# Transformers for Image-Goal Navigation

Nikhilanj V. Pelluri
Minnesota Robotics Institute, Department of Computer Science and Engineering
University of Minnesota - Twin Cities
United States
<EMAIL_ADDRESS>

###### Abstract

Visual perception and navigation have emerged as major focus areas in the
field of embodied artificial intelligence. We consider the task of image-goal
navigation, where an agent is tasked to navigate to a goal specified by an
image, relying only on images from an onboard camera. This task is
particularly challenging since it demands robust scene understanding, goal-
oriented planning and long-horizon navigation. Most existing approaches
typically learn navigation policies reliant on recurrent neural networks
trained via online reinforcement learning. However, training such policies
requires substantial computational resources and time, and performance of
these models is not reliable on long-horizon navigation. In this work, we
present a generative Transformer based model that jointly models image goals,
camera observations and the robot’s past actions to predict future actions. We
use state-of-the-art perception models and navigation policies to learn robust
goal conditioned policies without the need for real-time interaction with the
environment. Our model demonstrates capability in capturing and associating
visual information across long time horizons, helping in effective navigation.

> Keywords: Visual Navigation, Transformers, Behavior Cloning

##  1 Introduction

Autonomous navigation in environments is a critical capability for modern
mobile robots, and has been extensively studied over several decades.
Classical approaches to navigation rely on constructing detailed maps of the
environment and accurate localization of the robot within the map [1, 2, 3].
However, with increasing demand for deploying robots in novel uncontrolled
environments such as households, last-mile delivery, etc., constructing
accurate and fine-grained maps frequently is often impractical. Robots must
now be able to navigate without maps, which means efficient navigation
policies require accurate semantic understanding of the scene, efficient
exploration and episodic memory, and long-horizon planning with limited
knowledge of the environment. Advances in scene understanding and a have led
to semantic navigation tasks such as image-goal navigation [4, 5], object-goal
navigation [6, 7], etc. receiving significant focus in recent years.
In this work, we consider the specific task of image-goal navigation where
robot’s navigation objective is specified by an RGB image. We motivate the
task with the following scenario: Consider a mobile household robot equipped
with an onboard camera tasked with picking up a novel unseen object (say, a
new shirt). Since the robot has no prior knowledge about the novel object, it
would need other semantic information to understand the object – an image of
the object would serve this purpose effectively. However, in order to navigate
efficiently using just this image, the robot would need to accurately
understand the semantic information in the image, compare it against its
camera observations and navigate towards the goal, while maintain a history of
its previous observations and actions.

Figure 1: The Image-Goal navigation task. The robot receives a goal image
(shown in the green dotted box), and must navigate to find the goal image
using only visual observations. In the figure above, the red camera marks the
initial viewpoint, with “S” marking the robot’s initial position. The robot
navigates to the point marked “G” described by the goal image, i.e., the view
from the green camera. The dashed lines out of the camera indicate the
camera’s approximate field of view. The yellow arrows indicate one possible
path from “S” to “G”. The figure above shows the “Pablo” scene from the Gibson
training dataset, obtained from [8].

Advances in learning based methods for perception and planning have led to the
popularity of learnt end-to-end models that takes images as inputs and
directly predict actions. These models generally consists of a visual encoder
stage to convert input images into features, while a recurrent neural network
(RNN) stage maintains history of these features and predicts actions [9, 10,
11]. However, most RNN based implementations compress history into a single
state vector, leading to degraded performance in capturing long-range
dependencies [12]. Furthermore, most end-to-end models learn navigation
policies using online reinforcement learning and suffer from sample
inefficiency - training usually requires extensive interactions in the
environment, often in the order of hundreds of millions of episodes. Hence,
training these models requires significant compute and time, and is sensitive
to hyperparameter tuning [13].
In recent years, Transformer based models have significantly outperformed RNN
based models on sequential modeling tasks, particularly in tasks demanding
long horizon understanding. Furthermore, the flexible nature of Transformers
enables it to jointly model different modalities such as images, text,
actions, etc., which unlocks a powerful capability to correlate input
observations and actions [14, 15]. Propelled by advances in language
generation, Transformer models have shown an excellent capability to
autoregressively generate very long sequences of tokens [16, 17, 18]. These
models are also easier to train and are able to scale learning with more data.
In our current work, we leverage the effectiveness of Transformer based models
to learn policies for image-goal navigation. We jointly model observations,
goals and the robot’s previous actions to learn goal-conditioned navigation
policies, and generate arbitrarily long sequences of actions to reach image
goals.
We use state-of-the-art models in image-goal navigation to create datasets of
expert trajectories to solve the task, and then mimic these expert policies
using a behavior cloning approach. This approach circumvents the need to
access an environment online and enables us to train a model with
significantly lower compute and time requirements.

##  2 Related Work

###  2.1 Image-Goal Navigation

Visual Navigation, i.e., robot navigation using visual input has been
extensively studied in literature [19]. We particularly focus on image-goal
navigation in our work. Several different approaches have been suggested to
solve this task, but they can be primarily categorized into:

  * •

Map Based Approaches: These approaches focus on first generating a full map of
the environment, and using the map for localizing the agent and planning a
path to the goal. While some works focus on building accurate metric maps with
embedded semantic information [20, 7, 21, 22] of the environment, some other
works focus on learning a topographical map instead, and use graph-based
planning methods for global navigation [23, 24, 25]. However, map-based
approaches are heavily reliant on accurate mapping and precise localization of
the robot, which may not be feasible, especially in dynamic scenarios.

  * •

Learning Based Approaches: The development of high-fidelity physics simulators
such as Habitat [26], AI2-THOR [27] and photorealistic scene datasets such as
Gibson [28], Matterport-3D [29], etc. has enabled agents to be trained in
simulation using deep reinforcement learning techniques. Mishkin et al. [30]
showed that while classical mapping and localization based approaches worked
well with depth information, learned policies performed better with RGB-only
input. [4, 31, 32, 11, 33, 34] all utilize the similar architecture of a
visual encoder for image representation and a recurrent network for episodic
memory.

Our approach falls under the learning based approach. We do not generate an
explicit map of the environment and instead rely on the learned policy to
guide the agent towards the goal. However, as detailed earlier in the Section
1, recurrent networks do not perform well with long horizon tasks. We believe
new state-of-the-art sequential architectures, particularly Transformer models
hold the key to efficient long horizon image-goal navigation.

###  2.2 Transformers

Sequence models have seen rapid progress in recent times, particularly since
the introduction of the Transformer model [35]. Transformer based models have
shown remarkable ability to model several different types of inputs including
vision [36, 37], vision and language [38], multi-modal inputs including
actions [39, 40], etc. This versatility of Transformer is useful to model
observations and actions together, which could potentially help build better
“world knowledge” in the model.
Some works such as [41] use the Transformer architecture for visual
navigation, but do not specifically target the image-goal or object-goal
navigation tasks. Several works such as [42, 43] apply Transformers to visual-
language navigation. However, these works generally learn navigation using
language instructions, whereas we focus on goal-oriented vision-only
navigation.
In this work, we primarily draw from the work of Decision Transformer [44],
Trajectory Transformer [45] and PACT [46]. We adapt the formulation of the
Decision Transformer model to fit our image-goal navigation task by
conditioning the model to pay attention to the goal. We achieve this by
inserting goal tokens into the input tokens sequence. We use a similar
decoder-only architecture that autoregressively generates action tokens.
Training with online reinforcement learning (RL) requires access to rewards
from a simulated environment over hundreds of millions of episodes, which
demands humongous compute resources and time. Furthermore, it has been shown
that Transformer models are considerably difficult to train in an online RL
setting [47].
To circumvent this, we adopt a goal-conditioned behavior cloning (GCBC)
approach. This allows us to learn scalably using an expert agent’s experience.

###  2.3 Behavior Cloning

Behavior Cloning (BC) has been used to learn several kinds of robotic tasks,
particularly where online RL techniques cannot be applied or is too expensive,
such as autonomous driving [48], real robot manipulation [49], social
navigation [50], etc. Specifically, we follow an approach similar to [51, 52]
to learn “goal-conditioned” policies from behavior cloning.
Most BC works rely on large datasets of inputs and corresponding actions taken
by human experts [53]. However, collecting such a large number of human
navigation trajectories entail significant effort and time. While the Habitat-
Web dataset [54], with 77k human demonstrations, has been created for the
object-goal navigation task, we are not aware of a similar dataset for image-
goal navigation.
Hence, we rely on an expert agent – an agent trained on episodes in the
training set, to generate training data for BC. We collect all trajectories
from this pre-trained agent, filter out unsuccessful episodes, and create an
expert dataset containing trajectories that reach the image goals for the
episodes. RLBench [55] generates expert trajectory data for robot manipulation
using hand-engineered motion planners, whereas we use a learned policy to
generate new trajectory data.
We train our decoder model to mimic expert trajectories, closely related to
the approach taken by [56, 57].

