# Neighbor-view Enhanced Model for Vision and Language Navigation

Dong An1,2, Yuankai Qi3, Yan Huang1*, Qi Wu3, Liang Wang1,4,5, Tieniu Tan1,4 1
Center for Research on Intelligent Perception and Computing, Institution of
Automation, Chinese Academy of Sciences  2School of Future Technology,
University of Chinese Academy of Sciences, 3University of Adelaide  4 Center
for Excellence in Brain Science and Intelligence Technology (CEBSIT)  5
Chinese Academy of Sciences, Artificial Intelligence Research (CAS-AIR)
dong.an@cripac.ia.ac.cn,qykshr@gmail.com,yhuang,wangliang,tnt<EMAIL_ADDRESS>
(2021)

###### Abstract.

Vision and Language Navigation (VLN) requires an agent to navigate to a target
location by following natural language instructions. Most of existing works
represent a navigation candidate by the feature of the corresponding single
view where the candidate lies in. However, an instruction may mention
landmarks out of the single view as references, which might lead to failures
of textual-visual matching of existing methods. In this work, we propose a
multi-module Neighbor-View Enhanced Model (NvEM) to adaptively incorporate
visual contexts from neighbor views for better textual-visual matching.
Specifically, our NvEM utilizes a subject module and a reference module to
collect contexts from neighbor views. The subject module fuses neighbor views
at a global level, and the reference module fuses neighbor objects at a local
level. Subjects and references are adaptively determined via attention
mechanisms. Our model also includes an action module to utilize the strong
orientation guidance (e.g., “turn left”) in instructions. Each module predicts
navigation action separately and their weighted sum is used for predicting the
final action. Extensive experimental results demonstrate the effectiveness of
the proposed method on the R2R and R4R benchmarks against several state-of-
the-art navigators, and NvEM even beats some pre-training ones. Our code is
available at https://github.com/MarSaKi/NvEM.

Vision and Language Navigation, Visual context modeling, Modular attention
networks

††journalyear: 2021††copyright: acmcopyright††conference: Proceedings of the
29th ACM International Conference on Multimedia; October 20–24, 2021; Virtual
Event, China††booktitle: Proceedings of the 29th ACM International Conference
on Multimedia (MM ’21), October 20–24, 2021, Virtual Event, China††price:
15.00††doi: 10.1145/3474085.3475282††isbn: 978-1-4503-8651-7/21/10††ccs:
Computing methodologies Knowledge representation and reasoning††ccs:
Information systems Information systems applications Figure 1. (a)
Illustration of panorama view and navigation graph in VLN tasks. The yellow
node is the current location. Candidate nodes are blue and green. Note that
the green one is the ground-truth. (b) Single-view-based candidates used in
previous works, which contain limited visual contexts and might be
insufficient for action prediction. (c) Our neighbor-view scheme enriches the
visual context of each candidate, leading to better textual-visual matching.
Numbers above each view denote (heading, elevation). Figure 2. Examples of
action-, subject\- and reference-related phrases in instructions. Some
instructions do not contain references.

##  1\. Introduction

Vision and Language Navigation (VLN) has drawn increasing interest in recent
years, partly because it represents a significant step towards enabling
intelligent agents to interact with the realistic world. Running in a 3D
simulator (Anderson et al., 2018) rendered with real-world images (Chang et
al., 2017), the goal of VLN is to navigate to a target location by following a
detailed natural language instruction, such as “Walk around the table and exit
the room. Walk down the first set of stairs. Wait there.”. There are two kinds
of simulators, which render continuous navigation trajectories (Krantz et al.,
2020) and discrete trajectories (Anderson et al., 2018) respectively. In this
paper we focus on the discrete one, where the agent navigates on a discrete
graph (see Figure 1 (a)).

A variety of approaches have been proposed to address the VLN problem (Wang et
al., 2018; Fried et al., 2018; Landi et al., 2019; Tan et al., 2019; Ma et
al., 2019a; Wang et al., 2019; Li et al., 2019; Qi et al., 2020a; Hao et al.,
2020; Majumdar et al., 2020; Wang et al., 2020a, b; Hong et al., 2020a; Deng
et al., 2020; Hong et al., 2020b). Most of them adopt panoramic action space
(Fried et al., 2018), where the agent select a navigable candidate from its
observations to transport at each step. However, the context of navigable
candidates is rarely discussed in existing works, and the commonly used
single-view candidates are of limited visual contexts which may hamper the
matching between instructions and the visual representations of candidates.
Figure 1 shows such an example, where there are three candidate “archways”,
each of which is represented by a single-view visual perception (Figure 1
(b)). According to the instruction, only the archway “to the left of mirror”
leads to the correct navigation. However, most of existing agents may fail
because they cannot find the referred “mirror” in any single-view-based
candidate.

Thus, we propose to enhance the textual-visual matching by fusing visual
information from candidates’ neighbor views as shown in Figure 1 (c), which is
rarely explored before. It is non-trivial to fuse neighbor views for visual
contexts modeling, because many unmentioned visual clues exist which may
interfere with the agent’s decision (e.g., the lamp in Figure 1 (c)). In
addition, some instructions even do not involve visual clues in neighbor
views, such as “go through the doorway”. To handle this challenging problem,
we propose to decompose an instruction into: action-, subject\- and reference-
related phrases as shown in Figure 2. Generally, the action and subject are
necessary, and the optional reference helps to distinguish the desired
candidate from other similar ones.

Based on the above mentioned three types of instruction phrases, we further
design a multi-module Neighbor-view Enhanced Model (NvEM) to adaptively fuse
neighbor visual contexts in order to improve the textual-visual matching
between instructions and candidates’ visual perceptions. Specifically, our
NvEM includes a subject module, a reference module and an action module, where
subjects and references are determined via attention mechanisms. On one hand,
the subject module aggregates neighbor views at a global level based on
spatial information. On the other hand, the reference module aggregates
related objects from neighbor views at a local level. The action module makes
use of the orientation guidance (i.e., “turn left”) in instructions. Each
module predicts navigation action separately and their weighted sum is used to
predict the final action. Note that the combination weights are trainable and
predicted based on the decomposed subject-, reference- and action-related
phrases.

The contributions of this work are summarized as follows:

$\bullet$ To improve the textual-visual matching between instructions and
navigable candidates, we propose to take into account the visual contexts from
neighbor views for the first time.

$\bullet$ We propose a subject module and a reference module to adaptively
fuse visual contexts from neighbor views at both global level and local level.

$\bullet$ Extensive experimental results demonstrate the effectiveness of the
proposed method with comparisons against several existing state-of-the-art
methods, and NvEM even beats some pre-training ones.

