# Symmetry Duality: Exploring Exotic Oscillators And Dissipative Dynamics
Through The Glass Of Newton-Hooke

Sayan Kumar Pal 1<EMAIL_ADDRESS>Partha Nandi 2<EMAIL_ADDRESS>1Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata,
203 B.T. Road, West Bengal 700108, India.  2Department of Physics, University
of Stellenbosch, Stellenbosch-7600, South Africa.

###### Abstract

Motivated by the symmetry in the non-relativistic limit of anti-de Sitter
geometry, we employ planar dynamical models featuring exotic (deformed)
harmonic oscillators, presented through direct and indirect Lagrangian
representations. The latter introduces Bateman dissipative oscillator system.
Analyzing these dynamic systems with a first-order Lagrangian scheme, our
phase-space-based approach utilizes the moment map components to reveal the
underlying symmetry algebra. This obtained algebra, interpreted as an extended
version of Newton-Hooke (NH) cosmological symmetry algebras, has the potential
to cast an augmented non-relativistic shadow over the expanding universe,
offering an insightful perspective on extended NH spacetime in 2+1 dimensions
through our dynamical realizations.

##  I Introduction

Studies in nonrelativistic spacetimes with a special emphasis on their
associated symmetries have particularly gained importance and popularity due
to some exciting discoveries in recent times. These studies have provided a
foundation for studies of transport phenomena in nonrelativistic fluids prabhu
, Chern-Simons gravity theories obers , phonons moving in a perfect
nonrelativistic fluid visser , nonrelativistic Nambu-Goldstone modes kobayashi
, Newton-Cartan gravity and supergravity theories rbnew ; bergshoeff , Dirac
fermions in curved space-times annalenderphysik , etc. All these studies are
broadly directed towards one of the two major themes: the AdS-CFT
correspondence sachdev  and the analogue gravity theories barcelo ; nat . At
this point, we would want to clarify to the readers that by space-times, we
may mean here even a degenerate metric between space and time, which is
typically the case in nonrelativistic scenarios. As a matter of fact,
nonrelativistic spacetimes of relevance in the present context such as Newton-
Hooke spacetimes or the similar Galilei-Maxwell spacetime serve as examples of
non-metric but curved spacetimes as their canonical connections have non-
vanishing Riemann curvature tensors, in contrast to the Galilean space-time.
Associated with these nonrelativistic spacetimes, one has the corresponding
symmetry groups which have become a major thrust of research in recent times
due to their cosmological and statistical mechanical applications ellis ;
duval2017 ; dtson ; martelli , of which the Newton-Hooke group in (2+1)
dimension ($\mathcal{NH}_{3}$) becomes important in our present context. In
fact, the Newton-Hooke (NH) group in n+1 dimensions is the symmetry group of
the n-dimensional harmonic oscillator (H.O.) and can be obtained by a group
contraction of the anti-de Sitter group in n+1 dimensions, $AdS_{n}$ zhang1 ;
zhang2 ; gibbons1 ; NHspacetime1 . Newton-Hooke spacetimes have been
interpreted as flat and torsionless Newton-Cartan spacetime with a quadratic
Newtonian potential and have also been described as a non-relativistic FLRW
geometry with flat spatial slices grosvenor . They are therefore considered as
cosmological nonrelativistic groups, and as such may be regarded as dynamical
groups of (free and nonrelativistic) Galilean systems, where the frequency of
the system plays the role of a Hubble’s constant in an accelerated
expanding/contracting universe aldrovandi ; gibbons . Just recently in
gibbonsnewest , using intuitive arguments from Newton-Hooke theory and
subsequently invoking numerical solutions of Einstein’s equations, the
existence of static black hole binaries has been reported in de Sitter
spacetime thereby providing the first four-dimensional example of
nonuniqueness in general relativity without matter. A classification of
systems in (2+1)D, both classical and quantum, having Newton Hooke symmetry
has been presented in olmo , which has also discussed the exotic central
extensions of the group in (2+1) dimensional context. Here we aim to provide a
dynamical realization of the extended Newton-Hooke group
$\overline{\mathcal{NH}_{3}}$, where we borrow the notation from olmo
mentioned just above. This extended Newton-Hooke group
$\overline{\mathcal{NH}_{3}}$ will be a nine-dimensional group containing
three central extensions. In addition to that, the extended
$\overline{\mathcal{NH}_{3}}$ resembles partly the reduced Galilei-Maxwell
group corresponding to the magnetic Landau group limit negro06 ; negro2002 .
In newNH4 , a study of (1+1)D Newton-Hooke group for the simple and the damped
H.O. has been carried out within the framework of the orbit method, while here
we present a study corresponding to the more physical and interesting (2+1)D
case using moment maps simalg ; cariglia  which, roughly speaking, are
generalizations of the notion of Noether charges in a symplectic manifold.

Besides, there have been studies that indicate non-trivial geometries or
space-times to be associated with the noncommutativity of dynamical operators
at the quantum level or symplectic deformations of phase space degrees of
freedom at the classical level, see for example shahn ; glik101 ; glik102 ;
cortes ; pjhep . Alternatively, the presence of a Berry curvature in momentum
space deforms the phase-space bracket relations xiao . This motivates us to
present here a symmetry study of deformed systems by taking the exotic
oscillators say2 ; stichel ; sir2  as our basic system, which is basically a
2D H.O. embedded in a deformed symplectic space where position coordinates
satisfy a non-zero Poisson structure. The deformed oscillators are thought to
represent a low-energy nonrelativistic effective system that is believed to be
originally derived from physics near Planck scales. More significantly, from
the symplectic viewpoint, deformations in the symplectic structure naturally
arise inside solids when electron dynamics is described in terms of Wannier
coordinates and Bloch wavefunctions xiao . It is worth mentioning at this
point that there exists a mapping between exotic oscillators and the planar
motion of an electron in a normal magnetic field together with a harmonic
trapping say2 . The related problems of the planar Landau quantum systems have
attracted much interest over the last few decades as they display integer and
fractional quantum hall effects, which is debatably the first and the simplest
manifestation of the topological phase of quantum matter. In this light of
view, symmetry studies of these kinds of fundamental systems become very
important which shed deep clues into many intriguing features and therefore
have been undertaken in negro2002 ; Gomis ; geracieson , etc. Indeed, as we
now well know, the Galilean group in 2+1 dimensions admits quite peculiarly
two central extensions - mass and anyonic spin levy . These 2D anyonic
systems, arising in the effective description of the fractional quantum Hall
effect, have also been given a symplectic phase-space description in terms of
noncommutativity in the position variables and such a description can readily
give rise to the Laughlin wavefunctions ady . Without much ado, we return to
our prime focus and remind that the Newton-Hooke group is distinctively
characterized by noncommutativity between time and spatial translation
generators, unlike in the Galilean algebra. Thus, analogously to the (anti) de
Sitter cases, energy is not invariant under spatial translations. Now here in
the present investigation, we will show that the boosts don’t commute which
can be ascribed to the anyonic spin jackiwnair ; duval . Moreover, the
obtained symmetry algebra strikingly contains spatial translation generators
that also no longer commute, a fact reminiscent of the anti-de Sitter or the
similar Galilei-Maxwell algebra, and suggests the existence of a non-zero
curvature.

Now, dissipation is a ubiquitous aspect of real-life physical systems. Even
so, dissipation is believed to play a crucial role in the physics of quantum
systems around Planck scales hoo ; hooftar . If general relativity is true at
all scales, black holes that absorb information inevitably arise in the
description of physics at very small scales viz at the Planck scale. Thus
speculations have always been there on the role of dissipation at such high
energy scales hooftbook ; visco ; mairi ; ameli . Furthermore, in skp1 , an
interplay between dissipation and deformation has been observed through the
path-integral scheme. Therefore, it becomes interesting to have a comparative
study from the symmetry perspective between these two systems, which has not
been undertaken previously. Thus, the above considerations make our purpose
two-fold in the present context. In this paper, we show at the classical level
the emergent damping mechanism in an otherwise conservative system of 2D
harmonic oscillators (H.O.) in the so-called indirect representation. This
observation is then strengthened by studying the symmetry algebra of the
relevant Bateman oscillators (a canonical model for treating dissipative
mechanics) bate  and the exotic (deformed) oscillators, which opens up a
fairly new door for investigating such systems. We find the symmetry algebra
identical in both systems.
The paper is organized in the following manner: In section II, we first study
the symmetry group of the exotic oscillators using moment map components. In
section III, we then consider a system of 2D harmonic oscillators in indirect
representation embedded in a deformed position space and show the appearance
of damping and anti-damping in the two modes. Later, we analyze the symmetry
of the resulting damped and anti-damped system. In section IV, we discuss how
the standard extended $\overline{\mathcal{NH}_{3}}$ symmetry algebra can be
obtained from the $AdS_{3}$ algebra through a suitable group contraction.
Finally, we summarise our results and conclude in section V. We have also
provided an appendix to recapitulate the symmetry group of the 1D H.O., which
is the firstly extended Galilei algebra with the four parameters: Boost (K),
momentum (T), Hamiltonian (H), and the mass (m) as the central extension. A
contrast of the Galilean algebra with the Newton-Hooke algebra has been
included in appendix A, while appendix B contains the Jacobi identities in
terms of the redefined generators of $\overline{\mathcal{NH}_{3}}$ showing the
algebraic consistency of the obtained symmetry algebra.