##  3 Methods

###  3.1 Problem Statement

####  3.1.1 Image-Goal Navigation

We consider the task of image-goal navigation. An agent is initialized at a
random pose $T_{0}$ in an unseen environment, and is tasked with reaching a
goal pose $T_{G}$, described by the agent’s observation at the goal pose,
i.e., RGB image $I_{G}$. The agent take an action $a_{t}$ in the environment
using egocentric observations $o_{t}$ at every timestep $t$, and receives a
reward $r_{t}$, including a bonus reward for finishing within a radius $d_{g}$
of the goal.
The problem can be formalized as a Partially Observable Markov Decision
Process (POMDP) defined by the tuple
$<\mathcal{S},\mathcal{A},\mathcal{T},\mathcal{R},\mathcal{O}>$ with states
and goals ${s,g}\in\mathcal{S}$, actions $a\in\mathcal{A}$, transition
dynamics $p(s^{\prime}|a,s)\in\mathcal{T}$, rewards $r(s,a,g)\in\mathcal{R}$,
sensor observations $o\in\mathcal{O}$. It is worth noting here that in our
task, states and goals belong to the same space, since any state can be
considered a goal.
Solving the task consists of learning a policy $\pi(a|o,G)$ that maximizes the
cumulative return over an episode.

$J(\pi)=\mathbb{E}_{g\sim p_{g}\mathcal{S},\tau\sim
d_{\pi}(.|g)}[\sum_{t}\gamma^{t}r(s_{t},a_{t},g)]$

####  3.1.2 Goal-Conditioned Supervised Learning/Behavior Cloning

We follow the formulation of [58] for Goal-Conditioned Supervised Learning
(GCSL), also known as Goal-Conditioned Behavior Cloning (GCBC).
We assume access to a dataset $D^{*}$ of demonstrations
$\\{\tau_{0},\tau_{1}....\tau_{N}\\}$ from an expert policy $\pi^{*}$. Each
demonstration $\tau_{i}$ contains a goal image $I_{G}$ and a sequence of
observations $o_{t}$, actions $o_{t}$ taken by the expert policy, and the
corresponding rewards received $r_{t}$.

$\tau_{i}=[I_{G};\\{o_{t},a_{t},r_{t}\\}_{t=1}^{T}]_{i=0}^{N}$

The objective of GCBC is to learn a policy $\pi_{\theta}(a|o,I_{G})$ by
minimizing the error between the output actions and the expert’s actions, such
that

$\pi_{\theta}=\arg\min_{\theta}\sum_{\tau\in
D^{*}}\sum_{\\{o,I_{G}\\}\in\tau}L(\pi(a|o,I_{G}),\pi^{*}(a|{o,I_{G}}))$

###  3.2 Model

Our model is built as a decoder-only Transformer model, drawing from the works
of GPT [59], Decision Transformer (DT) [44], ViNT [56], RT-1 [57], etc.
The model takes as input the goal image ($I_{G}$), a sequence of observation
images (${I_{i}....T_{i+T}}$, $T$ being the sequence length) and the actions
taken by the expert (${a_{i}....a_{i+T}}$). Notably, unlike DT, we do not
condition the model on the returns-to-go and instead condition the model’s
actions on the goal token.
Our model has 27M trainable parameters, excluding the frozen parameters of the
image encoder model.
The full architecture is shown in Figure 2.

Figure 2: The full architecture of our model. The goal image $I_{G}$ and
observation images $I_{1},I_{2},...I_{T}$ are embedded into tokens $E_{I_{G}}$
and $E_{I_{1}},E_{I_{2}},....E_{I_{T}}$ respectively using a pre-trained
DINOv2 model, actions $a_{1},a_{2},...a_{T}$ are passed to a specific lookup
embedding. To enable autoregressive generation, the input actions are shifted
right by one step. Image and action embeddings are interleaved to form input
embeddings $E_{1},E_{2},...E_{T}$. Learnable position embeddings
$p_{1},p_{2},...p_{T}$are added before passing all inputs to a Transformer
decoder with a causal attention mask. The decoder outputs logits,
corresponding to the most likely action. An actor head samples the actions
from these logits.

####  3.2.1 Input Representation

Image Encoding: We first encode the goal image $I_{G}$ and the observation
images $I_{t}$ to corresponding embeddings $E_{I_{i}}\in\mathbb{R}^{d}$ using
an image encoder. We use the “ViT-B/14” version of the DINOv2 model [37] with
registers [60] as the encoder, which reports robust performance on a variety
of downstream vision tasks. The DINOv2 model is based on the Vision
Transformer architecture [36], which splits images into patches (here, of size
14x14), before passing linear projections of the patches to a Transformer
encoder consisting of multiple layers of multi-head self-attention (MHSA)
blocks. We freeze the pre-trained weights for the model and do not train the
feature extraction layers of the model.
To improve learning from finer features, we concatenate the patch tokens from
the final four layers of the DINOv2 model, and along with the output of the
[CLS] token, which contains global context about the image. The concatenated
feature vector is passed to a 2-layer multi-layer perceptron module (MLP) with
an output dimension $d$.

Action Embedding: The categorical input actions are encoded into
$d$-dimensional embeddings $E_{a_{i}}\in\mathbb{R}^{d}$ using a lookup table
with learnable weights. It is important to note that during training, the
actions are shifted to the right, to condition the model for autoregressive
action generation.
Following [44], we first interleave the input observation and action
sequences, such that the combined input sequence becomes

$\mathbf{e}=[E_{I_{1}},E_{a_{1}},E_{I_{2}},E_{a_{2}},....]$

Position Embedding: We add a learnable position embedding
$p_{i}\in\mathbb{R}^{d}$ to each element sequence. We add the same position
embedding to tokens of each timestep, i.e., the same embedding $p_{i}$ is
added to $E_{I_{i}}$ and $E_{a_{i}}$.Hence, the position embedding
$\mathbf{p}\in\mathbb{R}^{2T\times d}=[p_{1},p_{1},p_{2},p_{2},...]$
Since we want to learn policies conditioned on the goal image $I_{G}$, we
insert the goal image embedding $E_{I_{G}}$ at the beginning of the sequence.
This is similar to the approach taken by [58] and [45]. Inserting the goal
embedding at the beginning enables all embeddings in the sequence to attend to
the goal, since we use a causal attention mask in the decoder.
The full input to the Transformer is

$\mathbf{x}=[E_{I_{G}};\mathbf{e}+\mathbf{p}]$

####  3.2.2 Transformer Decoder

Our decoder-only Transformer sequence model consists of $L=12$ layers, with
each layer composed of $H=8$ multi-head self-attention (MHSA) blocks and feed-
forward network dimension $d_{FFN}=1024$. Following [61], the LayerNorm (LN)
operation is applied before each block. We apply a causal self-attention mask,
which prevents the model from “looking ahead” and copying future actions. The
decoder accepts embeddings of dimension $d=384$ as input.
Scaled dot-product attention operation of a single head $h$ for an input token
sequence $\mathbf{x}\in\mathbb{R}^{T\times d}$ is defined as:

$\displaystyle
Q=\mathbf{x}{W_{h}}^{Q};K=\mathbf{x}{W_{h}}^{K};V=\mathbf{x}{W_{h}}^{V}$
$\displaystyle\text{where}\
\\{{W_{h}}^{Q},{W_{h}}^{K},{W_{h}}^{V}\\}\in\mathbb{R}^{d\times d}$
$\displaystyle\textsc{Attn}_{h}(Q,K,V)=\textit{softmax}(\frac{Q\cdot
K^{T}}{\sqrt{d_{k}}})\cdot V$

The attention values from each head are concatenated together to form the MHSA
output:

$\displaystyle\textsc{MHSA}(\mathbf{x})=\textsc{Concat}[\textsc{Attn}_{1},\textsc{Attn}_{2}.....\textsc{Attn}_{H}]W^{O}$
$\displaystyle\text{where}\ W^{O}\in\mathbb{R}^{Hd\times d}$

The computations in a single layer $l$ of the decoder for the input token
sequence $\mathbf{x}$ can be represented as:

$\displaystyle\mathbf{y}_{l}=\textsc{MHSA}(\textsc{LN}(\mathbf{x}_{l}))+\mathbf{x}_{l}$
$\displaystyle\mathbf{z}_{l}=\textsc{FFN}(\textsc{LN}(\mathbf{y}_{l}))+\mathbf{y}_{l}$

####  3.2.3 Actor model

The actor model has a multi-layer perceptron (MLP) network that maps the
$d$-dimensional output from the decoder to logits of the actions space.
During training, we apply a softmax operation over the logits, and the highest
probability tokens are chosen as the output actions.
However, during autoregressive action generation in 4.6.2, we find empirically
that greedily picking the highest probability token leads to degenerate
scenarios such as endlessly repeating the same token or the agent getting
stuck in closed spaces. To resolve this, we apply a top-$k$ ($k=2$) sampling
strategy to pick the next action. Sampling actions improves the exploration
capability of the agent, while slightly increasing the risk of higher
distribution shift away from the expert policy.

##  4 Experimental Results and Analysis

###  4.1 Environment Setup

We use v0.2.1 of the Habitat simulator [26].
Following [33], our agent contains only an egocentric camera sensor with a
field-of-view of 90°, which returns observations as RGB images of size 128
pixels x 128 pixels. We do not use depth observations or GPS+Compass
information.
The action space of the agent is composed of 4 discrete actions: [STOP,
MOVE_FORWARD, TURN_LEFT, TURN_RIGHT]. The agent moves forward by 0.25m on the
MOVE_FORWARD action, and turns 30°on TURN_LEFT or TURN_RIGHT.

###  4.2 Dataset details

We run the agent in scenes from the Gibson dataset for Habitat-Sim [28]. We
use the training split from [31], which consists of 9000 episodes sampled from
each of the 72 scenes in the Gibson training set. The test split from [24] is
used for evaluation. In both training and test splits, the episodes are split
into “straight” and “curved” episodes. We refer to [24] for the categorization
criteria: “in ‘straight’ episodes, the ratio of shortest path geodesic-
distance to euclidean-distance between the start and goal locations is $<$1.2
meters and rotational difference between the orientation of the start position
and goal image is $<$45°. All other start-goal location pairs are labeled as
curved episodes.”
Episodes in each category are further categorized depending on the geodesic
distance between start and goal locations into easy (1.5m-3m), medium (3m-5m)
and hard (5m-10m) difficulties.

