# Neuro-mimetic Task-free Unsupervised Online Learning with Continual Self-
Organizing Maps

Hitesh Vaidya
Department of Computer Science and Engineering
University of South Florida
Tampa, FL 33620
<EMAIL_ADDRESS>
&Travis Desell
Department of Software Engineering
Rochester Institute of Technology
Rochester, NY 14623
<EMAIL_ADDRESS>
&Ankur Mali
Department of Computer Science and Engineering
University of South Florida
Tampa, FL 33620
<EMAIL_ADDRESS>
&Alexander Ororbia
Department of Computer Science
Rochester Institute of Technology
Rochester, NY 14623
<EMAIL_ADDRESS>

###### Abstract

An intelligent system capable of continual learning is one that can process
and extract knowledge from potentially infinitely long streams of pattern
vectors. The major challenge that makes crafting such a system difficult is
known as _catastrophic forgetting_ – an agent, such as one based on artificial
neural networks (ANNs), struggles to retain previously acquired knowledge when
learning from new samples. Furthermore, ensuring that knowledge is preserved
for previous tasks becomes more challenging when input is not supplemented
with task boundary information. Although forgetting in the context of ANNs has
been studied extensively, there still exists far less work investigating it in
terms of unsupervised architectures such as the venerable self-organizing map
(SOM), a neural model often used in clustering and dimensionality reduction.
While the internal mechanisms of SOMs could, in principle, yield sparse
representations that improve memory retention, we observe that, when a fixed-
size SOM processes continuous data streams, it experiences concept drift. In
light of this, we propose a generalization of the SOM, the continual SOM
(CSOM), which is capable of online unsupervised learning under a low memory
budget. Our results, on benchmarks including MNIST, Kuzushiji-MNIST, and
Fashion-MNIST, show almost a two times increase in accuracy, and CIFAR-10
demonstrates a state-of-the-art result when tested on (online) unsupervised
class incremental learning setting.

##  1 Introduction

A major hurdle in designing autonomous, continuously learning agents is
ensuring that such systems effectively preserve previously acquired knowledge
when faced with new information. The difficulty that agents face in retaining
information acquired over time is known as catastrophic forgetting (or
interference) [44, 53, 44] and is a significant challenge in the problem
setting known as lifelong or continual learning [60, 10, 43, 46, 47]. In
continual learning, an agent must embed knowledge learned from data in a
sequential manner without compromising prior knowledge. Much as humans do,
this agent should process samples from these sources online, consolidating and
transferring the knowledge acquired over time without forgetting previously
learned tasks.

Notably, catastrophic interference has been investigated with respect to deep
neural networks (DNNs), which are often trained to solve supervised prediction
tasks, especially in efforts such as [29, 38, 57, 46, 47]. However, a relative
dearth of work exists concerning less mainstream neural systems, particularly
those that conduct unsupervised learning such as self-organizing maps (SOMs)
[30]. In this work, we seek to rectify this gap by, first, studying the degree
to which forgetting occurs in a classical model such as the SOM and, second,
developing a generalization of this system, which we call the Continual SOM or
Continual Kohonen map (_CSOM_), which is robust to the interference
encountered in the context of online continual learning.

In essence, SOMs are a type of brain-inspired (or neuro-mimetic) unsupervised
neural system where its neuronal units compete for the right to activate in
the presence of particular input patterns, and the synaptic parameters
associated with the winning unit(s) are adjusted via a form of Hebbian
learning [25, 42, 49]. Notably, the SOM’s units are often arranged in either a
spatial format, e.g., in a Cartesian plane/grid, or in a topological fashion,
e.g., in a neighborhood/field based on the Euclidean distance between the
activation values of units themselves. A useful property of the SOM is that
its topologically-arranged neuronal units effectively learn to construct a
low-dimensional “semantic” map of the more complicated sensory input space,
where similar data patterns are grouped more closely together (around
particular “prototypes”) and farther apart from more dissimilar ones. This
makes SOMs quite useful for clustering and dimensionality reduction tasks [62,
4, 7].

At first glance, it would appear that a competitive learning model such as the
SOM might offer a natural immunity to forgetting since any particular neuronal
unit tends to activate more often than others (by virtue of the distance
function) in response to similar sensory input patterns throughout the
learning process. This specialization would, in principle, result in non-
overlapping, sparse representations, which have been argued to be a key way
for reducing neural cross-talk [58, 48], a source of forgetting [44, 53, 16].
Furthermore, some approaches, such as [18], have advocated for using SOMs in
the continual learning setting. Nevertheless, as uncovered in the experiments
of this paper, the SOM, in its purest form, appears to be prone to forgetting,
thus motivating our particular model generalization and computational
framework [45].

In service of the problem of online continual learning, the core contributions
of this paper can be summarized as:

  * •

We adapt the classical SOM model, which was initially formulated for fitting
to single static datasets, to the online continual learning setting;
specifically, we study its memory retention in tandem with its ability to
adapt to pattern streams. Our experiments show that the SOM experiences
substantial interference across all benchmarks examined.

  * •

We develop a generalization of the SOM, termed the CSOM, which experiences
significantly less forgetting by introducing mechanisms such as specialized
decay functionality and running variance to select the best matching unit
(BMU) for an input at a given time step.

  * •

We present experimental results for four class-incremental datasets, i.e.,
MNIST, Fashion MNIST, Kuzushiji-MNIST, and CIFAR-10, and empirically
demonstrate the robustness of the proposed CSOM to forgetting, yielding a
promising, unsupervised neuromimetic system for the lifelong learning setting.

##  2 Related Work

In this section, we review prior work related to the central topics that drive
this paper: general principles of competitive learning, self-organizing maps,
and continual unsupervised learning.

Competitive Learning. In competitive learning [23], neurons compete with each
other to best match with encountered input sample patterns. This form of
neural computation is often built on top of Hebbian learning [25, 11, 49]
(where an adjustment made to a synapse depends on only local information that
is available to it both spatially and temporally) and is typically used to
perform clustering on input data. Vector quantization [21] and self-organizing
maps (Kohonen maps) [30] are prominent examples within this family of models.
Unlike modern-day DNNs, where all internal neurons participate in every step
of inference and learning, in competitive learning, only the neurons that
satisfy certain criteria “win” the right to compute [58] and update their
connection weight strengths. This form of learning can be useful in
identifying and extracting useful features within a dataset. According to [56,
49], three fundamental elements define competitive learning in general: 1) the
model starts with a set of units that are highly similar, except for some
random noise, which makes each of them respond slightly differently to a set
of inputs; 2) there is a limit to the “strength” of each unit (which motivates
the notion of neighborhood functions, as discussed later); and 3) the units
are allowed to compete in some way for the right to respond to a particular
subset of inputs.

Self-Organizing Maps. The self-organizing map (SOM) [30], and its many
variants [28, 55, 31, 19], is an unsupervised clustering neural model that
adjusts its connection strengths via a Hebbian update rule [25]. During
training, spatially arranged clusters gradually form around the best-matching
neurons within the SOM. This allows the SOM to be useful as a data exploration
tool and even as an effective minimally supervised learner [40], wherein its
internal units represent a summary of the latent patterns found within a
dataset. In contrast to clustering algorithms, such as K-means [15, 37], SOMs
perform a soft clustering over inputs, which means that the connection weight
update for the best matching neuron is the strongest while the updates made to
the others decay/fade gradually as one moves farther away to other neurons
within the best matching neuron’s neighborhood.

SOMs, in the view of this study, make potentially invaluable memory systems of
sensory input patterns given that they iteratively compress their information
into compact parameter vectors (or neural “templates”) as opposed to the raw
data buffers [8, 63] generally used for storing clusters or exemplars of
specific task datasets, much like those often used in many continual learning
methods [24, 38]. Furthermore, as has been shown in prior work [5, 51], the
SOM’s learning process is better equipped to facilitate the capture of the
subtle differences (or variance) across a task’s constituent data patterns.

Continual Unsupervised Learning. One of the applications of continual learning
is to design agents capable of performing intelligent operations on low-
resource or edge-computing devices, such as those generally found in self-
driving cars [36] or robotic control systems. Like many other real-world
sources, self-driving cars generate enormous quantities of data that are often
unlabeled or unannotated. Therefore, the value of unsupervised learning is
relatively high for these kinds of applications. A consequence is that effort
will be required to obtain optimized solutions to tackle the problem of
catastrophic forgetting in unsupervised learning.

Crafting probabilistic generative models [57, 69] that are capable of
synthesizing data samples, which can be used to refresh the memory of task-
specialized neural models (e.g., a classifier/regressor), is one prevalent
approach. [3] employed (neural) encoder models to store the centroids of task
data, which were later used to generate data from previous tasks in order to
induce replay. However, this approach is not computationally feasible for
large quantities of data – storing multiple encoder models per task results in
an increase in memory complexity as well as incurs additional computational
time needed to train each additional new encoder network for every newly
encountered task. This greatly hinders the scalability of such an approach. In
general, rehearsal-based approaches [54, 26], of which some schemes could
technically be considered to fall under the umbrella of unsupervised continual
learning, often store data samples in some form of explicit memory. However,
low-resource devices often do not have a large memory capacity to store
(enough) samples that adequately capture the variance of every task’s
distribution; therefore, this reduces such schemes’ effectiveness given that
explicitly storing more data to facilitate effective (memory-based)
refreshing/retraining of more complex neural predictor model is not feasible
[34].

In this work, we will show that our proposed CSOM can readily and usefully
capture the variance inherent to each task’s dataset (within a sequence) and
yet not need constant, expensive retraining or refreshing itself. Finally,
although out of scope for this paper, we remark that the SOM models we study
could also be made to expand much as in (growing, [67, 69, 68]) dynamical
neural gas models [17, 61]

##  3 Methodology

Notation. We start by defining the notation that will be used throughout this
paper. $\odot$ indicates a Hadamard product, $\cdot$ indicates a matrix/vector
multiplication (or dot product if the two objects it is applied to are vectors
of the same shape). $||\mathbf{v}||_{p}$ is used to represent the $p$-norm
(distance function w.r.t. the difference between two vectors), i.e., $p=2$
selects the $2$-norm or Euclidean distance. $\mathbf{W}[:,i]$ is the slice
operator, meant to extract the $i$th column vector of matrix $\mathbf{W}$;
$\mathbf{W}[i,:]$ is meant to extract the $i$th row. $\cos(\theta)$ indicates
cosine similarity.

Problem Definition. Consider a sequence of $T$ tasks, which we formally denote
by $\mathcal{S}=\bigcup_{k=1}^{T}\mathcal{T}_{k}$. Each task $\mathcal{T}_{k}$
has a training dataset (each containing $C$ classes),
$\mathcal{D}_{train}^{(k)}=\bigcup_{i=1}^{N_{k}}\\{(\mathbf{x}_{i}^{(k)},\mathbf{y}_{i}^{(k)})\\}$,
where $\mathbf{x}_{i}^{(k)}\in\mathcal{R}^{D\times 1}$ is a data pattern and
$\mathbf{y}_{i}^{(k)}\in\\{0,1\\}^{C\times 1}$ is its label vector.
Furthermore, we let $N_{k}$ be the number of patterns in task $T_{k}$ and
$\mathcal{D}_{test}^{(k)}$ to represent the test dataset for task
$\mathcal{T}_{k}$. We remark that, although the datasets investigated in this
work come with labels, our models will never use them since they are
unsupervised. However, we will use the labels for external analysis, as we
will see in our defined metric(s). Finally, all models process each data point
in $\mathcal{D}^{(k)}_{train}$ only once (online adaptation) and time steps
are tracked in simulation within the variable $t$, i.e., each time that a data
point $\mathbf{x}^{(k)}_{i}$ is sampled from $\mathcal{D}^{(k)}_{train}$ of
task $\mathcal{T}_{k}$, $t$ is incremented as $t\leftarrow t+1$.

When a continual learning agent is finished training on task $\mathcal{T}_{k}$
(using $\mathcal{D}_{train}^{(k)}$), the data $\mathcal{D}_{train}^{(k)}$ will
be lost as soon as the model proceeds to task $\mathcal{T}_{k+1}$ (up to
$\mathcal{T}_{T}$). Furthermore, note that $\mathcal{D}_{test}^{(k)}$ is only
used for external model evaluation. The general objective/goal is to maximize
the agent’s generalization performance on task $\mathcal{T}_{k}$ while
minimizing how much its performance degrades on prior tasks $\mathcal{T}_{1}$
to $\mathcal{T}_{k-1}$.

Algorithm 1 The inference and learning processes for the (classical) SOM,
formulated for the online stream setting.