##  II H.O. in 2D deformed space: The exotic oscillator

We start by considering a 2D H.O. system but here we intend to introduce
spatial deformation (non-commutativity) in the system along the lines of
deriglazov . For this, we work in the first-order Lagrangian formalism. The
Lagrangian is written as:

${}L=p_{i}\dot{x}_{i}-\frac{p_{i}^{2}}{2m}-\frac{1}{2}m\omega^{2}x_{i}^{2}+\frac{\theta}{2}\epsilon_{ij}p_{i}\dot{p}_{j}$
(1)

where we treat $(x,p)$ as configuration space variables. For the sake of
clarity, we provide here the constraint analysis of the above system to
explicitly display the deformation of position space of the system of 2D H.O.
To that end, computing the canonical momenta, we get-

$\pi_{i}^{(x)}=\frac{\partial L}{\partial\dot{x}_{i}}=p_{i}$

and,

$\pi_{i}^{(p)}=\frac{\partial
L}{\partial\dot{p}_{i}}=-\frac{\theta}{2}\epsilon_{ij}p_{j}$

These yield the primary constraints and it is a 2nd class constrained system.
The set of constraints identified are,

$\phi_{i}^{1}=\pi_{i}^{(x)}-p_{i}\approx 0$

and,

$\phi_{i}^{2}=\pi_{i}^{(p)}+\frac{\theta}{2}\epsilon_{ij}p_{j}\approx 0$

The constraint matrix is

$C_{ij}^{\alpha\beta}=\begin{pmatrix}0&-\delta_{ij}\\\
\delta_{ij}&\theta\epsilon_{ij}\ \end{pmatrix}$ (2)

We construct the Dirac bracket defined by the following equation:

$\\{M_{i},N_{j}\\}_{DB}=\\{M_{i},N_{j}\\}_{PB}-\\{M_{i},\phi_{\alpha}^{k}\\}_{PB}(C^{-1})^{\alpha\beta}_{kl}\\{\phi_{\beta}^{l},N_{j}\\}_{PB}$
(3)

The Dirac brackets are computed to be,

$\\{x_{i},x_{j}\\}_{DB}=\theta\epsilon_{ij}~{}~{};~{}~{}\\{x_{i},p_{j}\\}_{DB}=\delta_{ij}~{};~{}\\{p_{i},p_{j}\\}_{DB}=0$
(4)

Next, computing the Euler-Lagrange equations of motion from (1), we get:

$\ddot{x}_{i}-m\omega^{2}\theta\epsilon_{ij}\dot{x}_{j}+\omega^{2}x_{i}=0$ (5)

Thus, the system in (1) can be interpreted as a 2-dim S.H.O. in NC position
space with the standard Hamiltonian-

$H=\frac{p_{i}^{2}}{2m}+\frac{1}{2}m\omega^{2}x_{i}^{2}$ (6)

but now with the following non-trivial symplectic structure:

$\\{x_{i},x_{j}\\}_{DB}=\theta\epsilon_{ij}$ (7)

The trajectories of the system (1) or equivalently (5) are given by,

$x_{i}=\cos{\beta t}(A_{i}\cos{\Omega t}+B_{i}\sin{\Omega t})-\sin{\beta
t}\epsilon_{ij}(A_{j}\cos{\Omega t}+B_{j}\sin{\Omega t})$

where, $\beta=-\frac{m\omega^{2}\theta}{2}$ , and
$\Omega=\omega\sqrt{1+\frac{m^{2}\omega^{2}\theta^{2}}{4}}=\sqrt{\omega^{2}+\beta^{2}}$.
Thus, the space of motion of the particular system $\mathcal{R}^{4}$ is
spanned by the parameters $A_{i},B_{i}$. After a straightforward lengthy
computation, the moment map components simalg ; cariglia  are found to be
given by,

$\displaystyle K_{i}=$ $\displaystyle\bigg{(}m\cos{\beta t}cos{\Omega
t}-\frac{m\beta}{\Omega}sin{\beta t}sin{\Omega
t}\bigg{)}x_{i}-\frac{1}{\Omega}\cos{\beta t}sin{\Omega t}~{}p_{i}$ (8)
$\displaystyle+\bigg{(}m\sin{\beta t}cos{\Omega
t}+\frac{m\beta}{\Omega}cos{\beta t}sin{\Omega
t}\bigg{)}\epsilon_{ij}x_{j}-\frac{1}{\Omega}\sin{\beta t}sin{\Omega
t}~{}\epsilon_{ij}p_{j}$

and,

$\displaystyle T_{i}=$ $\displaystyle\bigg{(}m\beta\sin{\beta t}cos{\Omega
t}+m\Omega cos{\beta t}sin{\Omega t}\bigg{)}x_{i}+\cos{\beta t}cos{\Omega
t}~{}p_{i}$ (9) $\displaystyle-\bigg{(}m\beta\cos{\beta t}cos{\Omega
t}-m\Omega sin{\beta t}sin{\Omega t}\bigg{)}\epsilon_{ij}x_{j}+\sin{\beta
t}cos{\Omega t}~{}\epsilon_{ij}p_{j}$

where K’s are the NH boost generators and T’s are the NH spatial translation
generators. Finally, we compute the Dirac brackets among the symmetry
generators to arrive at the following symmetry algebra :

$\\{K_{i},K_{j}\\}=\theta m^{2}\epsilon_{ij}$ (10) $\\{T_{i},T_{j}\\}=\theta
m^{2}\Omega^{2}\epsilon_{ij}$ (11)
$\\{K_{i},T_{j}\\}=m(1+\frac{m^{2}\omega^{2}\theta^{2}}{2})\delta_{ij}$ (12)
$\\{H,K_{i}\\}=-T_{i}+\beta\epsilon_{ij}K_{j}~{}~{};~{}\\{H,T_{i}\\}=\Omega^{2}K_{i}+\beta\epsilon_{ij}T_{j}$
(13)

As all the above generators are symmetry generators $G:=\\{K_{i},T_{i},H\\}$,
one may check that $\frac{dG}{dt}=0$ holds indeed. Note that the spatial
translations do not commute here unlike in the case of 2-D harmonic
oscillators and boosts also don’t commute. Particularly, the noncommutativity
of the boost generators is solely due to the deformation in the symplectic
structure w.r.t the position variables and also arises in anyonic systems
duval ; stichel . It is worthwhile to mention here that the above-obtained
symmetry algebra is reminiscent of the Newton-Hooke algebra $\mathcal{NH}_{3}$
which we will take up in section IV. As a passing remark, we mention here that
there obviously exists another conserved generator $J$ in the system, the
generator of rotations in the plane which we have not mentioned in (13) since
the commutations between $J$ and the other generators remain unchanged, which
can be verified easily. Having provided an explicit realization of the form of
the symmetry generators together with the symmetry algebra for the exotic
oscillators, let us now dive into the second part of the paper which presents
a novel analysis of 2D H.O. in indirect representation in a deformed position
space and a parallel study of its associated symmetries.

##  III 2D H.O. in indirect representation in symplectic deformed space:
Emergence of Bateman’s model of dissipation

The damped harmonic oscillator (dho) is characterized by the following
equation:

$\ddot{Y}+\gamma\dot{Y}+\omega^{2}Y=0$ (14)

which represents a non-conservative system as the Lagrangian, or so the
Hamiltonian is explicitly time-dependent. Here $\gamma>0$ and $\omega$ are
independent of time representing the damping parameter per unit mass and the
frequency respectively. Non-conservative systems are notorious to work with
but fortunately, H. Bateman gave a canonical formulation to handle dissipative
systems bate  in which one doubles the system by also considering additionally
the time-reversed image of equation (14) where the action functional for the
Bateman’s doubling scheme is given by:

$S=\int
dtL~{}~{}~{};~{}~{}L=m\dot{Y_{1}}\dot{Y_{2}}+m\frac{\gamma}{2}\left(Y_{1}\dot{Y}_{2}-\dot{Y}_{1}Y_{2}\right)-m\omega^{2}Y_{1}Y_{2}$
(15)

This form (15) has been referred to as the indirect representation in modern
literature Morse ; Sant ; Vit2 . The whole system is closed with damped
(source) and anti-damped (sink) harmonic oscillators along $Y_{1}$ and $Y_{2}$
axes on the 2D real plane. A thorough study of the quantum mechanics of the
d.h.o in the above-mentioned framework of the Bateman dual system (15) was
carried out in jizba . The analysis of Bateman is very similar to the notion
of Thermofield double (TFD) used widely in non-equilibrium thermodynamics Vit2
. Now, the case $\gamma=0$ leads us to a system of 2D H.O. in the indirect
representation and becomes interesting in the present sequel. Below, we put
forward a first-order Lagrangian for the two-dimensional H.O. in the indirect
representation that is embedded in a symplectic-deformed position space,

