11institutetext: Centre for Vision, Speech and Signal Processing (CVSSP),
University of Surrey, Guildford, United Kingdom 22institutetext: Children’s
National Hospital, Washington, DC, USA 33institutetext: Surrey Institute for
People-Centred AI

# SB-SSL: Slice-Based Self-Supervised Transformers for Knee Abnormality
Classification from MRI

Sara Atito 11 0000-0002-7576-5791 Syed Muhammad Anwar 22 0000-0002-8179-3959
Muhammad Awais 1133 0000-0002-1122-0709 Josef Kittler 11 0000-0002-8110-9205

###### Abstract

The availability of large scale data with high quality ground truth labels is
a challenge when developing supervised machine learning solutions for
healthcare domain. Although, the amount of digital data in clinical workflows
is increasing, most of this data is distributed on clinical sites and
protected to ensure patient privacy. Radiological readings and dealing with
large-scale clinical data puts a significant burden on the available
resources, and this is where machine learning and artificial intelligence play
a pivotal role. Magnetic Resonance Imaging (MRI) for musculoskeletal (MSK)
diagnosis is one example where the scans have a wealth of information, but
require a significant amount of time for reading and labeling. Self-supervised
learning (SSL) can be a solution for handling the lack of availability of
ground truth labels, but generally requires a large amount of training data
during the pretraining stage. Herein, we propose a slice-based self-supervised
deep learning framework (SB-SSL), a novel slice-based paradigm for classifying
abnormality using knee MRI scans. We show that for a limited number of cases
(<1000), our proposed framework is capable to identify anterior cruciate
ligament tear with an accuracy of 89.17% and an AUC of 0.954, outperforming
state-of-the-art without usage of external data during pretraining. This
demonstrates that our proposed framework is suited for SSL in the limited data
regime.

###### Keywords:

Self-Supervised Learning Group Masked Model Learning Masked Autoencoders Knee
Abnormality Transformers MRI.

##  1 Introduction

Knee abnormality can arise from a variety of factors including aging, physical
injury, and joint disease. MRI is the standard-of-care for diagnosis of knee
abnormalities [21], where the image contains a wealth of information and the
scanning protocols are safe from a clinical perspective. Knee MRI exams are
among the most widely performed scans in MSK radiology [14]. MSK conditions
arise from a variety of reasons (including sports injury and lifestyle
choices) effecting adults and pediatrics. Both the amount of information
within a knee MRI scan, and the number of such scans performed on a daily
basis put a huge burden on the radiologist and the clinical workforce dealing
with MSK related conditions and knee abnormalities In recent years, machine
learning is the technology of choice in radiology for automated image analysis
and abnormality identification [2]. However, the clinical translation of this
technology is facing challenges such as lack of adequate annotations and
training data. In particular, manual segmentation and data labeling is a labor
intensive and tedious task, which is also effected by inter-rater variability.
The probability of error, accounting for the day-to-day workload on
radiologists, is high and this is where machine learning can benefit the most
by identifying the most critical cases needing immediate attention.

In contrast to Convolutional Neural Networks (CNNs), transformer-based deep
learning models have shown to perform better due to an inherent design
incorporating attention and parallel computing [17]. The success of
transformer based networks in the field of natural language processing (NLP)
is phenomenal and became the default choice in most recent NLP applications.
The recent introduction of vision transformer [10], has resulted in the
translation of some of this success to vision tasks. Training self-supervised
vision transformers for medical applications could alleviate some of the
problems associated with acquiring high quality ground truth labels and hence,
accelerate the research in computer aided diagnosis. However, such networks
require a large training data. Therefore, in Computer Vision (CV) problems,
the default practice is to use a pretrained model on a large supervised data
like ImageNet-1K, before fine tuning for a specific downstream task with
limited data [4].

Recently, self-supervised pretraining of deep neural networks without using
any labels has outperformed supervised pretraining in CV [3, 5]. This
phenomenal shift in CV is less investigated in medical image analysis domain.
We argue that recent SSL approaches are ideally suited for medical image
analysis, since medical data are an order of magnitude smaller than natural
images due to several reasons, including privacy concerns, expensive
annotation, rarity of certain diseases, etc. Hence for medical applications,
SSL can lead the way for a wider adoption of such techniques in domains where
labels are not available or are difficult to acquire [1]. Therefore, the
purpose of this study is to investigate: 1) is ImageNet-1K pretraining needed
for medical imaging? 2) can we perform self-supervised pretraining on a small
medical data and outperform large scale out of distribution supervised
pretraining? If successful this will form the basis for SSL for medical
imaging in limited data regimes. Towards this, we propose a slice-based self-
supervised deep learning framework (SB-SSL) for abnormality classification
using knee MRI, where our main contributions are:

  * •

We propose a novel slice based self-supervised transformer model (SB-SSL) for
knee abnormality classification using magnetic resonance imaging data.

  * •

The model is pretrained from scratch on limited data without labels and fine
tuned for the downstream knee abnormality classification task with state-of-
the-art performance.

  * •

Our experimental results show that, when trained using the group masked model
learning (GMML) paradigm, SSL can be successfully applied for medical image
analysis with limited data/label.

##  2 Related Works

In [6], a deep learning based method was presented for the detection of
abnormalities in knee MRI. The publicly available MRNet data was presented,
along with an AlexNet [16] based model for classifying abnormalities, meniscal
tear, and anterior cruciate ligament (ACL) tear. This was among the first
approaches where deep learning was applied to this task, and since then has
been used in multiple studies to further improve the classification
performance [20, 24, 13, 11].

A CNN based self supervised training paradigm was developed, where solving the
jigsaw puzzle was used as the pre-text task [20]. In the downstream task, ACL
tear was classified with an accuracy of $76.62\%$ and an area under the curve
(AUC) of 0.848 using the sagittal plane. In [24], efficiently-layered network
(ELNet) was proposed where the model reduced the number of parameters compared
to AlexNet, and utilized individual slice views for classification of meniscus
(coronal) and ACL (axial) tears. An accuracy of 0.904 with an AUC of 0.960 was
achieved in detecting the ACL tear. This performance was improved by adding a
feature pyramid network and pyramidal detail pooling to ELNet [11]. An AUC of
0.976 and an accuracy of 0.886 was achieved in ACL tear classification task.
However, both these methods are based on supervised training. Meniscus tears
were identified using a deep learning model and compared with manual
evaluation [13]. An accuracy of $95.8\%$ was achieved for an internal
validation set, however the model was not evaluated on any of the publicly
available data.

