# Discretization of continuous input spaces in the hippocampal autoencoder

Adrián F. Amil Ismael T. Freire Paul F.M.J. Verschure
Donders Institute for Brain, Cognition and Behaviour - Radboud University
Nijmegen, The Netherlands
{adrian.fernandezamil, ismael.titofreire<EMAIL_ADDRESS>

###### Abstract

The hippocampus has been associated with both spatial cognition and episodic
memory formation, but integrating these functions into a unified framework
remains challenging. Here, we demonstrate that forming discrete memories of
visual events in sparse autoencoder neurons can produce spatial tuning similar
to hippocampal place cells. We then show that the resulting very high-
dimensional code enables neurons to discretize and tile the underlying image
space with minimal overlap. Additionally, we extend our results to the
auditory domain, showing that neurons similarly tile the frequency space in an
experience-dependent manner. Lastly, we show that reinforcement learning
agents can effectively perform various visuo-spatial cognitive tasks using
these sparse, very high-dimensional representations.

##  1 Introduction

Ever since the seminal work of [41] on place cells in rats, decades of animal
research have established the hippocampus as a neural system that learns a
cognitive map of the environment and uses it for spatial navigation [38].
Further experimental studies have also identified the hippocampus as a key
structure in episodic memory formation [39]. Although several attempts have
been made to unify these perspectives [47, 16], a coherent framework that
mechanistically relates them remains elusive. Moreover, numerous experiments
reporting the instability of place cell activity and their modulation by non-
spatial variables [18, 25] raise an open question: what are these cells truly
encoding?

Efforts to answer this question have shown that place cell-like activity can
emerge under various conditions: when agents optimize a predictive coding
objective [46, 54, 45, 24, 29, 10], when networks optimize temporal stability
and pairwise decorrelation in processing visual inputs [55], or when building
sparse, compressed representations of environmental states [49, 50, 6, 27].
Notably, the approach where information compression leads to spatial tuning
aligns with the earlier hippocampal autoencoder model [23], and has been shown
to replicate several distinct place cell phenomena following environmental
manipulations [49, 50].

In this work, we further investigate the mechanisms behind episodic memory
formation and the emergence of place cells in the hippocampal autoencoder
model. We demonstrate that sparse autoencoders can create discontinuities in
the neural population manifold, discretizing arbitrary input spaces into non-
overlapping receptive fields, whereby subsets of similar inputs collapse onto
single neurons. When applied to visual images, this process generates place
cells operating on a very high-dimensional population code. Furthermore, we
show that reinforcement learning agents can effectively use such sparse and
high-dimensional hippocampal-like representations to solve spatial cognitive
tasks.

##  2 Model and Results

###  2.1 Hippocampal-like place cells emerge in sparse autoencoders

We studied the learning of spatial representations by training autoencoders
(Figure 1a) with randomly sampled images from four different tasks in the
Animal-AI environment [8]: Double T-maze, Cylinder, Object Permanence, and
Thorndike. We trained two types of autoencoders. "Dense" autoencoders aimed
solely to reconstruct the input images, thus preserving input information in
their latent space $Z$. In contrast, "sparse" autoencoders had an additional
objective beyond input reconstruction: to develop sparse activity patterns in
the latent space $Z$. This was achieved using the following loss function:

$\mathcal{L}=\frac{1}{m}\|\textbf{X}-\hat{\textbf{X}}\|_{2}^{2}+\frac{\lambda}{mn}\|\textbf{I}_{n}-\textbf{Z}^{\text{T}}\textbf{Z}\|_{\text{F}},$
(1)

where $m$ denotes the batch size, $n$ the number of neurons in $Z$, and the
first term is the mean squared error (MSE) between inputs X and their
reconstructions $\hat{\textbf{X}}$, encouraging $Z$ to preserve input
information. The second term is an orthonormal activity regularization term,
whose strength is controlled by $\lambda$, pushing the Gramian
$\textbf{Z}^{\text{T}}\textbf{Z}$ towards the identity matrix
$\textbf{I}_{n}$. Since $\textbf{Z}^{\text{T}}\textbf{Z}$ captures the co-
activation strengths between neurons in a training sample batch, the
orthonormal activity regularization promotes pairwise decorrelation while
ensuring equal contribution across neurons. We found this approach yields
improved and more reliable results compared to the L1 activity regularization
term typically used in sparse autoencoders, particularly in alleviating the
dead ReLU problem [34]. For dense autoencoders, $\lambda$ was set to zero,
leaving only the reconstruction error. We refer the reader to the Detailed
methods section in the Appendix for a complete description of the
environments, dataset generation, and parameters used in this study.

Training both types of autoencoders yielded significantly different internal
representations of space in their latent space. Dense autoencoders developed
many neurons that fired almost everywhere in space, with no defined place
fields. In contrast, sparse autoencoders developed a majority of neurons with
one or two localized place fields, similar to place cells in the hippocampus
(Figure 1b, c). The spatial specificity of sparse autoencoder neurons was also
reflected in significantly higher spatial information scores compared to dense
autoencoder neurons (Figure 1d). These results demonstrate that single-unit
spatial tuning emerges in sparse autoencoders but not in dense autoencoders,
despite both types of networks containing the same amount of positional
information at the population level, as shown by linear decoding analyses
(Figure 1e).

Figure 1: Hippocampal-like place cells emerge in sparse autoencoders. (a)
Autoencoder architecture, featuring the hidden layer or latent space $Z$
(denoted as Fc1) with 1000 neurons. (b) Representative examples of the
neurons’ spatial ratemaps for sparse and dense autoencoders. (c) Probability
distribution of place field number across environments. (d) Average spatial
information per neuron across environments. (e) Normalized average distance
error of linear decoding of position with the ratemaps’ population vectors,
across environments. The grey dots represent the expected linear decoding
errors after performing 1000 random permutations of the ratemaps’ values.

###  2.2 Sparse autoencoders discretize and tile the image space with
interpretable neurons

Identifying neurons with spatial selectivity similar to hippocampal place
cells allowed us to investigate what these neurons encode. Given that their
spatial selectivity must arise from some form of visual selectivity, we
explored whether they also exhibit localized receptive fields in image space.

We created an image space by extracting semantically-relevant image embeddings
of all samples using CLIP and further reducing the dimensionality to a 2D
space with UMAP. We then searched for clusters in this 2D image space by
running the DBSCAN algorithm on the points corresponding to images that
maximally activated a particular neuron (see Figure 2a and Detailed methods in
the Appendix). These clusters formed convex hulls (i.e., patches) that
corresponded to the neuron’s receptive fields in the image space. When pooled
together, receptive fields across neurons partitioned and covered the entire
image space (see example in Figure 2b). Furthermore, we computed an overlap
metric to estimate the redundancy across the neurons’ receptive fields. We
observed that sparse autoencoder neurons tiled the image space in a minimally-
overlapping manner, in contrast to dense autoencoder neurons, whose overlap
tended to be significantly higher (Figure 2c).

Additionally, we performed unit clamping experiments, setting neurons to their
maximal recorded value while others were set to zero, and then decoded their
activity back to images. The generated images showed a striking resemblance to
the training images, making these neurons highly interpretable (Figure 2d).
These results establish a solid relationship between episodic memory formation
and spatial coding.

Figure 2: Sparse autoencoders discretize and tile the image space with
interpretable neurons. (a) Images taken in one of the environments
(’Cylinder’), encoded with CLIP and further reduced to two dimensions with
UMAP. Points of different colors correspond to the images that maximally
activate each example neuron (above the 50% threshold of the maximum neuron’s
recorded activity). Clusters of maximally activated images are extracted with
DBSCAN, making up the convex hulls. (b) Convex hulls for all neurons in a
sparse autoencoder trained with images from the ’Cylinder’ environment. The
overlap metric corresponds to the expected overlapping area (in %) of two
randomly chosen hulls (see Detailed methods in the Appendix for further
details). (c) Average overlap in 2D image space of sparse and dense
autoencoders, across tasks and for a range of threshold values of maximal
activation. (d) Example interpretable neurons in the sparse autoencoder. The
corresponding neuron in latent space $Z$ is set to its maximum recorded value
across the dataset, while all other neurons are set to zero. Then, the
enforced activity vector $Z$ is deconvolved into an image by passing it
through the decoder.