###  4.3 Evaluation Metrics

We refer to [6] which introduces two standard metrics for goal navigation :

  * •

Success Rate: The agent is considered to have successfully completed the
episode if it calls the ‘STOP’ action within a radius of 1 meter from the goal
position.

  * •

Success weighted by Path Length (SPL): This metric measures the ratio of the
shortest path versus the path taken by the agent for successful episodes,
thereby giving significant weightage to reaching the goal efficiently.

###  4.4 Collecting expert trajectories for Behavior Cloning

We collect a dataset $D^{*}$ of expert trajectories using the agent from
OVRL-v2 (“OVRL agent”).The agent with pre-trained weights (from [62]) gathers
trajectories from episodes in the training split. Similarly, we collect the
agent’s trajectories on the test split for evaluation.

For training and evaluation, we filter out episodes where the OVRL agent was
unsuccessful. The trajectories where the OVRL agent successfully navigates to
the goal are taken as expert trajectories.

The success rate of the OVRL agent in each category of training and test
episodes is given in Table 1.

Split | Path Type | Number of episodes (Success% of total)
---|---|---
|  | Easy | Medium | Hard
Train [31] | Curved | 117755 (87.0%) | 176525 (81.7%) | 172133 (80.8%)
Test [24] | Curved | 475 (47.5%) | 441 (44.1%) | 319 (31.9%)
Test [24] | Straight | 584 (58.4%) | 495 (49.5%) | 323 (40.1%*)
Table 1: The number of successful episodes and their proportion out of all
episodes, collected by the pre-trained agent from OVRL-v2 [33] on different
categories of episodes in the Gibson dataset. *From 805 total collected
episodes for this split.

We were unable to access the OVRL agent’s actions during the local navigation
stage of [62], since it uses a local geometric planner disconnected from the
global navigation stage. Hence, the final observation and action in the expert
dataset, i.e., $o_{T}$ may, in a few cases, not correspond exactly to the goal
image $I_{G}$.

###  4.5 Training

We train the model on 96K expert trajectories randomly sampled from a mix of
“easy”, “medium” and “hard” difficulty trajectories from the “curved” split.
For each trajectory during training, we extract a contiguous subsequence of
length $T=8$, with a randomly sampled start position.
The model is trained using the standard categorical cross-entropy loss. We use
the AdamW optimizer with a weight decay of 0.01 and a custom epoch-wise
decaying learning rate schedule, starting at 1e-4.
We train the model for 20 epochs on an NVIDIA 1080Ti GPU with a batch size of
16.

###  4.6 Evaluation

We evaluate the model’s performance on two questions: (1) how accurately did
our model learn the expert policy [4.6.1] ? and (2) how does the learned
policy perform in an online setting, i.e., with autoregressive action
generation [4.6.2] ?

####  4.6.1 Learning the expert policy

We measure the accuracy between the actions taken by the expert and the
actions taken by our learned agent, given the same goals and observations. We
first run inference over non-overlapping sliding windows on expert
trajectories from the test set, and measure the accuracy cumulatively across
all time windows for a trajectory. The accuracy on various test splits is
reported in Table 2.

Test Split | Easy | Medium | Hard
---|---|---|---
Curved | 75.45% | 77.98% | 79.31%
Straight | 77.44% | 79.83% | 80.49%
Table 2: Accuracy of our model’s actions vs expert actions at various
difficulty levels of test episodes from the Gibson dataset, when evaluated
over non-overlapping windows. We were unable to collect all trajectories on
the “Straight-Hard” split.

We further analyze this question: does the model’s performance differ as the
distance between goal and observations changes, i.e., as move further along
the trajectory ? To answer this question, we measured the accuracy separately
in each non-overlapping window, and analyzed it against the start index. The
results are shown in Figure 3.

(a) Performance on curved episodes.

(b) Performance on straight episodes.

Figure 3: Model accuracy at different start indices of non-overlapping windows
of input trajectory, for curved (3(a)) and straight episodes (3(b)). Results
indicate model accuracy is higher at the episode’s beginning and lower near
its middle.

We see that the average accuracy is higher in the initial windows, and
decreases marginally at the middle of the trajectory. This could be attributed
to the distribution of the input trajectory data, since the expert
trajectories are biased to be shorter. This behavior is particularly
pronounced in the easy trajectories, which have a shorter episode length. We
also observe that the variance in accuracy at the middle of the trajectory is
higher than at the start. We show the distribution of expert trajectory
lengths across various difficulty splits from the “Gibson-curved” training set
in figure 4.

Figure 4: Distribution of episode lengths across various difficulty splits
from the “Gibson-curved” training set.“Easy” episodes are considerably shorter
than “Medium” and “Hard” episodes.

A perplexing question that arises from table 2 and figure 3 is that the
accuracy on the “Hard” difficulty trajectories is higher than the “Easy” and
“Medium” trajectories. We posit that this could be due to the learned policy
having “learnt” the exploratory policy of the expert – “Hard” trajectories
require considerably more exploration than the others, and exploration
strategy is generally similar across different episodes. A similar observation
is made with “Medium” trajectories, where the accuracy is higher than “Easy”
trajectories.

####  4.6.2 Autoregressive action generation

We evaluate the action generation capability of our learned agent on unseen
environments from the Gibson environment, following the same test split from
[24] as detailed in section 4.2. We refer to section 4.3 for details on the
success criteria.
The agent is initialized in the environment with the goal image. The agent
maintains a “first-in-first-out” (FIFO) buffer of observations and actions,
with a buffer size equal to the decoder model’s context length ($T=8$).

We evaluate our model against the following baselines, with results and pre-
trained weights provided by [62]:

  1. 1.

Behavior Cloning with Spatial Memory: Observation and goal images are encoded
using a ResNet-18 [63] model, while the depth $D_{t}$ is used to construct a
spatial metric map, and navigation learnt using behavior cloning (BC).

  2. 2.

Behavior Cloning with GRU: Similar to the above model, observations and goal
are encoded using the ResNet-18 model, while a Gated Recurrent Unit [64]
network is used to maintain state. BC is used to train the navigation policy.

  3. 3.

OVRL-v2 [33]: This model uses a Vision Transformer [36] model to encode
observation and goal images, while an LSTM-based [65] neural network provides
state memory. Navigation policy is trained using policy gradient methods over
500M episodes. We use this model to generate expert trajectories data for our
model.
We were able to independently verify the results provided by [62] for this
model.

The results of our evaluation are shown in table 3. We observe that our
behavior cloned model outperforms other behavior cloned models on most data
splits. We further note that we use an input image size of 128x128, which is
significantly smaller than other BC models, which use an input image size of
480x640 [24].

Method | Overall | Easy | Medium | Hard
---|---|---|---|---
Success↑ | SPL↑ | Success↑ | SPL↑ | Success↑ | SPL↑ | Success↑ | SPL↑
Dataset split: Gibson-curved
Behavior Cloning w/ Spatial Memory† | 1.30 | 1.10 | 3.10 | 2.50 | 0.80 | 0.70 | 0.20 | 0.10
Behavior Cloning w/ GRU State† | 1.70 | 1.30 | 3.60 | 2.80 | 1.10 | 0.90 | 0.50 | 0.30
Ours* | 6.78 | 6.37 | 16.00 | 15.08 | 3.67 | 3.49 | 0.67 | 0.54
OVRL-v2† (Expert) | 45.60 | 28.00 | 53.60 | 31.70 | 47.60 | 30.20 | 35.60 | 21.90
Dataset split: Gibson-straight
Behavior Cloning w/ Spatial Memory† | 12.50 | 12.10 | 24.80 | 23.90 | 11.50 | 11.20 | 1.30 | 1.20
Behavior Cloning w/ GRU State† | 19.50 | 18.70 | 34.90 | 33.40 | 17.60 | 17.00 | 6.00 | 5.90
Ours* | 15.06 | 14.04 | 28.33 | 26.13 | 11.33 | 10.53 | 4.00 | 3.65
OVRL-v2† (Expert) | 44.90 | 30.00 | 53.60 | 34.70 | 48.60 | 33.30 | 32.50 | 21.90
Table 3:  Results from evaluating our trained agent on test scenes from the
Gibson dataset. *Note that we evaluate on a random sample of 300 out of total
1000 episodes, and limit our episode length to 100 for easy episodes and 350
for medium and hard episodes. †Results from [62]

We show some qualitative results of the model’s performance in figures 5, 6.

Figure 5: Model’s performance on a successful “easy” difficulty episode from
the “curved” split of the Gibson test set.  Figure 6: Model’s performance on a
successful “medium” difficulty episode from the “curved” split of the Gibson
test set.

###  4.7 Failure Modes

We now analyze some cases where our agent fails:

  * •

Small/Invisible obstacles: In several cases, the agent encounters obstacles
that are not visible in its onboard camera. Since our model relies exclusively
on images from the camera, the model is unable to “understand” the presence of
a obstacle, particularly when the height of the obstacle is less than the
height of the camera (for example, a sofa or a coffee table).

  * •

Inability to recover from collisions: An extension of the above issue is that
since our model does not access rewards from the environment nor other
information such as pose or depth, the agent cannot recover from collisions
with the environment. For example, the agent may run into a wall partially,
but since it has no way of understanding this scenario, the agent continues
trying to go straight until it expends all its steps budget. Furthermore,
since we use successful expert trajectories, which are biased against
collisions, the model does not learn the behavior required to recover from
collisions.