In general, it should be noted that for methods that report higher
performance, training is based on the availability of ground truth labels.
Whereas for self supervised training, which could alleviate this burden, the
model performance drops. We propose, for the first time, a transformer based
self-supervised framework for knee abnormality classification using MRI. Our
innovative training paradigm use self-supervised training and shows that such
a framework can be effectively used even when the size of training data is
relatively small.

Figure 1: Proposed self-supervised learning approach.

##  3 Methodology

In this work, we introduce a general slice-based self-supervised vision
transformer for knee MRI medical records. The system diagram of the proposed
approach is shown in Figure 1. Transformers [25] have shown great success in
various NLP and CV tasks [9, 22, 7, 4, 3, 5, 27, 8] and are the basis of our
proposed framework.

###  3.1 Vision Transformer

Vision transformer [10] receives, as input, a feature map from the output of a
convolutional block/layer with $K$ kernels of size $p\times p$ and stride
$p\times p$. The convolutional block takes an input image
$\mathbf{x}\in\mathcal{R}^{C\times H\times W}$ and converts it to feature maps
of size $\sqrt{n}\times\sqrt{n}\times K$, where $C$, $H$, and $W$ are the
number of channels, height, and width, of the input image, ($p\times p$) is
the patch size, and $n$ is the number of patches, i.e.,
$n=\frac{H}{p}\times\frac{W}{p}$. Learnable position embeddings are added to
the patch embeddings as an input to the transformer encoder to retain the
relative spatial relation between the patches.

The transformer encoder consists of $L$ consecutive Multi-head Self-Attention
(MSA) and Multi-Layer Perceptron (MLP) blocks. The MSA block is defined by $h$
self-attention heads, where each head outputs a sequence of size $n\times d$.
The self attention mechanism is based on a trainable triplet (query, key, and
value). Each query vector in $\mathbf{Q}\in\mathcal{R}^{n\times d}$ for a
given head is matched against a set of key vectors
$\mathbf{K}\in\mathcal{R}^{n\times d}$, scaled by the square root of $d$ to
have more stable gradients as the dot product of $q$ and $k$ tend to grow
large in magnitude, resulting in vanishing gradients and a slowdown of
learning. After applying softmax, the output is then multiplied by a set of
values $\mathbf{V}\in\mathcal{R}^{n\times d}$. Thus, the output of the self-
attention block is the weighted sum of $\mathbf{V}$ as shown in equation 1.
The output sequences across heads are then concatenated into $n\times(d\times
h)$, and projected by a linear layer to a $n\times K$ sequence. The MLP block
consists of two point-wise convolution layers with GeLU [12] non-linearity.

${\rm SelfAttention}(\mathbf{Q},\mathbf{K},\mathbf{V})={\rm
Softmax}(\frac{\mathbf{Q}\mathbf{K}^{T}}{\sqrt{d}})\mathbf{V}.$ (1)

###  3.2 Self-supervised Pretraining

We leverage the strength of the transformers and train it as an autoencoder
with a light decoder employing GMML [4, 5]. Starting with the vanilla
transformer autoencoder, the model is pretrained as an autoencoder to
reconstruct the input image, i.e., $D(E(\textbf{x}))=\textbf{x}$, where x is
the input image, $E$ is the encoder which is vision transformer in our case,
and $D$ is a light reconstruction decoder. Due to the strength of
transformers, it is expected that the model will perfectly reconstruct the
input image after a few training epochs. Indeed, this is attributed to the
fact that without a proper choice of constraints, autoencoders are capable of
learning identity mapping, i.e., memorizing the input without learning any
useful discriminative features.

To promote the learning of context and learn better semantic representations
of the input images from the transformer-based autoencoder, we apply several
transformations to local patches of the image. The aim is to recover these
masked local parts at the output of the light decoder. In doing so, especially
with a high percentage of corruption (up to 70%), the model implicitly learns
the semantic concepts in the image and the underline structure of the data in
order to be able to recover the image back. Image in-painting is a simple but
effective pre-text task for self-supervision, which proceeds by training a
network to predict arbitrary transformed regions based on the context.

The objective of image reconstruction is to restore the original image from
the corrupted image. For this task, we use the $\ell 1$-loss between the
reconstructed image and the original image in an end-to-end self-supervised
trainable system as shown in equation 2. Although, $\ell 2$-loss generally
converges faster than $\ell 1$-loss, it is prone to over-smooth the edges for
image restoration [26]. Therefore, $\ell 1$-loss is more commonly used for
image-to-image processing.

$\mathcal{L}(\mathbf{W})=\sum_{k}^{b}\left(\sum_{i}^{H}\sum_{j}^{W}\mathds{1}_{[\textbf{M}^{k}_{i,j}=1]}|\mathbf{x}_{i,j}^{k}-\mathbf{\bar{x}}^{k}_{i,j}|\right),$
(2)

where $\mathbf{W}$ denotes the parameters to be learned during training, $b$
is the batch size, $\mathbf{M}$ is a binary mask with 1 indicating the
manipulated pixels, and $\mathbf{\bar{x}}$ is the reconstructed image. To
further improve the performance of the autoencoder, we introduced skip
connections from several intermediate transformer blocks to the decoder. These
additional connections can directly send the feature maps from the earlier
layers of the transformers to the decoder which helps to use fine-grained
details learned in the early layers to construct the image. Besides, skip
connections in general make the loss landscape smoother which leads to faster
convergence. Further, the reconstructed image $\mathbf{\bar{x}}$ is obtained
by averaging the output features from the intermediate blocks from the
transformer encoder ($E(.)$) and feeding the output to a light decoder
($D(.)$) represented mathematically as
$\mathbf{\bar{x}}=D\left(\textstyle\sum_{i\in\mathcal{B}}E_{i}(\mathbf{\hat{x}})\right)$,
where $E_{i}(.)$ is the output features from block $i$ and $\mathcal{B}$ is a
pre-defined index set of transformer blocks that are included in the decoding
process. Herein, we set $\mathcal{B}$ to $\\{6,8,10,12\\}$.

As for the decoder, unlike CNN-based autoencoders which require expensive
decoders consisting of convolutional and transposed convolution layers, the
decoder in the transformer autoencoder can be implemented using a light
decoder design. Specifically, our decoder consisted of two point-wise
convolutional layers with GeLU non-linearity and a transposed convolutional
layer to return back to the image space. Since the backbone, i.e., vision
transformer, and the light decoder are isotropic, some of the transformer
blocks may act as decoder and hence, heavy and computationally expensive type
of decoders are not required.