###  2.3 Population structure in sparse autoencoders is grounded on mixed
selectivity

Having linked the formation of episodic memories with the discretization of
the image space in sparse autoencoders, we explored the population structure
of the latent space representations. Inspired by [53] on the dimensionality of
the population code in the mouse visual cortex, we examined the dimensionality
in our autoencoders. Dimensionality was estimated by performing PCA on $Z$ and
computing the linear fit of the resulting eigenspectrum in log-log space,
yielding a power-law exponent, $\alpha$. High $\alpha$ values indicate low-
dimensional codes, while low $\alpha$ values suggest high-dimensional codes.
An $\alpha\approx 1$ indicates a high-dimensional but smooth (i.e., no
discontinuities) underlying manifold, as seen in neural responses in the
visual cortex [53].

We found that dense autoencoders had dimensionality scores close to 1, similar
to [53], whereas sparse autoencoders exhibited much higher dimensional
representations (Figure 3a), aligning more with the efficient coding
hypothesis [5]. The almost-flat eigenspectrum suggests that sparse
autoencoders’ population activity indeed encodes fine stimulus features.
Moreover, the orthonormal activity regularization also disrupted the input-
output similarity preservation typically seen in dense autoencoders (Figure
3b), further supporting the idea of a sharp discretization of the image space
by sparse autoencoders.

Borrowing concepts from sparse dictionary learning [30], we considered the
decoder weights to be the dictionary of kernels, and the sparse neuron
activities to be the coefficients that use the dictionary to reconstruct the
inputs. Dense autoencoders exhibited orthogonal kernels, while sparse
autoencoders showed non-orthogonal kernels despite highly uncorrelated
activity patterns (Figure 3c). This indicates that orthogonal activity does
not imply orthogonal kernels, and that sparse autoencoder neurons learned
similar feature combinations, indicative of mixed selectivity in neurons [21].
These findings suggest that mixed selectivity and high dimensionality are
closely linked to the formation of detailed episodic memories.

Figure 3: Population structure in sparse autoencoders is grounded on mixed
selectivity. (a) Eigenspectrum decay in latent space representations (first
two rows) and images from the environments (third row). Parameter $\alpha$
corresponds to the power law exponent from linear fitting in log-log space.
(b) Input-output similarity for sparse and dense autoencoders, with data
pooled across environments. Correlation scores correspond to Spearman’s rank
coefficients, and fitting curves have been generated with a locally weighted
scatter-plot smoother (LOWESS) for improved visualization. (c) Pairwise
Pearson correlation scores between all neurons’ activity in latent space,
pooled across environments (left) and pairwise kernel similarity in the
decoder weights (layer Fc2), representing the similarity density across
"words" in the learned "dictionary" (right).

###  2.4 Zero-shot learning of place cells in sparse autoencoders

A typical observation in the hippocampus is that place cells can be identified
within the first minutes of an animal being exposed to a new environment [19].
Given that we have shown that sparse autoencoders exhibit very high
dimensionality (Figure 3a) and single neurons tend to encode small clusters of
samples (Figure 2), we investigated the extent to which neurons that learned
to encode samples in one environment could generalize to encoding unseen
environments and exhibit zero-shot place cells. Therefore, we trained sparse
autoencoders in one environment and tested them across all others. Strikingly,
neurons developed place fields with distributions very similar to those in
their training environment (Figure 4a). Furthermore, the average spatial
information across neurons and the mean decoding error of the rate maps were
very similar, with no significant degradation compared to the training
environment (Figure 4b). These results suggest that the network’s circuitry
learned to cluster and collapse similar samples onto single neurons in a more
generic manner, beyond the specific details of the dataset.

Figure 4: Zero-shot place cells in sparse autoencoders. (a) Probability
distributions of place field number when testing a model within its training
environment (light blue) or across unseen environments (dark blue). (b)
Average spatial information per neuron, pooled across models and testing
environments. (c) Normalized average distance error of linear decoding of
position with the ratemaps’ population vectors, across models and testing
environments. The grey dots represent the expected linear decoding errors
after performing 1000 random permutations of the ratemaps’ values.

###  2.5 Sparse autoencoders discretize and tile the input frequency space in
an experience-dependent manner

If the hippocampus functions as a generic, modality-independent episodic
memory system, our findings with the sparse autoencoder should generalize to
other input modalities, such as sound. Indeed, "place cell"-like activity in
the hippocampus has been reported for tasks involving "navigating" the sound
frequency space, with neurons developing localized receptive fields around
particular sound frequencies [2]. To investigate whether a similar effect
could be observed within our framework, we trained autoencoders to compress
and encode sound waves uniformly sampled from a linearly-varying frequency
signal (Figure 5a).

We observed the emergence of frequency-specific receptive fields in sparse
autoencoders, but not in dense autoencoders (Figure 5b). These receptive
fields tiled the entire frequency space in a linear manner. However, when
sampling was biased towards certain frequencies, the neurons’ receptive fields
became denser and clustered around those frequencies (Figure 5d).
Additionally, similar to our previous findings with visually interpretable
neurons, we found that individual neurons encoded particular frequencies so
that these representations were readily decodable via clamping (Figure 5c).
These results demonstrate that sparse autoencoders can discretize arbitrary
input spaces to support episodic memory formation.

Figure 5: Sparse autoencoders discretize and tile the input frequency space in
an experience-dependent manner. (a) Data samples are generated by applying a
uniformly distributed sliding window to a linearly-varying frequency signal.
The samples are fed into a convolutional autoencoder, analogous to the one
used for vision (more details can be found in the Detailed Methods section of
the Appendix). (b) Unsorted and sorted receptive fields by peak activity
location for both sparse and dense autoencoders. Latent space activity $Z$
responding to pure tone test inputs was convolved with a Gaussian kernel
(sigma of 0.5 Hz), and then normalized by the maximum per neuron in the sorted
plots. Lanczos interpolation was applied to all plots for improved
visualization. (c) Decoded output signals after setting the corresponding
neuron in latent space to its maximum recorded value across the dataset, while
all other neurons were set to zero. (d) Sorted receptive fields in a sparse
autoencoder trained with an unbalanced dataset. The data samples were
generated with a sliding window that was not uniformly distributed in the
frequency space, but whose density followed a Gaussian distribution centered
at 45 Hz. Dashed vertical lines denote one standard deviation $\sigma$ from
the mean.

###  2.6 DQN agents learn effectively with sparse, very high-dimensional
representations

Representations used to build episodic memories are likely also employed for
behavioral learning in the brain. Therefore, we investigated whether
hippocampal-like representations emerging in sparse autoencoders would be
suitable for reinforcement learning. To test this, we employed DQNs
incorporating either sparse or dense autoencoders to solve a range of tasks in
the Animal-AI environment, which inherently require spatial navigation skills
(see Figure 6a, and Detailed methods in the Appendix for further details on
the tasks, model, and training parameters).

Very high-dimensional representations (such as those based on efficient
coding) are known to be highly sensitive to slight input perturbations and are
thought to generalize poorly to new, unseen samples [40]. Thus, one might
expect that DQNs would struggle with tasks requiring generalization to unseen
samples during training in non-stationary environments. Contrary to this
expectation, we found that DQNs using sparse autoencoders were not
systematically worse than those using dense autoencoders (Figure 6b). Although
sparse autoencoders seemed to perform worse than dense autoencoders in two of
the four tasks tested here (Object Permanence and Thorndike), they were
superior in one of them (Cylinder) and had matched performance in the
remaining one (Double T-maze). Therefore, while further testing is definitely
needed to obtain a more reliable picture of their relative performance, these
results suggest that, in practice, hippocampal-like representations can be
suitable for reinforcement learning, despite their very high dimensionality.