##  2\. Related Work

Vision and Language Navigation. Numerous approaches have been proposed to
address the VLN problem. Most of them are based on the CNN-LSTM architecture
with attention mechanisms: at each time step, the agent first grounds
surrounding observations to instructions, then chooses the most matched
candidate according to the grounded instructions as the next location. Early
work Speaker-Follower (Fried et al., 2018) develops a speaker model to
synthesize new instructions for randomly sampled trajectories. Additionally,
they design a panoramic action space for efficient navigation. Later on,
EnvDrop (Tan et al., 2019) increases the diversity of synthetic data by
randomly removing objects to generate “new environments”.

On the other line, Self-monitoring (Ma et al., 2019a) and RCM (Wang et al.,
2019) utilize the cross-modality co-attention mechanism to enhance the
alignment of instructions and trajectories. To learn generic linguistic and
visual representations for VLN, AuxRN (Zhu et al., 2020) designs several
auxiliary self-supervised losses. Very recently, large-scale pre-training
models for VLN are widely explored (Li et al., 2019; Hao et al., 2020;
Majumdar et al., 2020; Hong et al., 2020b), where they improve the agent’s
generation abilities dramatically by benefitting from priors of other
datasets. Different types of visual clues correspond to different phrases in
an instruction, OAAM (Qi et al., 2020a) and RelGraph (Hong et al., 2020a)
utilize decomposed phrases to guide more accurate action prediction. OAAM (Qi
et al., 2020a) adopts action and object specialized clues to vote action at
each time step, while RelGraph (Hong et al., 2020a) proposes a graph network
to model the intra- and inter-relationships among the contextual and visual
clues. The most relevant work to ours is RelGraph (Hong et al., 2020a), where
we both attempt to exploit view-level and object-level features. The key
difference is: we focus on enhancing each candidate’s representation with its
multiple neighbor views (namely inter-view), while the representation in
RelGraph is limited to a single view and thus it’s intra-view.

Modular Attention Networks. Modular networks are widely adopted in vision and
language models. It attempts to decompose sentences into multiple phrases via
attention mechanisms, as different phrases usually correspond to different
visual clues. MAttNet (Yu et al., 2018) decomposes a long sentence into
appearance, location and relationship three parts for referring expression.
LCGN (Hu et al., 2019b) utilizes a multi-step textual attention mechanism to
extract different object-related phrases, then modeling objects‘ contexts via
the relations among the phrases. LGI (Mun et al., 2020) utilizes a sequential
query attention module to decompose the query into multiple semantic phrases,
then using these phrases to interact with video clips for video grounding. To
the best of our knowledge, OAAM (Qi et al., 2020a) is the earliest attempt to
decompose instructions in VLN. They decompose instructions into “action” and
“object ” specialized phrases, and use these phrases to vote next action. Our
work has two key differences with OAAM: (I) Different modules: our modules are
(subject, reference, action) v.s. (object, action) of OAAM. (II) Our subject
module and reference module fuse information from neighbor views while OAAM
only use information within one single view.

Figure 3. Main architecture of the proposed multi-module Neighbor-view
Enhanced Model (NvEM). First, action-, subject\- and reference-related phrases
are attended via an attention mechanism (Section 4.2). Then, reference module
and subject module predict navigation actions via aggregating visual contexts
from candidates’ neighbor views at local and global levels. The action module
predicts navigation in terms of orientation information. (Section 4.3).
Lastly, the weighted sum of all three predictions predicts the final
navigation decision. (Section 4.4).

##  3\. Preliminary

In VLN (Anderson et al., 2018), given a natural language instruction with $L$
words, an agent navigates on a discrete graph to reach the described target by
following the instruction. At each time step $t$, the agent observes a
panorama which consists of 36 discrete views. Each view is represented by an
image $\mathbf{v}_{t,i}$, with its orientation including heading
$\mathbf{\theta}_{t,i}$ and elevation $\mathbf{\phi}_{t,i}$. Also, there are
$K$ candidates at time step $t$, and each candidate is represented by the
single view where the candidate lies in, with the candidate’s relative
orientation to the agent. Formally, for the $i$-th view:

(1) $\displaystyle\mathbf{f}_{t,i}$
$\displaystyle=[\textrm{ResNet}(\mathbf{v}_{t,i});E_{ori}(\mathbf{\theta}_{t,i},\mathbf{\phi}_{t,i})]$

where $\textrm{ResNet}(\cdot)$ represents ResNet (He et al., 2016) pooling
features, $E_{ori}(\cdot)$ is an embedding function for heading and elevation,
which repeats $[\cos\mathbf{\theta}_{t,i},\sin\mathbf{\theta}_{t,i}$,
$\cos\mathbf{\phi}_{t,i},\sin\mathbf{\phi}_{t,i}]$ 32 times following (Tan et
al., 2019). The $k$-th candidate $\mathbf{g}_{t,k}$ is encoded in the same
way.

Previous works show that data augmentation is able to significantly improve
the generalization ability in unseen environments (Fried et al., 2018; Tan et
al., 2019). We make use of this strategy by adopting EnvDrop (Tan et al.,
2019) as baseline, which first uses a bi-directional LSTM to encode the
instruction, and the encoded instruction is represented as
$\mathbf{I}=\\{\mathbf{u_{l}}\\}_{l=1}^{L}$. Then the agent’s previous
context-aware state $\tilde{\mathbf{h}}_{t-1}$ is used to attend on all views
to get scene feature:
$\tilde{\mathbf{f}}_{t}=\textrm{SoftAttn}(\tilde{\mathbf{h}}_{t-1},\\{\mathbf{f}_{t,i}\\}_{i=1}^{36})$.
The concatenation of $\tilde{\mathbf{f}}_{t}$ and the previous action
embedding $\mathbf{a}_{t-1}$ is fed into the decoder LSTM to update the
agent’s state:
$\mathbf{h}_{t}=\textrm{LSTM}([\tilde{\mathbf{f}}_{t};\mathbf{a}_{t-1}],\tilde{\mathbf{h}}_{t-1})$.
Note that the context-aware agent state is updated via the attentive
instruction feature $\mathbf{u}_{t}$:

(2)
$\displaystyle\tilde{\mathbf{h}}_{t}=\textrm{Tanh}(\mathbf{W}_{H}[\mathbf{u}_{t};\mathbf{h}_{t}]),\
\
\mathbf{u}_{t}=\textrm{SoftAttn}(\tilde{\mathbf{h}}_{t-1},\\{\mathbf{u}_{l}\\}_{l=1}^{L})$

where $\mathbf{W}_{H}$ is a trainable linear projection and
$\textrm{SoftAttn}(\cdot)$ mentioned above denotes soft-dot attention.
Finally, EnvDrop predicts navigation action by selecting the candidate with
the highest probability ($\mathbf{W}_{G}$ is a trainable linear projection):

(3)
$\displaystyle\mathbf{a}_{t}^{*}=\mathop{\arg\max}_{k}P(\mathbf{g}_{t,k}),\ \
P(\mathbf{g}_{t,k})=\textrm{softmax}_{k}(\mathbf{g}_{t,k}^{\top}\mathbf{W}_{G}\tilde{\mathbf{h}}_{t})$

##  4\. Methodology

In this section, we first briefly describe the pipeline of the proposed model
in training phase, then we detail the proposed Neighbor-view Enhanced Model
(NvEM). Note that in this section, we omit the time step $t$ to avoid
notational clutter in the exposition.