##  4 Experimental Results

To demonstrate the effectiveness of our proposed self-supervised vision
transformer on medical images, we employed the MRNet dataset [6]. The dataset
consists of 1,370 knee MRI records, split into a training set of 1,130 records
of 1,088 patients and a validation set of 120 records of 111 patients. Each
MRI is labeled according to the presence/absence of meniscus tear, ACL tear,
or any other abnormality in the knee. In this work, we tackled the ACL tear
identification problem using the Sagittal plane. The dataset is highly
imbalanced with only 208 MRIs representing ACL tear.

###  4.1 Implementation Details

In our work, we employed the ViT Small (ViT-S) variant of the transformer [23]
with $256\times 256$ input image size. For optimization of the transformer
parameters during self-supervised pre-training, we used the Adam optimizer
[19] with a momentum of $0.9$. The weight decay follows a cosine schedule [18]
from $0.04$ to $0.4$, and a base learning rate of $5e^{-4}$. All models were
pre-trained employing $4$ Nvidia Tesla V100 32GB GPU cards with $64$ batch
size per GPU.

Simple data augmentation techniques were applied like random cropping, random
horizontal flipping, random Gaussian blurring, and random adjusting of the
sharpness, contrast, saturation, and the hue of the image. The augmented image
was further corrupted by randomly replacing patches from the image with zeros,
with a replacement rate of up to 70% of the image pixels.

For fine-tuning, we drop the light decoder and fine tune the pre-trained model
by passing the volume, slice by slice, to the transformer encoder. The outputs
of the class tokens corresponding to each slice are then concatenated to
obtain $y\in\mathcal{R}^{f\times K}$, where $f$ is the number of slices. After
that, the features $y$ are fed to a fully connected layer with $K$ nodes
followed by GeLU non-linearity, followed by a linear layer with $2$ nodes
corresponding to the presence/absence of the ACL tear. As the dataset is
highly imbalanced, we used oversampling on the training set to balance the
dataset. Specifically, we over-sample the minority class, i.e., presence of
ACL tear, to match the number of the majority class. Finally, we applied the
same optimization parameters and data augmentations used for the self-
supervised training.

Further, we employed ensemble learning [15]. Generally, neural networks have
high variance due to the stochastic training approach that makes them
sensitive to the nature of the training data. The models may find a different
set of weights each time they are trained, which in turn may produce different
predictions. To mitigate this issue, for each experiment, we trained $5$
models with different weight initialization and combined the predictions from
these models. Not only this approach reduced the variance of the predictions,
but also resulted in predictions that were better than any single model.

###  4.2 Results

It is well known that transformers are data-hungry which make them hard to
train, mostly, due to the lack of the typical inductive bias of convolution
operations. Consequently, the common protocol for self-supervised learning
with transformers is to pretrain the model on a large scale dataset, such as
ImageNet or even larger datasets. We compare our proposed approach with the
state-of-the-art SSL methods when the pretraining and the fine-tuning are
performed only on the MRNet dataset. Table 1 shows that our method outperforms
the state-of-the-art with a large margin with an improvement of 12.6% top-1
validation accuracy on the ACL tears classification task employing the
sagittal plane. Most importantly, without using any external data, our
proposed approach outperforms the competitors that are pre-trained with
ImageNet-1K marking a milestone for the medical domain. The receiver operating
characteristic (ROC) curve for three transformer variants, ViT-Tiny, ViT-
Small, and ViT-Base are shown in Figure 2, where ViT-T performs the best.

Table 1: Comparison with SOTA on ACL tears classification employing sagittal
plane.

Method | Backbone | # params | ACL Tear (Sagittal plane)
---|---|---|---
|  Accuracy |  AUC
|  |  (%) |
Training using only the given dataset
Random Init |  CNN |  77M |  71.67 |  0.754
Random Init |  ViT-S |  21M |  70.00 |  0.721
[20] |  CNN |  77M |  76.62 |  0.848
[20] \+ noise |  CNN |  77M |  75.83 |  0.817
SB-SSL (Ours) |  ViT-T |  5M |  85.83 |  0.952
SB-SSL (Ours) |  ViT-S |  21M |  88.33 |  0.954
SB-SSL (Ours) |  ViT-B |  86M |  89.17 |  0.954
Transfer learning from ImageNet-1K dataset
MRNet [6] |  AlexNet |  61M |  86.63 |  0.963

Figure 2: ROC curves of the classification of ACL tears employing different
vision transformer architectures.

###  4.3 Ablation Studies

In this section, we investigate the effect of different recipes of the
proposed approach, such as the effect of longer pretraining, the size of the
model, and the type of image corruption during the pretraining stage. Further,
we show the interpretability of the system by visualizing the attention of the
trained models.

Effects of Longer Pretraining and Model Size. In Figure 3, we show the
performance of the proposed approach when pretrained for longer duration
across different vision transformer architectures. The x-axis represents the
number of self-supervised pretraining epochs, with zero indicating that the
model was not pretrained, i.e., training from scratch. From the reported
results, it is evident that the training from random initialization has
produced a lower accuracy as the amount of data available is insufficient to
train the transformer. The results significantly improved when the models were
pretrained without any external data by 25.8%, 18.3%, and 13.3% employing
ViT-T, ViT-S, and ViT-B, respectively, compared to training from scratch.
Another observation is that pre-training the self-supervised for longer and
employing bigger transformer architectures contribute positively to the
performance of the proposed approach.

(a) Accuracy

(b) AUC

Figure 3: Top-1 validation accuracies and AUC of the MRNet validation set
across different vision transformer architectures. The x-axis represents
number of epochs used for pretraining.

The Effects of Different Types of Corruption: We first investigate the effect
of training a vanilla transformer autoencoder, where the model is pretrained
as an autoencoder to reconstruct the input image. As expected, after
finetuning, the performance was similar to the performance of the model
trained from scratch. Following, we investigate the effect of applying
different types of image inpainting including: random masking by replacing a
group of connected patches from the image with zeros, ones, or noise. Samples
of the different types of corruption are shown in Figure 4 along with the
reconstructed images after the pretraining stage. The performance when
pretraining the models with different types of corruption is on par, with
noise being marginally better than others.

(a) Zeros

(b) Ones

(c) Noise

Figure 4: Samples of different types of corruption. The rows represent the
original images, corrupted images, and the reconstructed images after the
pretraining stage, respectively.