Figure 6: DQN agents learn effectively with sparse, very high-dimensional
representations. (a) Overhead images captured from above the virtual arena of
the four tasks (from left to right): Double T-maze, Cylinder, Object
Permanence, and Thorndike. (b) Performance (average reward across episodes) of
DQN agents using sparse and dense autoencoders across tasks.

##  3 Discussion

##### Optimization objectives underlying place cell emergence

We have demonstrated the distinct emergence of place cells in autoencoders
with orthonormal activity regularization (Figure 1). Notably, sparse
compression alone was sufficient to develop spatial tuning, without the need
for a predictive objective [46, 45, 54, 24, 29, 10]. While predictive coding
may explain other features of the hippocampus, such as place-cell theta
sequences [13], prediction does not appear to be necessary for the emergence
of realistic place cells. We speculate that models optimized for next-input
predictions likely learn compressed representations of the environment
implicitly as part of the predictive objective. Furthermore, by training the
autoencoders with randomly sampled and shuffled images, we have shown that
neither temporal contiguity of samples nor any temporal correlations are
required to develop place cells. This finding suggests that while predictive
learning capturing temporal input correlations might correspond to experience-
dependent theta sequences in the hippocampus, the formation of compressed
state representations might correspond to time-independent learning processes
at the gamma frequency scale [33]. Additionally, we have demonstrated that
sparse autoencoders can learn localized receptive fields of the input space
while breaking the relationship between input similarity and latent space
similarity. This finding contrasts with previous research that emphasized the
preservation of input-output similarity matching for learning localized
receptive fields [51, 43]. Overall, it appears that sparse compression alone
is sufficient to learn localized receptive fields, that in turn manifest as
place cells when applied to the visual domain.

##### Sparse and very high-dimensional codes

A slowly-decaying eigenspectrum, where fine details are over-weighted,
represents codes that create discontinuities by disrupting the locality of the
manifold structure supporting the input space distribution [40]. We
demonstrated that this discontinuity can be induced by an orthonormal activity
regularization objective, facilitating the creation of event memories by
discretizing the image input space (Figure 2). Our results suggest that very
high-dimensional codes underlie the formation of place cells (Figure 3),
aligning with the efficient coding hypothesis [5], which posits that the brain
maximizes information by eliminating correlations in sensory inputs, leading
to sparse coding [42]. Indeed, it has been shown that hippocampal neurons in
rodents become sparser with prolonged exposure to the environment [45].
Moreover, the storage of social memories in mice has been linked to high-
dimensional representations in the hippocampus [9]. Importantly, sparsity has
been shown to control a generalization-discrimination trade-off [3], which
could explain why the hippocampus relies on sparse representations, well-
suited for progressive discrimination of similar environments and events. Our
study shows that smooth place cell maps can coexist with and emerge from
extremely high-dimensional sparse codes [11]. We therefore predict that the
dimensionality of the population code along the sensory hierarchy should
decrease to support the learning of invariant sensory representations [20],
and then increase sharply at the apex, in the hippocampus, to enable the
formation of detailed memories based on the specific combination of such
invariant representations. This role of the hippocampus aligns with our
observation of mixed selectivity, i.e., neurons learning similar feature
combinations (Figure 5c), which in turn has been proposed to enable high-
dimensional representations important for higher cognition areas [21, 7].

##### Circuit mechanisms underlying memory formation

The surprising observation of zero-shot learning of place cells (Figure 4)
suggests that the sparse autoencoders learned to cluster similar samples onto
single neurons in a generic manner. We hypothesize that the responsible
circuits might correspond to known hippocampal processes, mainly pattern
completion and pattern separation [48]. On the one hand, neurons are pushed to
collapse across-sample variability by clustering samples based on similarity,
an effect akin to pattern completion. On the other hand, sparsity also imposes
sharp discontinuities between clusters in neuronal space, even when they might
be close in input space, a process akin to pattern separation. The combination
of both processes is reminiscent of the "locality-sensitive hashing" (LSH)
algorithms used in the computer science field for fast image search [28].
Future work will shed light on the learned circuit mechanisms behind such a
LSH in sparse autoencoders, and their potential mapping to pattern separation
and completion.

##### Place cell over-representation near reward areas

We have shown that the development of localized receptive fields can be
generalized to other input modalities, such as sound. Crucially, we used this
simplified framework to demonstrate that receptive field distribution tends to
be modulated by the input sampling distribution (Figure 5d). Importantly, it
has been observed that the density of place fields increases near reward areas
[35]. This has led researchers to seek external reward signals in the
hippocampus that would modulate the place cell map [26]. However, based on our
results, we propose an alternative explanation based on oversampling: animals
tend to spend more time within rewarded areas due to consummatory behaviors,
hence biasing sensory sampling and learning. Additionally, in line with this
hypothesis, place cell trajectories leading to rewards or goals tend to be
replayed more often than unrewarded past trajectories [1], which would further
reinforce the sampling bias.

##### Biological plausibility

Although it has been claimed that error backpropagation and gradient descent
are mechanisms that could be implemented in the brain [31], particularly in
the hippocampus [50], we believe that such strong assumptions are unnecessary
to map our model and observations to the real hippocampus. The orthonormal
activity regularization term used in our sparse autoencoders could be realized
by combining strong lateral inhibition (promoting pairwise decorrelation) and
homeostatic plasticity (ensuring that neurons maintain equalized firing rates
over time). Additionally, the orthonormal term can be thought of as sparse
whitening in ReLU-like neurons (i.e., pairwise decorrelation and variance
normalization in the low-firing rate regime), a mechanism proposed to be
realized in the brain by an overcomplete basis of inhibitory interneuron
projections [15, 14, 32]. Therefore, we contemplate several alternative
mechanisms whereby the main objective driving our sparse autoencoders could be
realized in brain circuits.

##### Limitations

While our reinforcement learning experiments suggest that DQNs can make use of
very high-dimensional representations to solve complex tasks, the present
study is limited in scope (Figure 6). We tested only a few tasks (four tasks
within the Animal-AI testbed) and a single model (DQN). To gain a more
comprehensive understanding of the suitability of hippocampal-like
representations for behavioral and policy learning, further testing is
required with a broader range of tasks and models, especially in non-
stationary environments where unseen samples are the norm. Additionally,
future research should explore how these sparse autoencoders could enhance
reinforcement learning algorithms that rely on discrete representations,
potentially enabling algorithms based on, e.g., the successor representations
[12], to extend beyond simplified grid worlds.

## Code Availability

All the code necessary to reproduce the figures and results presented in this
paper has been made publicly available at:
https://github.com/adriamilcar/HashingBrain.

## Acknowledgments and Disclosure of Funding

We want to thank Raimon Bullich Vilarrubias for suggesting to apply the model
to the frequency domain, and Oscar Guerrero Rosado for aiding in the design of
the models’ diagrams.

## References

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##  Appendix A Appendix

###  A.1 Detailed methods

####  A.1.1 Model’s architecture and training

##### Visual autoencoder (Vis-AE)

To compress the images from the Animal-AI enviroment, we employed a
convolutional autoencoder (ConvAE) that maps the $3\times 84\times 84$ to a
latent space with 1000 neurons, effectively compressing the images by a factor
of $\sim$21\. The details on the architecture can be found in Table 1.

Table 1: Architecture of the Visual Autoencoder (Vis-AE) Layer | Type | Act. Func. | Filters/Units | Kernel Size | Stride/Padding
---|---|---|---|---|---
Input | Input | - | $3\times 84\times 84$ | - | -
Conv1 | 2D Conv. | ReLU | 16 | $4\times 4$ | 2/1
Conv2 | 2D Conv. | ReLU | 32 | $4\times 4$ | 2/1
Conv3 | 2D Conv. | ReLU | 64 | $4\times 4$ | 2/1
Reshape | Reshape | - | 6400 | - | -
Fc1 ($Z$) | Linear | ReLU | 1000 | - | -
Fc2 | Linear | ReLU | 6400 | - | -
Conv4 | Trans. 2D Conv. | ReLU | 32 | $4\times 4$ | 2/1 (out. pad. 1)
Conv5 | Trans. 2D Conv. | ReLU | 16 | $4\times 4$ | 2/1
Conv6 | Trans. 2D Conv. | Sigmoid | 3 | $4\times 4$ | 2/1
Output | Output | - | $3\times 84\times 84$ | - | -

##### Audio autoencoder (Aud-AE)

To compress the frequency signals from the synthetic sound dataset, we
employed another ConvAE that maps one-second time series data (with a
resolution of $10^{-3}$ s) to a latent space with 100 neurons, effectively
compressing the signals by a factor of 10. The details on the architecture can
be found in Table 2.