###  4.1. Overview

Figure 3 illustrates the main pipeline of our NvEM. First, action-, subject\-
and reference-related phrases are attended by three independent attention
schemes. Then, reference module and subject module predict navigation actions
via aggregating visual contexts from candidates’ neighbor views. The action
module predicts navigation in terms of orientation information. Lastly, the
final navigation action is determined by combining these three predictions
together with weights generated from phrases’ embeddings.

###  4.2. Phrase Extractor

Considering an instruction, such as “walk through the archway to the left of
the mirror…”, there are three types of phrases which the agent needs to
identify: the action describes the orientation of target candidate (e.g.,
“walk through”), the subject describes the main visual entity of the correct
navigation (e.g., “the archway”) and the reference which is referenced by the
subject (e.g., “to the left of the mirror”). Thus, NvEM first performs three
soft-attentions independently on the instruction, conditioned on the current
agent state $\mathbf{h}_{t}$, to attend on these three types of phrases:

(4)
$\displaystyle\tilde{\mathbf{h}}_{a}=\textrm{Tanh}(\mathbf{W}_{Ha}[\mathbf{u}_{a};\mathbf{h}_{t}]),\
\
\mathbf{u}_{a}=\textrm{SoftAttn}_{a}(\tilde{\mathbf{h}}_{t-1},\\{\mathbf{u}_{l}\\}_{l=1}^{L})$
$\displaystyle\tilde{\mathbf{h}}_{s}=\textrm{Tanh}(\mathbf{W}_{Hs}[\mathbf{u}_{s};\mathbf{h}_{t}]),\
\
\mathbf{u}_{s}=\textrm{SoftAttn}_{s}(\tilde{\mathbf{h}}_{t-1},\\{\mathbf{u}_{l}\\}_{l=1}^{L})$
$\displaystyle\tilde{\mathbf{h}}_{r}=\textrm{Tanh}(\mathbf{W}_{Hr}[\mathbf{u}_{r};\mathbf{h}_{t}]),\
\
\mathbf{u}_{r}=\textrm{SoftAttn}_{r}(\tilde{\mathbf{h}}_{t-1},\\{\mathbf{u}_{l}\\}_{l=1}^{L})$

where the subscripts denote the corresponding types of phrases,
$\textrm{SoftAttn}_{*}(\cdot)$ is the same with Eq (2). $\mathbf{u}_{*}$ and
$\mathbf{\tilde{h}}_{*}$ denote features of corresponding phrases and context-
aware agent states, and they are updated by different linear projections
$\mathbf{W}_{H*}$. The global context-aware agent state
$\mathbf{\tilde{h}}_{t}$ in Eq (2) is now calculated by averaging the three
specialized context-aware states $\mathbf{\tilde{h}}_{a}$,
$\mathbf{\tilde{h}}_{s}$ and $\mathbf{\tilde{h}}_{r}$.

###  4.3. Neighbor-view Enhanced Navigator

Corresponding to the attended three types of phrases in instructions, our
neighbor-view enhanced navigator contains three modules: a reference module, a
subject module and an action module. The reference module and subject module
predict navigation actions via aggregating visual contexts from neighbor views
at local and global levels, respectively. The action module predicts
navigation actions according to orientation information. We provide the
details below.

Reference Module. Reference usually exists as a landmark surrounding the
subject to clarify similar navigation candidates. In the example “walk through
the archway to the left of the mirror…”, the reference is “the mirror” and it
is referred with a spatial relationship to the subject (e.g., “to the left
of”). This motivates us to enhance the representation of a navigation
candidate by using features of local objects and their spatial relations.

Formally, for the $k$-th candidate $\mathbf{g}_{k}$, given its orientation
$\mathbf{d}_{k}$, the objects’ features $\\{\mathbf{o}_{k,i,j}\\}_{j=1}^{J}$
extracted by (Ren et al., 2015) and their orientations
$\\{\mathbf{d}_{k,i,j}\\}_{j=1}^{J}$ in the candidate’s neighbor views (assume
there are $J$ objects in each neighbor and $i$ denotes the $i$-th neighbor).
We then calculate the spatial relations of the neighbor objects to the $k$-th
candidate:

(5)
$\displaystyle\mathbf{e}_{k,i,j}=E_{ori}(\mathbf{d}_{k,i,j}-\mathbf{d}_{k})$

where $E_{ori}(\cdot)$ is the same as that in Eq (1). Then each neighbor
object is represented as the concatenation of its object feature and relative
spatial embedding:

(6)
$\displaystyle\mathbf{\bar{o}}_{k,i,j}=\mathbf{\Gamma}_{o}([\mathbf{o}_{k,i,j};\mathbf{e}_{k,i,j}])$

where $[:]$ denotes concatenation, $\mathbf{\Gamma}_{o}(\cdot)$ projects
objects’ features into a $\mathbf{D}_{k}$ dimensional space and all
$\mathbf{\Gamma}(\cdot)_{*}$ in this section represent trainable non-linear
projections, with $\textrm{Tanh}(\cdot)$ as the activation function. We use
the reference-related phrases $\mathbf{u}_{r}$ to highlight relevant objects
in neighbor views:

(7) $\displaystyle\mathbf{A}_{k,i,j}^{r}=\textrm{softmax}_{i,j}$
$\displaystyle(\frac{(\mathbf{W}_{q}^{r}\cdot\mathbf{u}_{r})^{\top}(\mathbf{W}_{k}^{r}\cdot\mathbf{\bar{o}}_{k,i,j})}{\sqrt{\mathbf{D}_{k}}})$
$\displaystyle\mathbf{F}_{k}^{r}=$
$\displaystyle\sum_{i,j}\mathbf{A}_{k,i,j}^{r}\cdot\tilde{\mathbf{o}}_{k,i,j}$

where $\mathbf{W}_{q}^{r}$ and $\mathbf{W}_{k}^{r}$ are trainable linear
projections. The reference module predicts the confidence of the candidate
$\mathbf{g}_{k}$ being the next navigation action using reference-related
state $\mathbf{\tilde{h}}_{r}$ and the neighbor reference enhanced candidate
representation $\mathbf{F}_{k}^{r}$:

(8)
$\mathbf{\tau}_{r,k}=\mathbf{\tilde{h}}_{r}^{\top}\mathbf{\widetilde{W}}_{r}\mathbf{F}_{k}^{r}$

where $\mathbf{\widetilde{W}}_{r}$ is a trainable linear projection.