  * •

Dataset specific considerations: As [33] observe in their failure analysis,
about 6% of episodes in the dataset have goal images that are semantically
meaningful - such as blank walls. Since we rely exclusively on image feature
extraction, not having distinguishable features significantly affects the
model’s ability to understand the scene. Another mode of failure is when the
agent needs to pass through narrow hallways to reach the goal. Accuracy in
such cases demands fine-grained control of the robot, which is difficult to
learn with a global navigation policy designed for long-horizon navigation.

##  5 Conclusion

In this work, we presented a decoder-only Transformer model for image-goal
navigation. We trained the model using a Goal-conditioned Behavior Cloning
(GCBC) approach, thereby avoiding the need for interactions with the
environment. Our model significantly outperforms other behavior cloning
baselines on the image-goal navigation task. While our GCSL model is easily
trained under computational constraints in a relatively short period of time,
we find that it underperforms state-of-the-art online RL based methods, which
were trained for millions of episodes. This leaves further room across various
axes of improvement to the current model. The flexibility of the Transformer
architecture allows the same model to be trained with other input modalities
such as text, speech, etc.

##  6 Future Work

Our work opens up several interesting areas of further research.
In our current work, the Transformer Decoder uses self-attention layers to
learn correlations between the goal features, observations and previous
actions. However, training such a model end-to-end might result in subpar
performance at low training scale since the priors for the model are too weak
– the model has to learn image matching as well as navigation at the same
time. We believe that pre-training or auxiliary training on explicit image
matching tasks such as wide-baseline stereo matching, optical flow, etc. would
help decouple learning in the perception and navigation modules. This could
potentially boost performance on the overall task, since the decoder can focus
on planning and navigation.
Another avenue for improvement is the fact that we used the vanilla
Transformer implementation from [35], where the self-attention computation
grows quadratically with input context length. However, in recent years,
several improvements have been suggested to this architecture, particularly
towards increasing the context length and reducing computational costs. Using
these improved architectures could potentially help navigate efficiently in
particularly complex and long-horizon scenarios.
Additionally, in this work, we focus on behavior cloning to train our policies
quickly and efficiently. However, there remains a significant gap to fill
between behavior cloned policies and efficient policies in the real world.
Online fine-tuning could help bridge this gap [66]. The gap between the
simulated environments and real world scenarios also points to the fact that
fine-tuning the model in the real world is an essential step towards robust
deployment of these models.

#### Acknowledgments

We thank Houjian Yu and members from CHOICE Lab for their inputs and support.
We would also like to thank Justin Wassermann from the University of Illinois
Urbana-Champaign for sharing their code and pre-trained weights for the
OVRL-v2 model. This work was made possible due to the continued support of the
Minnesota Robotics Institute and the Department of Electrical and Computer
Engineering at the University of Minnesota.

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##  Appendix A Example Expert Trajectories

The following shows some example expert trajectories from the test split of
Gibson dataset. The trajectories are collected using the pre-trained model
from OVRL-v2 [33].
Episodes in the dataset are split into “straight” and “curved” path types
depending on the path, which are further split into “easy”, “medium” and
“hard” difficulty depending on the geodesic distance between start and goal
locations.

###  A.1 Path type curved, difficulty hard

Figure 7: An expert trajectory on the “curved - easy” split of the Gibson test
set. The first image (left) shows the agent’s observation at its start
location. The second image (middle) shows the goal image, while the last image
(right) shows the trajectory taken by the expert, indicated by the blue line.
The green line indicates the shortest path between start and goal locations.
Figure 8: An expert trajectory on the “curved - easy” split of the Gibson test
set. The first image (left) shows the agent’s observation at its start
location. The second image (middle) shows the goal image, while the last image
(right) shows the trajectory taken by the expert, indicated by the blue line.
The green line indicates the shortest path between start and goal locations.

###  A.2 Path type curved, difficulty medium

Figure 9: An expert trajectory on the “curved - medium” split of the Gibson
test set. The second image (middle) shows the goal image, while the last image
(right) shows the trajectory taken by the expert, indicated by the blue line.
The green line indicates the shortest path between start and goal locations.
Figure 10: An expert trajectory on the “curved - medium” split of the Gibson
test set. The first image (left) shows the agent’s observation at its start
location. The second image (middle) shows the goal image, while the last image
(right) shows the trajectory taken by the expert, indicated by the blue line.
The green line indicates the shortest path between start and goal locations.


# Wandering domains in $\mathbb{C}\times\mathbb{R}$

Ramanpreet Kaur  Department of Mathematics, University of Delhi, Delhi–110
007, India<EMAIL_ADDRESS>
###### Abstract.

Approximation theory of entire functions has been used to demonstrate the
construction of a map on $\mathbb{C}\times\mathbb{R}$ having wandering
domains. We also present suitable modification to this construction that helps
in obtaining maps with similar characteristics such as infinitely many
attracting Fatou components, infinitely many wandering domains having common
paths, etc.

###### Key words and phrases:

transcendental entire function, approximation theory, wandering domains,
attracting domains, domains with common paths

###### 2020 Mathematics Subject Classification:

30D05, 37F10

##  1\. Introduction and Preliminaries

In this note, we shall consider the map
$F:\mathbb{C}\times\mathbb{R}\to\mathbb{C}\times\mathbb{R}$ which is of the
following form:

$F(z,w)=(h_{1}(z,w),h_{2}(w)),$ (1)

where $h_{1}(z,w)$ is a transcendental entire function in the variables $z,w$
and $h_{2}(w)$ is a real valued function. For the construction of the required
map, first we shall recall few definitions and results. The Fatou set of $F$,
denoted by $\mathscr{F}(F)$, is the largest open set where the family of
iterates $\\{F^{n}\\}_{n\in\mathbb{N}}$ of $F$ is normal. A connected
component of the Fatou set is called a Fatou component. The complement of the
Fatou set is called as Julia set. For more details about the Fatou-Julia
theory in several variables, one can refer to [3, 5] and the references
therein.

In one variable, for a rational function of degree $d\geq 2$, every Fatou
component is pre-periodic [9]. In particular, the periodic Fatou component,
say $U$, is one of the following forms.

  1. (1)

Attracting: every point of the component converges to periodic point $z_{0}\in
U$ under iteration of the function. In this case, the multiplier at the
periodic point has modulus less than 1;

  2. (2)

Parabolic: every point of the component converges to periodic point
$z_{0}\in\partial U$ under iteration of the function;

  3. (3)

Siegel disk: the function on $U$ is conformally conjugate to an irrational
rotation of a unit disk. In this case, the multiplier at the periodic point
has modulus 1;

  4. (4)

Herman ring: the function on $U$ is conformally conjugate to an irrational
rotation of a standard annulus.

In contrast to this, a transcendental entire function does not have Herman
rings. Indeed they may have

  1. (1)

Baker domains: a Fatou component $U$ for which the orbit of every point of $U$
converges to $z_{0}$ and the function is not analytic at $z_{0}$. It is clear
that for transcendental entire functions, $z_{0}=\infty$;

  2. (2)

Wandering domains: a Fatou component $U$ which is not pre-periodic [2].

This led to the study of the classification of Fatou components in higher
variables. Several authors have discussed about these components for maps of
certain types, for example, recurrent Fatou components of a transcendental
Hénon map [1], Fatou components of attracting skew products [6], etc.

In this note, we shall restrict ourselves to wandering domains corresponding
to a map $F$ mentioned in Equation 1. We make use of approximation theory to
construct an example of wandering domains in $\mathbb{C}\times\mathbb{R}$. In
fact, the construction of such a map is inspired from the construction of a
Carleman set used by Singh [7]. We construct a general Carleman set in one
variable. This leads us to construct an example of wandering domains in
$\mathbb{C}\times\mathbb{R}$. We shall prove the following

######  Theorem 1.

There exists a mapping
$F:\mathbb{C}\times\mathbb{R}\to\mathbb{C}\times\mathbb{R}$ having escaping
wandering domains.

To apply approximation theory, let us first recall the definition of a
Carleman set (in $\mathbb{C}$). For a relatively closed subset $G$ of
$\mathbb{C}$, and

$\mathbb{C}(G)=\\{f:G\to\mathbb{C}\mid f\text{ is continuous on }G\text{ and
analytic on }G^{\circ}\\}.$

We call $G$ as a Carleman set in $\mathbb{C}$ if for any $f\in\mathbb{C}(G)$
and for any positive, continuous function $\epsilon$ on $G$, there exists an
entire function $g$ such that

$|f(z)-g(z)|<\epsilon(z)\text{ for every }z\in G.$

The following is a sufficient condition for a set to be a Carleman set.

######  Theorem 2 (Nersesjen, [4]).

Suppose $G$ is a closed proper subset of $\mathbb{C}$. Then $G$ is a Carleman
set in $\mathbb{C}$ if and only if $G$ satisfies the following conditions:

  * (i)

$\hat{\mathbb{C}}\setminus G$ is connected.

  * (ii)

$\hat{\mathbb{C}}\setminus G$ is locally connected at $\infty$.

  * (iii)

For every compact set $K\subset\mathbb{C}$, there exists a neighborhood $V$ of
$\infty$ in $\hat{\mathbb{C}}$ such that no components of $G^{\circ}$
intersects both $K$ and $V$.