Attention Visualization. To verify that the model is learning pertinent
features, in Figure 5, we provide visualizations of the self-attention
corresponding to the class token of the $10^{\text{th}}$ layer of the vision
transformer. To generate the attention for an image, we compute the normalized
average over the self-attention heads to obtain a $16\times 16$ tokens. The
tokens are then mapped to a color scheme, up-sampled to $256\times 256$
pixels, and overlaid with the original input image. For visualization, we
selected the mid slice of randomly selected MRI volumes from the MRNet
validation set. We observe that the attention is clearly focusing on the area
of interest, corresponding to the main part of the MRI slice on which the
detection of ACL tears is performed.

Figure 5: Self-attention visualizations from the ViT-S model finetuned on the
ACL tears task employing the sagittal plane.

##  5 Conclusion

We proposed a novel framework SB-SSL, pre-trained in a self-supervised manner
for knee abnormality classification. We established a new benchmark in SSL for
MRI data, where pretraining on a large supervised data was not required. The
state-of-the-art performance, with an accuracy of 89.17% in ACL tear
classification, shows that our proposed method can be employed in MR image
classification even when the data are limited and ground truth labels are not
available.

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# Learning by Cheating : An End-to-End Zero Shot Framework for Autonomous
Drone Navigation

Praveen Venkatesh, Viraj Shah*, Vrutik Shah*, Yash Kamble*, Joycee Mekie
Indian Institute of Technology, Gandhinagar, India

###### Abstract

This paper proposes a novel framework for autonomous drone navigation through
a cluttered environment. Control policies are learnt in a low-level
environment during training and are applied to a complex environment during
inference. The controller learnt in the training environment is tricked into
believing that the robot is still in the training environment when it is
actually navigating in a more complex environment. The framework presented in
this paper can be adapted to reuse simple policies in more complex tasks. We
also show that the framework can be used as an interpretation tool for
reinforcement learning algorithms.

††*Equal contribution

##  1 Introduction

Piloting a drone through a crowded environment is a very complex task. Expert
human pilots use first-person-view images and are able to navigate through a
given environment successfully, even at high speeds. The inherent instability
of the quadrotor platform, combined with moving through a cluttered
environment, makes navigation safety of utmost importance. Even slight
collisions with the environment can cause irrecoverable failures, making the
task of navigating even harder. The need for such systems is emphasised in
situations such as disasters, construction, surveying, among others. Expert
pilots often train for several years before gaining mastery over their
technique, leading to high training costs. In this work, we attempt to explore
the possibility of autonomously navigating through a cluttered environment
safely.

Traditionally, the problem of drone navigation is tackled through separate
state estimation and control. Conventional perception methods such as SLAM [1]
for state estimation are often unreliable in dynamically changing
environments. Deep learning-based perception methods have shown significantly
improved robustness compared to traditional methods leading to their
widespread adoption [2]. However, the dual-stage pipeline leads to a
decoupling between perception and control. This decoupling may lead to
unrecoverable failures in the event of errors in either of the two stages.
Hence, there is a need for an end-to-end framework to improve safety and
improve system performance.

Our framework takes inspiration from human flying. Pilots maintain a rough
estimate of the drone within the world, paying attention to essential aspects
of the environment to maintain a safe and smooth flight. These estimates are
not always entirely accurate but are merely imprecise estimates of potentially
safe vs unsafe areas. We propose a novel end-to-end framework based on this
idea to learn control policies based on representations in one environment
that are later transferred to different environments. This is where the learnt
policy network is cheated into believing that it still exists in the
environment where it was trained, when it is operating on a real environment.

##  2 Our Approach

Figure 1: Perception - Variational Autoencoder compresses observed inputs into
a low dimensional space on which a control policy is learnt

Figure 2: Cheating the network - We replace the encoding stage with a new
trainable encoder that learns to map inputs from observed space to the fake
environment.

Figure 3: Controller block - A 3 layer MLP is provided with concatenated
inputs of 1) Sample from the latent space of the encoder 2) Hidden state of an
unrolled LSTM

In this work, we present a novel zero shot, end-to-end framework for drone
navigation. We formulate the problem as a mapping from $\mathrm{R}^{H\times
W\times 3}\rightarrow\mathrm{R}^{4}$, where the inputs are a three-channel
front-facing RGB image, and the outputs are three-dimensional velocity
commands and the desired yaw rate.

We divide our approach into three steps: 1) Learning a compressed
representation of a simulated environment (referred to as fake from here on)
2) Learning a control policy based on compressed representations & 3) Cheating
the learnt controller by learning to predict the compressed features on a
different real environment. By cheating the learnt controller, we essentially
force the network to believe that it is still navigating through the fake
environment, whereas in reality, it is navigating through a real environment.
Each of the steps is trained independently, freezing appropriate weights as
explained in the following subsections.

###  2.1 Learning Compressed Representations

Representation learning for robotic applications has gained increasing
interest in recent years to reduce the dimensionality of a given problem
robustly. In this work, we utilize AirSim [3], a high fidelity drone-based
environment, to train our representations. We spawn rectangular gates
procedurally (as seen in fig 2) and use segmented images of these gates to
learn a low dimensional representation from RGB images. This low dimensional
representation is learnt as a smooth $k$-dimensional space using a
convolutional variational auto-encoder(VAE). i.e., We learn a
$\mathrm{R}^{H\times W\times 3}\rightarrow\mathrm{R}^{k}$ mapping from the
input RGB images. This forms our compressed representation space on which a
control policy is later learnt. A block diagram of the network can be seen in
fig 2. The VAE is trained separately on the fake environment, and all
parameters are trainable.

Figure 4: a) Real environment to which the policy is transferred. b) FPV view
of the drone in the real environment. c) Fake environment on which the policy
is learnt. d) FPV view of the drone in the fake environment

###  2.2 Learning a Control Policy

A diagram of the control network can be seen in fig 3. We use an unrolled
single-cell LSTM to encode any time-dependent data. The LSTM takes as input a
sample from the latent space $z$ of the VAE. The hidden state of the LSTM -
$h$ along with the latent sample $z$ is concatenated to form the inputs to a
simple three-layered MLP (Multilayer Perceptron). The MLP outputs a
$\mathrm{R}^{4}$ dimensional output representing the velocities and desired
yaw rate of the drone. The parameters of the encoder of the VAE are fixed
during training. We collect expert trajectories from the fake environment for
training. Due to the low dimensional nature of the problem, we train the
network using genetic evolution.