$L=\pi_{1}\dot{Y}_{1}+\pi_{2}\dot{Y}_{2}-\frac{1}{m}\pi_{1}\pi_{2}-m\omega^{2}Y_{1}Y_{2}+\frac{\theta}{2}\left(\pi_{1}\dot{\pi}_{2}-\dot{\pi}_{1}\pi_{2}\right)$
(16)

where the last term is a non-relativistic Chern-Simons like term in the
momentum (here $\pi_{i}$’s are the canonical momenta) variables and is
specifically responsible for the deformation of the associated configuration
space. This Lagrangian (16) also gives rise to a second-class constrained
system and the analysis follows similarly to the previous section which we
therefore omit here. It is seen that the coordinates of the system described
by (16) satisfy $\\{Y_{i},Y_{j}\\}=\theta\epsilon_{ij}$, whereas the equations
of motion obtained from the above Lagrangian are:

$\ddot{Y_{1}}-\lambda\dot{Y_{1}}+\omega^{2}Y_{1}=0~{};~{}~{}~{}~{}~{}~{}\lambda=m\omega^{2}\theta$
(17)

and,

$\ddot{Y_{2}}+\lambda\dot{Y_{2}}+\omega^{2}Y_{2}=0$ (18)

which interestingly represent anti-damping (if $\theta$ is chosen positive)
and damping respectively. These two equations together constitute the Bateman
doubled system111These pair of equations are sometimes alternatively described
Vit2 ; mairi  in terms of light-cone coordinates $y_{i}=(Y_{1}\pm
Y_{2})/\sqrt{2}$ yielding,
$\ddot{y_{1}}-\lambda\dot{y_{2}}+\omega^{2}y_{1}=0~{},~{}~{}~{}\ddot{y_{2}}-\lambda\dot{y_{1}}+\omega^{2}y_{2}=0$
(19) to handle dissipative dynamics just like in the standard equation (14)
and its time-reversed image, where $\lambda=m\omega^{2}\theta$ plays the role
of the damping parameter $\gamma$. We now look forward to extending our
present analysis of the Bateman system, obtained from an appropriate
symplectic deformation as discussed above, to find the symmetry properties and
to further discern the similarities or dissimilarities with the exotic
oscillator of the previous section.

###  III.1 Symmetry algebra for the Bateman’s dual model of dissipation

The associated trajectories for the Bateman pair (17) and (18) result in:

$Y_{i}=e^{\pm\frac{\lambda
t}{2}}\left(A_{i}\cos{\tilde{\omega}t}+B_{i}\sin{\tilde{\omega}t}\right)~{};~{}~{}\text{$+$
for $Y_{1}$ and $-$ for $Y_{2}$}$

where $\tilde{\omega}^{2}=\omega^{2}-\frac{\lambda^{2}}{4}$. The components of
the moment map are again computed for the present system and given to be,

$K_{1}=e^{-\frac{\lambda
t}{2}}\left(m\cos{\tilde{\omega}t}-\frac{m\lambda}{2\tilde{\omega}}\sin{\tilde{\omega}t}\right)Y_{1}-e^{-\frac{\lambda
t}{2}}~{}\frac{\sin{\tilde{\omega}t}}{\tilde{\omega}}\pi_{2}$
$K_{2}=e^{\frac{\lambda
t}{2}}\left(m\cos{\tilde{\omega}t}+\frac{m\lambda}{2\tilde{\omega}}\sin{\tilde{\omega}t}\right)Y_{2}-e^{\frac{\lambda
t}{2}}~{}\frac{\sin{\tilde{\omega}t}}{\tilde{\omega}}\pi_{1}$

and,

$T_{1}=e^{-\frac{\lambda
t}{2}}\left(m\tilde{\omega}\sin{\tilde{\omega}t}+\frac{m\lambda}{2}\cos{\tilde{\omega}t}\right)Y_{1}+e^{-\frac{\lambda
t}{2}}~{}\cos{\tilde{\omega}t}~{}\pi_{2}$ $T_{2}=e^{\frac{\lambda
t}{2}}\left(m\tilde{\omega}\sin{\tilde{\omega}t}-\frac{m\lambda}{2}\cos{\tilde{\omega}t}\right)Y_{2}+e^{\frac{\lambda
t}{2}}~{}\cos{\tilde{\omega}t}~{}\pi_{1}$

Thus, the symmetry algebra between the symmetry generators for the Bateman
pair turns out as follows,

$\\{K_{i},K_{j}\\}=\theta m^{2}\epsilon_{ij}$ (20) $\\{T_{i},T_{j}\\}=\theta
m^{2}\tilde{\omega}^{2}\epsilon_{ij}$ (21)
$\\{K_{i},T_{j}\\}=m\bigg{(}1-\frac{m^{2}\omega^{2}\theta^{2}}{2}\bigg{)}(\delta_{i1}\delta_{j2}+\delta_{j1}\delta_{i2})$
(22)
$\\{H,K_{i}\\}=-T_{i}+(-1)^{i}\frac{\lambda}{2}K_{i}~{}~{};~{}\\{H,T_{i}\\}=\tilde{\omega}^{2}K_{i}+(-1)^{i}\frac{\lambda}{2}T_{i}$
(23)

Appealingly in this case too, the symmetry algebra is very similar to the
algebra of symmetries of the exotic oscillator system and is another extended
$\overline{\mathcal{NH}}_{3}$ algebra. However, at this point, one might
notice slight differences in the last terms of the r.h.s of (23) with (13).
These differences stem from the indirect representation of the Bateman
oscillator system in terms of the standard Bateman damped and anti-damped
variables ($Y_{2},Y_{1}$) and can be eliminated by working with the light-cone
coordinates $y_{1},y_{2}$ (see equation 19 in footnote previously, also see
mairi ; Vit2 ). This is also the reason for the difference in the algebra
between boost generator and translation where we can notice that it is
$\\{K_{1},T_{2}\\}\neq 0$ in (22) rather than $\\{K_{1},T_{1}\\}\neq 0$
typical for most dynamical systems. It is pertinent here to mention that the
whole analysis of this section can be worked out in a similar consistent
fashion by working with the light-cone coordinates $y_{1},y_{2}$. Therefore,
we conclude that the two algebras are overall identical - apparent differences
are solely due to the choice of coordinates. Nonetheless, they merely differ
only in the relative signs of the deformation terms of (22) and (12); being
negative for the former and positive for the latter. Actually, these two
bracket relations respectively characterize their systems and contain
information on the normal mode frequencies of their respective systems. As an
instance, we note that the negative sign in the modification term of (22)
provides a footprint of the existence of different dynamical phases in a d.h.o
system (over-critical, sub-critical, and critical damping phases). Therefore
from the perspective of symmetries, dissipative systems obtained as a result
of symplectic deformation follow that of the exotic (deformed) oscillators.
This result corroborates and provides another paradigm to our previous studies
in skp1 . This newly obtained unifying symmetry algebra is one of the
important results of our paper. We will now identify in the following section
the form of this obtained algebra as an extended version of the Newton-Hooke
algebra.

##  IV $AdS_{3}$ and extended $\overline{\mathcal{NH}}_{3}$ algebra

In this section, the extended Newton-Hooke algebra in 2+1 dimensions
($\overline{\mathcal{NH}}_{3}$) will be obtained as a group contraction of
$AdS_{3}$ which is :

$\\{M_{\mu\nu},M_{\lambda\sigma}\\}=(\eta_{\mu\lambda}M_{\nu\sigma}-\eta_{\mu\sigma}M_{\nu\lambda}+\eta_{\nu\sigma}M_{\mu\lambda}-\eta_{\nu\lambda}M_{\mu\sigma})$
$\\{M_{\mu\nu},T_{\lambda}\\}=(\eta_{\mu\lambda}T_{\nu}-\eta_{\nu\lambda}T_{\mu})$
$\\{T_{\mu},T_{\nu}\\}=\Lambda M_{\mu\nu}$

where $~{}\mu=0,1,2~{}and~{}\eta_{\mu\nu}=diag(-,+,+)$.
To show this, we re-write explicitly in components,

$\\{T_{0},M_{0i}\\}=T_{i}~{};~{}\\{M_{0i},T_{j}\\}=-\delta_{ij}T_{0}~{};~{}\\{T_{0},T_{i}\\}=\Lambda
M_{0i}$
$\\{M_{0i},M_{0j}\\}=-\epsilon_{ij}M_{12}~{}~{};~{}~{}\\{T_{i},T_{j}\\}=\Lambda\epsilon_{ij}M_{12}~{};~{}~{}~{}\text{where}~{}i,j=1,2$

Defining the following _rescaled_ generators -

$T_{0}\longrightarrow-\xi m-\frac{1}{\xi}H$ $M_{0i}\longrightarrow\xi
K_{i}~{}~{};~{}~{}M_{12}\longrightarrow\xi^{2}\sigma+J~{}~{};~{}~{}\Lambda\longrightarrow\frac{1}{\xi^{2}}\Lambda$

Next, we calculate the brackets with the rescaled generators and then finally
take the limit $\xi\rightarrow\infty$ to obtain -