##  2\. Preparation for the Main Result

To construct the desired map, first we construct an infinite number of
pairwise disjoint domains with certain properties, so that no two domains are
subsets of a single Fatou component of $F$. We prove that our desired map $F$
with wandering domains have the following form:

$F(z,w)=(e^{f_{1}(z)}+\delta
e^{\text{Re}f_{2}(w)},e^{\text{Re}f_{2}(w)}),\text{ for every
}(z,w)\in\mathbb{C}\times\mathbb{R}.$

Here, $f_{1},f_{2}$ are entire functions (existence of such functions is shown
during the construction of a Carleman set) and $\delta\in(0,1)$ is chosen
suitably (one of the conditions is $\frac{17\delta}{16}<\frac{9}{8}$).

### Construction of a Carleman set in one variable:

For every $k\in\mathbb{N}$, define

$B_{k}:=\\{z\in\mathbb{C}:|z-\delta_{k}|\leq l_{k}\\},$

where the sequences $\\{\delta_{k}\\}$ and $\\{l_{k}\\}$ are chosen such that

  * •

$\\{\delta_{k}\\}$ and $\\{l_{k}\\}$ are strictly increasing real sequences
converging to infinity.

  * •

$l_{1}$ satisfies $\frac{17\delta}{16}<\frac{l_{1}}{8}$.

  * •

$B_{i}\cap B_{j}=\emptyset$ for every $i\not=j$.

  * •

$B_{1}\cap G_{1}=\emptyset$, where $G_{1}=\\{z\in\mathbb{C}:|z-1|\leq 3\\}$.

Now, choose a sequence $\\{m_{k}\\}$ such that for every $k\in\mathbb{N}$, we
have

$\delta_{k}+l_{k}<m_{k}<\delta_{k+1}-l_{k+1}.$

For such a sequence, define

$M_{k}:=\\{z\in\mathbb{C}:\text{Re}z=m_{k}\\},k\in\mathbb{N}.$

Further, define

$\displaystyle A_{k}$ $\displaystyle:=\\{z\in\mathbb{C}:|z+r_{k}|\leq
s_{k}\\},k\in\mathbb{N}$ $\displaystyle\text{ and, }L_{k}$
$\displaystyle:=\\{z\in\mathbb{C}:\text{Re}z=-t_{k}\\},k\in\mathbb{N}$

where,

  * •

$\\{-r_{k}\\}$ is chosen to be a decreasing real sequence converging to
$-\infty$.

  * •

$\\{s_{k}\\}$ is chosen such that $A_{i}\cap A_{j}=\emptyset$ for every
$i\not=j$.

  * •

$\\{t_{k}\\}$ is chosen such that $(-r_{k+1}+s_{k+1})<-t_{k}<(-r_{k}+s_{k})$,
for every $k\in\mathbb{N}$.

Using the continuity of the exponential function $z\mapsto e^{z}$, we have

  * (a)

for every $\epsilon_{k}=\frac{l_{k+1}}{4}$, there exists
$\delta^{\prime}_{k}>0$ such that

$|z-\log(\delta_{k+1})|<\delta^{\prime}_{k}\text{ implies that
}|e^{z}-\delta_{k+1}|<\frac{l_{k+1}}{4}.$

  * (b)

As $\frac{s_{1}}{8}>0$, there exists $\delta^{{}^{\prime\prime}}_{1}>0$ such
that

$|z-\pi\iota+\log(r_{1})|<\delta^{{}^{\prime\prime}}_{1}\text{ implies that
}|e^{z}+r_{1}|<\frac{s_{1}}{8}.$

  * (c)

there exists $\delta^{{}^{\prime\prime}}_{2}>0$ such that

$|z|<\delta^{{}^{\prime\prime}}_{2}\text{ implies that
}|e^{z}-1|<\frac{1}{16}.$

Defining the functions $g(z)$ and $\epsilon(z)$: Let,

$g(z)=\left\\{\begin{array}[]{ll}\log(\delta_{k+1}),&z\in
B_{k},k\in\mathbb{N}\\\ \pi\iota+\log(r_{1}),&z\in A_{k},k\in\mathbb{N}\\\
\log 1,&z\in G_{1}\cup\left(\cup_{k\in\mathbb{N}}(M_{k}\cup
L_{k})\right),\end{array}\right.$

and

$\epsilon(z)=\left\\{\begin{array}[]{ll}\delta^{\prime}_{k},&z\in
B_{k},k\in\mathbb{N}\\\ \delta^{{}^{\prime\prime}}_{1},&z\in
A_{k},k\in\mathbb{N}\\\ \delta^{{}^{\prime\prime}}_{2},&z\in
G_{1}\cup\left(\cup_{k\in\mathbb{N}}(M_{k}\cup
L_{k})\right).\end{array}\right.$

Further, let $G=G_{1}\cup\left(\cup_{k\in\mathbb{N}}(B_{k}\cup M_{k}\cup
A_{k}\cup L_{k})\right)$ (See Figure 1). By 2, $G$ is a Carleman set. Since
$g\in\mathbb{C}(G)$, there exists an entire function $f_{1}(z)$ such that

$|f_{1}(z)-g(z)|<\epsilon(z),\text{ for every }z\in G.$

In particular, we get that

  * (d)

$|f_{1}(z)-\log(\delta_{k+1})|<\delta^{\prime}_{k}$, whenever $z\in
B_{k},k\in\mathbb{N}$.

  * (e)

$|f_{1}(z)-\pi\iota+\log(r_{1})|<\delta^{{}^{\prime\prime}}_{1}$, whenever
$z\in A_{k},k\in\mathbb{N}.$

  * (f)

$|f_{1}(z)-1|<\delta^{{}^{\prime\prime}}_{2}$, whenever $z\in
G_{1}\cup\left(\cup_{k\in\mathbb{N}}(M_{k}\cup L_{k})\right)$.

This further implies that

  * (g)

$|e^{f_{1}(z)}-\delta_{k+1}|<\frac{l_{k+1}}{4}$, whenever $z\in
B_{k},k\in\mathbb{N}$.

  * (h)

$|e^{f_{1}(z)}+r_{1}|<\frac{s_{1}}{8}$, for every $z\in A_{k},k\in\mathbb{N}$.

  * (i)

$|e^{f_{1}(z)}-1|<\frac{1}{16},$ for $z\in
G_{1}\cup(\cup_{k\in\mathbb{N}}(M_{k}\cup L_{k}))$.

Now, using continuity of $e^{w}$ at origin, there exists $d_{1}>0$ such that
whenever $|w-\log 1|<d_{1}$, we have $|e^{w}-1|<\frac{1}{16}.$
Further, define $g_{1}(w)=\log 1$ and $\epsilon_{1}(w)=d_{1},\text{ for
}w\in\mathbb{R}$. Since $\mathbb{R}$ is a Carleman set [4], there exists an
entire function $f_{2}$ such that

$|f_{2}(w)-g_{1}(w)|<\epsilon_{1}(w),\text{ for every }w\in\mathbb{R}.$

In particular,

$|f_{2}(w)-\log 1|<d_{1},\text{ for every }w\in\mathbb{R}.$

This further implies that

$|e^{f_{2}(w)}-1|<\frac{1}{16},\text{ for every }w\in\mathbb{R}.$

From here, we can easily observe that

$|e^{\text{Re}f_{2}(w)}|=|e^{f_{2}(w)}|<\frac{17}{16},\text{ for every
}w\in\mathbb{R}.$

Figure 1.  The arrows represent mapping of $L_{k},M_{k},B_{k}$ (in the proof
of 1) and that of $A_{k}$ (in the proof of Proposition 4) under the function
$e^{f_{1}(z)}$, for $k\in\mathbb{N}$.

###### Proof of 1.

Let $f_{1}$ and $f_{2}$ be defined as above. Let

$F(z,w)=(e^{f_{1}(z)}+\delta
e^{\text{Re}f_{2}(w)},e^{\text{Re}f_{2}(w)}),\text{ for
}(z,w)\in\mathbb{C}\times\mathbb{R}.$

For wandering domains of the above map, define

$\displaystyle B_{k}^{\prime}$
$\displaystyle:=\\{(z,w)\in\mathbb{C}\times\mathbb{R}:|z-\delta_{k}|\leq
l_{k},|w|<\frac{l_{k}}{8\delta}\\},k\in\mathbb{N}$ $\displaystyle
M_{k}^{\prime}$
$\displaystyle:=\\{(z,w)\in\mathbb{C}\times\mathbb{R}:\text{Re}z=m_{k},w\in\mathbb{R}\\},k\in\mathbb{N}$
$\displaystyle A_{k}^{\prime}$
$\displaystyle:=\\{(z,w)\in\mathbb{C}\times\mathbb{R}:|z+r_{k}|\leq
s_{k},|w|\leq\frac{l_{1}}{4}\\},k\in\mathbb{N}$ $\displaystyle L_{k}^{\prime}$
$\displaystyle:=\\{(z,w)\in\mathbb{C}\times\mathbb{R}:\text{Re}z=-t_{k},w\in\mathbb{R}\\},k\in\mathbb{N}$
$\displaystyle\text{and }G_{2}$
$\displaystyle:=\\{(z,w)\in\mathbb{C}\times\mathbb{R}:|z-1|\leq
3,|w|\leq\frac{9}{8\delta}\\}.$

First, we shall show that for every $(z,w)\in B_{k}^{\prime}$, we have
$F(z,w)\in B_{k+1}^{\prime}$. For this, suppose that $(z,w)\in
B_{k}^{\prime}$. By definition of $F$, we have $F(z,w)=(e^{f_{1}(z)}+\delta
w,e^{f_{2}(w)})$. Consider,