Subject Module. Instruction subject describes the main visual entity of the
correct navigation, such as “the archway” in “walk through the archway to the
left of the mirror…”. However, sometimes there are multiple candidates
containing the different instances of the subject. To alleviate the ambiguity
in visual side, we propose to enhance the visual representation of subject by
incorporating contexts from neighbor views. Specifically, we aggregate
neighbor views at a global level, with the help of the spatial affinities
neighbor views to the candidate.

Formally, for the $k$-th candidate $\mathbf{g}_{k}$, given its orientation
$\mathbf{d}_{k}$, its neighbor views’ orientations
$\\{\mathbf{d}_{k,i}\\}_{i=1}^{I}$ and ResNet features
$\\{\mathbf{v}_{k,i}\\}_{i=1}^{I}$ (assume it has $I$ neighbor views), we
first embed all neighbor views using a trainable non-linear projection:

(9)
$\displaystyle\mathbf{\tilde{v}}_{k,i}=\mathbf{\Gamma}_{s}({\mathbf{v}}_{k,i})$

Then we compute the spatial affinities among the candidate and its neighbor
views based on their orientations in a query-key manner (Vaswani et al.,
2017). Then the enhanced subject visual representation $\mathbf{F}_{k}^{s}$ is
obtained by adaptively aggregating neighbor views’ embeddings:

(10) $\displaystyle\mathbf{A}_{k,i}^{s}=\textrm{softmax}_{i}$
$\displaystyle(\frac{(\mathbf{W}_{q}^{s}\cdot\mathbf{d}_{k})^{\top}(\mathbf{W}_{k}^{s}\cdot\mathbf{d}_{k,i})}{\sqrt{\mathbf{D}_{k}}})$
$\displaystyle\mathbf{F}_{k}^{s}$
$\displaystyle=\sum_{i}\mathbf{A}_{k,i}^{s}\cdot\mathbf{\tilde{v}}_{k,i}$

where $\mathbf{W}_{q}^{s}$ and $\mathbf{W}_{k}^{s}$ are trainable linear
projections. Similar to reference module, the subject module predicts the
confidence of candidate $\mathbf{g}_{k}$ being the next navigation action via:

(11)
$\mathbf{\tau}_{s,k}=\mathbf{\tilde{h}}_{s}^{\top}\mathbf{\widetilde{W}}_{s}\mathbf{F}_{k}^{s}$

where $\mathbf{\widetilde{W}}_{s}$ is a trainable linear projection.

Action Module. Action related phrases serve as strong guidance in navigation
(Hu et al., 2019a; Qi et al., 2020a), such as “go forward”, “turn left” and
“go down”. Inspired by (Qi et al., 2020a), for the $k$-th candidate
$\mathbf{g}_{k}$, the action module predicts the confidence of candidate
$\mathbf{g}_{k}$ being the next navigation action, using the candidate’s
orientation $\mathbf{d}_{k}$ and action-related state $\tilde{h}_{a}$:

(12) $\displaystyle\mathbf{F}_{k}^{a}=$
$\displaystyle\mathbf{\Gamma}_{a}(E_{ori}(\mathbf{d}_{k}))$
$\displaystyle\mathbf{\tau}_{a,k}=$
$\displaystyle\mathbf{\tilde{h}}_{a}^{\top}\mathbf{\widetilde{W}}_{a}\mathbf{F}_{k}^{a}$

where $\mathbf{\widetilde{W}}_{a}$ is a trainable linear projection.

###  4.4. Adaptive Action Integrator

The action, subject and reference modules usually contribute at different
degrees to the final decision. Thus, we propose to adaptively integrate
predictions from these three modules. We first calculate the combination
weights conditioned on action-, subject\- and reference-specialized phrases
$\mathbf{u}_{a}$, $\mathbf{u}_{s}$ and $\mathbf{u}_{r}$. Take the action
weight $\mathbf{w}_{a}$ as an example:

(13)
$\displaystyle\mathbf{w}_{a}=\frac{exp(\mathbf{W}_{a}\cdot\mathbf{u}_{a})}{exp(\mathbf{W}_{a}\cdot\mathbf{u}_{a})+exp(\mathbf{W}_{s}\cdot\mathbf{u}_{s})+exp(\mathbf{W}_{r}\cdot\mathbf{u}_{r})}$

where the $\mathbf{W}_{a}$, $\mathbf{W}_{s}$ and $\mathbf{W}_{r}$ are
trainable linear projections. Then the final action probability of the $k$-th
candidate is calculated by weighted summation of above confidences:

(14)
$\displaystyle\mathbf{P}_{k}=\textrm{softmax}_{k}(\mathbf{w}_{a}\cdot\mathbf{\tau}_{a,k}+\mathbf{w}_{s}\cdot\mathbf{\tau}_{s,k}+\mathbf{w}_{r}\cdot\mathbf{\tau}_{r,k})$

In the inference phase, the agent selects the candidate with the maximum
probability as shown in Eq (3) at each step.

###  4.5. Training

We apply the Imitation Learning (IL) + Reinforcement Learning (RL) objectives
to train our model following (Tan et al., 2019). In imitation learning, the
agent takes the teacher action $\mathbf{a}_{t}^{*}$ at each time step to learn
to follow the ground-truth trajectory. In reinforcement learning, the agent
samples an action $\mathbf{a}_{t}^{S}$ via the probability $\mathbf{P}_{t}$
and learns from the rewards. Formally:

(15)
$\displaystyle\mathbf{L}=\mathbf{\lambda}\sum_{t=1}^{\mathbf{T}^{*}}-\mathbf{a}_{t}^{*}\log(\mathbf{P}_{t})+\sum_{t=1}^{\mathbf{T}^{S}}-\mathbf{a}_{t}^{S}\log(\mathbf{P}_{t})\mathbf{A}_{t}$

where $\mathbf{\lambda}$ is a coefficient for weighting the IL loss,
$\mathbf{T}^{*}$ and $\mathbf{T}^{S}$ are the total numbers of steps the agent
takes in IL and RL respectively. $\mathbf{A}_{t}$ is the advantage in A2C
algorithm (Mnih et al., 2016). We apply the summation of two types of rewards
in RL objective, goal reward and fidelity reward following (Jain et al.,
2019).