$\\{H,K_{i}\\}=-T_{i}~{}~{};~{}\\{H,T_{i}\\}=\Lambda
K_{i}~{};~{}\\{K_{i},T_{j}\\}=m\delta_{ij}$
$~{}~{}~{}\\{K_{i},K_{j}\\}=\sigma\epsilon_{ij}~{}~{};~{}~{}\\{T_{i},T_{j}\\}=\Lambda\sigma\epsilon_{ij}~{}~{}~{}~{};~{}where~{}i,j=1,2.$
(24)

This is the second centrally-extended Newton-Hooke algebra
$\overline{\mathcal{NH}}_{3}$ which is a larger symmetry algebra. Furthermore,
if one now takes the limit $\Lambda\rightarrow 0$ above, the well-known
secondly extended Galilean algebra in 2+1 dimensions is obtained -

$\\{H,K_{i}\\}=-T_{i}~{}~{};~{}\\{H,T_{i}\\}=0~{};~{}\\{K_{i},T_{j}\\}=m\delta_{ij}$
$~{}~{}~{}\\{K_{i},K_{j}\\}=\sigma\epsilon_{ij}~{}~{};~{}~{}\\{T_{i},T_{j}\\}=0~{}~{}~{}~{};~{}where~{}i,j=1,2.$
(25)

and finally, on taking the commutative (no deformation) limit
$\theta\rightarrow 0$, one arrives at the trivial firstly extended
($\emph{Gal}_{3}$), the Galilean algebra in 2+1 dimensions.

Now, comparing (IV) with (10-13) or (20-23), we observe that the symmetry
algebra for the exotic and the Bateman oscillators is similar to
$\overline{\mathcal{NH}}_{3}$ upon identifying $\sigma=\theta m^{2}$ and
$\Lambda=\Omega^{2}$. However, our reported symmetry algebra differs by the
presence of an additional second term on the r.h.s. in the Lie brackets
between $H$ and $K_{i}$ and $H$ and $T_{i}$. Additionally, the Lie bracket
between $\\{K_{i},T_{j}\\}$ is also modified in our case, which contains
information on the energy spectrum of the problem as discussed after (23). In
the literature, the noncommutative parameter $(\sigma)$ of the boost-boost
sector (IV) has been regarded to represent the anyonic spin in two dimensions
which can take any real value jackiwnair ; wilczek0 ; wilc . The radius of
curvature $(R=\frac{1}{\sqrt{-\Lambda}})$ of the $AdS$ space is survived in
the nonrelativistic limit by the inverse of frequency $\Omega$ of the
nonrelativistic exotic (or $\tilde{\omega}$ of Bateman) oscillator and
together with the deformation parameter $\theta$ is responsible for the
noncommuting translations which possibly indicate to a presence of curvature
in the space of motions. The Jacobi identities of the algebra have been
provided in appendix B, where we demonstrate by using a general basis a
mapping to the symmetry algebra.

##  V Conclusion

We first summarize our key findings of the paper. In the beginning, we have
explicitly obtained the time-dependent symmetry generators of the exotic
deformed 2D oscillator through the computation of moment maps and found the
associated symmetry algebra. Next, we have provided a Lagrangian that
effectively describes the Bateman system (a canonical model for treating
dissipative mechanics) in which the deformation parameter of position sector
(in the symplectic structure) accounts for the induction of dissipation. This
observation corroborates and provides fresh insights on the opinions made in
the literature mairi ; skp1 ; rbmpl  regarding a connection between
dissipation and deformation through the presented Lagrangian formulation (16).
This opens up the possibility of a new paradigm in the analysis of dissipative
systems in certain situations. Then we extended the analysis to compute moment
map components for the modeled Bateman system to find the associated
symmetries. Remarkably, the symmetry algebras are found to be almost identical
in both systems. This obtained symmetry algebra has then been identified as a
new kind of generalized (extended) Newton-Hooke algebra in 2+1 dimensions,
$\overline{\mathcal{NH}_{3}}$. And importantly, on taking the zero limit of
the symplectic deformation parameter $\theta$ in the obtained algebra, we
recover the standard Newton-Hooke algebra in (2+1) dimensions,
$\mathcal{NH}_{3}$. This provides a benchmark to our analysis as well because
in such a limit, physically both the systems reduce to the standard two-
dimensional harmonic oscillator. Thus, the present work provides a dynamical
realization of the extended Newton-Hooke symmetry in terms of the dynamical
models discussed. Not unexpectedly, the extended NH symmetry also consists of
rotational invariance in terms of a deformed rotation generator (angular
momentum) frederikapp . However, the rotation sector of the algebra will
remain unchanged and therefore has not been considered.

Most interestingly, we find that noncommuting translations (11,21) appear in
the obtained algebra which is rather special in the usual domain of
nonrelativistic spacetime. Apart from this, we obtain boost generators that do
not commute with each other - a fact that is usually ascribed to anyonic spin
particles. It is worthwhile to observe that the bracket relations in (12) and
(22) contain traces of their respective dimensionless characteristic
frequencies multiplied with mass in the r.h.s. In light of all the above
observations, we infer that symplectic deformations induce structural and
kinematical changes in the geometry of dynamical systems viz. nonrelativistic
spacetimes. One might refer to our case as generalized NH spacetimes.
Previously in skp1 , we have shown that deformed oscillators have a close
connection with the problem of a nonrelativistic charged particle moving in a
two-dimensional plane in a background magnetic field acting normally to the
plane. Our obtained algebra therefore also shares a resemblance with the
reduced Galilei-Maxwell group negro06 , it being a symmetry group of the
nonrelativistic charged particle moving in an electromagnetic field background
and can be obtained as a specific limit of the anti-de Sitter Maxwell group
gomis ; gilkman .

As mentioned before, the primary distinction between Newton-Hooke and Galilei
kinematics stems from the non-commutativity between temporal and spatial
translations. This leads to alterations in standard time-evolution equations
where the Laplacian is time-invariant, and hence has consequences in
nonrelativistic cosmology gibbons ; freed ; duval2017 . Therefore, this newly
obtained (generalized) extended Newton-Hooke spacetimes may give rise to
further and significant changes to cosmic nonrelativistic physics. It will be
curious to analyze studies related to anyonic black holes in cosmological
backgrounds marsot ; mirza  as the symmetry algebra hints at the anyonic
nature of the corresponding particles due to noncommutativity in the boost
sector.

## Acknowledgments

One of the authors, S.K.P. gratefully acknowledges the support from the Indian
Statistical Institute Kolkata, India for a visiting scientist fellowship, and
the other author, P.N. acknowledges the support from a postdoctoral fellowship
grants from Stellenbosch University, South Africa. S.K.P. extends his thanks
to Subir Ghosh for several discussions and helpful suggestions during the
course of this work. P.N. would like to express his appreciation to Frederik
G. Scholtz for a careful reading and feedback on the manuscript.

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## Appendix A: A short review on 1D H.O. symmetries

Here, we provide a brief outline of the Galilean and the Newton-Hooke group
symmetries for the one-dimensional harmonic oscillator. We recall that any
transformation of a system in classical mechanics will be called a symmetry
transformation iff:

$\delta L=L^{\prime}-L=\frac{dF}{dt}$ (26)

where F is a function of the configuration space variables.

Both $L$ and $L^{\prime}$ give rise to the same Euler-Lagrange equations of
motion. The generator $\mathcal{B}$ which gives rise to the above symmetry
transformation is the conserved charge associated with the transformation and
is given by the Noether prescription-

$\mathcal{B}=p_{i}\delta q_{i}-F$ (27)

The Hamiltonian of the 1-dim. H.O. is given by-

$H=\frac{p^{2}}{2m}+\frac{1}{2}m\omega^{2}x^{2}$ (28)

The second-order Newton’s equation of motion of the system is given by:

$\ddot{x}+\omega^{2}x=0$ (29)

which has the solution,

$x=x_{0}\cos{\omega t}+\frac{p_{0}}{m\omega}\sin{\omega t}$ (30)

The boost and the spatial translation generators are respectively given by the
following moment maps -

$K=mx\cos{\omega t}-\frac{p}{\omega}\sin{\omega t}$ (31) $T=m\omega
x\sin{\omega t}+p\cos{\omega t}$ (32)

The conserved time translation generator or “Hamiltonian" is then expressed in
terms of the invariant Casimir, $H=\frac{T^{2}}{2m}$ following the orbit
method. Thus, the symmetry algebra of the 1D H.O. is given by:

$\\{K,T\\}=m~{}~{};~{}~{}\\{H,K\\}=-T~{}~{};~{}~{}\\{H,T\\}=0$ (33)

This is a four-parameter group comprising of $K$, $T$, $H$, and the central
extension $m$. All other commutators vanish. This defines the Galilean algebra
in one spatial dimension ($\emph{Gal}_{1}$) which is the Lie algebra
corresponding to the four-parameter centrally extended Galilean group,
sometimes also referred to as the Bargmann algebra. On the other hand, the
Newton-Hooke $\mathcal{NH}_{2}$ algebra is given by the following:

$\\{K,T\\}=m~{}~{};~{}~{}\\{H,K\\}=-T~{}~{};~{}~{}\\{H,T\\}=\omega^{2}K$ (34)

where the time translation generator, denoted with $H$ as before, is given by
(28). It is instructive now to write down the phase-space brackets between
$x,p$, and $H$:

$\\{x,p\\}=1~{}~{};~{}~{}\\{H,x\\}=-\frac{p}{m}~{}~{};~{}~{}\\{H,p\\}=m\omega^{2}x$
(35)

Replacing $K=mx$ and $p=T$ in the above reduces to Newton-Hooke algebra (34),
where $m$ is a constant function (central generator) to be interpreted
physically as the mass. Hence, the set of Lie group generators {K,T,H,m} spans
the Poisson sub-algebra of the Poisson algebra of well-defined functions of
canonical phase-space coordinates $x$ and $p$. This provides a glimpse into
the phase-space approach to symmetry principles and symmetry groups in non-
relativistic physics.