Input: sample $\mathbf{x}^{(k)}_{i}(t)$, synaptic weights $\mathbf{M}$,
topology $\mathcal{G}$, simulation time step $t$
Parameter:
$\lambda,\sigma,\tau_{\lambda},\tau_{\sigma},\lambda_{t=0}\leftarrow\lambda,\sigma_{0}\leftarrow\sigma$

function Update($\mathbf{x}^{(k)}_{i}(t)$, $\mathbf{M}$, $\mathcal{G}$,
$\sigma_{t}$, $\lambda_{t}$) $\triangleright$ Compute L2 distances and perform
weighted updates

$\mathbf{\Delta_{2}}=\mathbf{x}^{(k)}_{i}(t)-\mathbf{M}$,
$\mathbf{d}=||\mathbf{\Delta_{2}}[:,j]||_{2},j=0,1,...H$ $\triangleright$
Compute the BMU, $u$ and its neighbors, $\mathcal{N}_{u}$

$u=\arg\min_{j\in H}\mathbf{\mathbf{d}}$,
$\mathcal{N}_{u}=\textsc{GetNeigh}(u,\mathbf{M},\mathcal{G})$

for $v_{j}\in(u\cup\mathcal{N}_{u})$ do $\triangleright$ Update synapses of
the BMU and its neighbors

$\mathbf{M}[:,v_{j}]\leftarrow\mathbf{M}[:,v_{j}]+\lambda_{t}\phi(u,v_{j},\mathcal{G},\sigma_{t})(\mathbf{\Delta}[:,v_{j}])$

function Train($\mathbf{x}^{(k)}_{i}(t),\mathbf{M},\mathcal{G},t$)
$\triangleright$ Online training routine Train(.), at time step $t$

$\lambda_{t}=\lambda_{0}\exp{(-t/\tau_{\lambda})}$ $\triangleright$ or use
Equation 1

$\sigma_{t}=\sigma_{0}\exp{(-t/\tau_{\sigma})}$ $\triangleright$ or use
Equation 2

$\textsc{Update}(\mathbf{x}^{(k)}_{i}(t),\mathbf{M},\mathcal{G},\sigma_{t},\lambda_{t})$
$\triangleright$ inference step

$t\leftarrow t+1$ $\triangleright$ Advance simulation time forward

###  3.1 The Kohonen Map

In Algorithm 1, we present our formulation of the classical SOM (Kohonen Map)
for the online processing of data points from a (task) stream. Each data point
$\mathbf{x}^{(k)}_{i}$ is sampled from the task $\mathcal{T}_{k}$ (with
dataset $\mathcal{D}^{(k)}_{train}$). The SOM, with $H$ neuronal units, adapts
its synaptic matrix $\mathbf{M}\in\mathcal{R}^{D\times H}$ given input
$\mathbf{x}^{(k)}_{i}$. In this study, the topology $\mathcal{G}$ is a square
grid of dimensions $K\times L$ (i.e., $H=K*L,K=L$).
$\phi(u,v_{j},\mathcal{G},\sigma_{t})$ indicates a neighborhood weighting
function used to scale the update to unit $v_{j}$ with synaptic vector
$\mathbf{M}[:,v_{j}]$. Note that the weighting function depends on the
euclidean distance between the BMU ($u$) and the neighboring units ($v_{j}$)
in the cartesian plane.

Specifically, we see that inference is conducted in the $\textsc{Update}()$
routine of Algorithm 1. The SOM first computes the (Euclidean) distance
between the current sample $\mathbf{x}^{(t)}_{i}$ and all internal units in
$\mathbf{M}$ and then stores these distances in the distance (row) vector
$\mathbf{d}\in\mathcal{R}^{1\times H}$. The BMU is calculated by taking the
argmin over $\mathbf{d}$, returning the index of the BMU as integer $u$111Note
that $u$ further maps to a fixed two-dimensional coordinate $(k_{u},l_{u})$,
where $0<k_{u}\leq K$ and $0<l_{u}\leq L$, since $\mathcal{G}$ is rectangular
in our formulation of SOMs. Finally, the indices/coordinates of the
neighboring units of BMU – any such neighbor is indexed by $v_{j}$ – are
returned by the $\textsc{GetNeigh}()$ sub-routine and stored in the array
$\mathcal{N}_{u}$.

To update the relevant synaptic weight vectors of the SOM’s matrix
$\mathbf{M}$, a loop runs through unit indices $(u\bigcup\mathcal{N}_{u})$ and
a weighted Hebbian update is applied. Note that the Hebbian rule in Algorithm
1 reuses the difference vectors stored in $\mathbf{\Delta}$ and applies both a
dynamic learning rate $\lambda_{t}$ and a coefficient produced by
$\phi(u,v_{j},\mathcal{G},\sigma_{t})$. The neighborhood function
$\phi(u,v_{j},\mathcal{G},\sigma_{t})$ was set to be a Gaussian function,
centered around the BMU indexed by $u$, with radius $\sigma_{t}$. Note that
the classical SOM maintains one global copy of learning rate $\lambda_{t}$ and
radius $\sigma_{t}$ that is shared among all neurons. They are designed to be
dynamic, time-dependent functions and are shown in the $\textsc{Train}()$
routine (which also depicts the full training step performed by our online
SOM). We found, through empirical study, that the following set of equations
yielded better clusters as opposed to original decay functions of [30]:

$\displaystyle\lambda_{t}=\lambda_{0}(1+t*\exp{(t/\tau_{\lambda})})^{-1}$ (1)
$\displaystyle\sigma_{t}=\sigma_{0}(1+t*\exp{(t/\tau_{\sigma})})^{-1}\mbox{.}$
(2)

Figure 1: Illustration of the overall process for best-matching unit (BMU)
selection and synaptic updating of the CSOM for unsupervised online learning.
Algorithm 2 CSOM inference and learning processes.

1:data sample $\mathbf{x}^{(k)}_{i}(t)$, SOM weight matrix $\mathbf{M}$,
running variance matrix $\mathbf{M}^{\omega^{2}}\in\mathcal{R}^{D\times H}$,
radius $\sigma_{v},\sigma_{V}$, learning rate $\lambda_{h}$, BMU count for a
unit ($\eta_{u}$ is count for unit $u$)

2:Note: $\text{HTile}(\mathbf{x},H)$ tiles vector $\mathbf{v}$ $H$ times
horizontally; $\text{VTile}(\mathbf{x},H)$ tiles vector $\mathbf{x}$ $H$ times
vertically,

3: $d(a,b;\mathcal{G})$ is a distance over topology $\mathcal{G}$, where $a$
maps to $(a_{i},a_{j})$ (2D coordinate) and $b$ maps to $(b_{i},b_{j})$

4:function
Update($\mathbf{X}_{i},u,V,\sigma_{u}^{(t)},\lambda_{u}^{(t)},\mathbf{M},\mathbf{M}^{\omega^{2}},\mathcal{G}$)

5: $\delta=(2*\sigma_{u}^{2})^{-1}$

6: $\tau_{1}=-\delta^{-1}*\log{(\frac{10^{-8}}{\lambda_{u}})}$

7: $\mathbf{o}_{1,v_{j}}=\begin{cases}1&d(v_{j},u;\mathcal{G})<\sigma_{u}\\\
0&\text{otherwise}\end{cases}$ $\triangleright$ $v_{j}$ is any non-BMU neuron,
$\mathbf{o}\in\\{0,1\\}^{1\times H}$ (masking vector)

8: $\mathbf{s}_{1,j}=\begin{cases}\lambda_{u}&j=u\\\
\lambda_{v}&j=v_{j}\end{cases}$ $\triangleright$
$\mathbf{s}\in\mathcal{R}^{1\times H}$ learning rate vector

9:
$\phi=\text{VTile}\Big{(}\\{\mathbf{o}_{v_{j}}\mathbf{s}_{v_{j}}\exp{[-d(u,j;\mathcal{G})\delta]},\;\text{for}\;j=1,2,...,H\\},D\Big{)}$
$\triangleright$ $\phi\in\mathcal{R}^{D\times H}$

10: $\mathbf{M}=\mathbf{M}+\phi\odot(\mathbf{X}_{i}-\mathbf{M})$
$\triangleright$ CSOM update step

11: $\eta_{u}=\eta_{u}+1$

12:
$\lambda_{\omega}=\text{VTile}\Big{(}\\{(\lambda^{0}_{\omega}-0.5)+(1+\exp{(-d(u,j;\mathcal{G})/\tau_{1})})^{-1},\;\text{for}\;j=1,2,...,H\\},D\Big{)}$
$\triangleright$ $\lambda_{\omega}\in\mathcal{R}^{D\times H}$

13: $\lambda_{\omega}=\lambda_{\omega}\odot\mathbf{O}+(1-\mathbf{O})$, where
$\mathbf{O}=\text{VTile}(\mathbf{o},D)$ $\triangleright$ scaling factor for
updating running variance

14:
$\mathbf{M}^{\omega^{2}}=\lambda_{\omega}\mathbf{M}^{\omega^{2}}+(1-\lambda_{\omega})(\mathbf{X}_{i}-\mathbf{M})^{2}$
$\triangleright$ update running variance of all neurons

15: $\sigma_{u}^{(t)}=\sigma_{u}^{(t-1)}\exp{(\eta_{u}/\tau_{\sigma})}$
$\triangleright$ $\tau_{\sigma}\leftarrow$ constant

16: $\lambda_{u}^{(t)}=\lambda_{u}^{(t-1)}\exp{(\eta_{u}/\tau_{\lambda})}$
$\triangleright$ $\tau_{\lambda}\leftarrow$ constant

17:function
Train($\mathbf{x}^{(k)}_{i}(t),\mathbf{M},\mathbf{M}^{\omega^{2}},\mathcal{G},t$)

18: $\mathbf{X}_{i}\leftarrow\text{HTile}(\mathbf{x}^{(k)}_{i}(t),H)$
$\triangleright$ tile/repeat $\mathbf{x}^{(t)}_{i}$ $H$ times horizontally to
create $\mathbf{x}^{(k)}_{i}(t)=\mathbf{X}_{i}\in\mathcal{R}^{D\times H}$

19:
$\mathbf{\Delta_{\omega}}=(\mathbf{X}_{i}-\mathbf{M})^{2}/\mathbf{M}^{\omega}$
$\triangleright$ distance calculation specific to CSOM ,
$\mathbf{M}^{\omega}=\sqrt{\mathbf{M}^{\omega^{2}}}$

20: $\mathbf{d}=\Delta_{\omega}[i,j]$ $\triangleright$ $i\in\\{1,2,..K\\}$ and
$j\in\\{1,2,..L\\}$

21: $u=\operatorname*{arg\,min}_{h\in H}\mathbf{d}$ $\triangleright$ BMU
calculation

22:
Update($X_{i},u,V,\sigma_{u}^{(t)},\lambda_{u}^{(t)},\mathbf{M},\mathbf{M}^{\omega^{2}},\mathcal{G}$)
$\triangleright$ $V=H\setminus u$ (i.e., $V$ is set of all non-BMU indices)

###  3.2 The Continual Kohonen Map

In Algorithm 2, we present our proposed model, the continual SOM (CSOM 222All
the notations are summarized in Table 5, Figure 1), built to process and
generalize dynamically from samples drawn from a stream of tasks. Notice that,
first of all, the CSOM now maintains an additional (non-negative) matrix
$\mathbf{M}^{\omega^{2}}\in\mathcal{R}^{D\times H}_{+}$, which contains
synaptic weight parameters associated with the “running variance”
($\omega^{2}$) of each neuronal unit in the system. This means that each
unit/prototype in the CSOM is defined by two vectors of weights – one for
approximate unit means and another for approximate standard deviations. The
intuition is that each unit in our SOM model is generalized to maintain its
own learnable multivariate Gaussian distribution (inspired by the latent
variables of incremental Gaussian mixture models) with a diagonal covariance
matrix. Notice that, in the $\textsc{Update}()$ routine of our CSOM, the
variance parameters are adjusted using a Hebbian-like rule inspired by
Welford’s online algorithm [64] but modified to use the weighting provided by
our CSOM’s neighborhood function. We denote the initial scaling/update factor
for running variance of every unit as $\lambda^{0}_{\omega}$ as shown in step
12 of Algorithm 2. In SOMs, the ratio of weight update that each connection
goes through depends on its distance from the BMU. In CSOM, we empirically
found that the update/scaling factor of running variance of every neuron is
proportional to the magnitude of its weight update. Therefore, this update
factor for a neuron is decayed as per its distance from the BMU to obtain
$\lambda_{\omega}$ in step 12 of Algorithm 2. The running variance vector
coupled to every neuron in the CSOM can be used to generate samples belonging
to the class that matched it. Thus, the underlying structure of our neural
system could be likened to a simple, dynamic generative model.