Table 1. Comparison of single-run performance with the state-of-the-art
methods on R2R. ${\dagger}$ denotes works that apply pre-trained textual or
visual encoders.

| Val Seen | Val Unseen | Test Unseen
---|---|---|---
Agent | TL | NE$\downarrow$ | SR$\uparrow$ | SPL$\uparrow$ | TL | NE$\downarrow$ | SR$\uparrow$ | SPL$\uparrow$ | TL | NE$\downarrow$ | SR$\uparrow$ | SPL$\uparrow$
Random | 9.58 | 9.45 | 0.16 | - | 9.77 | 9.23 | 0.16 | - | 9.89 | 9.79 | 0.13 | 0.12
Human | - | - | - | - | - | - | - | - | 11.85 | 1.61 | 0.86 | 0.76
PRESS (Li et al., 2019) ${\dagger}$ | 10.57 | 4.39 | 0.58 | 0.55 | 10.36 | 5.28 | 0.49 | 0.45 | 10.77 | 5.49 | 0.49 | 0.45
PREVALENT (Hao et al., 2020) ${\dagger}$ | 10.32 | 3.67 | 0.69 | 0.65 | 10.19 | 4.71 | 0.58 | 0.53 | 10.51 | 5.30 | 0.54 | 0.51
VLNBert (init. OSCAR) (Hong et al., 2020b)${\dagger}$ | 10.79 | 3.11 | 0.71 | 0.67 | 11.86 | 4.29 | 0.59 | 0.53 | 12.34 | 4.59 | 0.57 | 0.53
VLNBert (init. PREVALENT) (Hong et al., 2020b)${\dagger}$ | 11.13 | 2.90 | 0.72 | 0.68 | 12.01 | 3.93 | 0.63 | 0.57 | 12.35 | 4.09 | 0.63 | 0.57
Seq2Seq (Anderson et al., 2018) | 11.33 | 6.01 | 0.39 | - | 8.39 | 7.81 | 0.22 | - | 8.13 | 7.85 | 0.20 | 0.18
Speaker-Follower (Fried et al., 2018) | - | 3.36 | 0.66 | - | - | 6.62 | 0.35 | - | 14.82 | 6.62 | 0.35 | 0.28
SM (Ma et al., 2019a) | - | 3.22 | 0.67 | 0.58 | - | 5.52 | 0.45 | 0.32 | 18.04 | 5.67 | 0.48 | 0.35
RCM+SIL (Wang et al., 2019) | 10.65 | 3.53 | 0.67 | - | 11.46 | 6.09 | 0.43 | - | 11.97 | 6.12 | 0.43 | 0.38
Regretful (Ma et al., 2019b) | - | 3.23 | 0.69 | 0.63 | - | 5.32 | 0.50 | 0.41 | 13.69 | 5.69 | 0.48 | 0.40
VLNBert (no init.) (Hong et al., 2020b) | 9.78 | 3.92 | 0.62 | 0.59 | 10.31 | 5.10 | 0.50 | 0.46 | 11.15 | 5.45 | 0.51 | 0.47
EnvDrop (Tan et al., 2019) | 11.00 | 3.99 | 0.62 | 0.59 | 10.70 | 5.22 | 0.52 | 0.48 | 11.66 | 5.23 | 0.51 | 0.47
AuxRN (Zhu et al., 2020) | - | 3.33 | 0.70 | 0.67 | - | 5.28 | 0.55 | 0.50 | - | 5.15 | 0.55 | 0.51
RelGraph (Hong et al., 2020a) | 10.13 | 3.47 | 0.67 | 0.65 | 9.99 | 4.73 | 0.57 | 0.53 | 10.29 | 4.75 | 0.55 | 0.52
NvEM (ours) | 11.09 | 3.44 | 0.69 | 0.65 | 11.83 | 4.27 | 0.60 | 0.55 | 12.98 | 4.37 | 0.58 | 0.54

Table 2. Comparison of single-run performance with the state-of-the-art
methods on R4R. goal and fidelity indicate goal and fidelity reward in
reinforcement learning. ${\ddagger}$ denotes our reimplemented R4R results.

| Val Seen | Val Unseen
---|---|---
Agent | NE$\downarrow$ | SR$\uparrow$ | SPL$\uparrow$ | CLS$\uparrow$ | nDTW$\uparrow$ | sDTW$\uparrow$ | NE$\downarrow$ | SR$\uparrow$ | SPL$\uparrow$ | CLS$\uparrow$ | nDTW$\uparrow$ | sDTW$\uparrow$
EnvDrop (Tan et al., 2019) | - | 0.52 | 0.41 | 0.53 | - | 0.27 | - | 0.29 | 0.18 | 0.34 | - | 0.09
RCM-a (goal) (Jain et al., 2019) | 5.11 | 0.56 | 0.32 | 0.40 | - | - | 8.45 | 0.29 | 0.10 | 0.20 | - | -
RCM-a (fidelity) (Jain et al., 2019) | 5.37 | 0.53 | 0.31 | 0.55 | - | - | 8.08 | 0.26 | 0.08 | 0.35 | - | -
RCM-b (goal) (Ilharco et al., 2019) | - | - | - | - | - | - | - | 0.29 | 0.15 | 0.33 | 0.27 | 0.11
RCM-b (fidelity) (Ilharco et al., 2019) | - | - | - | - | - | - | - | 0.29 | 0.21 | 0.35 | 0.30 | 0.13
OAAM (Qi et al., 2020a) | - | 0.56 | 0.49 | 0.54 | - | 0.32 | - | 0.31 | 0.23 | 0.40 | - | 0.11
RelGraph (Hong et al., 2020a)${\ddagger}$ | 5.14 | 0.55 | 0.50 | 0.51 | 0.48 | 0.35 | 7.55 | 0.35 | 0.25 | 0.37 | 0.32 | 0.18
NvEM (ours) | 5.38 | 0.54 | 0.47 | 0.51 | 0.48 | 0.35 | 6.85 | 0.38 | 0.28 | 0.41 | 0.36 | 0.20

##  5\. Experiments

In this section, we first describe the commonly used VLN datasets and the
evaluation metrics. Then we present implementation details of NvEM. Finally,
we compare against several state-of-the-art methods and provide ablation
experiments. Qualitative visualizations are also presented.

###  5.1. Datasets and Evaluation Metrics

R2R benchmark. The Room-to-Room (R2R) dataset (Anderson et al., 2018) consists
of 10,567 panoramic view nodes in 90 real-world environments as well as 7,189
trajectories described by three natural language instructions. The dataset is
split into train, validation seen, validation unseen and test unseen sets. We
follow the standard metrics employed by previous works to evaluate the
performance of our agent. These metrics include: the Trajectory Length (TL)
which measures the average length of the agent’s navigation path, the
Navigation Error (NE) which is the average distance between the agent’s final
location and the target, the Success Rate (SR) which measures the ratio of
trajectories where the agent stops at 3 meters within the target, and the
Success Rate weighted by Path Length (SPL) which considers both path length
and success rate. Note that the SR and SPL in unseen environments are main
metrics for R2R.