## Appendix B: General basis of extended planar Newton-Hooke algebra

In this section, our objective is to define an equivalent class of the
extended Newton-Hooke algebra $\overline{\mathcal{NH}_{3}}$ by introducing a
general basis for this algebra thus providing the algebraic consistency check.
Let us define,

$\displaystyle\tilde{K}_{i}$ $\displaystyle=$
$\displaystyle[a_{ij}K_{j}+b_{ij}T_{j}]~{},$ (36) $\displaystyle\tilde{T}_{i}$
$\displaystyle=$ $\displaystyle[c_{ij}T_{j}+d_{ij}K_{j}]~{}:~{}~{}i,j=1,2,$
(37)

where, $a_{ij}$ and $c_{ij}$ are dimensionless, while $b_{ij}$ and $d_{ij}$
are dimensionful real-valued 2$\times$2 non-singular transformation matrices,
respectively. Following this, the Newton-Hooke algebra is re-expressed in
terms of the new generators,

$\\{\tilde{K}_{i},\tilde{K}_{j}\\}=\alpha_{ij}=\sigma(a_{il}a_{jm}+\lambda
b_{il}b_{jm})\epsilon_{lm}+m(a_{il}b_{jl}-b_{il}a_{jl})$ (38)
$\\{\tilde{T}_{i},\tilde{T}_{j}\\}=\beta_{ij}=\sigma(\lambda
c_{ik}c_{jl}+d_{ik}d_{jl})\epsilon_{kl}+m(d_{ik}c_{jk}-c_{ik}d_{jk})$ (39)
$\\{\tilde{K}_{i},\tilde{T}_{j}\\}=h_{ij}=\sigma(\lambda
b_{ik}c_{jl}+a_{ik}d_{jl})\epsilon_{kl}+m(a_{ik}c_{jk}-b_{ik}d_{jk})$ (40)
$\\{H,\tilde{K}_{i}\\}=-\Sigma_{ij}\tilde{T}_{j}+\Theta_{ij}\tilde{K}_{j}~{}~{};~{}\\{H,\tilde{T}_{i}\\}=\Omega_{ij}\tilde{K}_{j}+\Phi_{ij}\tilde{T}_{j}$
(41)

where $\Sigma_{ij}=a_{ik}C_{kj}-\lambda b_{ik}B_{kj}$, $\Theta_{ij}=\lambda
b_{ik}A_{kj}-a_{ik}D_{kj}$, $\Omega_{ij}=\lambda c_{ik}A_{kj}-d_{ik}C_{kj}$,
and $\Phi_{ij}=\lambda c_{ik}B_{kj}-d_{ik}C_{kj}$. The coefficients are
defined as follows: $A_{ij}=-B_{ik}d_{kl}b^{-1}_{lj}$ and
$C_{ij}=-D_{il}c_{lm}a^{-1}_{mj}$, and they satisfy the conditions
$B^{-1}_{ij}=c_{ij}-d_{il}b^{-1}_{lk}a_{kj}$ and
$D^{-1}_{ij}=b_{ij}-c_{il}a^{-1}_{lk}d_{kj}$.

Now, the Jacobi identities for $H,\tilde{K}_{i},\tilde{K}_{j}$,
$H,\tilde{T}_{i},\tilde{T}_{j}$ and $H,\tilde{T}_{i},\tilde{K}_{j}$ are
respectively given by,

$\Sigma_{jk}h_{ik}-\Theta_{jk}\alpha_{ik}-\Sigma_{ik}h_{jk}+\Theta_{ik}\alpha_{jk}=0$
(42)
$\Omega_{jk}h_{ij}-\Phi_{jk}\beta_{ik}-\Omega_{ik}h_{jk}+\Phi_{ik}\beta_{jk}=0$
(43)
$\Sigma_{jk}\beta_{ik}+\Theta_{jk}h_{ki}+\Omega_{ik}\alpha_{jk}+\Phi_{ik}h_{jk}=0$
(44)

If we make specific choices for the structure constants as follows:

$\alpha_{ij}=\theta m^{2}\epsilon_{ij},~{}~{}\beta_{ij}=\theta
m^{2}\Omega^{2}\epsilon_{ij},~{}~{}\Sigma_{ij}=\delta_{ij},~{}~{}\Theta_{ij}=\beta\epsilon_{ij},~{}~{}\Omega_{ij}=\delta_{ij},~{}~{}\Phi_{ij}=\beta\epsilon_{ij},~{}~{}h_{ij}=m(1+\frac{m^{2}\omega^{2}\theta^{2}}{2})\delta_{ij},$
(45)

then the above Jacobi identities are satisfied for a suitable choice of
primitive two-by-two transformation matrices. Consequently, we can assert that
the algebra derived and expressed in terms of Dirac brackets (10-13) falls
into an equivalent class of algebra with extended Newton-Hooke symmetry
written in a different basis.


# Continual Learning Using Bayesian Neural Networks

Honglin Li, Payam Barnaghi, , Shirin Enshaeifar, , Frieder Ganz H. Li, P.
Barnaghi and S. Enshaeifar are with the Centre for Vision, Speesh and Signal
Processing (CVSSP) at the University of Surrey and also with Care Research and
Technology Centre at the UK Dementia Research Institute (UK DRI). P. Barnaghi
is also with the Department of Brain Sciences at Imperial College London.
email<EMAIL_ADDRESS><EMAIL_ADDRESS>Ganz is
with the Adobe, Germany. email<EMAIL_ADDRESS>
###### Abstract

Continual learning models allow them to learn and adapt to new changes and
tasks over time. However, in continual and sequential learning scenarios in
which the models are trained using different data with various distributions,
neural networks tend to forget the previously learned knowledge. This
phenomenon is often referred to as catastrophic forgetting. The catastrophic
forgetting is an inevitable problem in continual learning models for dynamic
environments. To address this issue, we propose a method, called Continual
Bayesian Learning Networks (CBLN), which enables the networks to allocate
additional resources to adapt to new tasks without forgetting the previously
learned tasks. Using a Bayesian Neural Network, CBLN maintains a mixture of
Gaussian posterior distributions that are associated with different tasks. The
proposed method tries to optimise the number of resources that are needed to
learn each task and avoids an exponential increase in the number of resources
that are involved in learning multiple tasks. The proposed method does not
need to access the past training data and can choose suitable weights to
classify the data points during the test time automatically based on an
uncertainty criterion. We have evaluated our method on the MNIST and UCR time-
series datasets. The evaluation results show that our method can address the
catastrophic forgetting problem at a promising rate compared to the state-of-
the-art models.

###### Index Terms:

Catastrophic forgetting, continual learning, incremental learning, Bayesian
neural networks, uncertainty

##  I Introduction

Deep learning models provide an effective end-to-end learning approach in a
variety of fields. One common solution in deep neural networks to solve a
complex task such as ImageNet Large Scale Visual Recognition Challenge
(ILSVRC) [1] is to increase the depth of the network [2]. However, as the
depth increases, it becomes harder for the training model to converge. On the
other hand, a shallower network is not able to solve a complex classification
task at once, but it may be able to find a solution for a smaller set of
classes and converges much faster. If a model can continually learn several
tasks, then it can solve a complex task by dividing it into several simple
tasks. In online learning scenarios [3], the model repeatedly receives new
data, and the training data is not complete at any given time. If we re-train
the entire model whenever there are new instances, it would be very
inefficient, and we have to store the trained samples [4]. The key challenge
in such continual learning scenarios in changing environments is how to
incrementally and continually learn new tasks without forgetting the previous
or creating highly complex models that may require accessing the entire
training data.

Most of the common deep learning models are not capable of adapting to
different tasks without forgetting what they have learned in the past. These
models are often trained via back-propagation where the weights are updated
based on a global error function. Updating and altering tasks of an already
learned model leads to the loss of the previously learned knowledge as the
network is not able to maintain the important weights for various
distributions. If hard constraints are applied to the model to prevent the
forgetting, it can retain the previously learned knowledge. However, the model
will not be able to acquire new information efficiently. This scenario is
referred to as the stability-plasticity dilemma [5, 6]. The attempt to
sequentially or continuously learn and adapt to various distributions will
eventually result in a model collapse. This phenomenon is referred as
catastrophic forgetting or interference [7, 8]. The catastrophic forgetting
problem makes the model inflexible. Furthermore, the need for a complete set
of training samples during the learning process is very different from the
normal biological systems which can incrementally learn and acquire new
knowledge without forgetting what is learned in the past [9, 10].