$\displaystyle|e^{f_{1}(z)}+\delta e^{\text{Re}f_{2}(w)}-\delta_{k+1}|$
$\displaystyle\leq|e^{f_{1}(z)}-\delta_{k+1}+\delta|e^{\text{Re}f_{2}(w)}|$
$\displaystyle\leq\frac{l_{k+1}}{4}+\delta\frac{17}{16}$
$\displaystyle\leq\frac{l_{k+1}}{4}+\frac{l_{k+1}}{8}$
$\displaystyle\leq\frac{l_{k+1}}{2}.$

Also,

$\displaystyle|e^{\text{Re}f_{2}(w)}|$ $\displaystyle\leq\frac{17}{16}$
$\displaystyle\leq\frac{l_{k+1}}{4\delta}.$

This shows that $F(B_{k}^{\prime})\subset B_{k+1}^{\prime}$, for every
$k\in\mathbb{N}$, i.e., $F$ maps $B_{k}^{\prime}$ into a smaller poly disc
inside $B_{k+1}^{\prime}$, for every $k\in\mathbb{N}$. Now, assume that
$(z,w)\in G_{2}\cup(\cup_{k\in\mathbb{N}}(M_{k}^{\prime}\cup
L_{k}^{\prime}))$. Hence, we have

$\displaystyle|e^{f_{1}(z)}+\delta e^{\text{Re}f_{2}(w)}-1|$
$\displaystyle\leq|e^{f_{1}(z)}-1|+\delta|e^{\text{Re}f_{2}(w)}|$
$\displaystyle\leq\frac{1}{16}+\delta\frac{17}{16}$ $\displaystyle<3.$

Also,

$\displaystyle|e^{\text{Re}f_{2}(w)}|$ $\displaystyle<\frac{17}{16}$
$\displaystyle<\frac{9}{8\delta}.$

This, in particular shows that
$F(M_{k}^{\prime})\subset\\{(z,w)\in\mathbb{C}^{2}:|z-1|<2,|w|<\frac{9}{8\delta}\\}\subset
G_{2}$, for every $k\in\mathbb{N}$. Since, each $B_{k}^{\prime}$ is disjoint
and $F(M_{k}^{\prime})\subset G_{2}$, for every $k\in\mathbb{N}$. Let $N_{k}$
be the Fatou component of $F$ containing $B_{k}^{\prime}$, for
$k\in\mathbb{N}$. By the construction of $F$, it is clear that each $N_{k}$ is
an escaping wandering domain of $F$ for every $k\in\mathbb{N}$. ∎

######  Proposition 3.

There exists a mapping $F$ on $\mathbb{C}\times\mathbb{R}$ having infinitely
many attracting Fatou components.

###### Proof.

Consider $B_{k}$ and $M_{k}$ for every $k\in\mathbb{N}$ as we defined earlier.
Now we shall follow the following steps:
Step1: Using continuity of $e^{z}$, there exists $h_{k}^{\prime}>0$ such that

$|z-\log(\delta_{k})|<h_{k}^{\prime}\text{ implies that
}|e^{z}-\delta_{k}|<\frac{l_{k}}{4}.$

Step2: Define functions $g(z)$ and $\epsilon(z)$ as follows:

$g(z)=\left\\{\begin{array}[]{ll}\log(\delta_{k}),&z\in
B_{k},k\in\mathbb{N}\\\ \log 1,&z\in
G_{1}\cup(\cup_{k\in\mathbb{N}}M_{k}),\end{array}\right.$

and

$\epsilon(z)=\left\\{\begin{array}[]{ll}\frac{l_{k}}{4},&z\in
B_{k},k\in\mathbb{N}\\\ \delta^{{}^{\prime\prime}}_{2},&z\in
G_{1}\cup(\cup_{k\in\mathbb{N}}M_{k}).\end{array}\right.$

For such functions, using the definition of a Carleman set, we have

$|f_{1}(z)-\log(\delta_{k})|<\delta^{\prime}_{k},\text{ whenever }z\in
B_{k},k\in\mathbb{N}.$

This further implies that

$|e^{f_{1}(z)}-\delta_{k}|<\frac{l_{k}}{4}.$

Similarly, for $z\in G_{1}\cup(\cup_{k\in\mathbb{N}}M_{k})$, we have

$|e^{f_{1}(z)}-1|<3.$

Now, define $B_{k}^{\prime}$ and $M_{k}^{\prime}$ for every $k\in\mathbb{N}$
as we did earlier. Consider,

$\displaystyle|e^{f_{1}(z)}+\delta e^{\text{Re}f_{2}(w)}-\delta_{k}|$
$\displaystyle\leq|e^{f_{1}(z)}-\delta_{k}|+\delta|e^{\text{Re}f_{2}(w)}|$
$\displaystyle\leq\frac{l_{k}}{4}+\frac{17\delta}{16}$
$\displaystyle\leq\frac{l_{k}}{4}+\frac{l_{k}}{4}$
$\displaystyle=\frac{l_{k}}{2},$

and

$\displaystyle|e^{\text{Re}f_{2}(w)}|$ $\displaystyle\leq\frac{17}{16}$
$\displaystyle<\frac{l_{k}}{8\delta}.$

These inequalities show that $F(B_{k}^{\prime})\subset B_{k}^{\prime}$ for
every $k\in\mathbb{N}$, i.e., every $B_{k}^{\prime}$ is being mapped by $F$
into a smaller poly disc inside $B_{k}^{\prime}$. For $k\in\mathbb{N}$, let
$C_{k}$ be the Fatou component containing $B_{k}^{\prime}$ which is an
attracting Fatou component. Since,
$F(G_{2}\cup(\cup_{k\in\mathbb{N}}M_{k}^{\prime}))\subset G_{2}$ and each
$B_{k}^{\prime}$ is disjoint, we have $C_{i}\cap C_{j}=\emptyset$ for
$i\not=j$. This gives us the existence of an infinite number of attracting
Fatou components. ∎

Recall that two wandering domains $C_{0}$ and $V_{0}$ are said to have a
common path if $C_{m}=V_{n}$ for some $n,m\in\mathbb{N}$, where $C_{m}$ and
$V_{n}$ are the Fatou components containing $F^{m}(C_{0})$ and $F^{n}(V_{0})$
respectively. If $C_{m}\not=V_{n}$ for any $n,m\in\mathbb{N}$, then $C_{0}$
and $V_{0}$ are said to have distinct paths.

######  Proposition 4.

There exists a mapping $F$ on $\mathbb{C}\times\mathbb{R}$ with infinitely
many wandering domains having a common path.

###### Proof.

For this, we shall make use of the domains $A_{k}^{\prime},k\in\mathbb{N}$.
Firstly, using continuity of $e^{z}$ there exists
$\delta^{{}^{\prime\prime}}_{3}>0$ such that

$|z-\log(\delta_{1})|<\delta^{{}^{\prime\prime}}_{3}\text{ implies that
}|e^{z}-\delta_{1}|<\frac{l_{1}}{4}.$

Define the function $g(z)$ and $\epsilon(z)$ as

$g(z)=\left\\{\begin{array}[]{ll}\log(\delta_{1}),&z\in
A_{k},k\in\mathbb{N}\\\ \log(\delta_{k+1}),&z\in B_{k},k\in\mathbb{N}\\\ \log
1,&z\in G_{1}\cup\left(\cup_{k\in\mathbb{N}}(M_{k}\cup
L_{k})\right),\end{array}\right.$

and

$\epsilon(z)=\left\\{\begin{array}[]{ll}\frac{l_{1}}{4},&z\in
A_{k},k\in\mathbb{N}\\\ \delta^{\prime}_{k},&z\in B_{k},k\in\mathbb{N}\\\
\frac{1}{4},&z\in G_{1}\cup\left(\cup_{k\in\mathbb{N}}(M_{k}\cup
L_{k})\right).\end{array}\right.$

Now, for every $k\in\mathbb{N}$, define

$A_{k}^{\prime}:=\\{(z,w)\in\mathbb{C}\times\mathbb{R}:|z-r_{k}|\leq
s_{k},|w|\leq\frac{l_{1}}{4}\\}.$

Arguing as in the previous manner, we can observe that $|e^{f_{1}(z)}+\delta
e^{f_{2}(z)}-\delta_{1}|<\frac{l_{1}}{2}$ and
$|e^{f_{2}(w)}|<\frac{l_{1}}{4}$, for every $(z,w)\in A_{k}^{\prime}$. This
means that $F(A_{k}^{\prime})\subset B_{1}^{\prime}$, for every
$k\in\mathbb{N}$. In particular, if $D_{k}$ is the Fatou component containing
$A_{k}^{\prime}$, $k\in\mathbb{N}$, then $C_{1}$ (the Fatou component
containing $B_{1}^{\prime}$) is a common path for these wandering domains. ∎

######  Remark 5.

From the above proposition, one can observe that by modifying functions
$g(z),\epsilon(z)$ and by suitably using domains, we can have

  * •

An $F$ with wandering domains each having a distinct path.

  * •

$F$ having wandering domains with a common path as well as having wandering
domains with distinct paths.

## Acknowledgement

I would like to thank my thesis advisor, Prof. Sanjay Kumar, for several
fruitful discussions. This work is supported by the National Board for Higher
Mathematics, India.