Table 2: Architecture of the Audio Autoencoder (Aud-AE) Layer | Type | Act. Func. | Filters/Units | Kernel Size | Stride/Padding
---|---|---|---|---|---
Input | Input | - | $1\times 1000$ | - | -
Conv1 | 1D Conv. | ReLU | 16 | $100$ | 2/1
Conv2 | 1D Conv. | ReLU | 32 | $100$ | 2/1
Conv3 | 1D Conv. | ReLU | 64 | $100$ | 2/1
Reshape | Reshape | - | $2624$ | - | -
Fc1 ($Z$) | Linear | ReLU | 100 | - | -
Fc2 | Linear | ReLU | $2624$ | - | -
Conv4 | Trans. 1D Conv. | ReLU | 32 | $100$ | 2/1
Conv5 | Trans. 1D Conv. | ReLU | 16 | $100$ | 2/1
Conv6 | Trans. 1D Conv. | Tanh | 1 | $100$ | 2/1
Output | Output | - | $1\times 1000$ | - | -

##### Loss function

The loss function to be minimized in both autoencoders (Vis-AE and Aud-AE)
includes a mean squared error (MSE) term as a reconstruction error to force
the latent space to preserve the input information, and a orthonormal activity
regularization term that promotes sparse representations in the latent space
$Z$:

$\mathcal{L}=\frac{1}{m}\|\textbf{X}-\hat{\textbf{X}}\|_{2}^{2}+\frac{\lambda}{mn}\|\textbf{I}_{n}-\textbf{Z}^{\text{T}}\textbf{Z}\|_{\text{F}},$
(2)

where X is the input, $\hat{\textbf{X}}$ is the output (i.e., the
reconstructed input), $\lambda$ is the regularization coefficient ($10^{3}$ by
default), $\textbf{I}_{n}$ is the identity matrix with shape $n\times n$, Z is
the middle layer’s activity matrix of shape $m\times n$, $m$ being the batch
size, and $n$ the number of hidden units. Therefore, the Gramian
$\textbf{Z}^{\text{T}}\textbf{Z}$ (with shape $n\times n$) captures the
pairwise co-activation strengths between neurons in latent space. The symbols
$\|\cdot\|_{2}$ and $\|\cdot\|_{\text{F}}$ denote the squared L2-norm and the
Frobenius norm, respectively. The orthonormal activity regularization term
promotes pairwise neuron decorrelation while achieving an equalized
contribution across neurons, alleviating the dying ReLU problem. For
comparisons with the dense AE, $\lambda$ was simply set to 0. We have found
this orthonormal activity regularization to give improved and more reliable
results than the L1 activity regulation term used in standard sparse
autoencoders, especially in preventing the dead ReLU problem.

##### Data generation and training of Vis-AE

For the visual experiments, datasets were generated by sampling a total of
10000 images in each of four Animal-AI environments: "doubleTmaze",
"permanence", "cylinder", and "thorndike", at random locations within the
arena (excluding the 10% of the space closest to each wall) and with random
angles, following uniform distributions. Training was conducted using batches
of 256 images for 10000 epochs, with independent training runs per
environment. The Adam optimizer with no weight decay was used to train the
network and the learning rate was set at $10^{-4}$. In addition, the
regularization strength $\lambda$ was set to $10^{3}$ for the sparse
autoencoder, and 0 for the dense autoencoder. All weights were initialized
using Xavier initialization except for $Z$ (i.e., Fc1), whose weights were
initialized following a random asymmetric initialization to minimize the dying
ReLU problem [34]. An early stop of 0.0005 in reconstruction loss was used to
compare sparse and dense autoencoders with similar reconstruction
capabilities.

##### Data generation and training of Aud-AE

For the synthetic audio experiments, training consisted of batches of 256 one-
second audio slices of varying frequencies. A sliding window of 1 second (with
1 ms shift) was applied to a linearly-varying frequency signal of total time
100 seconds, moving from 10 to 80 Hz, hence resulting in a total of 99001
samples. The sampling frequency was set at ($10^{4}$ Hz, so that the kernel
size (1000) matched to one full cycle at the lowest input frequency (10 Hz).
Training was conducted for 1000 epochs using the Adam optimizer with a
learning rate of $10^{-4}$, with no weight decay. Here, the regularization
strength $\lambda$ was set to $10^{4}$ for the sparse autoencoder and 0 for
the dense autoencoder. The weights were initialized as with the visual-AE,
with Xavier initialization in all layers except for the the latent space $Z$
that followed a random asymmetric initialization. An early stop of 0.002 in
reconstruction loss was used to have a fair comparison between sparse and
dense autoencoders.

####  A.1.2 Spatial tuning

##### Firing ratemaps

To generate ratemaps from latent space activity, we first created a grid of
$60\times 60$ bins (or $30\times 30$ for computing spatial information scores)
for each neuron. For each bin in the grid, we summed the neuron’s activity
values for images sampled within that bin, generating an activity map in
space. Then, an occupancy map was generated to account for the variability in
the number of images sampled at each spatial bin (sampling density), which was
used to normalize the values in the activity map. Finally, Gaussian smoothing
was applied to each neuron’s normalized activity map, using a standard
deviation of 3 bins. The resulting maps were normalized to their corresponding
maximum values, yielding smooth ratemaps representing spatially-distributed
neural activity.

##### Place field identification

To identify and quantify place fields in each neuron’s ratemap, we first
binarized the ratemap by setting pixels with activity below 20% of the maximum
activity to zero (inactive bins) and those above to one (active bins).
Clusters were identified by grouping adjacent active bins, forming a cluster
if a group of active bins was completely surrounded by inactive bins. Clusters
not meeting the size criteria for place cells (between 3% and 50% of the total
number of bins, 3600) were discarded. The remaining clusters were considered
place fields.

##### Spatial information

Spatial information (SI) scores measure the amount of information a neuron’s
firing rate ($\nu$) conveys about the agent’s position (r). For each neuron,
we first normalized its ratemap (using a $30\times 30$ bin grid) by the
overall mean activity $\bar{\nu}$. Then, we computed an occupancy map that was
normalized by the total number of samples to reflect the proportion of "time"
spent in each bin of the ratemap, denoted as $p(\textbf{r})$. Finally, we
applied the formula introduced in [52] to compute the SI scores:

$\text{SI}=\sum_{\textbf{r}\in\textbf{R}}\frac{\nu(\textbf{r})}{\bar{\nu}}\log_{2}\left(\frac{\nu(\textbf{r})}{\bar{\nu}}\right)p(\textbf{r}).$
(3)

The average SI across all neurons in the latent space $Z$ provides an estimate
of the degree of spatial tuning that the model has developed.

##### Spatial position decoding

The spatial decoding error measures the expected error of a linear decoder
using latent space activations $Z$ to predict the spatial position r. We fit a
linear regression model with $Z$ as the independent variables and r as the
dependent variables, predicting positions as $\hat{\textbf{R}}=Z\mathbf{W}$.
Then, we compute the mean squared error (MSE) between the predicted positions
$\hat{\textbf{R}}$ and the actual ones R:

$\text{MSE}=\frac{1}{n_{\text{samples}}}\|\textbf{R}-\hat{\textbf{R}}\|_{2}^{2}.$
(4)

Finally, the average spatial decoding error (MSE) is re-scaled by dividing it
by the maximum distance in the environment, that is, the diagonal of the
arena, computed as $d=s\sqrt{2}$, with $s$ being the side length.