After this step, the drone can successfully navigate through a series of gates
placed in the AirSim environment. However, we are interested in transferring
this learnt policy onto an entirely different environment, where there are no
gates. In essence, we want the drone to believe that it is still operating in
the fake environment with gates, but in actuality operating in a different
environment without gates.

###  2.3 Cheating the Network

A block diagram of the cheating process is illustrated in fig 2. Here, we
freeze all of the weights of the network thus far and replace the encoding
stage containing a different network with a pre-trained ResNet-18 [4]
backbone. We place gates in the new environment and train the encoding stage
in a supervised manner. Using this method, the new encoder network attempts to
cheat the controller, and hence the drone, into performing actions without
ever experiencing the actual environment. We use high fidelity indoor
environments provided by AirSim and PEDRA [5] to train the encoder network.
The transferred policy can be executed on all indoor environments available on
PEDRA, even if the drone has not explored any of them prior to testing.

##  3 Preliminary Results

Figure 5: Cheated network - RGB inputs from real environment (top) vs predicted state in fake environment (bottom). The bottom row indicates what the drone believes to observe when it is executing the policy in an unexplored environment Method | Mean Distance before crash (m)
---|---
Ours | $14.86$
Regression | $15.43$
Table 1: Experimental results

To experimentally validate our framework we provide some preliminary results
by training the original VAE on the fake environment generated from AirSim,
where a control policy for controlling the steering angle was learnt via
behavioural cloning. We then transferred this learnt policy into the
Building_99 environment on AirSim. We compare our framework with a policy
learnt directly via steering regression from RGB images using a pre-trained
ResNet-18 backbone. We compare the mean distance to crash to determine whether
the policy is transferred appropriately. Table 1 shows that our method is
comparable to a policy trained directly on the indoor environment, without
ever interfacing with that environment.

##  4 Related Work

Classical techniques utilize visual-inertial odometry for state estimation and
model-based path planning for control of the quadcopter. World Models [6]
explores learning separate networks for the environment representation and
controls, instead of the end-to-end paradigm. [7] uses a fully learning-based
pipeline for drone navigation by introducing an intermediate learnt
representation for perception. [8] replaced planning and control with
corresponding networks to navigate through a narrow gap. [9] have proposed a
general shallow network to replace the traditional ”map-localize-plan”
approach that generalizes well for indoor corridors and parking lots. [10]
have used a novel approach of negative learning for navigating drones in a
cluttered environment. Sim-to-real papers [11], [12] uses popular deep
learning technique of transfer learning [13] to transfer knowledge acquired in
simulation to real world. [13]

Although these methods attempt to solve the problem of end-to-end autonomous
drone navigation, to the best of our knowledge, our work is the first to
present a cheating based methodology for the problem by transferring learnt
policies through different domains.

##  5 Conclusion and Outlook

Figure 6: Cheated network in a more complex case - Depth map inputs (top) vs
predicted state in fake environment (bottom).

In this paper, we present a framework that aims to reuse learnt policies to
act in different environments. By cheating the learnt policy into believing
that it exists in an imagined world, the learnt policy can be directly
transferred onto an unseen environment. The strength of this method is twofold
- 1) Ability to transfer low-level control policies to more complex problems &
2) Ability to act as an interpretation tool (such as in Fig 6 and Fig 5).

For example, if we consider the toy problem of the popular CarRacing-v0
environment by OpenAI Gym [14], a policy learnt on this network can be
transferred directly to a more complex situation such as the CARLA [15]
environment using the method presented in this paper. Moreover, in its
converse scenario, it is possible to transfer a learnt policy on the CARLA
environment to the simpler CarRacing-v0 environment in order to interpret and
visualize the policy via the decoding stage.

The framework presented in this paper can also be extended to work as a semi-
guided navigation system for drones. For example, by conditioning the VAE on a
3D intent vector, we can direct where to place a particular gate in 3D space.
This intent vector can be computed based on a map as a local guiding vector in
a global space. The method can also be adapted to function in tandem with a
text-based guidance system, where guiding intent vectors are generated from
visual-textual based inputs via captioning networks.

## References

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The gauge-fixed action for the Schwarzschild mechanics is given by (4.10). The
potential term being a mere constant, one can omit it and consider the
modified mechanical action

$S_{\epsilon}[V_{1},V_{2};\tau]=\epsilon
c\ell_{P}\int\textrm{d}\tau\left[\frac{V_{2}\dot{V}^{2}_{1}-2V_{1}\dot{V}_{1}\dot{V}_{2}}{2V^{2}_{1}}\right]$
(C.1)

Then, the ED lift is a four dimensional manifold with line element

$\textrm{d}s^{2}=g_{AB}\textrm{d}X^{A}\textrm{d}X^{B}=2\textrm{d}u\textrm{d}w+\epsilon
c\ell_{P}\left(\frac{V_{2}}{V^{2}_{1}}\textrm{d}V^{2}_{1}-\frac{2\textrm{d}V_{1}\textrm{d}V_{2}}{V_{1}}\right)$
(C.2)

Just as for the FLRW cosmological system, this manifold turns out to be
conformally flat, i.e its Weyl and Cotton tensors vanish, and it is thus
maximally symmetric, i.e. it possesses fifteen CKVs333Notice that if we had
kept the constant term in the action, the ED lift would have been given by
$\textrm{d}s^{2}=\frac{2\epsilon}{\ell^{2}_{s}}\textrm{d}u^{2}+2\textrm{d}u\textrm{d}w+\epsilon
c\ell_{P}\left(\frac{V_{2}}{V^{2}_{1}}\textrm{d}V^{2}_{1}-\frac{2\textrm{d}V_{1}\textrm{d}V_{2}}{V_{1}}\right)$
(C.3) One can see that the potential term being constant, $\partial_{u}$ is
still a Killing vector. Moreover, one can show that this additional term does
not spoil the conformally flatness of the lift such that this alternative
version admits again fifteen CKVs. While the form of these CKVs is slightly
different starting from this lift, the conserved charges one builds from the
CKVs are the same.. They can be organized w.r.t. the commutation properties
with the constant null vector field $N^{A}\partial_{A}=\partial_{w}$.