Figure 1: The network architectures. In Continual Bayesian Learning networks,
the weights are Gaussian mixture distributions with an arbitrary number of
components. e.g. As shown above, two Bayesian Neural Networks which have
learned Task A and B respectively. Each of them contains multiple Gaussian
distributions. The CBLN merges these Gaussian distributions into one Gaussian
mixture distribution. The number of components in the mixture distribution in
this example can be 2, which means task A and task B have different weight
distributions (shown in red and blue), or it could be 1, which means tasks A
and B have the same weight distributions (shown in green).

Terminology: In this paper, the term task refers to the overall function of a
model; e.g. classification, clustering and outlier detection. A task has an
input distribution and an output distribution. A dataset is used to train and
evaluate a model for a task. A dataset follows a certain distribution. The
distribution of a dataset that is used to train a specific task can change
over time. We can train a model with different tasks. Each task can be trained
on its own individual dataset. In other words, each task can be trained based
on different input and output distribution.

To address the catastrophic forgetting problem, there are mainly three
approaches [11]:

Regularisation Approaches: Regularisation based approaches re-train the model
with trading off the learned knowledge and new knowledge. Kirkpatrick et. al
[12] propose Elastic Weights Consolidation (EWC), which uses sequential
Bayesian inference to approximate the posterior distribution by taking the
learned parameters as prior knowledge. EWC finds the important parameters to
the learned tasks according to Fisher Information and mitigates their changes
by adding quadratic items in the loss function. Similarly, Zenke et. al [13]
inequitably penalise the parameters in the objective function. Zenke et. al
define a set of influential parameters by using the information obtained from
the gradients of the model. The idea of using a quadratic form to approximate
the posterior function is also used in Incremental Moment Matching (IMM) [14].
In IMM, there are three transfer techniques: weight-transfer, L2-transfer and
drop-transfer to smooth the loss surface between the different tasks. The IMM
method approximates the mixture. Recently, the variational inference has drawn
attention to solving the continual learning problem [15]. The core idea of
this method is to approximate the intractable true posterior distribution by
variational learning. Regularisation approach can continually learn new tasks
without saving the trained samples or adding new neuron resources. However,
when the number of tasks increases, regulating the model becomes very complex.

Memory Replay: The core idea of memory replay is to interleave the new
training data with the previously learned samples. The recent developments in
this direction reduce the memory of the old knowledge by leveraging a pseudo-
rehearsal technique [16]. Instead of explicitly storing the entire training
samples, the pseudo-rehearsal technique draws the training samples of the old
knowledge from a probabilistic distribution model. Shin et. al [17] propose an
architecture consisting of a deep generative model and a task solver.
Similarly, Kamra et. al [18] use a variational autoencoder to regenerate the
previously trained samples. The performance of memory replay approaches is
high [17, 18]. However, it is a memory consuming approach. Furthermore, the
computational resources required to train a generative model can also be very
high.

Dynamic Networks: Dynamic Networks allocate new neuron resources to learn new
tasks. For example, ensemble methods build a network for each task. As a
result, the number of models grows linearly with respect to the number of
tasks [19, 20, 21]. This is not always a desirable solution because of its
high resource demand and complexity [22]. One of the key issues in the dynamic
methods is that whenever there is a new task, new neuron resources will be
created without considering the possibility of generating redundant resources.
In [23], the exponential parameter and resource increases are avoided by
selecting part of the existing neurons for training new tasks. However, during
the test process, the model has to be aware of which test task is targeted to
choose the appropriate parameters to perform the desired task [24]. In the
most dynamic methods [24, 23], the model will not forget the learned knowledge
because of the trained parameters are fixed. However, the major issues in
dynamic networks are how to prevent the parameters growing exponentially and
how to decide which parameters should be used at the testing stage.

The works mentioned above focus on supervised-learning. There are also other
exiting works that concentrate on unsupervised learning methods. For example,
Bianchi et. al[25] mix a supervised Convolutional Neural Network (CNN) with a
bio-inspired unsupervised learning component. Similarly, Muñoz-Martín et. al
[26] present a novel architecture that combines CNN with unsupervised learning
by spike-timing-dependent plasticity (STDP) to overcome the forgetting problem
in learning models.

In this paper, we propose a Continual Bayesian Learning Network (CBLN) to
address the forgetting problem and to allow the model to adapt to new
distributions and learn new tasks. The CBLN trains an entirely new model for
each task and merges them into a master model. The master model finds the
similarities and distinctions among these sub-models. For the similarities,
the master model merges them and produces a general representation. For the
distinctive parameters, the master model does not merge them and retains them.
CBLN is based on Bayesian Neural Networks (BNNs) [27], see Figure 1. Based on
BNNs, we assume that the weights in our BNN model have a Gaussian distribution
and the covariance matrix is diagonal. The distribution of the weights in
different tasks are independent of each other. Based on this assumption, we
can assume that the combined posterior distribution of all the training tasks
is a mixture of Gaussian distributions. We then use an Expectation-
Maximisation (EM) [28] algorithm to approximate the posterior mixture
distributions and remove the components that are redundant or less
significant. The final distribution of the weights can be a Gaussian mixture
distribution with an arbitrary number of components. At the test stage, we
produce an epistemic uncertainty [29] measure for each set of components. The
set which has minimal uncertainty will be used to give the final prediction.

In general, there are two main challenges while solving the continual learning
problem with dynamic methods: 1). How to prevent the exponential increase in
the number of parameters; 2). How to choose the parameters corresponding to
the test task. Firstly, we address the issue of the exponential increase in
the number of parameters by using variational learning and a clustering
algorithm. These allow us to decrease the number of parameters significantly.
Secondly, we address the issue of choosing the parameters by using an
uncertainty criterion. Instead of averaging the prediction of different models
corresponding to different tasks, or explicitly indicating the task identified
in the test state, the proposed model can evaluate the uncertainty of the test
points.

##  II Continual Bayesian Learning Networks (CBLN)

###  II-A Training Process

The training process in CBLN is similar to BNNs. At the beginning of the
training for each task, we initialise all the training parameters and train
the model. However, at the end of the training for each task, we store the
solution for the current task. We used the loss function shown in Equation
(1):

$\begin{split}\mathbb{L}(D,\theta)=\mathbb{E}_{q(w|\theta)}[\log
P(D|\textbf{w}^{(i)}]&-\\\
\frac{\lambda}{2}*D_{KL}&[q(\textbf{w}^{(i)}|\theta)||\log
P(\textbf{w}^{(i)}]\end{split}$ (1)

Where $\theta$ refers to the training parameters, $\textbf{w}^{(i)}$ is the
$i_{th}$ Monte Carlo sample [30] drawn from the variational posterior
$q(\textbf{w}^{(i)}|\theta)$, $D$ is the training data, $\lambda$ is a hyper-
parameter to regular the training of models. The common used $\lambda$ is 1.
We attempt to obtain weight parameters that have a similar Gaussian
distribution, which is close to the prior knowledge. After training $K$ tasks,
we can obtain $K$ sets of parameters that construct the posterior mixture
Gaussian distribution in which each component is associated with a different
task.

###  II-B Merging Process

The merging process in this method reduces the components in the posterior
mixture distribution. Taking one mixture Gaussian distribution as an example,
we approximate the posterior mixture distribution with an arbitrary number of
Gaussian distributions, see Equation (2), where $K$ is the number of tasks,
$n$ is the number of components in the final posterior mixture distribution,
$q_{1:K}$ is the posterior mixture distribution with the component $q_{k}$
associated with $k_{th}$ task, $\alpha$ and $\beta$ are the weight parameters
where $\alpha=\frac{1}{K},\beta=\frac{1}{n}$. In the extreme case, when $n=1$,
this process can be interpreted as a special case of IMM which merges several
models into a single one. When $n=K$, this process can be interpreted as a
special case of ensemble methods since there are $K$ set of parameters without
being merged.

$q_{1:K}=\sum_{j}^{K}\alpha q_{j}\approx q_{1:n}=\sum_{j}^{n}\beta q_{j}\text{
where }n<=K$ (2)

To obtain the final posterior distribution $q_{1:n}$ and restrict the sudden
increase in the number of parameters, we approximate the $q_{1:K}$ by using a
Gaussian Mixture Model (GMM) [31] with EM algorithm to get $q^{*}_{1:K}$. We
then remove the redundant distributions in $q^{*}_{1:K}$.

The EM algorithm contains an Estimation step (E-step) and a Maximisation step
(M-step). For each weight, we first sample $N$ data points $x_{[1:N]}$ from
the posterior mixture distribution and initialise a GMM model with $K$
components. Then, the E-step estimates the probability of each data point
generated from each $K$ random Gaussian distribution, see Equation (3). For
the $i_{th}$ data point $x_{i}$, we assume that it is generated from the
$m_{th}$ Gaussian distribution and calculate the probability $\pi_{m}$. We can
obtain a matrix of membership weights after applying Equation (3) to each data
point and determine the mixture of Gaussian distributions. The M-step modifies
the parameters of these $K$ random Gaussian distributions by maximising the
likelihood according to the weights generated from the first step; see
Equation (4).