####  A.1.3 Interpretability

##### Visualizing and quantifying the network’s tiling of the image space

We employed the CLIP neural network [44] to encode images (resized from
$3\times 84\times 84$ to $3\times 224\times 224$) into 512-dimensional
vectors. These vectors were subsequently reduced to a two-dimensional
representation using UMAP [36], with 10 neighbors and a minimum distance of
0.1, enabling the visualization of the high-dimensional image space.

Neurons in the hidden layer of the autoencoder that exhibited strong
activation in response to specific images—those triggering activations
exceeding a certain % of their maximum activation across the dataset—were
mapped to points in the 2D image space. We then identify clusters of points
using the DBSCAN algorithm [17], with radius $\epsilon$ of 1 and minimum
samples of 4. These parameters are very dataset-dependent and were thus
selected and validated via extensive visual inspection to ensure reliable
cluster identification. Convex hulls were constructed around these clusters
using the Quickhull algorithm [4] to delineate their spatial boundaries. This
allowed us to identify the regions of the input space that each neuron encodes
in their activations, i.e., their receptive fields.

Let $\\{H_{i}\\}$ denote the set of convex hulls corresponding to each
neuron’s activated image space. The average overlap metric, $\overline{O}$,
was calculated as follows:

$\overline{O}=\frac{1}{\binom{k}{2}}\sum_{i<j}\frac{\text{Area}(H_{i}\cap
H_{j})}{\text{Area}(H_{i}\cup H_{j})},$ (5)

where $\text{Area}(H_{i}\cap H_{j})$ represents the area of intersection
between hulls $H_{i}$ and $H_{j}$, $\text{Area}(H_{i}\cup H_{j})$ is the area
of the union of hulls $H_{i}$ and $H_{j}$, and $k$ is the total number of
hulls. The hull calculations were performed using the Shapely Python library
[22]. The metric $\overline{O}$ thus represents the average proportion of
overlap relative to the union for each pair of hulls and ranges from 0 (no
overlap) to 1 (complete overlap), thereby providing a quantitative measure of
the redundancy in the neurons’ receptive fields across the image space.

##### Neuron clamping and decoding

To test whether neurons in $Z$ were directly interpretable based on their
single-neuron activity (therefore obviating population codes), we conducted
clamping experiments. This involved setting the activation of a specific
neuron $i$ in $Z$ to its maximum activation value observed across the dataset
$\mathcal{X}$, while setting the activations of all other neurons to zero.
This is represented as $z_{i}^{\prime}=(0,\dots,0,x_{\text{max}},0,\dots,0)$
where $x_{\text{max}}=\max(\\{z_{i}|z=f(x),x\in\mathcal{X}\\})$ and $f(x)$
represents the encoding function mapping $\mathcal{X}$ to $z$. Then,
$z_{i}^{\prime}$ is processed by the decoder $g(z_{i}^{\prime})$ (with $g(x)$
representing the decoding function mapping $z$ to $\mathcal{\hat{X}}$) to
yield an output signal (image of audio wave, depending on the AE).

##### Population code dimensionality

The dimensionality of the population code was estimated by computing the
power-law exponent $\alpha$ of the latent space activity $Z$ [53]. We
performed PCA on $Z$ and computed the linear fit of the resulting
eigenspectrum in log-log space over the range of the first 10 to 100 principal
components. Since the exponent $\alpha$ provides an estimate of how fast the
population activity eigenspectrum decays as new dimensions are added, high
$\alpha$ values are indicative of low-dimensional codes, whereas low $\alpha$
values indicate high-dimensional codes.

####  A.1.4 Reinforcement learning experiments

##### Animal-AI Testbed

The Animal-AI testbed is a comprehensive platform designed for evaluating the
cognitive and learning capabilities of AI agents in a variety of tasks that
simulate real-world challenges [8]. This testbed provides diverse environments
where agents must use visual cues and navigate complex structures to achieve
specific goals. The visual inputs from these environments are standardized to
a resolution of 84 by 84 pixels, and agents can perform actions defined by a
2-dimensional vector of integers: the first component goes from 0 to 2 and
corresponds to not moving, moving forward, or moving backwards, respectively;
and the second component also goes from 0 to 2 and corresponds to not rotate,
rotate left, or rotate right, respectively. To encourage efficient behavior, a
standard frameskip of 4 is applied, and the reward value decreases by 0.001 at
each step. Episodes terminate either when the agent obtains the reward or
after 1000 frames.

We evaluate our reinforcement learning agents using four distinct benchmarks
within the Animal-AI testbed: the Double T-maze, Object Permanence, Cylinder,
and Thorndike tasks. Each of these tasks presents unique challenges that
require the agent to apply different strategies and cognitive abilities.

  * •

Double T-maze. Each episode starts with the agent positioned randomly at one
of the corners of the maze, and the objective is to navigate to the center to
obtain the reward. The center contains the only positive reward (+3) available
in the environment. Due to the high and opaque maze walls, the agent cannot
directly see the reward and must explore the maze to find it.

  * •

Object Permanence. At the beginning of each episode, the agent observes a
large reward (+3) falling behind a wall until it is completely occluded. The
agent must then navigate to the hidden reward, avoiding a small and visible
reward (+1) along the way.

  * •

Cylinder. This task involves an opaque cylinder with a medium-sized reward
hidden inside. The agent begins outside the cylinder and must navigate into
the cylinder to obtain the reward (+2).

  * •

Thorndike. The task tests the agent’s ability to escape from a closed box to
reach a reward located outside the box. The box is semi-transparent, allowing
the agent to see the reward from inside. The only exit is blocked by a movable
obstacle that the agent must push to escape. A medium reward (+2) outside the
box is the sole positive reward available.

##### Model

To evaluate the performance of our sparse autoencoders in reinforcement
learning scenarios, we used a standard Deep Q-Network (DQN) architecture [37]
with modifications to the input layer. Instead of feeding raw pixel data from
the Animal-AI environments, we used the compressed representations of 1000
units generated by the Visual Autoencoder (Vis-AE).

The loss function optimized by the Deep Q-Network (DQN) is the Mean Squared
Error (MSE) between the predicted Q-values and the target Q-values, calculated
using the Bellman equation:

$\mathcal{L}(\theta)=\mathbb{E}_{(s,a,r,s^{\prime},d)\sim\text{ReplayBuffer}}\left[\left(r+\gamma\cdot(1-d)\cdot\max_{a^{\prime}}Q_{\text{target}}(s^{\prime},a^{\prime};\theta^{-})-Q_{\text{main}}(s,a;\theta)\right)^{2}\right]$
(6)

where $Q_{\text{main}}$ is the main Q-network with parameters $\theta$,
$Q_{\text{target}}$ is the target Q-network with parameters $\theta^{-}$, $s$
is the current state, $a$ is the action taken, $r$ is the reward received,
$s^{\prime}$ is the next state, $d$ is a boolean indicating whether
$s^{\prime}$ is a terminal state, and $\gamma$ is the discount factor. This
loss function aims to minimize the difference between the Q-value predicted by
the main network and the target Q-value, which is computed based on the reward
and the maximum Q-value of the next state predicted by the target network. The
training of the DQN was performed by using the RMSprop optimizer. The target
network was periodically updated with the weights of the main DQN to stabilize
training. The DQN was trained with the following hyperparameters: a learning
rate of 0.00025, a discount factor ($\gamma$) of 0.99, an update frequency of
4 steps, and a target network update frequency of 2500 steps. The $\epsilon$
for the epsilon-greedy policy started at 1 and decayed linearly to 0.1 over
25000 steps. The replay buffer size was set to 25000, with a batch size of 32
for experience replay. The details on the architecture can be found in Table
3.