The CKVs which commute with $\partial_{w}$ are given by

$\displaystyle Q_{0}^{A}\partial_{A}$
$\displaystyle=V_{2}\partial_{V_{2}}+u\partial_{u}\,,$ (C.4) $\displaystyle
Q_{-}^{A}\partial_{A}$ $\displaystyle=2V_{2}\partial_{w}+\frac{2u}{\epsilon
c}\left(V_{1}\partial_{V_{1}}+V_{2}\partial_{V_{2}}\right)+\frac{1}{\epsilon
c}u^{2}\partial_{u}\,,$ (C.5)

and act on the metric of the lift as

$\mathcal{L}_{Q_{0}}g_{\mu\nu}=g_{\mu\nu}\;,\qquad\mathcal{L}_{Q_{-}}g_{\mu\nu}=\frac{4u}{c}g_{\mu\nu}$
(C.6)

We also identify six Killing vectors which are given by

$\displaystyle\begin{array}[]{c}\left|\begin{array}[]{l}P_{+}^{A}\partial_{A}=2\sqrt{V_{1}}\partial_{V_{1}}+\frac{V_{2}}{\sqrt{V_{1}}}\partial_{V_{2}}\,,\\\
P_{-}^{A}\partial_{A}=\sqrt{V_{1}}\partial_{V_{2}}\,,\\\
\end{array}\right.\qquad\left|\begin{array}[]{l}B_{+}^{A}\partial_{A}=\frac{2V_{2}}{\sqrt{V_{1}}}\partial_{w}+\frac{u}{\epsilon
c}\left(2\sqrt{V_{1}}\partial_{V_{1}}+\frac{V_{2}}{\sqrt{V_{1}}}\partial_{V_{2}}\right)\,,\\\
B_{-}^{A}\partial_{A}=2\sqrt{V_{1}}\partial_{w}+\frac{u}{\epsilon
c}\sqrt{V_{1}}\partial_{V_{2}}\,,\\\ \end{array}\right.\\\ \\\
Q_{+}^{\alpha}\partial_{\alpha}=\partial_{u}\,,\qquad
J^{\alpha}\partial_{\alpha}=V_{1}\partial_{V_{1}}\;,\end{array}$ (C.14)

Together with the null vector $N^{A}\partial_{A}=\partial_{w}$, they form the
Schrödinger algebra from which one can derive the Schrödinger observables
presented in Section 4.2.1.

Three other CKVs are easily identified as

$\displaystyle\tilde{Q}_{-}^{A}\partial_{A}$
$\displaystyle=2V_{2}\partial_{u}+\frac{2w}{\epsilon
c}\left(V_{1}\partial_{V_{1}}+V_{2}\partial_{V_{2}}\right)+\frac{1}{\epsilon
c}w^{2}\partial_{u}\,,$ (C.15) $\displaystyle\tilde{B}_{+}^{A}\partial_{A}$
$\displaystyle=\frac{2V_{2}}{\sqrt{V_{1}}}\partial_{u}+\frac{w}{\epsilon
c}\left(2\sqrt{V_{1}}\partial_{V_{1}}+\frac{V_{2}}{\sqrt{V_{1}}}\partial_{V_{2}}\right)$
(C.16) $\displaystyle\tilde{B}_{-}^{A}\partial_{A}$
$\displaystyle=2\sqrt{V_{1}}\partial_{u}+\frac{w}{\epsilon
c}\sqrt{V_{1}}\partial_{V_{2}}.$ (C.17)

The first one acts on the lift as

$\mathcal{L}_{\tilde{Q}_{-}}g_{\mu\nu}=\frac{4w}{c}g_{\mu\nu}\;,$ (C.18)

They are directly obtained as the dual of the CKVs $Q_{-}^{A}\partial_{A}$ and
$B^{A}_{\pm}\partial_{A}$ under the switch $u\leftrightarrow w$. The dual of
the CKV $Q_{0}^{A}\partial_{A}$ is recovered thanks to the generator of
dilatation in the $(u,v)$-plane $\mathcal{O}=u\partial_{u}-w\partial_{w}$.
Finally, the last two CKVs are

$\displaystyle Y^{A}_{+}\partial_{A}$
$\displaystyle=\frac{2V_{2}}{\sqrt{V_{1}}}\left(u\partial_{u}+w\partial_{w}\right)+2uw\sqrt{V_{1}}\partial_{V_{1}}+\left(uw+2V_{2}\right)\frac{V_{2}}{\sqrt{V_{1}}}\partial_{V_{2}}$
(C.19) $\displaystyle Y^{A}_{-}\partial_{A}$
$\displaystyle=2\sqrt{V_{1}}\left(u\partial_{u}+w\partial_{w}\right)+4V_{1}\sqrt{V_{1}}\partial_{V_{1}}+\left(uw+2V_{2}\right)\sqrt{V_{1}}\partial_{V_{2}}$
(C.20)

Altogether, these fifteen vectors fields from the ${\mathfrak{so}}(4,2)$ Lie
algebra associated to the Schwarzschild black hole mechanics.

###  CKVs of the superspace

Let us now compare with the conformal isometries of the superspace. The super-
metric of the Schwarzschild mechanics is given by

$\textrm{d}s^{2}=g_{ab}\textrm{d}x^{a}\textrm{d}x^{b}=\frac{V_{2}}{V^{2}_{1}}\textrm{d}V^{2}_{1}-\frac{2\textrm{d}V_{1}\textrm{d}V_{2}}{V_{1}}$
(C.21)

Notice that the Schwarzschild dynamics does not correspond to a null geodesic
on this geometry because the action contains a constant potential term.
Therefore, depending on the sign of this contribution, the dynamics is either
mapped to time-like or space-like geodesics of the super-metric (C.21). Thus,
as explained in the first section, the CKVs of this geometry do not provide
directly conserved charges for the system, and one needs to add corrective
terms proportional to the mass of the system, i.e. proportional to the
constant potential in the present case. With this subtlety in mind, we now
present the KVs and the CKVs of this super-metric.

First, one can check that this superspace has constant curvature such that it
admits three Killing vectors. They are given by

$\displaystyle
P^{a}_{+}\partial_{a}=2\sqrt{V_{1}}\partial_{V_{1}}+\frac{V_{2}}{\sqrt{V_{1}}}\partial_{V_{2}}\;,\qquad
P^{a}_{-}\partial_{a}=\sqrt{V_{1}}\partial_{V_{2}}\;,\qquad
J^{a}\partial_{a}=V_{1}\partial_{V_{1}}$ (C.22)

which correspond to the KVs of the ED lift. Being Killing vectors, they are
directly associated to strong Dirac observables of the system and no
corrective terms need to be added.