$\pi_{m}(x_{i})=\frac{\alpha_{m}\mathbb{N}(x_{i}|\mu_{m},\Sigma_{m})}{\sum_{j=1}^{K}\alpha_{j}\mathbb{N}(x_{i}|\mu_{j},\Sigma_{j})}$
(3)

$\begin{split}\alpha_{j}&=\frac{1}{N}\sum_{i=1}^{N}\pi_{m}(x_{i}),\\\
\mu_{j}&=\frac{\sum_{i=1}^{N}\pi_{m}(x_{i})x_{i}}{\sum_{i=1}^{N}\pi_{m}(x_{i})},\\\
\Sigma&=\frac{\sum_{i=1}^{N}\pi_{m}(x_{i})(x_{i}-\mu_{j})(x_{i}-\mu_{j})^{T}}{\sum_{i=1}^{N}\pi_{m}(x_{i})}\end{split}$
(4)

After the algorithm is converged, we can obtain an approximated posterior
mixture distribution $q^{*}_{1:K}=\sum_{j}^{K}\alpha^{*}q^{*}_{j}$, where
$\sum_{j}^{K}\alpha_{j}^{*}=1$. We then remove $q^{*}_{j}$, if
$\alpha_{j}^{*}$ is smaller than a threshold which is set to $t=\frac{1}{2K}$.
These distributions can be regarded as redundant components which overfit the
model. Since the EM algorithm clusters similar data points into one cluster,
we can merge the distributions if they are similar to each other and get the
final posterior mixture distribution $q_{1:n}$. We use the trained GMM to
cluster the mean value of each component in $q_{1:K}$. If the mean values of
two distributions are in the same cluster, these two distributions are merged
into a single Gaussian distribution. The training and merging processes are
recursive. In other words, the model saves information about learned mixture
distributions after learning several tasks. When there is a new task, the
model learns the new tasks and then merges the new distributions with existing
Gaussian mixture distributions.

###  II-C Testing Process

After the training and merging processes, we obtain sets of parameters to
construct the mixture posterior distribution with several components. Since
the CBLN contains the information from different learned tasks, we need to
identify which information is suitable to give a prediction in the test state.
We obtain several Monte Carlo samples of the weights drawn from the
variational posterior to determine the uncertainty. For this purpose, we
calculate the variance of the predictive scores. The set of parameters which
has minimal uncertainty is chosen to give the final prediction. We use the
epistemic uncertainty in our calculation. There are also other uncertainty
measurements such as computing the entropy of the predictive scores [32] or
Model Uncertainty as Measured by Mutual Information (MUMMI) [33], see Equation
(5). The trade-off between these uncertainty measures is discussed in Section
IV.

$\begin{split}\textbf{Entropy}&=\mathbb{H}[y^{*}|x,D]\\\
\textbf{MUMMI}&=\mathbb{H}[y^{*}|x,D]+\mathbb{E}_{q(w^{(i)}|\theta)}[\mathbb{H}[y^{*}|x,w^{(i)}]]\end{split}$
(5)

Where $y^{*}$ is the predicted distribution, $q(\textbf{w}^{(i)}|\theta)$ is
the variational posterior distribution, and $x$ is the test input.

##  III Experiments

We evaluated our method on the MNIST [34] image datasets and the UCR Two
Patterns time-series dataset [35]. MNIST is a handwritten dataset consist of
10 digits. UCR is an archive that contains batches of time-series datasets.
The MNIST and Two Patterns contain 60000 and 1000 training samples, 10000 and
4000 test samples, 10 and 4 classes respectively.

In our experiments, we do not re-access the samples after the first training
but let the model know that it needs to train for a new task. However, the
difference in our method compared with the existing works is that we do not
tell the model which task is being tried. Furthermore, the output nodes refer
to the appropriate number of classes that the task is trained for. The overlap
between the output classes, which are trained at different times, are also
taken into consideration. This means that at the time of the training for each
task, we do not know which other tasks the new samples could also be
associated with. The settings in our experiments are similar to [14], which is
more strict than other settings in the existing works. For example, in
contrast to our experiments, the other existing experiments are allowed to re-
access the training samples [17], tell the model which task is the test data
comes from [24], or use different classifiers for different tasks [15].

In CBLN, we randomly choose 200 test data from the test task and draw 200
Monte Carlo samples from the posterior distribution and measured the
uncertainty to decide which parameters should be used in the model for each
particular task.

We compare our model with state-of-the-art methods including Neural Networks
(NN), Incremental Moment Matching (IMM) [14] and Synaptic Intelligence (SI)
[13]. In the IMM model, we perform all the IMM algorithms combining with all
the transfer techniques mentioned in [14]. We also search for the best hyper-
parameters and choose the best accuracy according to [14]. In the SI model, we
search the best hyper-parameters as well. For the SI, Multiple-Head (MH)
approach is used in the original paper. The MH approach is used to divide the
output layer into several sections. For different tasks, each section will be
activated in which the overlap between different classes in different tasks is
also avoided. MH approach requires the model to be told about the test tasks.
We perform our evaluation based on the SI approach with and without using the
MH approach. In CBLN, we search for the best model that can distinguish the
test data. Since CBLN is based on BNN, we use the BNN as a baseline for
comparison.

###  III-A Split MNIST

The first experiment is based on the split MNIST dataset. This experiment is
to evaluate the ability of the model to learn new tasks continually. In this
experiment, we split the MNIST dataset into several sub-sets; e.g. when the
number of tasks is one, the networks are trained on the original MNIST; when
the number of tasks is two, the network is trained on the digits 0 to 4, 5 to
9 sequentially. The rest of the process follows the same manner. As the number
of tasks increasing, we keep splitting the MNIST into multiple sub-sets. To
implement the other methods, we follow the optimal architecture described in
the original papers. In IMM, we use two hidden layers with 800 neurons each.
In SI and NN, we use two hidden layers with 250 neurons each. We use a BNN,
which contains two hidden layers with 25 neurons in each layer. The total
number of parameters in the BNN is 41070 (the number of parameters in BNNs is
doubled). In CBLN, we use two hidden layers with only ten neurons each. To
evaluate the performance, we compute the average of test accuracy on all the
tasks.

(a) Test Accuracy

(b) Number of Parameters

(c) Uncertainty

Figure 2: Split Mnist Experiment. (a) Average of test accuracy of all the
tasks. (b) The number of parameters in CBLN before and after the merging
process. (c) Uncertainty of the model on the test tasks when the number of
tasks is set to 5.

As shown in Figure 2a, the average test accuracy of all the tasks in CBLN
keeps increasing, while the performance of other methods decreases over time.
As long as we divide the MNIST model training into several simpler tasks, the
performance of CBLN keeps increasing since CBLN can learn the new tasks
without forgetting the previously learned ones. The accuracy after training
five tasks sequentially reaches the performance of SI with the MH approach.
Shown in Figure 2a, the method using the MH approach avoids the interference
between the tasks with different classes at the output layer (i.e. by
interference we mean the situation in which learning a new task causes
changing the parameters in a way that the model forgets the previously learned
ones). However, we need to tell the model which task the test data refers to
in both training and test processes. The grey line in Figure 2a represents the
accuracy of training a BNN with the original MNIST. The performance of CBLN
which continually learns five different tasks outperforms the BNN.

The parameters used in CBLN are less than the BNN. Figure 2b illustrates the
number of parameters used in CBLN. The orange line shows the number of
parameters before the merging process. The blue line shows the number of
parameters after the merging process, and the green lines illustrate the
number of merged parameters. The number of parameters used while training five
tasks is 35094, which is significantly lower than the parameters used in other
state-of-the-art methods. The CBLN only doubles the number of parameters
during the experiment (which is 16140 at the beginning). The more tasks are
trained, the more parameters are merged because CBLN finds the similarity
among the solutions for all the tasks and merges them.

Figure 2c illustrates the uncertainty measure in the test process when the
number of tasks is five. In each block, the x-axis shows the prediction score
for that particular task; the y-axis shows the variance. If the density of
highlighted points is close to the lower right corner, the model has low
uncertainty and high prediction score. The blocks shown in the diagonal line
are the results with the lowest level of uncertainty for each particular task.

###  III-B Permuted MNIST

The second experiment is based on the permuted MNIST to evaluate the ability
of the model to learn new tasks incrementally. This experiment is different
from the split MNIST experiment since the number of classes in each task is
always 10. We follow the same setting in the previous work done by Kirkpatrick
et. al, Lee et. al in [12, 14]. The first task is based on the original MNIST.
In the rest of the tasks, we shuffle all the pixels in the images with
different random seeds. Therefore, each task requires a different solution.
However, the difficulty level of all the tasks is similar. In this experiment,
CBLN contains two hidden layers, with each having 50 neurons.

(a) Test Accuracy

(b) Number of Parameters

Figure 3: Permuted MNIST experiment.