R4R benchmark. The Room-for-Room (R4R) dataset (Jain et al., 2019) is an
extended version of R2R, which has longer instructions and trajectories. The
dataset is split into train, validation seen, validation unseen sets. Besides
the main metrics in R2R, R4R includes additional metrics: the Coverage
Weighted by Length Score (CLS) (Jain et al., 2019), the Normalized Dynamic
Time Wrapping (nDTW) (Ilharco et al., 2019) and the nDTW weighted by Success
Rate (sDTW) (Ilharco et al., 2019). In R4R, SR and SPL measure the accuracy of
navigation, while CLS, nDTW and sDTW measure the fidelity of predicted paths
and ground-truth paths.

###  5.2. Implementation Details

We use ResNet-152 (He et al., 2016) pre-trained on Places365 (Zhou et al.,
2018) to extract view features. We apply Faster-RCNN (Ren et al., 2015) pre-
trained on the Visual Genome Dataset (Krishna et al., 2017) to obtain object
labels in reference module, then encoded by Glove (Pennington et al., 2014).
To simplify the object vocabulary, we retain the top 100 most frequent classes
mentioned in R2R training data following (Hong et al., 2020a). Our method
exploits neighbor views to represent a navigation candidate, thus the number
of neighbors and objects are crucial for the final performance. Our default
setting adopts 4 neighbor views and the top 8 detected objects in each
neighbor view (we also tried other settings, please see Sec 5.4). For
simplicity, the objects’ positions are roughly represented by their
corresponding views’ orientations in Eq (5). This is reasonable as
instructions usually mention approximate relative relations of objects and
candidates.

Our model is trained with the widely used two-stage strategy (Tan et al.,
2019). At the first stage, only real training data is used. At the second
stage, we pick the model with the highest SR at the first stage, and keep
training it with both real data and synthetic data generated from (Fried et
al., 2018). As for R4R experiment, we only apply the first stage training. We
set the projection dimension $\mathbf{D}_{k}$ in Sec 4.3 as 512, the glove
embedding dimension as 300, and train the agent using RMSprop optimizer
(Ruder, 2016) with $1\times 10^{-4}$ learning rate. We train the first stage
for 80,000 iterations and the second stage will continue training up to
200,000 iterations. All our experiments are conducted on a Nvidia V100 Tensor
Core GPU.

###  5.3. Comparison with the State-of-The-Art

We compare NvEM with several SoTA methods under the single-run setting on both
R2R and R4R benchmarks. Note that VLN mainly focus on the agent’s performance
on unseen splits, so the performance we report is based on the model which has
the highest SR on the validation unseen split.

As shown in Table 1, on the R2R benchmark our NvEM outperforms the baseline
EnvDrop (Tan et al., 2019) by a large margin, which obtains $7\%$ absolute
improvements in terms of SPL on both Val Unseen and Test splits. Compared to
the state-of-the-art method RelGraph (Hong et al., 2020a), our model obtains
$2\%$ absolute improvements on Val Unseen and Test splits. Moreover, NvEM even
beats some pre-training methods, such as PRESS (Li et al., 2019), PREVALENT
(Hao et al., 2020), and VLNBert (init. OSCAR) (Hong et al., 2020b). We also
note that our method does not surpass VLNBert pre-trained on PREVALENT, which
uses in-domain data for pretraining. As shown later in the ablation study, the
success of our model mainly benefits from the incorporating of visual
contexts, where NvEM effectively improve the textual-visual matching leading
to more accurate actions.

On the R4R benchmark, we observe similar phenomenon with that on R2R. As shown
in Table 2, NvEM not only significantly outperforms the baseline EnvDrop (Tan
et al., 2019) with $10\%$ absolute improvement, but also sets the new SoTA. In
particular, NvEM achieves 0.41 CLS, 0.36 nDTW and 0.20 sDTW, which are largely
higher than the second best RelGraph (Hong et al., 2020a) by 4%, 4% and %2,
respectively.

###  5.4. Ablation Study

We conduct ablation experiments over different components of NvEM on R2R
dataset. Specifically, we study how the action-, subject\- and reference-
module contribute to navigation. Then we compare the single-view subject
module against the neighbor-view one. Lastly, we study how the numbers of
neighbor views and objects in the subject-and reference-module affect the
performance. All our ablation models are trained from scratch by two stage
training.

The importance of different modules. Our full model utilizes three modules:
the action-, subject\- and reference-module, which correspond to orientations,
global views and local objects. To study how they affect the performance of
navigation, we conduct an ablation experiment by removing corresponding
module. The results are shown in Table 3. In model #1, we remove the action-
module, and it achieves the worst performance compared to others in terms of
the main metric SPL. This indicates that orientations are strong guidance for
navigation. The same phenomenon is observed in (Hu et al., 2019a; Qi et al.,
2020a). In model #2, we remove the subject-module, and it performs slight
better than model#1 but still far behind the full model. This indicates the
global view information is important for VLN. In model #3, we remove the
reference-module, and it performs slight worse than the full model. This
indicates reference contains some useful information but not as important as
subject. Another reason is that the local reference can also be included in
global views of the subject-module.

Table 3. Ablation experiment about importance of different modules.

| Modules | Val Seen | Val Unseen
---|---|---|---
model | action | subject | reference | SR | SPL | SR | SPL
1 |  | ✓ | ✓ | 0.623 | 0.584 | 0.486 | 0.440
2 | ✓ |  | ✓ | 0.580 | 0.523 | 0.504 | 0.446
3 | ✓ | ✓ |  | 0.666 | 0.636 | 0.579 | 0.539
4 | ✓ | ✓ | ✓ | 0.686 | 0.645 | 0.601 | 0.549

Single-view subject vs neighbor-view subject. In the subject-module, we
aggregate neighbor view features based on their spatial affinities to the
candidate, which raises the following questions: is neighbor-view more
effective than single-view? How about aggregating views in other manners? To
answer the above questions, we test different types of subject-modules, and
the results are shown in Table 4. Note that we remove the reference module to
exclude the affect from reference. In table 4, single uses single-view global
features to represent subject, lang uses subject-aware phrases
$\mathbf{u}_{s}$ as query and view features as keys to ground views in Eq
(10), while spa is our default setting which is based on spatial affinity. The
results show that both model #2 and #3 perform better than the single view
based model #1, which indicates the superior of neighbor-view based models.
Moreover, we observe that language grounded neighbor-view fusion (model #2)
performs worse than the spatial based one (model #3). This may be caused by
the gap between language embeddings and visual embeddings.