In addition to the local running variance parameters, the CSOM is designed to
promote a form of neuronal competition driven by unit-centric learning and
distance weighting parameters. Specifically, each neuron $h\in H$ arranged in
topology $\mathcal{G}$ is assigned an independently-controlled, dynamic radius
$\sigma_{h}$ and learning rate $\lambda_{h}$ parameter (as shown in Figure 2).
As shown in Algorithm 2, in the routine $\textsc{Update}()$, we particularly
decay learning parameters only for the BMU ($u$), i.e., only $\sigma_{u}$ and
$\lambda_{u}$ are decayed at time $t$. This localized decay is furthermore a
function of the number of times that unit $u$ has been selected as the BMU,
i.e., unit $u$ adjusts $\sigma_{u}$ and $\lambda_{u}$ as a function of its BMU
count $\eta_{u}$. We do not decay $\sigma_{u}$ and $\lambda_{u}$ after they
reach certain infinitesimally small threshold to ensure their values do not
plummet to 0 and some positive weight update is always achieved as
$t\rightarrow T$. The neighborhood function of the CSOM is also notably a
function of the running variance parameters $\mathbf{M}^{\omega^{2}}$, further
facilitating the calculation of a per-unit region of influence (treating each
neuron as its own weighted multivariate Gaussian distribution). As a result,
whenever a weight update is triggered, synaptic values are adjusted on a
linear scale for the BMU $u$ while, for any non-BMU units $v_{j}$, the update
is adjusted on a scale that decreases as the euclidean distance from $u$
increases. The CSOM utilizes a masking matrix ($\mathbf{o}$) to control the
scale of its synaptic weight updates. The mask matrix also prevents any leaky
weight updates caused by infinitesimally small values passed by the
neighborhood function ($\phi$).

Crucially, the proposed CSOM is a task-free model [43, 2], which means that it
does not require any information about task boundaries (in the form of task
descriptors). In addition, the CSOM, much like our reformulation of the SOM
described earlier, is constructed to process data in an online, iterative
fashion, adapting its per-unit parameters as a function of simulation time.

Figure 2: Unit centric parameter update in CSOM inspired from competitive
learning

##  4 Mathematical Analysis

In this section, we formally show that the CSOM reaches an equilibrium state
or fixed point [22, 59, 9], meaning that it exhibits stability, a quality that
improves its generalization ability as well as helps it avoid catastrophic
forgetting.

Given the variance update rule in the continual self-organizing map (CSOM)
described in the previous section, reproduce below as follows (for an
arbitrary pattern vector sampled at time $t$):

$\omega_{i}^{2}(t+1)=\lambda_{\omega}\cdot\omega_{i}^{2}(t)+(1-\lambda_{\omega})\cdot(\mathbf{x}(t)-\mathbf{M}_{i}(t))^{2},$

where $\lambda_{\omega}\in(0,1)$ is the learning rate, we analyze the
convergence of the series defining $\omega_{i}^{2}(t)$.

### Lemma: Convergence of the Variance Series

###### Lemma 4.1

The series for $\omega_{i}^{2}(t)$ converges under the assumption that the
sequence of squared deviations ${(\mathbf{x}(t)-\mathbf{M}_{i}(t))^{2}}$ is
bounded.

##### Proof:

Expanding the recurrence relation yields the following:

$\omega_{i}^{2}(t+1)=\lambda_{\omega}^{t}\cdot\omega_{i}^{2}(1)+(1-\lambda_{\omega})\sum_{k=0}^{t}\lambda_{\omega}^{k}\cdot\big{(}\mathbf{x}(t-k)-\mathbf{M}_{i}(t-k)\big{)}^{2}.$

Assuming the sequence $\big{(}\mathbf{x}(t)-\mathbf{M}_{i}(t)\big{)}^{2}$ is
bounded by some value $M>0$, we analyze the convergence of the resultant
geometric series:

$\sum_{k=0}^{\infty}\lambda_{\omega}^{k}\cdot(\mathbf{x}(t-k)-\mathbf{M}_{i}(t-k))^{2}.$

Given $0<\lambda_{\omega}<1$, the following series:

$\sum_{k=0}^{\infty}\lambda_{\omega}^{k}$

is a convergent geometric series. By the comparison test, our series converges
since each term is bounded by $\lambda_{\omega}^{k}\cdot M$.

Hence, the series defining $\omega_{i}^{2}(t)$ converges, particularly
indicating that $\omega_{i}^{2}(t)$ stabilizes as $t\to\infty$, contributing
to the SOM’s capability to mitigate catastrophic forgetting by maintaining an
adaptive and stable measure of variance. $\square$

Next, we move to showing the convergence of the synaptic weight vectors of the
proposed CSOM.

###### Proposition 4.2

For the weight vectors $\mathbf{M}_{i}(t)$ in a continual self-organizing map
(CSOM) updated by the rule:

$\mathbf{M}_{i}(t+1)=\mathbf{M}_{i}(t)+\lambda_{i}\cdot\phi(t,u,i)\cdot(\mathbf{x}(t)-\mathbf{M}_{i}(t)),$

where $\lambda_{i}\in(0,1)$ is a fixed learning rate, $u$ is the (integer)
index of the best matching unit, and $\phi(t,u,i)$ is a neighborhood
function333Note that, in Algorithm 2, we set this to be $\phi(t,u,i)=d(u,i)$.,
it holds that

$\|\mathbf{M}_{i}(t+1)-\mathbf{M}_{i}(t)\|\rightarrow 0$

as $t\rightarrow\infty$, indicating the convergence of the model’s synaptic
weight vectors.

##### Proof:

The difference between successive weight vectors is given by:

$\Delta\mathbf{M}_{i}(t)=\mathbf{M}_{i}(t+1)-\mathbf{M}_{i}(t)=\lambda_{i}\cdot\phi(t,u,i)\cdot\big{(}\mathbf{x}(t)-\mathbf{M}_{i}(t)\big{)}.$

To demonstrate convergence, we need to show that
$\|\Delta\mathbf{M}_{i}(t)\|\rightarrow 0$ as $t\rightarrow\infty$.

1\. Bounding the Update Magnitude: Notice that $\|\Delta\mathbf{M}_{i}(t)\|$
is bounded above by
$\lambda_{i}\cdot\|\phi(t,i)\cdot(\mathbf{x}(t)-\mathbf{M}_{i}(t))\|$. Since
$\lambda_{i}\in(0,1)$ and assuming $\phi(t,u,i)$ is designed to decrease over
time and as the distance from the BMU increases, then
$\|\phi(t,u,i)\cdot(\mathbf{x}(t)-\mathbf{M}_{i}(t))\|$ also decreases.

2\. Decrease Over Time: As $t$ increases, the influence of $\phi(t,u,i)$
diminishes, which in turn reduces the magnitude of $\Delta\mathbf{M}_{i}(t)$.
This reduction is due to the adaptation of the weight vectors to the
distribution of the input pattern vectors, leading to smaller corrections
required over time as the neural map stabilizes.

3\. Limiting Behavior: Given that the sequence of input vectors
$\\{\mathbf{x}(t)\\}$ is bounded (a common assumption in many applications of
the SOM), and the learning rate $\lambda_{i}$ is fixed or changes at steady
rate ($\lambda_{i}>0$), the product $\lambda_{i}\cdot\phi(t,u,i)$ ensures that
the updates to $\mathbf{M}_{i}(t)$ decrease in magnitude. Hence,
$\|\Delta\mathbf{M}_{i}(t)\|\rightarrow 0$ as $t\rightarrow\infty$.

Therefore, the synaptic weight vectors $\mathbf{M}_{i}(t)$ converge as
evidenced by the decrease in the magnitude of the updates between successive
time steps, confirming the proposition. $\square$

Next, we show that the CSOM reaches an equilibrium state as well as reaches a
fixed point, indicating its continual adaptation of knowledge:

###### Theorem 4.3

Equilibrium in a continual self-organizing map (CSOM) is achieved through the
calibration of the learning rate values $\lambda_{\omega}$ and $\lambda_{i}$,
alongside the topological considerations enforced by the neighborhood radius
$\sigma_{u}$ as well as the neighborhood function $\phi(t,u,i,\sigma_{u})$.
This calibrated mechanism facilitates the CSOM’s adaptation to new input
patterns while allowing it retain critical information about the distribution
of past inputs, thus mitigating catastrophic forgetting.

##### Proof:

The proof of the above theorem focuses on the stabilization brought about by
these parameters:

1\. Learning Rate Impact: The learning rates $\lambda_{\omega}$ and
$\lambda_{i}$ directly influence the rate of adaptation for variance and
weight vectors, respectively. A careful balance of these rates ensures that
the CSOM network remains sensitive to new inputs without rapidly discarding
historical data.

2\. Neighborhood Radius and Function: The neighborhood radius $\sigma_{u}$,
which typically decreases over time, determines the extent to which the
neighborhood surrounding the BMU is affected by each input pattern. The
neighborhood function $\phi(t,u,i,\sigma_{u})$ modulates the update magnitude
based on $\sigma_{u}$ and the neuron’s distance from the BMU, ensuring a
cohesive topological adaptation across the network.

Mathematical Justification: The equilibrium condition, characterized by
diminishing updates to both variance and weight vectors, is achieved when
$\|\omega_{i}^{2}(t+1)-\omega_{i}^{2}(t)\|\rightarrow 0$ and
$\|\mathbf{M}_{i}(t+1)-\mathbf{M}_{i}(t)\|\rightarrow 0$ as
$t\rightarrow\infty$. The decrease in $\sigma_{u}$ over time reduces the
influence of new inputs on distant neurons, contributing to the network’s
overall stability.

Conclusion: Through the interplay of $\lambda_{\omega}$, $\lambda_{i}$,
$\sigma_{u}$, and $\phi(t,u,i,\sigma_{u})$, the SOM achieves a dynamic
equilibrium. This balance between sensitivity to new data and the retention of
historical patterns allows the CSOM to adaptively map its input space (to
neuronal unit space) while mitigating the risk of catastrophic forgetting.

$\square$

Finally, we formally show that the CSOM and its proposed synaptic update rule
help the model to mitigate forgetting and learn new features without
compromising prior knowledge:

###### Theorem 4.4

Given a sequence of input vectors $\\{\mathbf{x}(t)\\}_{t=1}^{\infty}$ where
$\mathbf{x}(t)\in\mathbb{R}^{D}$, and a continual self-organizing map (CSOM)
with $H$ neurons, each neuron $i$ characterized by a weight vector
$\mathbf{M}_{i}\in\mathbb{R}^{D}$ and a running variance
$\omega_{i}^{2}\in\mathbb{R}$, updated at each time $t$ by:

1\. Variance Update Rule:

$\omega_{i}^{2}(t+1)=\lambda_{\omega}\cdot\omega_{i}^{2}(t)+(1-\lambda_{\omega})\cdot(\mathbf{x}(t)-\mathbf{M}_{i}(t))^{2},$

2\. Synaptic Weight Update Rule:

$\mathbf{M}_{i}(t+1)=\mathbf{M}_{i}(t)+\lambda_{i}\cdot\phi(t,u,i)\cdot(\mathbf{x}(t)-\mathbf{M}_{i}(t)),$

with $\lambda_{\omega},\lambda_{i}\in(0,1)$ as learning rates, and
$\phi(t,u,i)$ as the neighborhood function, then the SOM achieves a dynamic
equilibrium, enhancing its stability and mitigating catastrophic forgetting.

##### Proof:

The proof consists of three key parts:

Part 1: Variance Stability. The exponential decay factor
$(1-\lambda_{\omega})$ ensures that older inputs have a diminishing influence
on the running variance $\omega_{i}^{2}(t)$. For large $t$, this stabilizes,
indicating the SOM’s ability to incorporate new variance information without
losing the significance of past data variance. Mathematically, this can be
shown by analyzing the series convergence of $\omega_{i}^{2}(t)$, which was
demonstrated in Lemma 4.1

Part 2: Convergence of Synaptic Weight Vectors. The weight vectors
$\mathbf{M}_{i}(t)$ are updated towards (the direction of) new inputs with a
learning rate that is modulated by $\phi(t,i)$, ensuring that neurons closer
to the BMU have a higher rate of adaptation. This promotes the stability and
convergence of $\mathbf{M}_{i}(t)$, which is crucial for maintaining knowledge
of learned patterns over time. The convergence can be argued by demonstrating
that the updates lead to a decrease in the difference between successive
weight vectors, $\|\mathbf{M}_{i}(t+1)-\mathbf{M}_{i}(t)\|\rightarrow 0$ as
$t\rightarrow\infty$, as was formally proven in Proposition 4.2.

Part 3: Dynamic Equilibrium and the Mitigation of Catastrophic Forgetting.
Combining the effects of the variance and synaptic weight updates, the CSOM
maintains a balance between adapting to new (sensory) data and preserving
information about the distribution of the past inputs it has encountered. This
equilibrium is achieved through the careful calibration of $\lambda_{\omega}$
and $\lambda_{i}$, alongside the topological considerations enforced by
$\phi(t,u,i)$, as shown in Theorem 4.3.