(a) Test Accuracy

(b) Number of Parameters

(c) Uncertainty Changes

Figure 4: Time Series Experiment. (a) Average of test accuracy of all the
tasks. (b) The number of parameters in CBLN before and after the merging
process. (c) Uncertainty changes before and after the merging process.

###  III-C Time-Series data

In the last experiment, we use the Two-Patterns dataset from UCR time-series
archive. In this experiment, CBLN uses two hidden layers, each containing 200
neurons. The other methods with two hidden layers, each containing 800 neurons
with Dropout layers. While training the CBLN model with the entire Two-
Patterns dataset, the best accuracy is around 0.8. If we split the dataset
into two parts, the accuracy is above 0.9. The accuracy of CBLN outperforms
other methods by continually learning Two-Patterns dataset divided into
smaller tasks rather than learning it as an entire model.

(a) Task 1

(b) Task 2

(c) Task 1 and Task 2

Figure 5: From split MNIST experiments. Orange points represent the merged
weights. (a) Scatter plot of the weights in Task 1. (b) Scatter plot of the
weights in Task 2. (c) Density of the merged weights for task 1 and 2.

(a) Absolute Difference

(b) Time Cost

(c) Variance. Correct: 4/10

Figure 6: Based on the split MNIST experiment. (a) Absolute difference between
the test accuracy before and after the merging process. The maximum value in
the y-axis is 0.01. (b) The running time of merging and testing process. 10
and 25 are the number of neurons in each layer. (c) Uncertainty of test task
when the number of tasks is ten. Each task contains only one class.

##  IV Discussion

Merged weights: We start the discussion with analysing how the weights are
merged. We visualise the weights in the Split MNIST experiment that was
carried out with two tasks. Shown in Figure 5, the orange points represent the
merged weights. In Figure 5a,5b, the x-axis shows the mean of weights; the
y-axis shows the variance of the weights. Figure 5c shows the density of the
merged parameters. If the mean of the weight distribution is closer to 0, the
weight has a larger chance to be merged because our prior knowledge is a
Gaussian distribution with a mean of 0. For the weights which the mean values
are higher, they have less chance to be merged because these weights can be
regarded as to have larger contributions to finding the solution for the
training tasks. For each task, the solution could be different. Hence these
weights are the distinctions among different tasks.

Sequential Updating: The settings of experiments above are the same as
described in Lee et. al[14]. These experiments inform the model about the
number of tasks is it will perform. The experiments also merge all into a
single model. This strategy can be viewed as training the tasks in parallel.
We perform further evaluations on giving the tasks in sequential order. In
this case, the model learns a new task and merges it with existing knowledge
at each time slot. We train the 5 tasks of split MNIST and Permuted MNIST
sequentially and compare to the parallel learning manner. The results are
shown in Table I.

TABLE I: Comparison the parallel learning with sequential learning. | Parallel | Sequential
---|---|---
Split MNIST | 97.14% | 97.69%
Permuted MNIST | 96.02% | 95.86%

Ablation study: Inspired by [22], we have evaluated our model with and without
the merging process. To evaluate the performance decreases after merging the
models, we calculate the absolute difference of test accuracy before and after
the merging process. Shown in Figure 6a, the absolute difference is almost 0.
Therefore, all the similar parameters have been merged perfectly, and the
distinct parameters are maintained very well. To evaluate the uncertainty
changes before and after the merging process, we track the uncertainty changes
in the time-series experiment. Shown in Figure 4c, the uncertainties are
decreased after the merging process, but it can still help the model to choose
the correct parameters to predict the test data. Furthermore, the merging
process significantly decreases the number of parameters needed to learn a
mode for multiple tasks as shown in Figure 2b,3b,4b. The merging process can
significantly prevent the exponential increase in the number of parameters
required to learn the model without degrading the performance. The main
advantages of CBLN compared with other dynamic methods are choosing suitable
parameters to classify the test samples and preventing an exponential increase
in the number of parameters. We describe Progressive Neural Networks (PNNs) as
an example. In PNNs, if the initial number of parameters in the model is $N$,
after learning $K$ tasks, the number of parameters in the model increases to
$N*K$. The number of parameters needed are shown in orange lines in Figure
2b,3b,4b. Furthermore, the model needs to be informed about which task is
currently given in advance at the test stage.

Complexity: We then evaluate the time complexity of the merging and testing
process with respect to the initial neuron resources. We ran the experiments
on a Macbook Pro (2015) with 2.7 GHz Intel Core i5. Shown in Figure 6b, where
10 represents the CBLN contains two hidden layers with ten neurons each, 25
represents the CBLN contains two hidden layers with 25 neurons each. CBLN is
time-consuming during the test state, especially when the number of trained
tasks grows. To produce the uncertainty measure, the computational complexity
of CBLN is $O(n^{2})$ while the BNNs are $O(n)$ for each test data. We assume
the model does not know in advance which task the test data is associated
with. This means that the model needs to identify and chooses the correct
solution for each test task. This is a key advantage of CBLN compared to other
existing methods that assume the model knows in advance which test task is
being performed; e.g. [24, 15, 13]. The test stage could be the same as a
conventional neural network if we informed the model which task is being
tested. However, in real-world applications, this information is not available
to the model in advance. The CBLN uses the uncertainty measure to choose the
appropriate learned solution for each particular task. The number of tasks
does not have much effect on the merging process. The main effect on the
merging process is the number of parameters of the model at the
initialisation. According to our experiments, we can initialise CBLN with a
much smaller number of parameters to solve a complex task as long as it can
solve it as a set of simpler tasks. Furthermore, CBLN does not need to
evaluate the importance of parameters by measures such as computing Fisher
Information (second derivative of loss function) [14, 12] which are
computationally expensive and intractable in large models. In summary, the
testing process of CBLN is increasing with respect to the number of tasks
having been learned; the merging process of CBLN is increasing with respect to
the initial neuron resources.

Uncertainty measure: In this section, we discuss the epistemic uncertainty
measure that is computed by the model given test data. The CBLN uses epistemic
uncertainty measure to identify the current task form the distribution of
training data. We evaluate the variance, entropy and MUMMI in different
experiments. To see which measurement of uncertainty is suitable to be used in
CBLN for choosing the learned solution, we run each experiment for ten times
and calculate the average selection rate. Shown in Table II, in the permuted
MNIST experiments, although the number of tasks is increasing, the model can
choose the correct solutions. In the split MNIST experiment, the rate of
uncertainty decreases, if the number of tasks increases. In other words, the
model cannot distinguish the tasks that the test data is associated with when
the number of classes in each task decrease. We analyse this as a Rare Class
Scenario in Epistemic Uncertainty (RCSEU). RCSEU means that when the number of
classes in each task is very small, the model will overfit the training data
quickly and will become over-confident with the result of classifying the test
data.

To illustrate the RCSEU, we visualise the uncertainty information in the split
MNIST experiment, when the number of tasks is ten. In Figure 6c, the blue
blocks are the correct solution (in the diagonal line), the green blocks
represent that the model identify the test data correctly and the red blocks
represent that the model identifies the test data incorrectly and the black
blocks represent very small uncertainty.

TABLE II: The average rate of correct selection.  Experiment | Split MNIST | Permuted
---|---|---
Number of Tasks | 2 | 3 | 4 | 5 | 10 | 10
Variance | 1.0 | 1.0 | 0.95 | 0.866 | 0.3 | 1.0
Entropy | 1.0 | 0.8 | 0.7 | 0.736 | 0.29 | 1.0
MUMMI | 1.0 | 0.87 | 0.925 | 0.894 | 0.32 | 1.0

##  V Conclusions

This paper proposes the Continual Bayesian Learning Networks (CBLN) to solve
the forgetting problem in continual learning scenarios. CBLN is based on
Bayesian Neural Networks (BNNs). Different from BNNs, the weights in the CBLN
are mixture Gaussian distributions with an arbitrary number of distributions.
The CBLN can solve a complex task by dividing it into several simpler tasks
and learning each of them sequentially. Since CBLN uses mixture Gaussian
distribution models in its network, the number of additionally required
parameters decreases as the number of tasks increases. The CBLN identifies
which solution should be used for which test data by using an uncertainty
measure. More importantly, our proposed model can overcome the forgetting
problem in learning models without requiring to re-access previously seen
training samples. We have evaluated our method based on MNIST image and UCR
time-series datasets and have compared the results with the state-of-the-art
models. In the split MNIST experiment, our method outperforms the Incremental
Moment Matching (IMM) model by 25%, and the Synaptic Intelligence (SI) model
by 80%. In the permuted MNIST experiment, our method outperforms IMM by 16%
and achieves the same accuracy as the SI model. In the time-Series experiment,
our method outperforms IMM by 40% and the SI model by 47%. The future work
will focus on developing solutions to let the model determine when it needs to
train for a new task given a series of samples. This will be achieved by
analysing the drifts and changes in the distribution of the training data. The
work will also focus on developing methods to group the neurons during the
merging process to construct regional functional areas in the network specific
to a set of similar tasks. This will allow us to reduce the complexity of the
network and create more scalable and generalisable models.

## Acknowledgment

This work is partially supported by the Care Research and Technology Centre at
the UK Dementia Research Institute (UK DRI).

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