Table 3: Architecture of the Deep Q-Network (DQN) Layer | Type | Act. Func. | Units
---|---|---|---
Input | Input | - | 1000
Fc1 | Linear | ReLU | 100
Fc2 | Linear | ReLU | 50
Fc3 | Linear | ReLU | 25
Fc4 | Linear | ReLU | 9
Output | Output | - | 9

##### Training and performance metrics

Each reported experiment tested two DQN agents, Sparse and Dense, which differ
only in their use of different Vis-AE models (sparse and dense autoencoders,
respectively) to obtain compressed representations from the environment
observations as input. The two agents were evaluated across the four Animal-AI
tasks described earlier. Each model run lasted 5000 episodes, and to ensure
statistical reliability, each model played each task between 20 and 27 times.
The reported average performance metric was calculated using a sliding window
of 20 episodes.


# Some results on Continuous dependence of fractal functions on the Sierpiński
gasket

*Vishal Agrawal  Department of Mathematical Sciences, IIT(BHU), Varanasi, India 221005<EMAIL_ADDRESS>,  Ajay Prajapati  Department of Computer science, Banaras Hindu University, Varanasi, India 221005<EMAIL_ADDRESS>,  Abhilash Sahu  Department of Mathematics Indian Institute of Technology Guwahati, Guwahati, Assam, India, 781039<EMAIL_ADDRESS>and  Tanmoy Som  Department of Mathematical Sciences, IIT(BHU), Varanasi, India 221005
<EMAIL_ADDRESS>
###### Abstract.

In this article, we show that $\displaystyle\alpha$-fractal functions defined
on Sierpiński gasket (denoted by $\displaystyle\triangle$) depend continuously
on the parameters involved in the construction. In the latter part of this
article, the continuous dependence of parameters on
$\displaystyle\alpha$-fractal functions defined on $\displaystyle\triangle$ is
shown graphically.

###### Key words and phrases:

fractal dimension, fractal interpolation, Sierpiński Gasket, continuous
dependence

###### 2010 Mathematics Subject Classification:

Primary 28A80; Secondary 41A10

##  1\. Introduction

In the field of Numerical Analysis, the computational implications of
interpolation have been a major concern for many years. The theory of
interpolation has evolved throughout the development of classical
approximation theory. Various interpolation techniques are used in Numerical
Analysis and Classical Approximation Theory, and they are based on polynomial,
trigonometric, spline, and rational functions. Based on the underlying idea of
the model under investigation, these techniques can be applied to a specific
data set. While nonrecursive interpolation techniques in the literature almost
always produce smooth interpolants, it should be emphasized that they are not
all recursive. There are a great number of real-world phenomena and
experimental signals that are confusing, and sometimes their traces appear
smooth. Due to their non-differentiability and complexity, a simple
mathematical framework may not adequately describe their smallest geometric
complexity. In order to produce an interpolant with a more complex geometric
structure, it is necessary to develop a novel interpolation approach.
Univariate real-valued interpolation functions constructed on a compact
interval in $\displaystyle\mathbb{R}$ were first proposed by Barnsley [3].
These are referred to as Fractal Interpolation Functions (FIFs for short), and
their construction is based on the Iterated Function System (IFS) theory [10].
In [3], the groundbreaking research on fractal interpolation has attracted a
lot of interest in the literature and is currently going strong. The author of
[16] demonstrated that the FIFs theory can be used to construct a family of
continuous functions with fractal properties from a given continuous function.

A number of studies have exposed several significant characteristics of FIFs,
including their smoothness, stability, one-sided approximation property and
constraint approximation property, as well as their box dimension and
Hausdorff dimension. There are several research articles available on various
types of FIFs, see, for instance, [5, 8, 12, 13, 17, 19]. Numerous studies
reveal several significant aspects of FIFs, such as their smoothness,
stability, one-sided approximation property and constrained approximation
properties, as well as box dimension and Hausdorff dimension of their graphs.
Celik et al. [4] expanded the notion of FIFs to incorporate interpolation of a
data set on $\displaystyle\triangle$. Following this article, Ruan [24]
developed FIFs on post critically finite self-similar sets. Kigami [14] is
credited with having introduced and analysed these sets. On
$\displaystyle\triangle$, Ri and Ruan [23] defined several fundamental
characteristics of a space of FIFs. The fractal dimensions of FIFs defined on
various domains have been thoroughly investigated in numerous articles, see,
for instance, [6, 7, 9, 11, 18, 21, 25, 27, 28, 32, 31]. Recently, Mohapatra
et al. [15] introduced a concept that generalised the notions of the Kannan
map and contraction. In [20], Prasad and Verma have constructed the FIFs on
the product of two Sierpiński gaskets.

We denote the space of all the real-valued continuous functions defined on
$\displaystyle\triangle$ by $\displaystyle C(\triangle,\mathbb{R})$ and graph
of $\displaystyle f$ by $\displaystyle graph(f)$ throughout this paper.

##  2\. Fractal interpolation function on the Sierpiński gasket

We begin by providing a brief overview of the relevant concepts and an
introduction to $\displaystyle\triangle$. The reader may refer to [2, 22, 26,
30] for further information. We begin by recalling an established
$\displaystyle\triangle$ construction based on IFS. Consider a set
$\displaystyle V_{0}=\left\\{x_{1},x_{2},x_{3}\right\\}$ such that points in
$\displaystyle V_{0}$ have equal distances from each other. Corresponding to
each point of $\displaystyle V_{0}$, define the contraction map $\displaystyle
u_{i}$ on $\displaystyle\mathbb{R}^{2}$ as follows:

$\displaystyle u_{i}(t)=\frac{1}{2}\left(t+x_{i}\right),$

where $\displaystyle i=1,2,3$. Then, three contraction maps together with the
plane constitute an IFS, which produces $\displaystyle\triangle$ as an
attractor, i.e.,

$\displaystyle\triangle=\bigcup_{i=1}^{3}u_{i}(\triangle).$

Define $\displaystyle V_{1}$ by $\displaystyle
V_{1}=\left\\{x_{1},x_{2},x_{3},u_{1}\left(x_{2}\right),u_{2}\left(x_{3}\right),u_{3}\left(x_{1}\right)\right\\}$.
Let us consider a continuous function $\displaystyle
f:\triangle\rightarrow\mathbb{R}$. Let $\displaystyle
U=\triangle\times\mathbb{R}$ and define maps $\displaystyle H_{i}:U\rightarrow
U$ by

$\displaystyle H_{i}(t,x)=\left(u_{i}(t),M_{i}(t,x)\right),i=1,2,3,$

where $\displaystyle
M_{i}(t,x):\triangle\times\mathbb{R}\rightarrow\mathbb{R}$ is a contraction
map in the last variable, that is,

$\displaystyle\left|M_{i}(.,x)-M_{i}\left(.,x^{\prime}\right)\right|\leq
c\left|x-x^{\prime}\right|$

with $\displaystyle
M_{i}\left(p_{j},f\left(p_{j}\right)\right)=f\left(u_{i}\left(p_{j}\right)\right)$.
In particular, we take

$\displaystyle M_{i}(t,x)=\alpha_{i}x+f\left(u_{i}(t)\right)-\alpha_{i}b(t),$

where $\displaystyle b\in C(\triangle,\mathbb{R})$ is base function and
$\displaystyle f\in C(\triangle,\mathbb{R})$ original function such that
$\displaystyle b\left(p\right)=f\left(p\right)~{}\forall~{}p\in V_{0}$, and
scale vector $\displaystyle\alpha_{i}\in\mathbb{R}$ with
$\displaystyle\left|\alpha_{i}\right|_{\infty}<1$. Thus, We have an IFS
$\displaystyle\mathcal{J}:=\left\\{U,H_{i}:i=1,2,3\right\\}$.