Thus, this mathematical framework ensures that the CSOM’s neuronal units’
updates are such that neither new nor old information is disproportionately
favored, which mitigates catastrophic forgetting. Therefore, by the specified
update rules and their mathematical properties, the CSOM achieves a dynamic
equilibrium that effectively mitigates catastrophic forgetting. $\square$

##  5 Experiments

To test our proposed CSOM and determine its performance on competitive
datasets, we performed experiments across three databases under two continual
learning setups.

###  5.1 Datasets

To evaluate the CSOM, we employed three grayscale datasets along with one
containing natural image samples.

##### Split MNIST, Fashion MNIST, and KMNIST:

We used variations of MNIST to test our neural models – specifically, the
original MNIST database [33], Fashion MNIST (FMNIST) [66], and Kuzushiji-MNIST
(KMNIST) [12], all containing $28\times 28$ gray-scale pixel images.
Furthermore, we transformed these datasets by normalizing them, i.e., diving
them by $255.0$, which brought them into the value range of $[0.0,1.0]$, thus
making the learning process easier for the SOMs/CSOM models.

##### Split CIFAR-10:

To test the CSOM on images belonging to different (more complex)
distributions, we also utilized the CIFAR-10 dataset [32], which contains a
large number of samples with $32\times 32$ Red-Green-Blue (RGB) images. We
specifically converted the CIFAR-10 images to grayscale to test the neural
models of this study (and defer the use of separate color channels for future
work). Furthermore, we normalized the images as in the MNIST datasets,
bringing their pixel values into the range of $[0.0,1.0]$.

Figure 3: Class-incrementally adapted classical SOM (top row) versus continual
SOM (bottom row) on: MNIST (Left), FashionMNIST (Middle), and KMNIST (Right).

Figure 4: (Left) CSOM containing 100x100 neurons/units trained class
incrementally on grayscale images of CIFAR-10. (Right) Two snapshots of
trained CSOM clusters

##### Model Baselines:

We adapt the Dendritic SOM (DendSOM) [52], further utilizing their label
prediction method to evaluate the performance of our proposed model. The
DendSOM uses a hit matrix to determine the label of a trained unit in the
evaluation phase. We maintain a similar hit matrix (BMU count, $\eta_{u}$)
later used in the evaluation phase for label prediction. This ultimately helps
us to establish a fair comparison between our model and baselines.

###  5.2 Learning Simulation Setups

Class Incremental Learning: In this study, we focus on the continual learning
setup where tasks consist of data points belonging entirely to one single
class. Specifically, this means that for any given task $\mathcal{T}_{t}$, its
training dataset $\mathcal{D}^{(t)}_{train}$ contains all data points labeled
under one specific class $c\in C$ (where $C$ is the total number of unique
classes in the entire benchmark dataset). For instance, for the case of Split-
MNIST, this means that task $\mathcal{T}_{0}$ consists entirely of patterns
with the class label of the digit zero, and task $\mathcal{T}_{1}$ consists
entirely of the digit one, and so on and so forth. Furthermore, note that all
models and baselines studied are trained under the same setup with the same
seeding/shuffling of data points in order to ensure a fair comparison. All
models are trained to process the data points incrementally/online, one sample
at a time, and no data points are ever revisited (meaning that each model is
only allowed one single epoch or pass through an entire task data subset,
effectively simulating the online streaming learning setting [1, 6]).

Domain Incremental Learning:  In this setup we simulated a domain incremental
learning within a given dataset. This is similar to a task incremental setup;
however, here, we divided the datasets into five tasks, where each task had
data samples representing two digits, labeled in binary, i.e., $c\in[0,1]$
($0$ encoded class one, $1$ encoded class two).

###  5.3 Training and Architecture Parameters

We followed a square design for the topology underlying all of the models
examined in this work. This means that every SOM/CSOM had $K\times L$ units in
its topology, ($K=L$). In Table 1, the column for ($K=L$) indicates values of
$K$ for MNIST, Fashion-MNIST, KMNIST, respectively. The initial value of the
running variance for all pixels in every SOM unit ($\omega^{2}_{0}$) was
decided based on the distribution of pixels in the input dataset. We found
empirically that setting $\omega^{2}_{0}$ slightly higher than the variance of
input distribution gave us a good starting point. Similarly, setting
$\lambda^{0}_{\omega}$ within the range $[0.9,0.99]$ helped to maintain an
effective balance between the old and new values of the running variance at
every simulation step. For the DendSOM, we modeled/instantiated four copies of
SOM ($M$) where the unit topology size in each copy was [$14\times 14$]. Since
the DendSOM breaks the input image into patches for every copy of SOM, the
easiest way to divide an MNIST-type of image having dimensions [$28\times 28$]
would be to have its four patches be of shape [$14\times 14$]. We followed
this for the implementations of DendSOM presented in our work. Through a brute
force approach, we found that setting $\tau_{\sigma}=8$ and
$\tau_{\lambda}=45$ performed a gradual decay of $\sigma$ and $\lambda$
throughout training.

Model | ($K=L$) | $\sigma_{0}$ | $\lambda_{0}$ | $\omega^{2}_{0}$ | $\lambda^{0}_{\omega}$
---|---|---|---|---|---
Vanilla SOM | [15, 20, 20] | 0.6 | 0.07 | 1 | 0
DendSOM | [10, 10, 10] | 1.5 | 0.07 | 0 | 0
CSOM | [15/20, 25, 35] | 1.5 | 0.07 | 0.5 | 0.9
Table 1: Parameter setting for the architectures and their training

###  5.4 Evaluation

For model evaluation, we adapted the point-wise mutual information (PMI) [52]
measure, which loosely follows the Hebbian rule of learning to suit the design
logic of CSOM. Formally, the PMI is calculated as follows:

$\displaystyle PMI(l;BMU)=\log{\frac{P(l|BMU)}{P(l)}}$ (3) $\displaystyle
P(l|BMU)=\frac{\eta[l,BMU]}{\Sigma_{i\in Labels}\eta[i,BMU]}$ (4)
$\displaystyle P(l)=\frac{\Sigma_{h\in Units}\eta[l,h]}{\Sigma_{h\in
Units}\Sigma_{i\in Labels}\eta[i,h]}$ (5) $\displaystyle Predicted\
Label=\operatorname*{arg\,max}_{l\in Labels}PMI(l;BMU).$ (6)

The PMI of two entities could take on either a positive or negative value,
depending on whether they co-occur more frequently or less frequently compared
to an independence assumption. The PMI measure can become zero if the two
entities are independent of each other. This PMI measure is used to predict
the label of a trained unit based on a hit matrix (i.e. BMU count, $\eta_{u}$)
which stores the number of times that a unit was selected as the BMU for any
input sample. The PMI uses label information from the input samples to
maintain the hit matrix that is used only at the test time for accuracy
calculation; as a result, the CSOM is unsupervised in training even though its
test-time performance is evaluated through a supervised lens.

Note that we measure the average accuracy of SOM models based on whether the
predicted SOM/CSOM unit’s label matches the input sample’s expected label.
Unlike the L2 distance with running variance used at training time, we
employed the cosine similarity at test time in order to measure the BMU for a
test input sample and to determine its predicted label. Using this method, we
calculated distinct accuracy measures from test samples for each task and
stored these in what is known in the continual learning literature as the
accuracy task matrix (of shape $T\times T$ for $T$ tasks). This helped us to
calculate classical lifelong learning metrics such as average accuracy (ACC)
[50], backward transfer (BWT) [39], as well as the forgetting measure (FM) and
learning accuracy (LA), both of which were mentioned in [70]. Formally, these
evaluation metrics are defined as follows:

$\displaystyle\text{ACC}{\color[rgb]{0,1,0}(\uparrow)}$
$\displaystyle=\frac{1}{T}\Sigma_{i=1}^{T}a_{i,T}$ (higher is better) (7)
$\displaystyle\text{BWT}{\color[rgb]{0,0,1}(\downarrow)}$
$\displaystyle=\frac{1}{T-1}\Sigma_{i=1}^{T-1}a_{T,i}-a_{i,i}$ (lower is
better) (8) $\displaystyle\text{FM}{\color[rgb]{0,0,1}(\downarrow)}$
$\displaystyle=\frac{1}{T}\Sigma_{i=1}^{T}\lvert a_{i,T}-a_{i}^{*}\rvert$
(lower is better) (9) $\displaystyle\text{LA}{\color[rgb]{0,1,0}(\uparrow)}$
$\displaystyle=\frac{1}{T}\Sigma_{i=1}^{T}a_{i,i}$ $\displaystyle\text{(higher
is better)}.$ (10)

In the above equations, $a_{i,j}$ is the accuracy for task $i$ after training
on task $j$. The BWT measure helps us quantify how training on a new task
$\mathcal{T}_{k}$ affects the accuracy of CSOM on the previously trained task
$\mathcal{T}_{k-1}$. In other words, the BWT quantifies the amount of
forgetting for a task as the training progresses incrementally for new tasks
in time for any given algorithm. FM is the measure of the difference between a
model’s final performance and its best performance ($a_{i}^{*}$) for a task.
LA measures the average of the current task accuracy of CSOM.

###  5.5 Simulation Results

In Table 2, we present our full experimental results for the three grayscale
datasets. In addition, we present quantitative benchmark measurements on the
CIFAR-10 dataset in Table 3, comparing to several prior, performant continual
learning neural models.

MNIST | ACC ($\uparrow$) | LA ($\uparrow$) | FM ($\downarrow$) | BWT ($\downarrow$)
---|---|---|---|---
vanilla SOM | 22.89 $\pm$ 0.02 | 46.4 $\pm$ 0.01 | 34.99 $\pm$ 0.01 | -26 $\pm$ 0.04
DendSOM | 17.13 $\pm$ 0.01 | 40.34 $\pm$ 0.03 | 23.33 $\pm$ 0.03 | -25.78 $\pm$ 0.04
CSOM | 85.03 $\pm$ 4.28 | 92.26 $\pm$ 2.49 | 7.23 $\pm$ 2.42 | -8.02 $\pm$ 2.7

FMNIST | ACC ($\uparrow$) | LA ($\uparrow$) | FM ($\downarrow$) | BWT ($\downarrow$)
---|---|---|---|---
vanilla SOM | 32.07 $\pm$ 0.03 | 57.52 $\pm$ 0.03 | 37.01 $\pm$ 0.03 | -28.27 $\pm$ 0.02
DendSOM | 16.62 $\pm$ 0.03 | 27.36 $\pm$ 0.03 | 11.34 $\pm$ 0.03 | -11.92 $\pm$ 0.03
CSOM | 75.13 $\pm$ 2.68 | 88.23 $\pm$ 0.56 | 13.11 $\pm$ 2.93 | -14.56 $\pm$ 3.25

KMNIST | ACC ($\uparrow$) | LA ($\uparrow$) | FM ($\downarrow$) | BWT ($\downarrow$)
---|---|---|---|---
vanilla SOM | 23.65 $\pm$ 0.03 | 54 $\pm$ 0.05 | 35.28 $\pm$ 0.01 | -33.79 $\pm$ 0.02
DendSOM | 10.95 $\pm$ 0.01 | 12.92 $\pm$ 0.01 | 2.02 $\pm$ 0 | -2.18 $\pm$ 0.01
CSOM | 81.602 $\pm$ 2.82 | 88.61 $\pm$ 1.6 | 7.04 $\pm$ 1.54 | -7.78 $\pm$ 1.73

((a)) Unsupervised online (epochs = 1) class incremental training

MNIST | ACC ($\uparrow$) | LA ($\uparrow$) | FM ($\downarrow$) | BWT ($\downarrow$)
---|---|---|---|---
vanilla SOM | 55.03 $\pm$ 0.01 | 80.07 $\pm$ 0.02 | 25.59 $\pm$ 0.02 | -30.71 $\pm$ 0.04
DendSOM | 58.40 $\pm$ 0.03 | 61.37 $\pm$ 0.05 | 7.30 $\pm$ 0.02 | -3.7 $\pm$ 0.04
CSOM | 89.09 $\pm$ 3.32 | 93.02 $\pm$ 2.03 | 4.03 $\pm$ 1.71 | -4.92 $\pm$ 2.11

FMNIST | ACC ($\uparrow$) | LA ($\uparrow$) | FM ($\downarrow$) | BWT ($\downarrow$)
---|---|---|---|---
vanilla SOM | 79.36 $\pm$ 0.06 | 89.45 $\pm$ 0.01 | 12.83 $\pm$ 0.06 | -12.61 $\pm$ 0.06
DendSOM | 50.11 $\pm$ 0 | 53.13 $\pm$ 0.03 | 3.98 $\pm$ 0.03 | -3.76 $\pm$ 0.04
CSOM | 95.61 $\pm$ 0.92 | 97.14 $\pm$ 0.4 | 1.57 $\pm$ 0.89 | -1.92 $\pm$ 1.12