Now, let us derive the general expression of the CKVs and the associated
charges. Just as the flat FLRW cosmological model discussed in Section 3.4.2,
the super-metric is two dimensional and therefore possesses an infinite tower
of CKVs. To derive the associated charges, let us introduce the new
coordinates

$x_{+}=\sqrt{V_{1}}\;,\qquad x_{-}=-\frac{V_{2}}{\sqrt{V_{1}}}$ (C.23)

such that the (rescaled) super-metric reads

$\textrm{d}s^{2}=\textrm{d}x_{+}\textrm{d}x_{-}$ (C.24)

The general expressions for the CKVs are given by

$X^{a}_{F}\partial_{a}=F(x_{+})\partial_{+}\;,\qquad
X^{a}_{G}\partial_{a}=G(x_{-})\partial_{-}$ (C.25)

where $F$ and $G$ are arbitrary functions. In order to compute the associated
charges, we first write the momenta $p_{\pm}$ which are given by

$\displaystyle
p_{+}=2\sqrt{V_{1}}P_{1}+\frac{V_{2}P_{2}}{\sqrt{V_{1}}}\;,\qquad
p_{-}=-\sqrt{V_{1}}P_{2}$ (C.26)

They are nothing else but the translational charges entering in the
Schrödinger algebra. Then, one obtains the following set of observables

$X^{+}_{F}=p_{+}F(\sqrt{V_{1}})\;,\qquad
X^{-}_{G}=p_{-}G\left(\frac{V_{2}}{\sqrt{V_{1}}}\right)$ (C.27)

As expected, these phase space functions are not conserved on shell. A
straightforward computation shows that

$\\{X^{+}_{F},h\\}=F^{\prime}\tilde{h}\;\,\hat{=}\,-\frac{c\ell_{P}F^{\prime}}{\ell^{2}_{s}}\;,\qquad\\{X^{-}_{G},h\\}=2G^{\prime}\tilde{h}\;\,\hat{=}\,-\frac{2c\ell_{P}G^{\prime}}{\ell^{2}_{s}}$
(C.28)

where $\tilde{h}$ is the shifted hamiltonian whose value can be interpreted as
the effective mass of the test particle moving on the superspace. To obtain
well defined weak Dirac observables, we slightly correct the charges as
follows. Consider

$\displaystyle W^{+}_{F}$
$\displaystyle=X_{F}-\frac{\alpha}{2\sqrt{V_{1}}P_{2}}F(\sqrt{V_{1}})\;,\;\;\qquad\text{with}\qquad\alpha=2\epsilon
c^{2}\frac{\ell^{2}_{P}}{\ell^{2}_{s}}\;,$ (C.29) $\displaystyle W^{-}_{G}$
$\displaystyle=X_{G}\left[2-\frac{\beta}{\tilde{h}}\right]G\left(\frac{V_{2}}{\sqrt{V_{1}}}\right),\qquad\text{with}\qquad\beta=-\frac{c\ell_{P}}{\ell^{2}_{s}}$
(C.30)

With these corrections, the modified charges now satisfy

$\\{W^{+}_{F},h\\}=F^{\prime}\left(\tilde{h}+\frac{c\ell_{P}}{\ell^{2}_{s}}\right)\,\hat{=}\,\;0\;,\qquad\\{W^{-}_{G},h\\}=2G^{\prime}\left(\tilde{h}+\frac{c\ell_{P}}{\ell^{2}_{s}}\right)\,\hat{=}\,\;0$
(C.31)

and one recovers the charges presented in Section 4.3. Let us further point
that the corrective terms spoil the commutation between the two set of charges
as shown by (4.47). This concludes the discussion on the CKVs of the black
hole superspace.

##  Appendix D Geometrizing the Bianchi I symmetry generators

In this last appendix, we present the explicit expressions of the CKVs of the
Bianchi I superspace and the ones derived from its ED lift. To simplify the
notation, we use the new rescaled variable
$\textrm{d}\varphi=\ell_{P}^{2}\textrm{d}\phi$.

###  CKVs of the superspace

The superspace metric is given by

$\displaystyle\textrm{d}s^{2}=\frac{c\ell_{P}}{2}\left[\left(\textrm{d}\varphi^{2}+\textrm{d}\beta_{1}^{2}+\textrm{d}\beta_{2}^{2}\right)z^{2}-\textrm{d}z^{2}\right]$
(D.1)

This four dimensional geometry has vanishing Weyl and Cotton tensors and is
thus conformally flat. It admits therefore the maximal number of CKVs,
fifteen, among which ten are KVs. The first obvious KVs are given by

$T^{\alpha}_{1}\partial_{\alpha}=\partial_{1}\;,\qquad
T^{\alpha}_{2}\partial_{\alpha}=\partial_{2}\;,\qquad
T_{\varphi}^{\alpha}\partial_{\alpha}=\partial_{\varphi}$ (D.2)

We have further

$T_{0}^{\alpha}\partial_{\alpha}=z\partial_{z}\;,\qquad
D^{\alpha}\partial_{\alpha}=\log\\!z\,z\partial_{z}+\beta_{1}\partial_{1}+\beta_{2}\partial_{2}+\varphi\partial_{\varphi}$
(D.3)

and

$\displaystyle\left|\begin{array}[]{l}M_{1,2}^{\alpha}\partial_{\alpha}=\beta_{1}\partial_{2}-\beta_{2}\partial_{1}\;,\\\
M_{1,\varphi}^{\alpha}\partial_{\alpha}=\beta_{1}\partial_{\varphi}-\varphi\partial_{1}\;,\\\
M_{2,\varphi}^{\alpha}\partial_{\alpha}=\beta_{2}\partial_{\varphi}-\varphi\partial_{2}\;,\end{array}\right.\qquad\left|\begin{array}[]{l}M_{1,0}^{\alpha}\partial_{\alpha}=\beta_{1}z\partial_{z}+\log\\!z\,\partial_{1}\;,\\\
M_{2,0}^{\alpha}\partial_{\alpha}=\beta_{2}z\partial_{z}+\log\\!z\,\partial_{2}\;,\\\
M_{\varphi,0}^{\alpha}\partial_{\alpha}=\varphi
z\partial_{z}+\log\\!z\,\partial\varphi\end{array}\right.$ (D.10)