Table 4. Different types of subject-modules, single denotes single-view, lang
denotes aggregating views using language as query while spa denotes using
spatial affinity.

| Subject | Val Seen | Val Unseen
---|---|---|---
model | single | lang | spa | SR | SPL | SR | SPL
1 | ✓ |  |  | 0.641 | 0.605 | 0.556 | 0.507
2 |  | ✓ |  | 0.637 | 0.608 | 0.564 | 0.522
3 |  |  | ✓ | 0.666 | 0.636 | 0.579 | 0.539

The analysis of number of neighbors and objects. In our default setting, the
subject-module uses 4 neighbor views and the reference-module uses 8 objects
in each neighbor view. However, it is naturally to consider more neighbors
(e.g., 8 neighbors) and more objects. In this experiment, we test 4, 8
neighbors, and 4, 8, 12 objects in each view. The results are shown in Table
5. In model #1 and model #2, we adjust the number of objects and keep 4
neighbors. They perform worse than our default setting (model #4). The reason
may be that fewer objects might not contain the objects mentioned by
instructions, and more objects could be redundant. To study the number of
neighbors, we keep the object number as 8. Comparing model #3 with model #4,
the results show 4 neighbors is the better. The reason may be that more
neighbors contain more redundant information. Not only will they affect the
aggregation for subject context, but also increase the difficulty to highlight
relevant objects. An example is shown in Figure 4, intuitively, 8 neighbors
have more redundant visual information than 4 neighbors one.

Table 5. Ablation experiments about the number of neighbor views and objects.
Views denote the number of neighbors, and Objects denote the number of objects
in a view.

| Views | Objects | Val Seen | Val Unseen
---|---|---|---|---
model | 4 | 8 | 4 | 8 | 12 | SR | SPL | SR | SPL
1 | ✓ |  |  |  | ✓ | 0.655 | 0.617 | 0.564 | 0.517
2 | ✓ |  | ✓ |  |  | 0.667 | 0.638 | 0.572 | 0.532
3 |  | ✓ |  | ✓ |  | 0.644 | 0.615 | 0.564 | 0.523
4 | ✓ |  |  | ✓ |  | 0.686 | 0.645 | 0.601 | 0.549

Figure 4. A schematic diagram of 4 neighbors and 8 neighbors.

Figure 5. Visualizations of two success cases. The mask on the panorama
denotes neighbor-view spatial affinities of the predicted candidate in subject
module (note that neighbor views overlap with the center view). The boxes are
objects with top three scores in reference module. Attentions on three types
of phrases are also showed. Figure 6. Visualization of a failure case. Green
arrow denotes ground-truth action, while red arrow denotes predicted action.
The mask on the panorama denotes neighbor-view spatial affinities of the
predicted candidate in subject module (note that neighbor views overlap with
the center view). The boxes are objects with top three scores in reference
module. Attentions on three types of phrases are also showed.

###  5.5. Qualitative Visualizations

Here we present some success and failure cases. Figure 5 shows two success
cases. Taking the bottom one as an example, each module attends on correct
phrases, such as “go up”, “stairs” and “to the left of the refrigerator”
corresponding to action, subject and reference modules respectively. More
visual information of the subject “stairs” from neighbor views are
incorporated by our subject-module. In addition, taking advantage of the
reference-module, our model could perceive the mentioned “refrigerator” in
neighbor views as reference, and suppress unmentioned ones. We note that the
attention scores of objects are not that high, which may mainly caused by the
fact that we consider totally 32 objects for each candidate (4 neighbors and 8
objects in each neighbor).

Figure 6 visualizes a failure case. The model mainly focus on the “bathroom”
in the subject-module, which leads to a wrong direction. This indicates that
the attentive phrases of NvEM are not always correct, and thus it still could
be improved by studying how to extract more accurate phrases.

##  6\. Conclusion

In this paper, we present a novel multi-module Neighbor-view Enhanced Model to
improve textual-visual matching via adaptively incorporating visual
information from neighbor views. Our subject module aggregates neighbor views
at a global level based on spatial-affinity, and our reference module
aggregates neighbor objects at a local level guided by referring phrases.
Extensive experiments demonstrate that NvEM effectively improves the agent’s
performance in unseen environments and our method sets the new state-of-the-
art.

Considering the similarity between R2R task and other VLN tasks, such as
dialogue navigation (Thomason et al., 2019; Nguyen and III, 2019), remote
object detection (Qi et al., 2020b) and navigation in continuous space (Krantz
et al., 2020), we believe the neighbor-view enhancement idea could also
benefit agents in other embodied AI tasks. We leave them as future works to
explore.

##  7\. Acknowledgments

This work was jointly supported by National Key Research and Development
Program of China Grant No. 2018AAA0100400, National Natural Science Foundation
of China (61525306, 61633021, 61721004, 61806194, U1803261, and 61976132),
Beijing Nova Program (Z201100006820079), Shandong Provincial Key Research and
Development Program (2019JZZY010119), Key Research Program of Frontier
Sciences CAS Grant No.ZDBS-LY-JSC032, and CAS-AIR.

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11institutetext:  Department of Physics and Astronomy, FI-20014 University of
Turku, Finland 22institutetext: Nordita, KTH Royal Institute of Technology and
Stockholm University, Hannes Alfvéns väg 12, SE-10691 Stockholm, Sweden
33institutetext: INAF Istituto di Astrofisica e Planetologia Spaziali, Via del
Fosso del Cavaliere 100, 00133 Roma, Italy 44institutetext: Dipartimento di
Fisica, Università degli Studi di Roma “Tor Vergata”, Via della Ricerca
Scientifica 1, 00133 Roma, Italy 55institutetext: Dipartimento di Fisica,
Università degli Studi di Roma “La Sapienza”, Piazzale Aldo Moro 5, 00185
Roma, Italy 66institutetext: Astronomical Institute of the Czech Academy of
Sciences, Boční II 1401/1, 14100 Praha 4, Czech Republic 77institutetext:
Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences, Bartycka
18, PL-00-716 Warszawa, Poland 88institutetext: Astrophysics Group, Cavendish
Laboratory, 19 J. J. Thomson Avenue, Cambridge, CB3 0HE, UK 99institutetext:
Harvard-Smithsonian Center for Astrophysics, 60 Garden St, Cambridge, MA
02138, USA 1010institutetext: Astrophysics, Department of Physics, University
of Oxford, Denys Wilkinson Building, Keble Road, Oxford, OX1 3RH, UK
1111institutetext: Dipartimento di Matematica e Fisica, Università degli Studi
Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy 1212institutetext: Max
Planck Institute for Astrophysics, Karl-Schwarzschild-Str 1, D-85741 Garching,
Germany 1313institutetext: Space Research Institute, Russian Academy of
Sciences, Profsoyuznaya 84/32, 117997 Moscow, Russia