KMNIST | ACC ($\uparrow$) | LA ($\uparrow$) | FM ($\downarrow$) | BWT ($\downarrow$)
---|---|---|---|---
vanilla SOM | 57.18 $\pm$ 0.01 | 78.91 $\pm$ 0.01 | 21.72 $\pm$ 0.01 | -27.16 $\pm$ 0.01
DendSOM | 52.3 $\pm$ 0.02 | 54.55 $\pm$ 0.03 | 3.62 $\pm$ 0.01 | -2.8 $\pm$ 0.02
CSOM | 88.11 $\pm$ 1.93 | 91.97 $\pm$ 1.04 | 3.88 $\pm$ 1.04 | -4.83 $\pm$ 1.27

((b)) Unsupervised online (epochs = 1) domain incremental learning

Table 2: Summary of (n=10 trials; mean and standard deviation of scores) results for class-incremental and domain incremental variant of vanilla SOM, DendSOM, and contSOM. For DendSOM, we created $n=4$ soms with unit size = 14  Cifar-10 Model | ACC ($\uparrow$) | LA ($\uparrow$) | FM ($\downarrow$) | BWT ($\downarrow$)
---|---|---|---|---
Finetune | 10 $\pm$ 0 | 37.49 $\pm$ 0 | 27.83 $\pm$ 0 | -30.54 $\pm$ 0
Replay | 14.84 $\pm$ 1.7E-15 | 87.86 $\pm$ 1.42E-14 | 73.64 $\pm$ 0 | -81.13 $\pm$ 0
LwF | 9.87 $\pm$ 0.17 | 11.37 $\pm$ 1.04 | 1.56 $\pm$ 1.31 | -1.66 $\pm$ 1.36
iCarl | 10 $\pm$ 0 | 90.65 $\pm$ 1.42E-14 | 80.65 $\pm$ 0 | -89.61 $\pm$ 1.42E-14
EWC | 10.11 $\pm$ 0 | 43.93 $\pm$ 7.1E-15 | 35.69 $\pm$ 0 | -37.57 $\pm$ 0
Memo | 11.42 $\pm$ 0.48 | 96.59 $\pm$ 2.22 | 85.17 $\pm$ 1.74 | -94.64 $\pm$ 1.94
Podnet | 12.55 $\pm$ 1.7E-15 | 77.47 $\pm$ 0 | 64.92 $\pm$ 0 | -72.13 $\pm$ 0
BiC | 12.86 $\pm$ 0.72 | 96.77 $\pm$ 0.99 | 83.91 $\pm$ 1 | -93.24 $\pm$ 1.11
SCALE | 29.54 $\pm$ 0.6 | 47.31 $\pm$ 3.21 | 20.23 $\pm$ 2.11 | -19.75 $\pm$ 3.52
CSOM |  31.3 $\pm$ 0.3 | 46.14 $\pm$ 0.48 | 14.92 $\pm$ 0.24 | -16.49 $\pm$ 0.25
Table 3: Summary (trials=10) of online (epochs=1) class incremental training on split-Cifar10 Papers | convolution | buffer | Data | single | class | task
---|---|---|---|---|---|---
|  | storage | Augmentation | pass | labels | boundaries
Finetune | ✓ | ✗ | ✗ | ✗ | ✓ | ✗
Replay | ✓ | ✓ | ✗ | ✗ | ✓ | ✓
LwF [35] | ✓ | ✗ | ✗ | ✗ | ✓ | ✓
iCarl [54] | ✓ | ✓ | ✗ | ✗ | ✓ | ✓
EWC [29] | ✗ | ✗ | ✗ | ✗ | ✓ | ✓
Memo [73] | ✓ | ✓ | ✗ | ✗ | ✓ | ✓
Podnet [14] | {✓, ✗} | ✓ | ✗ | ✗ | ✓ | ✓
BiC [65] | ✓ | ✓ | ✗ | ✗ | ✓ | ✓
SCALE [71] | ✓ | ✓ | ✓ | ✓ | ✗ | ✗
CSOM | ✗ | ✗ | ✗ | ✓ | ✗ | ✗
Table 4: Training schemes followed by all approaches

##### The Classical SOM:

As observed in Table 2, the standard/classical SOM, or “Vanilla SOM” (which
indicates that no special task-driven mechanism was integrated, thus meaning
that we used the online model presented in Algorithm 1), observably forgets
quite strongly across all three benchmarks/datasets. In short, our results
empirically confirm that, indeed, the standard SOM forgets information despite
the potential (for reduced neural cross-talk) offered by its competitive
internal activities. We present visual samples of the memories acquired by our
SOM model(s) in Figure 3, which qualitatively demonstrates the types of
memories these specific models acquire during learning – for datasets such as
Split-MNIST, the SOM only has stored in its internal memories
prototypes/templates that match the last few digits/tasks of the underlying
task sequence, i.e., largely the digit “8” and “9”.

##### The CSOM:

The conventional distance metrics, such as euclidean distance or cosine
similarity, often identify one of the previously trained units as the BMU for
the current incoming task samples due to the semantic overlap present among
different task samples in datasets like MNIST (as explained in section 5.1).
This means that the images from different classes in MNIST-like datasets
appear to be quite similar due to the common black background and white
foreground. This increases the probability of an already trained SOM unit to
be selected as BMU for an input sample from new class instead of an untrained
unit being selected as BMU. Thus, a vanilla SOM fails to use its full capacity
and ultimately does not train on all the available units in a continual
learning setting. The distance function shown in the $\textsc{Train}()$
subroutine of algorithm 2, divides the squared terms of the L2 distance by the
running standard deviation of units obtained from $\mathbf{M}^{\sigma}$. This
helps to preserve the trained units’ memories (with respect to previously
encountered classes) and successfully allocates the untrained units in the SOM
to new incoming tasks. In other words, such a distance metric for identifying
BMU helps to balance the stability and plasticity of the neural system. More
importantly, it eliminates the need for a task vector (refer to Table 4),
which sets up context for the incoming tasks in most modern-day continual
learning models. As a result, the CSOM can learn in a streaming manner without
needing task boundaries which makes it a powerful, unsupervised lifelong
learning system.

Experimentally, we noticed that the Gaussian function used to scale the
synaptic weight updates in a neighbourhood was liable to cause leaky weight
updates in the units that were very far away from the BMU in topology. This
caused an explosion of weight update in the available untrained units, thus
undesirably increasing the size of cluster for current incoming task. To
control this behaviour, we enforced a hard bound over the synaptic weight
update in the neighbourhood using the current $\sigma_{u}$ value of the BMU.
For this we create a binary mask as shown in step 7 of $\textsc{Update}()$
subroutine in Algorithm 2.

In Table 3, we present a comparison of model performance measurements on the
split-CIFAR10 benchmark. We used PyCIL [72] to obtain all the benchmarks for
standard continual learning models except SCALE [71]. For them, we kept the
same setting of total memory size=$2000$ and memory per class=$20$. These were
the standard parameter settings that [72] used to obtain their baseline
results. We conducted $10$ experimental trials for all the benchmarks and,
after every trial, we constructed a task matrix in order to calculate the ACC,
LA, FM, and BWT values for any model. Among all the benchmarks, [71] had the
exact same setting (shown in Table 4) as ours and had state-of-the-art (SOTA)
ACC values in a semi-supervised learning setting. Although it had marginally
higher mean LA value, their standard deviation was almost $7$ times higher
than our CSOM after $10$ trials. Except for EWC [29], all of the models used
convolution and most of them used a storage buffer (memory). Nevertheless, our
CSOM model yielded the best average accuracy among all others without
maintaining any buffer storage or requiring convolution to obtain better
performance, despite operating in an online setting (epochs=$1$, each input
sample processed once).

As shown in Figure 3, as a result of the neural clusters formed, our model
appeared to acquire a good latent representation of every input task. This
means that there is a good amount of variance among the trained units
allocated for every task. However, despite our good experimental results, we
could not obtain an equal number of trained units per class. Moreover, some
neuronal units (especially those at the line of separability of clusters) have
fuzzy representations. This may occur if a neuronal unit receives a weight
update that corresponds to multiple different classes.

Nevertheless, as indicated in Table 2 and 3, the CSOM achieves the best
performance compared to all of the other models/variants of SOM on all four
benchmarks/datasets and successfully beats the SOTA model on split-CIFAR10
dataset. This empirical result indicates that our CSOM framework, leveraging a
competitive learning scheme, can yield the potentially best memory retention
(or greatest reduction in forgetting) when processing classes incrementally
from a data stream. The visualization of the samples synthesized for this
particular model – see Figure 3 – also corroborate this result qualitatively;
the samples, in this case, look the clearest and the model’s internal units
seem to represent most of the individual classes/tasks (including those
presented at the start of the task sequence).

Discussion: The nature of the distance function and the hyperparameter decay
method used in all the SOM models induces an inherent bias towards the
initially encountered classes in the class incremental setting. Specifically,
it assigns a larger cluster or more trained units in the SOM to initial
classes compared to classes that appear later in the incremental order.
Although the proposed CSOM produced the best results out of all of the model
variants, further improvement is certainly possible in order to obtain
clearer/crisper representations of input samples in a trained model final
output of a trained SOM (this will be the subject of future work). By
augmenting the SOM’s internal dynamics with additional synaptic parameters
that maintained the running variance for all of its neuronal units, we show
that the CSOM can further be used as a useful generative model, facilitating a
natural internal memory replay mechanism that could further aid in memory
retention in more complex continual learning setups. Notably, this could be an
important component to improve the performance of memory-augmented neural
networks [13, 20, 27, 41, 59].

##  6 Conclusions

In this paper, we investigated catastrophic forgetting – a phenomenon where a
neural-based intelligent agent forgets the information/knowledge that it
acquired on previous datasets/tasks whenever it starts processing a new
dataset/task – in the context of continual unsupervised learning utilizing the
classical self-organizing map (SOM). Specifically, we designed an adaptation
of the classical model to this setting and empirically found that it exhibited
severe interference and low memory retention. In light of this, we proposed a
novel generalization of the model – the continual SOM (CSOM) – which we
theoretically and experimentally demonstrated, across several data benchmarks,
that it exhibited vastly improved memory retention ability notably through the
use of running variance and decay mechanisms, locally embedded into each
neuronal unit. Future work will include an examination of the integration of
the cross-task memory retention ability of our model into supervised and semi-
supervised neural systems on additional, larger-scale data problems, as well
as further improving the quality of the CSOM’s internally acquired neural
prototypes through additional extensions/generalizations of its local unit-
variance parameters.