The remaining CKVs are given by

$\displaystyle\left|\begin{array}[]{l}K_{0}^{\alpha}\partial_{\alpha}=-2\log\\!z\,\left[\left(\log\\!z\,+\frac{F}{2\log\\!z\,}\right)z\partial_{z}+\beta_{1}\partial_{1}+\beta_{2}\partial_{2}+\varphi\partial_{\varphi}\right]\;,\\\
\\\
K_{1}^{\alpha}\partial_{\alpha}=2\beta_{1}\left[\log\\!z\,z\partial_{z}+\left(\beta_{1}-\frac{F}{2\beta_{1}}\right)\partial_{1}+\beta_{2}\partial_{2}+\varphi\partial_{\varphi}\right]\;,\\\
\\\
K_{2}^{\alpha}\partial_{\alpha}=2\beta_{2}\left[\log\\!z\,z\partial_{z}+\beta_{1}\partial_{1}+\left(\beta_{2}-\frac{F}{2\beta_{2}}\right)\partial_{2}+\varphi\partial_{\varphi}\right]\;,\\\
\\\
K_{\varphi}^{\alpha}\partial_{\alpha}=2\varphi\left[\log\\!z\,z\partial_{z}+\beta_{1}\partial_{1}+\beta_{2}\partial_{2}+\left(\varphi-\frac{F}{2\varphi}\right)\partial_{\varphi}\right]\\\
\end{array}\right.$ (D.18)

where

$F(z,\beta_{1},\beta_{2},\varphi)\coloneqq\beta_{1}^{2}+\beta_{2}^{2}+\varphi^{2}-\log^{2}\\!z\,$
(D.19)

These fifteen vectors represent the maximal set of conformal isometries of the
superspace. Being conformally flat, these vectors fields form the standard
$\mathfrak{so}(4,2)$ Lie algebra which stands as the conformal structure of 4d
Minkowski spacetime. Indeed, taken the ususal Minkowski metric
$\eta_{\alpha\beta}$ mostly positive and $x^{\alpha}$ to be

$\eta_{\alpha\beta}=\begin{pmatrix}-1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\
0&0&0&1\end{pmatrix}\,,\qquad\qquad\qquad
x^{\alpha}=\begin{pmatrix}\log\\!z\,\\\ \beta_{1}\\\ \beta_{2}\\\
\varphi\end{pmatrix}$ (D.20)

so that

$\displaystyle\begin{array}[]{c}x_{\alpha}=\begin{pmatrix}-\log\\!z\,,&\beta_{1},&\beta_{2},&\varphi\end{pmatrix}\,,\qquad\quad\partial_{\alpha}=\frac{\partial}{\partial
x^{\alpha}}=\begin{pmatrix}z\partial_{z},&\partial_{1},&\partial_{2},&\partial_{\varphi}\end{pmatrix}\end{array}$
(D.22)

it is possible to write the fifteen CKVs in the more compact and common way

4 translations $\displaystyle T_{\alpha}=\partial_{\alpha}$ (D.23) 6 rotations
$\displaystyle
M_{\alpha\beta}=x_{\alpha}\partial_{\beta}-x_{\beta}\partial_{\alpha}$ (D.24)
1 dilation $\displaystyle D=x^{\alpha}\partial_{\alpha}$ (D.25) 4 inversions
$\displaystyle
K_{\alpha}=2x_{\alpha}x^{\beta}\partial_{\beta}-x^{\beta}x_{\beta}\partial_{\alpha}$
(D.26)

These indeed satisfy the $\mathfrak{so}(4,2)$ algebra

$\begin{split}&\left[M_{\alpha\beta},M_{\gamma\delta}\right]=\eta_{\alpha\delta}M_{\beta\gamma}+\eta_{\beta\gamma}M_{\alpha\delta}+\eta_{\alpha\gamma}M_{\delta\beta}+\eta_{\beta\delta}M_{\gamma\alpha}\,;\\\
&\left[T_{\alpha},T_{\beta}\right]=0\,;\qquad\left[T_{\alpha},M_{\beta\gamma}\right]=\eta_{\beta\alpha}T_{\gamma}-\eta_{\gamma\alpha}T_{\beta}\,;\\\
&\left[K_{\alpha},K_{\beta}\right]=0\,;\qquad\left[K_{\alpha},M_{\beta\gamma}\right]=\eta_{\beta\alpha}K_{\gamma}-\eta_{\gamma\alpha}K_{\beta}\,;\\\
&\left[D,K_{\alpha}\right]=K_{\alpha}\,;\qquad\left[D,M_{\alpha\beta}\right]=0\,;\qquad\left[T_{\alpha},D\right]=T_{\alpha}\,;\\\
&\left[T_{\alpha},K_{\beta}\right]=2\left(\eta_{\alpha\beta}D-M_{\alpha\beta}\right)\end{split}$
(D.27)

These CKVs automatically provide a ${\mathfrak{so}}(4,2)$ algebra of Dirac
observables for the Bianchi I model. Let us now compare with the conformal
isometries of the ED lift.

###  CKVs of the lift

The Eisenhart-Duval lift of the Bianchi I system is given by

$\textrm{d}s^{2}=2\textrm{d}u\textrm{d}w+c\ell_{P}\left[\left(\textrm{d}\varphi^{2}+\textrm{d}\beta_{1}^{2}+\textrm{d}\beta_{2}^{2}\right)z^{2}-\textrm{d}z^{2}\right]$
(D.28)

which is six dimensional. Contrary to the superspace, the ED lift is not
conformally flat because its Cotton tensor is not vanishing. This crucial
difference implies that the lift is not maximally symmetric. We find only
twelve CKVs for this geometry. The first three are given by

$\displaystyle Q_{-}$ $\displaystyle=\partial_{u}$ (D.29) $\displaystyle
Q_{0}$ $\displaystyle=\frac{z}{2}\partial_{z}+u\partial_{u}$ (D.30)
$\displaystyle Q_{+}$
$\displaystyle=\frac{c\ell_{P}z^{2}}{2}\partial_{w}+uz\partial_{z}+u^{2}\partial_{u}$
(D.31)

and are identical to the ones found for the FLRW model. We have also the
generators dual under the $u\leftrightarrow w$ change for $Q_{+}$ and $Q_{-}$;
$Q_{0}$ changes thanks to the generator

$\displaystyle\mathcal{O}$
$\displaystyle=w\partial_{w}-u\partial_{u}\hphantom{=\frac{c\ell_{P}z^{2}}{2}\partial_{w}}$
(D.32)

while the six other CKVs coincide with the generators $T_{1}$, $T_{2}$,
$T_{\varphi}$, $M_{1,2}$, $M_{1,\varphi}$ and $M_{2,\varphi}$ given in (D.2)
and (D.10) and that fully commute with the $Q$’s since they are independent
from $u,w,z$. The fact that the CKVs of the superspace are not all CKVs of the
ED lift can be understood by the fact that while the superspace is conformally
flat, the lift is not.

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