## References

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  * [4] D. Gaier, Lectures on Complex Approximation, Birkhaüser, Boston, 1987.
  * [5] J. E. Fornaess and N. Sibony, Fatou and Julia sets for entire mappings in ${\bf C}^{k}$, Math. Ann. 311 (1998), no. 1, 27–40.
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A clinical validation of VinDr-CXR, an AI system for detecting abnormal chest
radiographs

Ngoc Huy Nguyen1,☯, Ha Quy Nguyen2,3,☯, Nghia Trung Nguyen2, Thang Viet
Nguyen2, Hieu Huy Pham2,3,*, Tuan Ngoc-Minh Nguyen4

1 Phu Tho Department of Health, Phu Tho, Vietnam

2 Medical Imaging Department, Vingroup Big Data Institute, Hanoi, Vietnam

3 College of Engineering and Computer Science, VinUniversity, Hanoi, Vietnam

4 Training and Direction of Healthcare Activities Center, Phu Tho General
Hospital, Phu Tho, Vietnam

These authors contributed equally to this work.

* Corresponding author. Email<EMAIL_ADDRESS>
## Abstract

Computer-Aided Diagnosis (CAD) systems for chest radiographs using artificial
intelligence (AI) have recently shown a great potential as a second opinion
for radiologists. The performances of such systems, however, were mostly
evaluated on a fixed dataset in a retrospective manner and, thus, far from the
real performances in clinical practice. In this work, we demonstrate a
mechanism for validating an AI-based system for detecting abnormalities on
X-ray scans, VinDr-CXR, at the Phu Tho General Hospital–a provincial hospital
in the North of Vietnam. The AI system was directly integrated into the
Picture Archiving and Communication System (PACS) of the hospital after being
trained on a fixed annotated dataset from other sources. The performance of
the system was prospectively measured by matching and comparing the AI results
with the radiology reports of 6,285 chest X-ray examinations extracted from
the Hospital Information System (HIS) over the last two months of 2020. The
normal/abnormal status of a radiology report was determined by a set of rules
and served as the ground truth. Our system achieves an F1 score–the harmonic
average of the recall and the precision–of 0.653 (95% CI 0.635, 0.671) for
detecting any abnormalities on chest X-rays. Despite a significant drop from
the in-lab performance, this result establishes a high level of confidence in
applying such a system in real-life situations.

## Introduction

Chest radiograph, or chest X-ray (CXR), remains one of the most common, yet
hard to interpret, imaging protocols in medicine. It is hoped that a Computer-
Aided Diagnosis (CAD) system using artificial intelligence (AI) can
effectively assist radiologists and help mitigate the misdiagnosis rate on
CXRs. Leveraging recent advances in deep learning [1], such systems have
achieved a great success in detecting a wide range of abnormalities on CXRs
[2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. Most of the existing systems are
supervised-learning models that were trained and validated on different parts
of a dataset that was collected and labeled in a retrospective fashion. For
example, several deep learning models were developed [3, 5] on the ChestX-
ray14 dataset [13] for classifying 14 common thoracic pathologies. Recently,
most algorithms for detecting abnormalities on CXRs were trained and validated
on the CheXpert [4, 8, 6] and MIMIC-CXR [14] datasets, which include the same
set of 14 findings that are slightly different from the labels of ChestX-
ray14. The performances of the aforementioned AI systems in differentiating
multiple findings on CXRs were reported to be comparable with radiologists.
Other works were devoted to detecting a specific lung disease such as
pneumonia [2], pulmonary tuberculosis [10, 12] and lung cancer [15]. Notably,
Rajpurkar et al. [2] trained a convolutional neural network (CNN) for
detecting pneumonia that achieved an F1 score of 0.435 (95% CI 0.387, 0.481)
on the ChestX-ray14 dataset, which performance was claimed to exceed those of
practicing radiologists. Tang et al. proposed [16] to train an abnormality
classifier, which is closely related to our work, with various CNN
architectures over 3 CXR different datasets: the ChestX-ray14, the RSNA
Pneumonia Detection Challenge [17] and the Indiana University Hospital Network
[18]. Although reaching impressive AUC (Area under receiver operating
characteristic Curve) performances of 0.9x, those models were again evaluated
on retrospective curated datasets that might be drastically different from the
real data in clinical settings.

Unlike existing works, our study does not focus on the development and the
retrospective evaluation of an AI-based CAD system for CXR. Instead, we
propose a framework to validate such a system while being deployed at a
clinical site for a significantly long period. In particular, we integrate our
system, VinDr-CXR, directly to the Picture Archiving and Communication System
(PACS) of the Phu Tho General Hospital–a provincial hospital in Vietnam. The
systems consists of 3 AI models that were trained on our own dataset [19]
collected from other sources. All CXRs generated by the PACS during two months
are prospectively automatically analyzed by the VinDr-CXR. The obtained AI
results are then matched and compared with the radiology reports that are
extracted from the Hospital Information System (HIS) to compute the
performance of the system in distinguishing abnormal versus normal CXR
studies. Despite the ability of the system to localize multiple classes of
lesions, we only measure its performance as a binary classifier. The reason
for doing so is that it is much more reliable to decide if a radiology report
is abnormal than to interpret its subtle details. We also propose simple
template matching rules to determine the normal/abnormal status of a report.

Over the last two months of 2020, the VinDr-CXR system generated AI results
for 6,687 CXR studies taken at the Phu Tho General Hospital, 6,285 of which
were matched with corresponding radiology reports from the HIS. The matching
was nontrivial since the PACS and the HIS were not linked by accession
numbers. Instead, we had to rely on the patient ID and other attributes of the
DICOM files and the radiology reports. By treating the normal/abnormal status
of the 6,285 matched reports as a ground-truth reference, the abnormality
classifier of the VinDr-CXR yielded an F1 score of 0.653 (95% Confidence
Interval (CI) 0.635, 0.671). The 95% CI of the F1 score statistic was obtained
by bootstrapping [20], a method that was also used in [2]. The F1 score
obtained in this clinical setting is significantly below the one achieved
while training and validating the model “at home” on a retrospective dataset.
Nonetheless, the reported performance still gives us a high level of
confidence in deploying the VinDr-CXR system in clinical practice. It also
serves a good baseline for similar AI-based CAD systems to be clinically
validated.

## Materials and methods

We propose an overall scheme for validating our CAD system, VinDr-CXR, as
illustrated in Fig 1. A set of AI results are obtained by directly integrating
VinDr-CXR into the PACS of the Phu Tho General Hospital, while the
corresponding radiology reports are extracted from the HIS via an XML parser.
These two sets of results are then pairwise matched and compared to each other
to determine the correctness of the system in detecting abnormal CXRs. The
final result of the validation will be reported as an F1 score, a metric that
balances the precision and recall of a binary classifier.

Figure 1: Validation scheme. PACS and HIS are linked by Patient ID.

### Development of AI models

As shown in Fig 2, the VinDr-CXR system is a concatenation of 3 AI models: the
PA classifier, the abnormality classifier, and, finally, the lesion detector.
This system takes as input a CXR from the PACS and returns the probability
that the image is abnormal and the locations of multiple classes of lesions,
if any. All constitutional models of VinDr-CXR were obtained by training deep
neural networks entirely on our own dataset, also called VinDr-CXR, part of
which was made publicly available [19]. This dataset was retrospectively
collected from our partner hospitals in Vietnam and got annotated by a team of
experienced radiologists. Each image in the dataset was manually labeled by at
least 1 radiologist with a list of 6 different diagnoses where 22 types of
lesions were annotated with bounding boxes. It is important to emphasize that
none of the training data was from the Phu Tho General Hospital. Each
individual model was trained and validated before being deployed in the real
clinical workflow of the hospital. We did not make any changes to the models
during the two months of the clinical trial. This is to ensure that our models
are not at all biased to the real-life validation setup. We briefly describe
here the development of the 3 AI models; details of the training will be
presented somewhere else.

Figure 2: VinDr-CXR pipeline. The system includes 3 concatenated AI models
that are integrated to the PACS via a CXR adapter. The output of the system is
the probability of the CXR being abnormal and the locations of the lesions, if
any.

#### PA classifier

This PA classifier is attached to the CXR adapter to guarantee that only CXRs
of Posterior-Anterior (PA) view will be passed to the abnormality classifier,
which was trained only on this type of images. The output of the PA classifier
is a probability of the input image being a PA-view CXR. If this probability
is greater than a normalized threshold of 0.5, the image will go through to
the abnormality classifier; otherwise, the system will output an indicator
that the image is invalid. The PA classifier adopted the ResNet-18
architecture [21] that was trained and validated on a dataset of total 9,864
scans where 4,329 of them are actual PA-view CXRs taken from the VinDr-CXR
dataset. The negative training examples included lateral-view CXRs and images
of other body parts that sometimes got through the CXR filter due to
mismatched DICOM tags. The trained PA classifier achieves an F1 score of 0.980
on a validation set of 4,192 images. Here, the F1 score metric is defined as

$F_{1}=\frac{TP}{TP+(FP+FN)/2},$ (1)

where $TP,FP,FN$ denote the numbers of true positive, false positive, and
false negative samples, respectively.

#### Abnormality classifier

The abnormality classifier separates abnormal CXRs from normal ones. It takes
as input a PA-view CXR and outputs the probability that the image contains
abnormal findings. Only images whose abnormal probabilities are above 0.5 will
go to the lesion detector. We trained the abnormality classifier as an
EfficientNet-B6 [22] on a dataset of 38,065 PA-view CXRs. All images that were
labeled with “No finding” by the radiologists were treated as negative
samples, while the rest are considered positive. This model was validated on
another dataset of 9,611 images with an F1 score of 0.831. In this study, only
the output of the abnormality classifier will be compared to the radiology
reports to measure the performance of the whole system.