## References

  * [1] Agarwal, S., Saradhi, V. V., and Karnick, H. Kernel-based online machine learning and support vector reduction.  In The European Symposium on Artificial Neural Networks (2008).
  * [2] Aljundi, R., Kelchtermans, K., and Tuytelaars, T. Task-free continual learning.  2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) (2018), 11246–11255.
  * [3] Ayub, A., and Wagner, A. {EEC}: Learning to encode and regenerate images for continual learning.  In International Conference on Learning Representations (2021).
  * [4] Bação, F., Lobo, V., and Painho, M. Self-organizing maps as substitutes for k-means clustering.  In Computational Science–ICCS 2005: 5th International Conference, Atlanta, GA, USA, May 22-25, 2005, Proceedings, Part III 5 (2005), Springer, pp. 476–483.
  * [5] Bashivan, P., Schrimpf, M., Ajemian, R., Rish, I., Riemer, M., and Tu, Y. Continual learning with self-organizing maps.  arXiv preprint arXiv:1904.09330 (2019).
  * [6] Beyazit, E., Alagurajah, J., and Wu, X. Online learning from data streams with varying feature spaces.  In Proceedings of the Thirty-Third AAAI Conference on Artificial Intelligence and Thirty-First Innovative Applications of Artificial Intelligence Conference and Ninth AAAI Symposium on Educational Advances in Artificial Intelligence (2019), AAAI’19/IAAI’19/EAAI’19, AAAI Press.
  * [7] Bigdeli, A., Maghsoudi, A., and Ghezelbash, R. Application of self-organizing map (som) and k-means clustering algorithms for portraying geochemical anomaly patterns in moalleman district, ne iran.  Journal of Geochemical Exploration 233 (2022), 106923.
  * [8] Boschini, M., Bonicelli, L., Buzzega, P., Porrello, A., and Calderara, S. Class-incremental continual learning into the extended der-verse.  IEEE Transactions on Pattern Analysis and Machine Intelligence 45, 5 (2023), 5497–5512.
  * [9] Burton, T. Stability by fixed point theory or liapunov’s theory: A comparison.  Fixed point theory 4, 1 (2003), 15–32.
  * [10] Chen, Z., and Liu, B. Lifelong machine learning.  Synthesis Lectures on Artificial Intelligence and Machine Learning 10, 3 (2016), 1–145.
  * [11] Choe, Y. Hebbian Learning.  Springer New York, New York, NY, 2013, pp. 1–5.
  * [12] Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., and Ha, D. Deep learning for classical japanese literature.  ArXiv abs/1812.01718 (2018).
  * [13] Das, S., Giles, C., and Sun, G.-Z. Using prior knowledge in a nnpda to learn context-free languages.  Advances in neural information processing systems 5 (1992).
  * [14] Douillard, A., Cord, M., Ollion, C., Robert, T., and Valle, E. Podnet: Pooled outputs distillation for small-tasks incremental learning.  In Computer Vision – ECCV 2020: 16th European Conference, Glasgow, UK, August 23–28, 2020, Proceedings, Part XX (Berlin, Heidelberg, 2020), Springer-Verlag, p. 86–102.
  * [15] Forgy, E. W. Cluster analysis of multivariate data: efficiency versus interpretability of classifications.  biometrics 21 (1965), 768–769.
  * [16] French, R. M. Catastrophic forgetting in connectionist networks.  Trends in cognitive sciences 3, 4 (1999), 128–135.
  * [17] Fritzke, B. A growing neural gas network learns topologies.  In Proceedings of the 7th International Conference on Neural Information Processing Systems (Cambridge, MA, USA, 1994), NIPS’94, MIT Press, p. 625–632.
  * [18] Gepperth, A., Hecht, T., Lefort, M., and Korner, U. Biologically inspired incremental learning for high-dimensional spaces.  In 2015 Joint IEEE International Conference on Development and Learning and Epigenetic Robotics (ICDL-EpiRob) (2015), pp. 269–275.
  * [19] Gliozzi, V., and Plunkett, K. Self-organizing maps and generalization: an algorithmic description of numerosity and variability effects.  arXiv preprint arXiv:1802.09442 (2018).
  * [20] Graves, A., Wayne, G., and Danihelka, I. Neural turing machines.  arXiv preprint arXiv:1410.5401 (2014).
  * [21] Gray, R. Vector quantization.  IEEE Assp Magazine 1, 2 (1984), 4–29.
  * [22] Haag, R., Kastler, D., and Trych-Pohlmeyer, E. B. Stability and equilibrium states.  Communications in Mathematical Physics 38 (1974), 173–193.
  * [23] Hartono, P. Competitive Learning.  Springer US, Boston, MA, 2012, pp. 671–677.
  * [24] Hayes, T. L., Cahill, N. D., and Kanan, C. Memory efficient experience replay for streaming learning.  In 2019 International Conference on Robotics and Automation (ICRA) (2019), pp. 9769–9776.
  * [25] Hebb, D. O., et al. The organization of behavior, 1949.
  * [26] Isele, D., and Cosgun, A. Selective experience replay for lifelong learning.  In AAAI (2018).
  * [27] Joulin, A., and Mikolov, T. Inferring algorithmic patterns with stack-augmented recurrent nets.  Advances in neural information processing systems 28 (2015).
  * [28] Khacef, L., Rodriguez, L., and Miramond, B. Improving self-organizing maps with unsupervised feature extraction.  In International Conference on Neural Information Processing (2020), Springer, pp. 474–486.
  * [29] Kirkpatrick, J., Pascanu, R., Rabinowitz, N., Veness, J., Desjardins, G., Rusu, A. A., Milan, K., Quan, J., Ramalho, T., Grabska-Barwinska, A., et al. Overcoming catastrophic forgetting in neural networks.  Proceedings of the national academy of sciences 114, 13 (2017), 3521–3526.
  * [30] Kohonen, T. Self-organized formation of topologically correct feature maps.  Biological cybernetics 43, 1 (1982), 59–69.
  * [31] Kopczyński, D. C.-K. E. Non-euclidean self-organizing maps.  arXiv preprint arXiv:2109.11769 (2021).
  * [32] Krizhevsky, A., and Hinton, G. Convolutional deep belief networks on cifar-10.  Unpublished manuscript 40, 7 (2010), 1–9.
  * [33] LeCun, Y., Bottou, L., Bengio, Y., and Haffner, P. Gradient-based learning applied to document recognition.  Proc. IEEE 86 (1998), 2278–2324.
  * [34] Li, X., Wang, S., Sun, J., and Xu, Z. Variational data-free knowledge distillation for continual learning.  IEEE Transactions on Pattern Analysis and Machine Intelligence 45, 10 (2023), 12618–12634.
  * [35] Li, Z., and Hoiem, D. Learning without forgetting.  IEEE Transactions on Pattern Analysis and Machine Intelligence 40, 12 (2018), 2935–2947.
  * [36] Liu, S., Liu, L., Tang, J., Yu, B., Wang, Y., and Shi, W. Edge computing for autonomous driving: Opportunities and challenges.  Proceedings of the IEEE 107, 8 (2019), 1697–1716.
  * [37] Lloyd, S. Least squares quantization in pcm.  IEEE transactions on information theory 28, 2 (1982), 129–137.
  * [38] Lopez-Paz, D., and Ranzato, M. Gradient episodic memory for continual learning.  Advances in neural information processing systems 30 (2017), 6467–6476.
  * [39] Lopez-Paz, D., and Ranzato, M. Gradient episodic memory for continual learning.  In Proceedings of the 31st International Conference on Neural Information Processing Systems (Red Hook, NY, USA, 2017), NIPS’17, Curran Associates Inc., p. 6470–6479.
  * [40] Lyu, Z., Ororbia, A., Li, R., and Desell, T. Minimally supervised learning using topological projections in self-organizing maps.  arXiv preprint arXiv:2401.06923 (2024).
  * [41] Mali, A. A., Ororbia II, A. G., and Giles, C. L. A neural state pushdown automata.  IEEE Transactions on Artificial Intelligence 1, 3 (2020), 193–205.
  * [42] Martinetz, T. Competitive hebbian learning rule forms perfectly topology preserving maps.  In International conference on artificial neural networks (1993), Springer, pp. 427–434.
  * [43] Masana, M., Liu, X., Twardowski, B., Menta, M., Bagdanov, A. D., and van de Weijer, J. Class-incremental learning: Survey and performance evaluation on image classification.  IEEE Transactions on Pattern Analysis and Machine Intelligence 45, 5 (2023), 5513–5533.
  * [44] McCloskey, M., and Cohen, N. J. Catastrophic interference in connectionist networks: The sequential learning problem.  The psychology of learning and motivation 24, 109 (1989), 92.
  * [45] Mermillod, M., Bugaiska, A., and Bonin, P. The stability-plasticity dilemma: investigating the continuum from catastrophic forgetting to age-limited learning effects.  Frontiers in Psychology 4 (2013), 504.
  * [46] Ororbia, A., Mali, A., Giles, C. L., and Kifer, D. Continual learning of recurrent neural networks by locally aligning distributed representations.  IEEE transactions on neural networks and learning systems 31, 10 (2020), 4267–4278.
  * [47] Ororbia, A., Mali, A., Giles, C. L., and Kifer, D. Lifelong neural predictive coding: Learning cumulatively online without forgetting.  Advances in Neural Information Processing Systems 35 (2022), 5867–5881.
  * [48] Ororbia, A. G. Continual competitive memory: A neural system for online task-free lifelong learning.  arXiv preprint arXiv:2106.13300 (2021).
  * [49] Ororbia, A. G. Brain-inspired machine intelligence: A survey of neurobiologically-plausible credit assignment.  arXiv preprint arXiv:2312.09257 (2023).
  * [50] Pham, Q., Liu, C., and Hoi, S. C. H. Continual learning, fast and slow.  IEEE Transactions on Pattern Analysis and Machine Intelligence 46, 1 (2024), 134–149.
  * [51] Pinitas, K., Chavlis, S., and Poirazi, P. Dendritic self-organizing maps for continual learning.  arXiv preprint arXiv:2110.13611 (2021).
  * [52] Pinitas, K., Chavlis, S., and Poirazi, P. Dendritic self-organizing maps for continual learning.  arXiv preprint arXiv:2110.13611 (2021).
  * [53] Ratcliff, R. Connectionist models of recognition memory: constraints imposed by learning and forgetting functions.  Psychological review 97, 2 (1990), 285.
  * [54] Rebuffi, S.-A., Kolesnikov, A., Sperl, G., and Lampert, C. H. icarl: Incremental classifier and representation learning.  In 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2017), pp. 5533–5542.
  * [55] Rougier, N. P., and Detorakis, G. I. Randomized self-organizing map.  Neural Computation 33, 8 (2021), 2241–2273.
  * [56] Rumelhart, D. E., McClelland, J. L., and PDP Research Group, C. Parallel distributed processing: Explorations in the microstructure of cognition, Vol. 1: Foundations.  MIT press, 1986.
  * [57] Shin, H., Lee, J. K., Kim, J., and Kim, J. Continual learning with deep generative replay.  In Proceedings of the 31st International Conference on Neural Information Processing Systems (Red Hook, NY, USA, 2017), NIPS’17, Curran Associates Inc., p. 2994–3003.
  * [58] Srivastava, R. K., Masci, J., Kazerounian, S., Gomez, F., and Schmidhuber, J. Compete to compute.  Advances in neural information processing systems 26 (2013).
  * [59] Stogin, J., Mali, A., and Giles, C. L. A provably stable neural network turing machine with finite precision and time.  Information Sciences 658 (2024), 120034.
  * [60] Thrun, S., and Mitchell, T. M. Lifelong robot learning.  Robotics and autonomous systems 15, 1-2 (1995), 25–46.
  * [61] Ventocilla, E., Martins, R. M., Paulovich, F., and Riveiro, M. Scaling the growing neural gas for visual cluster analysis.  Big Data Research 26 (2021), 100254.
  * [62] Vesanto, J., and Alhoniemi, E. Clustering of the self-organizing map.  IEEE Transactions on neural networks 11, 3 (2000), 586–600.
  * [63] Wang, Z., Shen, L., Duan, T., Suo, Q., Fang, L., Liu, W., and Gao, M. Distributionally robust memory evolution with generalized divergence for continual learning.  IEEE Transactions on Pattern Analysis and Machine Intelligence 45, 12 (2023), 14337–14352.
  * [64] Welford, B. Note on a method for calculating corrected sums of squares and products.  Technometrics 4, 3 (1962), 419–420.
  * [65] Wu, Y., Chen, Y., Wang, L., Ye, Y., Liu, Z., Guo, Y., and Fu, Y. Large scale incremental learning.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) (June 2019).
  * [66] Xiao, H., Rasul, K., and Vollgraf, R. Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms.  ArXiv abs/1708.07747 (2017).
  * [67] Xu, J., Ma, J., Gao, X., and Zhu, Z. Adaptive progressive continual learning.  IEEE Transactions on Pattern Analysis and Machine Intelligence 44, 10 (2022), 6715–6728.
  * [68] Yang, B., Lin, M., Zhang, Y., Liu, B., Liang, X., Ji, R., and Ye, Q. Dynamic support network for few-shot class incremental learning.  IEEE Transactions on Pattern Analysis and Machine Intelligence 45, 3 (2023), 2945–2951.
  * [69] Ye, F., and Bors, A. G. Dynamic self-supervised teacher-student network learning.  IEEE Transactions on Pattern Analysis and Machine Intelligence 45, 5 (2023), 5731–5748.
  * [70] Yin, H., Yang, P., and 0001, P. L. Mitigating forgetting in online continual learning with neuron calibration.  In Advances in Neural Information Processing Systems 34: Annual Conference on Neural Information Processing Systems 2021, NeurIPS 2021, December 6-14, 2021, virtual (2021), M. Ranzato, A. Beygelzimer, Y. N. Dauphin, P. Liang, and J. W. Vaughan, Eds., pp. 10260–10272.
  * [71] Yu, X., Guo, Y., Gao, S., and Simunic, T. Scale: Online self-supervised lifelong learning without prior knowledge.  2023 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW) (2022), 2484–2495.
  * [72] Zhou, D.-W., Wang, F.-Y., Ye, H.-J., and Zhan, D.-C. Pycil: a python toolbox for class-incremental learning.  SCIENCE CHINA Information Sciences 66, 9 (2023), 197101–.
  * [73] Zhou, D.-W., Wang, Q.-W., Ye, H.-J., and Zhan, D.-C. A model or 603 exemplars: Towards memory-efficient class-incremental learning.  In The Eleventh International Conference on Learning Representations (2023).

##  7 Appendix

Revisiting the hyperparameters defined:
Table 5 indicates key notations, symbols and abbreviations used in this paper.