######  Theorem 2.1.

[1] Let $\displaystyle f\in C(\triangle,\mathbb{R})$, and $\displaystyle b\in
C(\triangle,\mathbb{R})$. Then, above defined IFS $\displaystyle\mathcal{J}$
has a unique attractor $\displaystyle graph(f^{\alpha})$. The set
$\displaystyle graph(f^{\alpha})$ is the graph of a continuous function
$\displaystyle f^{\alpha}:\triangle\rightarrow\mathbb{R}$, which satisfies
$\displaystyle\left.f^{\alpha}\right|_{V_{1}}=\left.f\right|_{V_{1}}$.
Furthermore, we have the following functional equation

$f^{\alpha}(t)=f(t)+\alpha_{i}\left(f^{\alpha}-b\right)\left(u_{i}^{-1}(t)\right)~{}\forall~{}t\in
u_{i}(\triangle),i\in\\{1,2,3\\}.$ (1)

##  3\. Continuous dependence on parameter $\displaystyle\alpha$ AND
$\displaystyle b$

Let $\displaystyle f\in C(\triangle,\mathbb{R})$. Define a map $\displaystyle
W$ from $\displaystyle S$ to $\displaystyle C(\triangle,\mathbb{R})$ by

$\displaystyle W(\alpha)=f_{b}^{\alpha},$

where $\displaystyle S=\left\\{\alpha\in\mathbb{R}^{3}:|\alpha|_{\infty}\leq
r<1\right.$ and $\displaystyle r$ is a fixed number$\displaystyle\\}$ and
$\displaystyle f_{b}^{\alpha}$ is $\displaystyle\alpha$-fractal function
associated with $\displaystyle f$ with respect to $\displaystyle b$ and the
scale vector $\displaystyle\alpha$.

######  Theorem 3.1.

For fixed $\displaystyle f\in C(\triangle,\mathbb{R})$ and for suitable
$\displaystyle b\in C(\triangle,\mathbb{R})$, the map $\displaystyle
W:S\rightarrow C(\triangle,\mathbb{R})$ is continuous.

###### Proof.

The fixed point theory says that for a fixed a scale vector
$\displaystyle\alpha$, and $\displaystyle b$, the map $\displaystyle
f_{b}^{\alpha}$ is unique. Further, being fixed point of RB- operator,
$\displaystyle f_{b}^{\alpha}$ satisfies the functional equation:

$\displaystyle
f_{b}^{\alpha}(t)=f(t)+\alpha_{i}\left(f_{b}^{\alpha}-b\right)\circ
u_{i}^{-1}(t),~{}\forall~{}t\in u_{i}(\triangle),i\in\\{1,2,3\\}.$

It is obvious that $\displaystyle W$ is well defined. Let
$\displaystyle\alpha\in S$, then from the above functional equation, we have

$\displaystyle
W(\alpha)(t)=f_{b}^{\alpha}(t)=f(t)+\alpha_{i}\left(f_{b}^{\alpha}-b\right)\circ
u_{i}^{-1}(t),~{}\forall~{}t\in u_{i}(\triangle),i\in\\{1,2,3\\}$

and for $\displaystyle\beta\in S$,

$\displaystyle
W(\beta)(t)=f_{b}^{\beta}(t)=f(t)+\beta_{i}\left(f_{b}^{\beta}-b\right)\circ
u_{i}^{-1}(t),~{}\forall~{}t\in u_{i}(\triangle),i\in\\{1,2,3\\}.$

We shall show that $\displaystyle W$ is continuous at $\displaystyle\alpha$.
For this, subtract one from other of the above two equations, for
$\displaystyle t\in u_{i}(\triangle)$, we have

$\displaystyle f_{b}^{\alpha}(t)-f_{b}^{\beta}(t)$
$\displaystyle=\left[f(t)+\alpha_{i}\left(f_{b}^{\alpha}-b\right)\circ
u_{i}^{-1}(t)\right]-\left[f(t)+\beta_{i}\left(f_{b}^{\beta}-b\right)\circ
u_{i}^{-1}(t)\right]$ (2)
$\displaystyle=\left[\alpha_{i}f_{b}^{\alpha}\left(u_{i}^{-1}(t)\right)-\beta_{i}f_{b}^{\beta}\left(u_{i}^{-1}(t)\right)\right]+\left(\beta_{i}-\alpha_{i}\right)b\circ
u_{i}^{-1}(t)$
$\displaystyle=\left[\alpha_{i}f_{b}^{\alpha}-\beta_{i}f_{b}^{\alpha}+\beta_{i}f_{b}^{\alpha}-\beta_{i}f_{b}^{\beta}\right]\circ
u_{i}^{-1}(t)+\left(\beta_{i}-\alpha_{i}\right)b\circ u_{i}^{-1}(t)$
$\displaystyle=\left[\left(\alpha_{i}-\beta_{i}\right)f_{b}^{\alpha}+\beta_{i}\left(f_{b}^{\alpha}-f_{b}^{\beta}\right)\right]\circ
u_{i}^{-1}(t)+\left(\beta_{i}-\alpha_{i}\right)b\circ u_{i}^{-1}(t).$

Now, using triangle inequality and definition of uniform norm, we have

$\displaystyle\left|f_{b}^{\alpha}(t)-f_{b}^{\beta}(t)\right|$
$\displaystyle\leq\left|\left[\left(\alpha_{i}-\beta_{i}\right)f_{b}^{\alpha}+\beta_{i}\left(f_{b}^{\alpha}-f_{b}^{\beta}\right)\right]\circ
u_{i}^{-1}(t)\right|+\left|\left(\beta_{i}-\alpha_{i}\right)b\circ
u_{i}^{-1}(t)\right|$ (3)
$\displaystyle\leq|\alpha-\beta|_{\infty}\left\|f_{b}^{\alpha}\right\|_{\infty}+|\beta|_{\infty}\left\|f_{b}^{\alpha}-f_{b}^{\beta}\right\|_{\infty}+|\beta-\alpha|_{\infty}\|b\|_{\infty}$
$\displaystyle=|\alpha-\beta|_{\infty}\left(\left\|f_{b}^{\alpha}\right\|_{\infty}+\|b\|_{\infty}\right)+|\beta|_{\infty}\left\|f_{b}^{\alpha}-f_{b}^{\beta}\right\|_{\infty}.$

It follows that, for all $\displaystyle t\in\triangle$, we get

$\left|f_{b}^{\alpha}(t)-f_{b}^{\beta}(t)\right|\leq|\alpha-\beta|_{\infty}\left(\left\|f_{b}^{\alpha}\right\|_{\infty}+\|b\|_{\infty}\right)+|\beta|_{\infty}\left\|f_{b}^{\alpha}-f_{b}^{\beta}\right\|_{\infty}.$
(4)

The above implies that

$\left\|f_{b}^{\alpha}-f_{b}^{\beta}\right\|_{\infty}\leq|\alpha-\beta|_{\infty}\left(\left\|f_{b}^{\alpha}\right\|_{\infty}+\|b\|_{\infty}\right)+|\beta|_{\infty}\left\|f_{b}^{\alpha}-f_{b}^{\beta}\right\|_{\infty}.$
(5)

Using $\displaystyle 1-|\beta|_{\infty}\geq 1-r$, finally we have

$\|W(\alpha)-W(\beta)\|_{\infty}=\|f_{b}^{\alpha}-f_{b}^{\beta}\|_{\infty}\leq\frac{|\alpha-\beta|_{\infty}}{1-r}\left(\left\|f_{b}^{\alpha}\right\|_{\infty}+\|b\|_{\infty}\right).$
(6)

Since $\displaystyle\alpha$ is fixed and
$\displaystyle\left\|f_{b}^{\alpha}\right\|_{\infty}$ is bounded, we have
$\displaystyle W$ is continuous at $\displaystyle\alpha$. Since
$\displaystyle\alpha$ was taken arbitrarily, hence, $\displaystyle W$ is
continuous on $\displaystyle S$. ∎

######  Theorem 3.2.

Let $\displaystyle f\in C(\triangle,\mathbb{R})$ and scale vector
$\displaystyle\alpha\in\mathbb{R}^{3}$ with $\displaystyle|\alpha|_{\infty}<1$
and $\displaystyle X_{f}=\left\\{b\in
C(\triangle,\mathbb{R}):\left.b\right|_{V_{0}}=\left.f\right|_{V_{0}}\right\\}$.
Then the map $\displaystyle T:X_{f}\rightarrow C(\triangle,\mathbb{R})$
defined by $\displaystyle T(b)=f_{b}^{\alpha}$ is Lipschitz continuous.

