key component to the model’s
performance.

Figure 5: Plots of $\bm{\Omega}_{\text{all}}$ as a function of hyper-parameter
$\bm{\lambda}_{2}$ and different sampling strategies and replacement policies
for $(i)$ instance, $(ii)$ class-instance ordering on iCubWorld 1.0.

Choice Of Buffer Capacity. We perform an ablation for the different buffer
capacities, i.e., $|\bm{M}|$. The results are shown in Figure 6. It is evident
that, with the longer sequence of incoming data, the model’s (BaSiL)
performance improves with the increase in the buffer capacity, as it helps
minimize the confusion in the output prediction.

Figure 6: Plots of $\bm{\Omega}_{\text{all}}$ as a function of buffer capacity
$|\bm{M}|$ for class-i.i.d data-ordering on CIFAR10 and CIFAR100.

##  VI Conclusion

_Streaming continual learning_ (SCL) is the most challenging and realistic
framework for CL; most of the recent promising models for CL are unable to
handle this above setting. Our work proposes a dual regularization and loss-
aware buffer replacement to handle the SCL scenario. The proposed model is
highly efficient since it learns a joint likelihood from the current and
replay samples without leveraging any external finetuning. Also, to improve
the training efficiency further, the proposed model selects a few most
informative samples from the buffer instead of using the entire buffer for the
replay. We have conducted a rigorous experiment over several challenging
datasets and showed that BaSiL outperforms the recent state-of-the-art
approaches in this setting by a significant margin. To disentangle the
importance of the various components, we perform extensive ablation studies
and observe that the proposed components are essential to handle the SCL
setting.

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