Item | Explanation
---|---
$\mathcal{G}=(K\times L)$ | Network Topology (dimensions of the SOM)
|  NOTE: $K=L$ for CSOM
$D$ | unit/neuron size and size of individual input sample
$\mathbf{M}\in\mathcal{R}^{D\times H}$ | synaptic matrix of SOM
$u$ | Best Matching Unit (BMU)
$v_{j}$ | a non-BMU at location $j$ in CSOM
$V=\\{v_{j}\ |\ 0<j<K^{2}\\}$ | all non-BMU units in CSOM
$H=\\{u\\}\cup V$ | all units in the SOM
$\sigma_{0}$ | initial radius for all neurons
$\lambda_{0}$ | initial learning rate all neurons
$\sigma_{u}$ and $\sigma_{v}$ | radius of BMU and radius of non-BMU neuron
$\lambda_{u}$ and $\lambda_{v}$ | learning rate of BMU and learning rate of non-BMU neuron
$\omega^{2}$ | running variance of a neuron
$\mathbf{M}^{\omega^{2}}\in\mathcal{R}^{D\times H}$ | matrix of running variance of all neurons in SOM
$\sigma_{h}=\\{\sigma_{u}\\}\cup\\{\sigma_{v_{j}}\\}\in\mathcal{R}^{1\times H}$ | radius of all neurons for weight update
$\lambda_{h}=\\{\lambda_{u}\\}\cup\\{\lambda_{v_{j}}\\}\in\mathcal{R}^{1\times H}$ | learning rate for all neurons used in the weight update steps of CSOM
$\lambda^{0}_{\omega}$ | initial scaling/update factor for updating the running variance of every unit
$\lambda_{\omega}$ | adjusted scaling/update factor for updating the running variance of every unit
$\tau_{\sigma}$ | time constant for radius
$\tau_{\lambda}$ | time constant for learning rate
$\mathrm{p}$ | patch size (DendSOM)
$\mathrm{s}$ | stride length (DendSOM)
Table 5: Notations, Symbols, abbreviations used in this paper

Note that the DendSOM benchmarks are our implementation of the model, as the
original source code was not publicly available.

###  7.1 Class Incremental Learning

This section contains information about class incremental settings where each
task size had exactly one class in it.

####  7.1.1 MNIST

Hyperparameters: Table 6 contains the empirically obtained initial
hyperparameters for obtaining the best possible results as described in Table
2(a).

Model | $H$ | $D$ | $\sigma_{0}$ | $\lambda_{0}$ | $\omega^{2}_{0}$ | $\lambda^{0}_{\omega}$ | $\tau_{\sigma}$ | $\tau_{\lambda}$ | $\mathrm{p}$ | $\mathrm{s}$
---|---|---|---|---|---|---|---|---|---|---
Vanilla SOM | $15\times 15$ | 28 $\times$ 28 | 0.6 | 0.07 | 1 | 0.9 | 8 | 45 | - | -
DendSOM | 8 $\times$ 8 | 7 $\times$ [14 $\times$ 14] | 4 | 0.95 | 2 | 0.005 | - | - | 10 | 3
CSOM | $15\times 15$ | $28\times 28$ | 1.5 | 0.07 | 0.5 | 0.9 | 8 | 45 | - | -
Table 6: Hyperparameters for Class Incremental setting on MNIST

Task matrices: We performed 10 trials of CSOM trained on MNIST. Table 7
indicates the resultant accuracy values,

100.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0
---|---|---|---|---|---|---|---|---|---
99.77 $\pm$ 0.12 | 99.89 $\pm$ 0.14 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0
99.17 $\pm$ 0.4 | 99.41 $\pm$ 0.31 | 96.91 $\pm$ 1.2 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0
98.83 $\pm$ 0.54 | 99.23 $\pm$ 0.31 | 93.31 $\pm$ 2.98 | 96.01 $\pm$ 1.41 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0
98.78 $\pm$ 0.52 | 99.22 $\pm$ 0.32 | 92.26 $\pm$ 3.38 | 94.61 $\pm$ 1.8 | 96.34 $\pm$ 1.02 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0
96.78 $\pm$ 3.13 | 99.07 $\pm$ 0.43 | 92.12 $\pm$ 3.39 | 84.99 $\pm$ 9.97 | 96.05 $\pm$ 1.02 | 88.98 $\pm$ 5.21 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0
95.96 $\pm$ 3.0 | 98.5 $\pm$ 1.22 | 89.5 $\pm$ 6.05 | 84.52 $\pm$ 10.0 | 95.39 $\pm$ 1.19 | 86.19 $\pm$ 6.37 | 94.8 $\pm$ 2.46 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0
95.93 $\pm$ 2.99 | 98.5 $\pm$ 1.22 | 88.33 $\pm$ 6.72 | 83.39 $\pm$ 10.11 | 93.3 $\pm$ 2.96 | 85.32 $\pm$ 6.34 | 94.75 $\pm$ 2.48 | 92.54 $\pm$ 1.59 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0
95.93 $\pm$ 3.03 | 98.41 $\pm$ 1.19 | 87.09 $\pm$ 6.55 | 81.73 $\pm$ 9.47 | 93.05 $\pm$ 3.07 | 80.09 $\pm$ 9.43 | 93.6 $\pm$ 4.35 | 92.42 $\pm$ 1.6 | 84.72 $\pm$ 2.25 | 0.0 $\pm$ 0.0
95.89 $\pm$ 3.02 | 98.41 $\pm$ 1.19 | 87.14 $\pm$ 6.55 | 81.39 $\pm$ 9.33 | 79.17 $\pm$ 8.47 | 79.53 $\pm$ 9.35 | 93.36 $\pm$ 4.28 | 80.64 $\pm$ 7.77 | 82.43 $\pm$ 3.17 | 72.37 $\pm$ 14.48

Table 7: mean and standard deviation of accuracies in a task matrix

####  7.1.2 KMNIST

Hyperparameters: Table 8 shows the hyperparameters we used for obtaining the
results in Table 2(a) on KMNIST dataset.

Model | $H$ | $D$ | $\sigma_{0}$ | $\lambda_{0}$ | $\omega^{2}_{0}$ | $\lambda^{0}_{\omega}$ | $\tau_{\sigma}$ | $\tau_{\lambda}$ | $\mathrm{p}$ | $\mathrm{s}$
---|---|---|---|---|---|---|---|---|---|---
Vanilla SOM | $20\times 20$ | 28 $\times$ 28 | 0.6 | 0.07 | 1 | 0.9 | 8 | 45 | - | -
DendSOM | 10 $\times$ 10 | 7 $\times$ [14 $\times$ 14] | 6 | 0.95 | 2 | 0.005 | - | - | 4 | 2
CSOM | $35\times 35$ | $28\times 28$ | 1.5 | 0.07 | 0.5 | 0.9 | 8 | 45 | - | -
Table 8: Hyperparameters for Class Incremental setting on KMNIST

Task matrices: We performed 10 trials of CSOM trained on KMNIST. Table 9
indicates the resultant accuracy values,

100.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0
---|---|---|---|---|---|---|---|---|---
98.08 $\pm$ 0.85 | 98.72 $\pm$ 0.71 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0
97.86 $\pm$ 0.71 | 88.0 $\pm$ 2.71 | 93.25 $\pm$ 1.14 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0
97.44 $\pm$ 0.82 | 88.06 $\pm$ 2.81 | 87.59 $\pm$ 1.93 | 94.2 $\pm$ 1.67 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0
93.98 $\pm$ 1.22 | 86.21 $\pm$ 3.18 | 86.63 $\pm$ 1.83 | 93.53 $\pm$ 1.66 | 82.79 $\pm$ 1.95 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0
89.3 $\pm$ 3.37 | 85.93 $\pm$ 3.2 | 84.0 $\pm$ 3.27 | 91.73 $\pm$ 2.03 | 81.15 $\pm$ 3.03 | 82.6 $\pm$ 3.57 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0
88.97 $\pm$ 3.34 | 83.55 $\pm$ 3.24 | 79.92 $\pm$ 3.71 | 91.27 $\pm$ 2.14 | 79.93 $\pm$ 3.2 | 80.48 $\pm$ 3.16 | 85.91 $\pm$ 3.51 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0
86.32 $\pm$ 3.14 | 83.37 $\pm$ 3.24 | 79.42 $\pm$ 3.74 | 91.16 $\pm$ 2.19 | 79.43 $\pm$ 3.17 | 80.33 $\pm$ 3.13 | 85.68 $\pm$ 3.49 | 89.19 $\pm$ 1.55 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0
83.86 $\pm$ 3.45 | 81.31 $\pm$ 3.21 | 77.98 $\pm$ 3.72 | 90.45 $\pm$ 2.68 | 77.59 $\pm$ 3.58 | 79.34 $\pm$ 3.38 | 83.33 $\pm$ 4.9 | 87.31 $\pm$ 2.63 | 80.75 $\pm$ 4.35 | 0.0 $\pm$ 0.0
83.75 $\pm$ 3.41 | 80.35 $\pm$ 3.18 | 77.21 $\pm$ 3.98 | 90.39 $\pm$ 2.72 | 75.72 $\pm$ 4.39 | 79.28 $\pm$ 3.41 | 83.25 $\pm$ 4.92 | 86.62 $\pm$ 2.2 | 80.79 $\pm$ 4.36 | 78.66 $\pm$ 4.79

Table 9: mean and standard deviation of accuracies in a task matrix

####  7.1.3 Fashion-MNIST

Hyperparameters: Table 10 indicates the hyperparameters used for obtaining
results in Table 2(a) on Fashion-MNIST dataset.

Model | $H$ | $D$ | $\sigma_{0}$ | $\lambda_{0}$ | $\omega^{2}_{0}$ | $\lambda^{0}_{\omega}$ | $\tau_{\sigma}$ | $\tau_{\lambda}$ | $\mathrm{p}$ | $\mathrm{s}$
---|---|---|---|---|---|---|---|---|---|---
Vanilla SOM | $20\times 20$ | 28 $\times$ 28 | 0.6 | 0.07 | 1 | 0.9 | 8 | 45 | - | -
DendSOM | 10 $\times$ 10 | 7 $\times$ [14 $\times$ 14] | 5 | 0.95 | 2 | 0.005 | - | - | 8 | 4
CSOM | $25\times 25$ | $28\times 28$ | 1.5 | 0.07 | 0.5 | 0.9 | 8 | 45 | - | -
Table 10: Hyperparameters for Class Incremental setting on Fashion-MNIST

100.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0
---|---|---|---|---|---|---|---|---|---
97.39 $\pm$ 1.49 | 97.7 $\pm$ 0.59 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0
92.9 $\pm$ 1.91 | 96.72 $\pm$ 0.82 | 94.32 $\pm$ 0.71 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0
85.83 $\pm$ 3.25 | 88.44 $\pm$ 6.73 | 93.67 $\pm$ 0.72 | 89.49 $\pm$ 3.6 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0
85.71 $\pm$ 2.99 | 88.34 $\pm$ 6.68 | 73.42 $\pm$ 4.09 | 83.11 $\pm$ 4.61 | 76.2 $\pm$ 3.7 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0
85.64 $\pm$ 3.03 | 88.34 $\pm$ 6.68 | 73.49 $\pm$ 4.08 | 83.07 $\pm$ 4.58 | 75.93 $\pm$ 3.79 | 99.76 $\pm$ 0.11 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0
77.5 $\pm$ 3.31 | 88.34 $\pm$ 6.69 | 68.65 $\pm$ 3.46 | 82.88 $\pm$ 4.54 | 72.45 $\pm$ 3.56 | 99.6 $\pm$ 0.39 | 41.62 $\pm$ 2.74 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0
77.51 $\pm$ 3.32 | 88.34 $\pm$ 6.69 | 68.65 $\pm$ 3.48 | 82.88 $\pm$ 4.54 | 72.45 $\pm$ 3.56 | 66.87 $\pm$ 13.25 | 41.55 $\pm$ 2.73 | 95.77 $\pm$ 3.74 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0
77.24 $\pm$ 3.24 | 88.33 $\pm$ 6.68 | 68.57 $\pm$ 3.48 | 82.62 $\pm$ 4.61 | 72.11 $\pm$ 3.42 | 66.83 $\pm$ 13.1 | 39.95 $\pm$ 2.75 | 90.12 $\pm$ 9.74 | 94.84 $\pm$ 0.94 | 0.0 $\pm$ 0.0
77.23 $\pm$ 3.21 | 88.34 $\pm$ 6.66 | 68.53 $\pm$ 3.47 | 82.6 $\pm$ 4.59 | 72.08 $\pm$ 3.4 | 58.84 $\pm$ 14.3 | 39.87 $\pm$ 2.69 | 76.89 $\pm$ 11.12 | 94.28 $\pm$ 1.19 | 92.65 $\pm$ 2.89

Table 11: mean and standard deviation of accuracies in a task matrix

####  7.1.4 CIFAR10

Hyperparameters: Table 12 indicates the hyperparameters used to obtain results
in Table 3 the on split-CIFAR10 dataset.