#### Lesion detector

The role of the lesion detector is to localize all findings on an abnormal CXR
with bounding boxes and, at the same time, classify them into different types
of lesion. That is, the system can tell not only whether a CXR is abnormal but
also why it is and where the abnormalities come from. An example output of the
lesion detector is visualized in Fig 2 where a bounding box of the class
“Pulmonary fibrosis” is drawn around the lesion. Out of the 22 local classes
in the VinDr-CXR dataset, we only trained the lesion detector on the 17 most
prevalent ones as listed in Table 1. The training was performed on 23,524
abnormal CXRs with an EfficientDet-ED4 model [23]. The performance of the
lesion detector was evaluated on a validation set of 4,470 images using the
Average Precision (AP) metric at the Intersection-over-Union (IoU) threshold
of 0.4 or shortly AP @ 0.4. This is a standard metric for objection detection
models in computer vision [24, 25]. Table 1 reports the performance of the
lesion detector in terms of AP for each of the 17 classes with an average AP
(mAP) of 0.365. However, the model was not validated on the data the Phu Tho
General Hospital. This is due to the hardness in interpreting the lesion
locations in a radiology report.

Table 1:  Performance of the lesion detector on 17 classes of abnormality. Lesion class | AP @ 0.4
---|---
Aortic enlargement | 0.663
Atelectasis | 0.231
Calcification | 0.272
Cardiomegaly | 0.860
Clavicle fracture | 0.459
Consolidation | 0.281
Emphysema | 0.185
Enlarged PA | 0.256
Infiltration | 0.318
Interstitial lung disease (ILD) | 0.315
Nodule/Mass | 0.251
Opacity | 0.197
Pleural effusion | 0.387
Pleural thickening | 0.228
Pneumothorax | 0.579
Pulmonary fibrosis | 0.340
Rib fracture | 0.381
mAP | 0.365

### Integration of AI models to PACS

As shown in Fig 2, the VinDr-CXR system is integrated to the PACS of the
hospital through the CXR adapter. This is a web service that pulls all images
from the PACS via the DICOMweb protocol [26] and only passes CXRs to the AI
models of the VinDr-CXR. To check if a scan is a CXR, we rely on the
`MODALITY` and the `BODY_PART_EXAMINIED` attributes of the DICOM file. In
particular, the AI models are triggered only if the value of `MODALITY` is
either `CR`, `DR` or `DX` and the value of `BODY_PART_EXAMINIED` is either
`CHEST` or `THORAX`. These conditions were established by surveying the
imaging procedure at the Radiology Department of the Phu Tho General Hospital.
For deploying the VinDr-CXR at other clinical sites, the CXR adapter might be
slightly modified to catch all CXRs from the PACS.

### Extraction of radiology reports from HIS

The radiology reports have to be extracted from HIS according to a procedure
described in Fig 3. Each session of examination and treatment is stored in a
single Extensible Markup Language (XML) file that can be exported from HIS. A
session includes all information of the patient from the check-in time to the
check-out time. The XML parser is used to read all the reports within a
session, each of which includes the `SERVICE_ID`, `REPORT_TIME`, and
`DESCRIPTION` attributes. The CXR service filter only keeps the reports whose
`SERVICE_ID` matches a fixed value reserved for the CXR imaging by the
Vietnamese Ministry of Health. The XML parser can also read the header of a
session that includes `SESSION_ID`, `PATIENT_ID`, `CHECK_IN_TIME`, and
`CHECK_OUT_TIME`. These attributes are shared among all radiology reports
within the session and will be used, in addition to the `REPORT_TIME`, to
match a radiology report with an AI result of a CXR scan.

Figure 3: Procedure for extracting all radiology reports of CXR examinations
from HIS. The original names of the attributes, which are in Vietnammese, are
put inside square brackets.

### Matching AI results with radiology reports

After extracting the CXR radiology reports from HIS, we match each of them
with an AI result obtained from VinDr-CXR via the algorithm illustrated in Fig
4. Note that an AI result includes the `ABNORMAL_STATUS` (0/1) of a CXR study
and is associated with a `PATIENT_ID` and a `STUDY_TIME`, which attributes are
extracted from the DICOM file. Since HIS and PACS are linked by the
`PATIENT_ID`, the matching algorithm uses this key to check whether an AI
result and a radiology report are of the same patient. Next, the `STUDY_TIME`
has to be in between the `CHECK_IN_TIME` and the `CHECK_OUT_TIME`. Finally,
the `REPORT_TIME` must be within 24 hours from the `STUDY_TIME`, which is a
regulated protocol of the hospital. If all the aforementioned conditions are
satisfied, the AI result and the CXR radiology report are matched.

Figure 4: Algorithm for matching an AI result with a radiology report.

### Comparing AI results to radiology reports

The `ABNORMAL_STATUS` of an AI result is then compared to the `DESCRIPTION` of
the matched radiology report, if any, to measure the performance of VinDr-CXR
in detecting abnormal CXR scans. To that end, we propose a simple template-
matching rule to determine if a description is normal. In particular, we
observe that a CXR radiology report without any findings always includes 4
paragraphs that describe the 4 fixed anatomical regions of the thorax: chest
wall, pleura, lung, and mediastinum. The templates for normal descriptions of
these 4 regions are summarized in Table 2. A region is considered normal if
one of the corresponding templates exactly appears in the `DESCRIPTION` of a
radiology report. A report is normal if all the 4 regions are normal,
otherwise it is abnormal.

Table 2:  Templates for normal descriptions of the 4 anatomical regions in a CXR radiology report. Anatomical region | Templates for normal descriptions
---|---
Chest wall | `không thấy hình bất thường xương lồng ngực`
`không thấy hình tổn thương xương lồng ngực`
Pleura | `không thấy hình tràn dịch màng phổi`
`không thấy hình tràn dịch, khí màng phổi`
`không thấy hình tràn khí, tràn dịch màng phổi`
Lung | `nhu mô phổi không thấy bất thường`
Mediastinum | `hình tim và trung thất không thấy bất thường`
`hình tim và trung thất bình thường`

## Results and Discussion

A set of 6,585 AI results was obtained by running the VinDr-CXR on all DICOM
images in the PACS of the Phu Tho General Hospital throughout November and
December of 2020. Meanwhile, another set of 6,687 CXR radiology reports was
extracted from the HIS during this period. Applying the matching algorithm to
these two sets resulted in 6,285 studies of 5,989 patients that have both an
AI result and a radiology report. By matching the radiology reports of these
studies with the templates given in Table 2, we achieved a ground truth of
4,529 (72.4%) normal and 1,756 (27.6%) abnormal cases. The confusion matrix of
the VinDr-CXR abnormality classifier over the total 6,285 studies is plotted
in Fig 5. We follow [2] to compute the average F1 score on 10,000 bootstrap
samples drawn with replacement [20] from the 6,285 studies. We also use the
2.5th and 97.5th percentiles of the bootstrap distribution to establish the
95% confidence interval (CI). The bootstrap distribution of F1 scores is shown
in Fig 6, which gives a mean F1 score of 0.653 (95% CI 0.635, 0.671).

It can be seen that the F1 score of our AI system in detecting abnormal CXRs
significantly drops from 0.831 to 0.653 when shifting from the training
settings to the deployment phase in a clinical site. This might be caused by
the shift in the distribution of the CXR images or the additional clinical
information received by the radiologists in practice. Furthermore, it must be
admitted that the radiologists’ reports at the deployed site are not
necessarily all accurate. Nevertheless, the obtained F1 score still exhibits a
high level of confidence when deploying the VinDr-CXR in practice, as compared
to the 0.435 F1 score of the pneumonia detector presented in [2].

One limitation of this study is that it only provides a coarse evaluation of
the VinDr-CXR system for the task of classifying a CXR into normal and
abnormal categories. To validate the lesion detector of the system, we need a
more sophisticated interpretation of the radiology reports that can extract a
ground truth comparable to the output of the AI model. Another shortcoming of
this work is that the reported clinical validation is for the AI system
itself. It is even more important to assess the effect of such a system on
improving the quality of the radiologists’ diagnoses. Moreover, an ideal AI-
based CAD system should continuously learn from the daily feedback of the
doctors rather than staying stationary like our VinDr-CXR. We plan to address
all of these drawbacks in our future research agenda.

Figure 5: Confusion matrix of the VinDr-CXR abnormality classifier. Figure 6:
Bootstrap distribution of F1 scores of the VinDr-CXR abnormality classifier
over 10,000 samples drawn from 6,285 studies.

## Conclusion

We have discussed in this paper a mechanism for validating the performance of
the VinDr-CXR system in classifying normal/abnormal chest radiographs at the
Phu Tho General Hospital during the last two months of 2020. Once the AI
models of the system were trained on an annotated dataset from different
sources, they were directly integrated into the PACS of the hospital and never
got retrained during the validation period. The performance of the abnormality
classifier was prospectively measured by matching and then comparing the
obtained AI results with a set of radiology reports exported from the HIS.
Since the PACS and the HIS were linked only by patient IDs, we proposed an
algorithm to match an AI result of a study with a CXR radiology reports. We
also adopted a simple template matching rule to decide the abnormal status of
a radiology report, which served as a ground-truth reference. We obtained an
average F1 score of 0.653 (95% CI 0.635, 0.671) for the abnormality classifier
over 10,000 resamples drawn from the 6,285 studies of 5,989. We believe this
result has set a significant benchmark for deploying AI systems for chest
radiograph analysis in clinical practice.

## Acknowledgments

This work was funded by the Vingroup JSC.

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