###### Proof.

We know that for a scale vector $\displaystyle\alpha$ and a suitable function
$\displaystyle b:\triangle\rightarrow\mathbb{R}$, the function $\displaystyle
f_{b}^{\alpha}$ is unique. Further, being fixed point of
$\displaystyle\mathrm{RB}$ \- operator, $\displaystyle f_{b}^{\alpha}$
satisfies the functional equation:

$\displaystyle
f_{b}^{\alpha}(t)=f(t)+\alpha_{i}\left(f_{b}^{\alpha}-b\right)\circ
u_{i}^{-1}(t),~{}\forall~{}t\in u_{i}(\triangle),i\in\\{1,2,3\\}.$

It is obvious that $\displaystyle T$ is well defined. Let $\displaystyle
b,c\in X_{f}$ then from the above functional equation, we have

$\displaystyle
T(b)(t)=f_{b}^{\alpha}(t)=f(t)+\alpha_{i}\left(f_{b}^{\alpha}-b\right)\circ
u_{i}^{-1}(t),~{}\forall~{}t\in u_{i}(\triangle),i\in\\{1,2,3\\}.$

and

$\displaystyle
T(c)(t)=f_{c}^{\alpha}(t)=f(t)+\alpha_{i}\left(f_{c}^{\alpha}-c\right)\circ
u_{i}^{-1}(t),~{}\forall~{}t\in u_{i}(\triangle),i\in\\{1,2,3\\}.$

On subtracting one from other of the above two equations, we get for
$\displaystyle t\in u_{i}(\triangle)$

$\displaystyle f_{b}^{\alpha}(t)-f_{c}^{\alpha}(t)$
$\displaystyle=\left[f(t)+\alpha_{i}\left(f_{b}^{\alpha}-b\right)\circ
u_{i}^{-1}(t)\right]-\left[f(t)+\alpha_{i}\left(f_{c}^{\alpha}-c\right)\circ
u_{i}^{-1}(t)\right]$
$\displaystyle=\alpha_{i}\left(f_{b}^{\alpha}-f_{c}^{\alpha}\right)\circ
u_{i}^{-1}(t)+\alpha_{i}(c-b)\circ u_{i}^{-1}(t).$

Now using triangle inequality and definition of uniform norm, we have

$\displaystyle\left|f_{b}^{\alpha}(t)-f_{c}^{\alpha}(t)\right|$
$\displaystyle=\left|\alpha_{i}\left(f_{b}^{\alpha}-f_{c}^{\alpha}\right)\circ
u_{i}^{-1}(t)+\alpha_{i}(c-b)\circ u_{i}^{-1}(t)\right|$
$\displaystyle\leq\left|\alpha_{i}\left(f_{b}^{\alpha}-f_{c}^{\alpha}\right)\circ
u_{i}^{-1}(t)\right|+\left|\alpha_{i}(c-b)\circ u_{i}^{-1}(t)\right|$
$\displaystyle\leq|\alpha|_{\infty}\left\|f_{b}^{\alpha}-f_{c}^{\alpha}\right\|_{\infty}+|\alpha|_{\infty}\|c-b\|_{\infty}.$

The above inquality holds for all $\displaystyle t\in\triangle$, therefore, we
write

$\displaystyle\left\|f_{b}^{\alpha}-f_{c}^{\alpha}\right\|_{\infty}\leq|\alpha|_{\infty}\left\|f_{b}^{\alpha}-f_{c}^{\alpha}\right\|_{\infty}+|\alpha|_{\infty}\|c-b\|_{\infty}.$

This can be recasted as,
$\displaystyle\left\|f_{b}^{\alpha}-f_{c}^{\alpha}\right\|_{\infty}\leq\frac{|\alpha|_{\infty}}{1-|\alpha|_{\infty}}\|b-c\|_{\infty}$.
It follows that

$\displaystyle\|T(b)-T(c)\|_{\infty}\leq\frac{|\alpha|_{\infty}}{1-|\alpha|_{\infty}}\|b-c\|_{\infty},$

which shows that $\displaystyle T$ is a Lipschitz continuous map with
Lipschitz constant
$\displaystyle\frac{|\alpha|_{\infty}}{1-|\alpha|_{\infty}}$. ∎

Next, we plot the $\displaystyle graph(f_{b}^{\alpha})$ for different values
of parameters, that is, original function, base function and scale vector. One
can easily identify the variation in these graphs by changing the values of
parameters. Hence, $\displaystyle graph(f_{b}^{\alpha})$ depends on these
parameters.

$\displaystyle graph(f_{b}^{\alpha})$ at $\displaystyle\alpha=0.1$

$\displaystyle graph(f_{b}^{\alpha})$ at $\displaystyle\alpha=0.3$

$\displaystyle graph(f_{b}^{\alpha})$ at $\displaystyle\alpha=0.6$

$\displaystyle graph(f_{b}^{\alpha})$ at $\displaystyle\alpha=0.9$

Figure 1. $\displaystyle graph(f_{b}^{\alpha})$ for the various values of the
$\displaystyle\alpha$, where $\displaystyle
b(x,y)=\frac{x}{4}+\frac{y}{9}-1.3y(x-0.5)$, $\displaystyle
f(x,y)=\frac{x}{4}+\frac{y}{9}$.

$\displaystyle graph(f_{b}^{\alpha})$ at $\displaystyle\alpha=0.1$

$\displaystyle graph(f_{b}^{\alpha})$ at $\displaystyle\alpha=0.3$

$\displaystyle graph(f_{b}^{\alpha})$ at $\displaystyle\alpha=0.6$

$\displaystyle graph(f_{b}^{\alpha})$ at $\displaystyle\alpha=0.9$

Figure 2. $\displaystyle graph(f_{b}^{\alpha})$ for the various values of the
$\displaystyle\alpha$, where $\displaystyle
b(x,y)=sin(x+3.7)+1.3x-x^{2}y+0.866x^{2}+xy-0.866x$, $\displaystyle
f(x,y)=sin(x+3.7)+1.3x$.

$\displaystyle graph(f_{b}^{\alpha})$ at $\displaystyle\alpha=0.1$

$\displaystyle graph(f_{b}^{\alpha})$ at $\displaystyle\alpha=0.3$

$\displaystyle graph(f_{b}^{\alpha})$ at $\displaystyle\alpha=0.6$

$\displaystyle graph(f_{b}^{\alpha})$ at $\displaystyle\alpha=0.9$

Figure 3. $\displaystyle graph(f_{b}^{\alpha})$ for the various values of the
$\displaystyle\alpha$, where $\displaystyle
b(x,y)=cos(2x+5)+sin(x+2.7)-1.5+1.3x-x^{2}y+0.866x^{2}+xy-0.866x$,
$\displaystyle f(x,y)=cos(2x+5)+sin(x+2.7)-1.5+1.3x$.

$\displaystyle graph(f_{b}^{\alpha})$ at $\displaystyle\alpha=0.1$

$\displaystyle graph(f_{b}^{\alpha})$ at $\displaystyle\alpha=0.3$

$\displaystyle graph(f_{b}^{\alpha})$ at $\displaystyle\alpha=0.6$

$\displaystyle graph(f_{b}^{\alpha})$ at $\displaystyle\alpha=0.9$

Figure 4. $\displaystyle graph(f_{b}^{\alpha})$ for the various values of the
$\displaystyle\alpha$, where $\displaystyle
b(x,y)=cos(100x+5)+sin(x+2.7)-1.5+1.3x-x^{2}y+0.866x^{2}+xy-0.866x$,
$\displaystyle f(x,y)=cos(100x+5)+sin(x+2.7)-1.5+1.3x$.

##  4\. Declaration

Funding. Not applicable.

Conflicts of interest. We do not have any conflict of interest.

Availability of data and material. Not applicable.

Code availability. Not applicable.

Authors’ contributions. Each author contributed equally in this manuscript.

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