Model | $H$ | $D$ | $\sigma_{0}$ | $\lambda_{0}$ | $\omega^{2}_{0}$ | $\lambda^{0}_{\omega}$ | $\tau_{\sigma}$ | $\tau_{\lambda}$
---|---|---|---|---|---|---|---|---
CSOM | $100\times 100$ | $32\times 32$ | 1.5 | 0.2 | 0.6 | 0.9 | 6 | 45
Table 12: Hyperparameters for CSOM for Class Incremental setting on CIFAR10

Table 13 indicates hyperparameters set while implementing models from [72] in
a Class Incremental Online Learning setting.

Papers | Memory Size | Memory per Class | Model | epochs | Fixed Memory
---|---|---|---|---|---
Finetune, Replay, LwF, | 2000 | 20 | Resnet32 | 1 | ✗
iCarl, EWC, Memo, Podnet, BiC |  |  |  |  |
Table 13: Hyperparameters for benchmarks from PyCIL [72] on CIFAR10 Paper | Model | Data setting | lr | epochs | memory size
---|---|---|---|---|---
SCALE | Resnet18 | seq | 0.03 | 1 | 256
Table 14: Hyperparameters for benchmarks from SCALE [71] using SGD optimizer

Apart from the hyperparameters mentioned in Tables 13 and 14, the rest of the
hyperparameters were unchanged in their publicly available code. We added an
evaluation code for obtaining task matrices for all the models in Table 3.

Task matrices: We performed 8 trials of CSOM trained on CIFAR-10. Following
resultant accuracies values.

100.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0
---|---|---|---|---|---|---|---|---|---
84.2 $\pm$ 1.55 | 76.88 $\pm$ 1.22 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0
65.86 $\pm$ 1.37 | 73.09 $\pm$ 1.15 | 52.61 $\pm$ 2.5 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0
62.0 $\pm$ 1.42 | 67.05 $\pm$ 1.07 | 41.47 $\pm$ 2.12 | 46.31 $\pm$ 1.57 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0
55.48 $\pm$ 1.41 | 65.26 $\pm$ 1.14 | 36.68 $\pm$ 1.92 | 43.8 $\pm$ 1.55 | 22.42 $\pm$ 1.16 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0
53.9 $\pm$ 1.59 | 63.2 $\pm$ 0.91 | 32.28 $\pm$ 2.28 | 32.22 $\pm$ 1.68 | 22.2 $\pm$ 0.42 | 31.49 $\pm$ 1.44 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0
52.76 $\pm$ 1.34 | 61.82 $\pm$ 0.7 | 30.75 $\pm$ 2.13 | 29.7 $\pm$ 1.89 | 21.41 $\pm$ 0.83 | 31.01 $\pm$ 1.5 | 20.24 $\pm$ 1.21 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0
50.96 $\pm$ 0.77 | 59.95 $\pm$ 1.06 | 29.02 $\pm$ 1.74 | 28.08 $\pm$ 1.74 | 19.91 $\pm$ 0.96 | 29.31 $\pm$ 1.66 | 19.79 $\pm$ 1.17 | 37.3 $\pm$ 1.09 | 0.0 $\pm$ 0.0 | 0.0 $\pm$ 0.0
42.49 $\pm$ 1.07 | 53.13 $\pm$ 1.29 | 27.98 $\pm$ 1.74 | 27.06 $\pm$ 1.25 | 19.11 $\pm$ 1.05 | 28.64 $\pm$ 1.48 | 19.32 $\pm$ 1.05 | 37.14 $\pm$ 0.99 | 38.04 $\pm$ 1.61 | 0.0 $\pm$ 0.0
41.2 $\pm$ 1.11 | 42.96 $\pm$ 1.1 | 27.71 $\pm$ 1.59 | 26.19 $\pm$ 1.15 | 18.74 $\pm$ 1.12 | 28.51 $\pm$ 1.18 | 18.88 $\pm$ 0.96 | 36.25 $\pm$ 1.04 | 36.36 $\pm$ 1.65 | 36.01 $\pm$ 0.9

Table 15: mean and standard deviation of accuracies in a task matrix

###  7.2 Domain Incremental Learning

For the domain incremental setting, we set number of tasks = 5, number of
classes/task = 2

####  7.2.1 MNIST

Hyperparameters: Table 16 contains the initial hyperparameters used for
obtaining the results shown in Table 2(b).

Model | $H$ | $D$ | $\sigma_{0}$ | $\lambda_{0}$ | $\omega^{2}_{h}$ | $\lambda^{0}_{\omega}$ | $\tau_{\sigma}$ | $\tau_{\lambda}$ | $\mathrm{p}$ | $\mathrm{s}$
---|---|---|---|---|---|---|---|---|---|---
Vanilla SOM | $15\times 15$ | 28 $\times$ 28 | 0.6 | 0.07 | 1 | 0.9 | 8 | 45 | - | -
DendSOM | 8 $\times$ 8 | 7 $\times$ [14 $\times$ 14] | 4 | 0.95 | 2 | 0.005 | - | - | 10 | 3
CSOM | $15\times 15$ | $28\times 28$ | 1.5 | 0.07 | 0.5 | 0.9 | 8 | 45 | - | -
Table 16: Hyperparameters for Domain Incremental setting on MNIST

Task matrices: Table 17 indicates the resultant accuracy values obtained after
performing 10 trials.

99.85 $\pm$ 0.04 | 45.59 $\pm$ 5.94 | 45.63 $\pm$ 3.38 | 68.99 $\pm$ 4.3 | 39.82 $\pm$ 5.51
---|---|---|---|---
99.04 $\pm$ 1.04 | 93.57 $\pm$ 2.13 | 61.77 $\pm$ 3.29 | 73.77 $\pm$ 10.54 | 45.9 $\pm$ 4.1
97.55 $\pm$ 1.65 | 93.25 $\pm$ 2.75 | 93.88 $\pm$ 2.96 | 55.48 $\pm$ 7.08 | 18.41 $\pm$ 3.85
98.1 $\pm$ 1.65 | 92.6 $\pm$ 2.78 | 90.52 $\pm$ 5.21 | 96.94 $\pm$ 0.97 | 29.71 $\pm$ 4.53
97.99 $\pm$ 1.65 | 91.85 $\pm$ 2.77 | 77.5 $\pm$ 9.88 | 97.2 $\pm$ 0.87 | 80.92 $\pm$ 6.1
Table 17: mean and standard deviation of accuracies in a task matrix

####  7.2.2 KMNIST

Hyperparameters: Table 18 contains the initial hyperparameters used for
obtaining the results shown in Table 2(b).

Model | $H$ | $D$ | $\sigma_{0}$ | $\lambda_{0}$ | $\omega^{2}_{h}$ | $\lambda^{0}_{\omega}$ | $\tau_{\sigma}$ | $\tau_{\lambda}$ | $\mathrm{p}$ | $\mathrm{s}$
---|---|---|---|---|---|---|---|---|---|---
Vanilla SOM | $20\times 20$ | 28 $\times$ 28 | 0.6 | 0.07 | 1 | 0.9 | 8 | 45 | - | -
DendSOM | 12 $\times$ 12 | 7 $\times$ [14 $\times$ 14] | 6 | 0.95 | 2 | 0.005 | - | - | 4 | 2
CSOM | $35\times 35$ | $28\times 28$ | 1.5 | 0.07 | 0.5 | 0.9 | 8 | 45 | - | -
Table 18: Hyperparameters for Domain Incremental setting on KMNIST

Task matrices: Table 19 indicates the resultant accuracy values obtained after
performing 10 trials

98.6 $\pm$ 0.4 | 29.35 $\pm$ 1.24 | 54.72 $\pm$ 2.57 | 28.63 $\pm$ 2.45 | 47.21 $\pm$ 2.47
---|---|---|---|---
92.19 $\pm$ 2.46 | 91.48 $\pm$ 1.68 | 41.84 $\pm$ 3.3 | 55.72 $\pm$ 3.25 | 42.46 $\pm$ 2.05
89.91 $\pm$ 2.75 | 89.74 $\pm$ 1.99 | 89.85 $\pm$ 2.09 | 52.24 $\pm$ 2.62 | 39.38 $\pm$ 1.62
86.8 $\pm$ 2.5 | 89.55 $\pm$ 1.94 | 88.32 $\pm$ 2.21 | 93.01 $\pm$ 1.83 | 45.63 $\pm$ 1.43
86.1 $\pm$ 2.69 | 88.89 $\pm$ 2.32 | 86.72 $\pm$ 3.45 | 91.91 $\pm$ 2.87 | 86.92 $\pm$ 2.3
Table 19: mean and standard deviation of accuracies in a task matrix

####  7.2.3 Fashion-MNIST

Hyperparameters: Table 20 contains the initial hyperparameters used for
obtaining the results shown in Table 2(b).

Model | $H$ | $D$ | $\sigma_{0}$ | $\lambda_{0}$ | $\omega^{2}_{h}$ | $\lambda^{0}_{\omega}$ | $\tau_{\sigma}$ | $\tau_{\lambda}$ | $\mathrm{p}$ | $\mathrm{s}$
---|---|---|---|---|---|---|---|---|---|---
Vanilla SOM | $20\times 20$ | 28 $\times$ 28 | 0.6 | 0.07 | 1 | 0.9 | 8 | 45 | - | -
DendSOM | 10 $\times$ 10 | 7 $\times$ [14 $\times$ 14] | 5 | 0.95 | 2 | 0.005 | - | - | 8 | 4
CSOM | $25\times 25$ | $28\times 28$ | 1.5 | 0.07 | 0.5 | 0.9 | 8 | 45 | - | -
Table 20: Hyperparameters for Domain Incremental setting on Fashion-MNIST

Task matrices: Table 21 indicates the resultant accuracy values obtained after
performing 10 trials.

96.72 $\pm$ 1.44 | 53.15 $\pm$ 6.27 | 28.88 $\pm$ 4.84 | 40.7 $\pm$ 2.94 | 49.77 $\pm$ 0.27
---|---|---|---|---
92.63 $\pm$ 1.79 | 95.08 $\pm$ 0.61 | 41.42 $\pm$ 1.21 | 44.46 $\pm$ 1.51 | 47.21 $\pm$ 2.26
92.94 $\pm$ 1.72 | 92.22 $\pm$ 1.61 | 97.55 $\pm$ 0.87 | 95.75 $\pm$ 1.36 | 83.07 $\pm$ 3.94
93.72 $\pm$ 1.71 | 91.37 $\pm$ 1.49 | 97.58 $\pm$ 0.92 | 97.22 $\pm$ 0.86 | 86.17 $\pm$ 4.71
93.96 $\pm$ 1.73 | 90.77 $\pm$ 2.15 | 97.1 $\pm$ 0.85 | 97.06 $\pm$ 1.22 | 99.13 $\pm$ 0.13
Table 21: mean and standard deviation of accuracies in a task matrix

####  7.2.4 CIFAR-10

Hyperparameters: Table 22 shows hyperparameters used for training CSOM on
CIFAR10 in a Domain Incremental Setting.

Model | $H$ | $D$ | $\sigma_{0}$ | $\lambda_{0}$ | $\omega^{2}_{h}$ | $\lambda^{0}_{\omega}$ | $\tau_{\sigma}$ | $\tau_{\lambda}$
---|---|---|---|---|---|---|---|---
CSOM | $15\times 15$ | $28\times 28$ | 1.5 | 0.07 | 0.5 | 0.9 | 8 | 45
Table 22: Hyperparameters for Domain Incremental setting on CIFAR10

Table 23 shows task matrix of accuracy values obtained after performing 8
trials of CSOM on grayscale images from CIFAR-10.

99.52 $\pm$ 0.37 | 46.01 $\pm$ 4.66 | 43.63 $\pm$ 5.55 | 65.38 $\pm$ 5.11 | 38.67 $\pm$ 6.55
---|---|---|---|---
98.79 $\pm$ 0.59 | 93.15 $\pm$ 1.09 | 60.14 $\pm$ 2.78 | 74.98 $\pm$ 7.41 | 50.55 $\pm$ 2.84
97.15 $\pm$ 1.38 | 91.67 $\pm$ 1.84 | 94.41 $\pm$ 1.08 | 52.1 $\pm$ 7.59 | 18.99 $\pm$ 3.8
95.53 $\pm$ 7.62 | 91.35 $\pm$ 1.74 | 89.13 $\pm$ 5.01 | 95.36 $\pm$ 0.72 | 30.34 $\pm$ 3.01
97.25 $\pm$ 2.91 | 91.23 $\pm$ 0.92 | 68.91 $\pm$ 5.9 | 95.85 $\pm$ 0.77 | 79.57 $\pm$ 5.46
Table 23: mean and standard deviation of accuracies in a task matrix