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astronomical puzzleUltrasoft state of microquasar Cygnus X-3:  X-ray
polarimetry reveals the geometry of astronomical puzzle 0000-0001-9167-2790
Fabio La Monaca Ultrasoft state of microquasar Cygnus X-3:  X-ray polarimetry
reveals the geometry of astronomical puzzleUltrasoft state of microquasar
Cygnus X-3:  X-ray polarimetry reveals the geometry of astronomical
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the geometry of astronomical puzzleUltrasoft state of microquasar Cygnus X-3:
X-ray polarimetry reveals the geometry of astronomical puzzleUltrasoft state
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0000-0001-5418-291X Sergey S. Tsygankov Ultrasoft state of microquasar Cygnus
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state of microquasar Cygnus X-3:  X-ray polarimetry reveals the geometry of
astronomical puzzle 0000-0002-9679-0793 Andrzej A. Zdziarski Ultrasoft state
of microquasar Cygnus X-3:  X-ray polarimetry reveals the geometry of
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of microquasar Cygnus X-3:  X-ray polarimetry reveals the geometry of
astronomical puzzle 0000-0003-3189-9998 Fabio Muleri Ultrasoft state of
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astronomical puzzleUltrasoft state of microquasar Cygnus X-3:  X-ray
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of microquasar Cygnus X-3:  X-ray polarimetry reveals the geometry of
astronomical puzzle 0000-0002-4622-4240 Enrico Costa Ultrasoft state of
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state of microquasar Cygnus X-3:  X-ray polarimetry reveals the geometry of
astronomical puzzle 0000-0001-6894-871X Michael McCollough Ultrasoft state of
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0000-0002-7781-4104 Rashid Sunyaev Ultrasoft state of microquasar Cygnus X-3:
X-ray polarimetry reveals the geometry of astronomical puzzleUltrasoft state
of microquasar Cygnus X-3:  X-ray polarimetry reveals the geometry of
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microquasar Cygnus X-3:  X-ray polarimetry reveals the geometry of
astronomical puzzle 0000-0002-2764-7192

(Received …; Accepted …)

Cygnus X-3 is an enigmatic X-ray binary, that is both an exceptional accreting
system and a cornerstone for the population synthesis studies. Prominent X-ray
and radio properties follow a well-defined pattern, yet the physical reasons
for the state changes observed in this system are not known. Recently, the
presence of an optically thick envelope around the central source in the hard
state was revealed using the X-ray polarization data obtained with Imaging
X-ray Polarimetry Explorer (IXPE). In this work, we analyse IXPE data obtained
in the ultrasoft (radio quenched) state of the source. The average
polarization degree (PD) of $11.9\pm 0.5\%$ at a polarization angle (PA) of
$94\degr\pm 1\degr$ is inconsistent with the simple geometry of the accretion
disc viewed at an intermediate inclination. The high PD, the blackbody-like
spectrum, and the weakness of fluorescent iron line imply that the central
source is hidden behind the optically thick outflow and its beamed radiation
is scattered towards our line of sight. In this picture the observed PD is
directly related to the source inclination, which we conservatively determine
to lie in the range $26\degr<i<28\degr$. .pdfUsing the new polarimetric
properties, we propose the scenario that can be responsible for the cyclic
behaviour of the state changes in the binary.

###### Key Words.:

accretion, accretion discs – polarization – stars: black holes – X-rays:
individuals: Cyg X-3 – X-rays: binaries

##  1 Introduction

Accreting binary system Cygnus X-3 (hereafter Cyg X-3) is one of the first
detected X-ray sources (Giacconi et al., 1967). This persistent source shows
several exceptional observational properties. It is the brightest X-ray binary
in radio wavelengths, with fluxes that can reach 20 Jy (McCollough et al.,
1999; Corbel et al., 2012), and is one of a few Galactic binaries with
detectable $\gamma$-ray emission (Tavani et al., 2009; Atwood et al., 2009).
It is one of the shortest-period X-ray binaries, $P_{\rm orb}=4.8$ h, that is
also rapidly changing owing to the high mass loss rate from the system (van
der Klis & Bonnet-Bidaud, 1981; Antokhin & Cherepashchuk, 2019). Short period
implies that the luminous companion star is very compact, suggesting that the
donor is a Wolf-Rayet (WR) star, which is further supported by the hydrogen-
depleted infrared spectrum (van Kerkwijk et al., 1992, 1996; Fender et al.,
1999). This unique binary plays an important role in population synthesis
studies: a successful model is obliged to reproduce the presence of Cyg X-3 in
the Milky Way, the compact object in a tight binary orbit with the WR
companion; at the same time, current estimates suggest the presence of only
one such system in our Galaxy (Lommen et al., 2005).

Figure 1: Panel (a): Radio-X-ray track made by Cyg X-3 as it swings between
different spectral states (data from Zdziarski et al. 2016). Panel (b): a set
of average broadband spectra corresponding to the hard/radio quiescent (blue),
intermediate/minor flaring (orange), ultrasoft/radio quenched (black), and
soft non-thermal/flaring (magenta) states as introduced in Szostek et al.
(2008). IXPE data are shown with red symbols and the 2$\sigma$ upper limit at
30 keV (in red) is from Swift/BAT.  Figure 2: Average PD and PA as a function
of energy. The horizontal dashed lines correspond the energy-average values.

There is no detected optical counterpart due to high distance to the source
($D=9.7\pm 0.5$ kpc; Reid & Miller-Jones, 2023) and its location close to the
galactic plane resulting in high absorption along to the line of sight.
However, the orbital phase dependence of radio, infrared, X-ray and
$\gamma$-ray properties of the system have been studied extensively (e.g.,
McCollough et al. 1999; Fender et al. 1999; Zdziarski et al. 2018). The source
was found to regularly switch between several spectral states, characterised
by distinct X-ray and radio properties (Fig. 1 and Szostek et al., 2008;
Koljonen et al., 2010). Transitions between the spectral states are generally
thought to be related to the changes of accretion geometry in the system, but
the exact picture of plasma configuration and understanding of the physical
reasons for these changes are still missing. This makes the source an ideal
target for X-ray polarimetric studies.

The source is most often found in the hard X-ray, quiescent radio state
(Waltman et al., 1996; Hjalmarsdotter et al., 2009), when the X-ray continuum
can be roughly described as a cutoff power law with a peak at around 20 keV
(Fig. 1b), and prominent iron lines can be clearly identified. Spectral
decomposition in this state is complicated by the unknown absorption, in
particular its component that is intrinsic to the source (Hjalmarsdotter et
al., 2008; Zdziarski et al., 2010).

Cyg X-3 is found to make regular transitions towards the (ultra-)soft state,
when the hard X-ray flux drops dramatically and the spectrum resembles a
blackbody with $kT_{\rm bb}\approx 1.5$ keV. Radio fluxes are likewise
suppressed, motivating the name of the radio quenched state. This transition
is often followed by a major radio flare, when the highest radio fluxes are
detected, while the X-ray continuum is moderately soft and exhibits high-
energy tail (Szostek et al., 2008; Koljonen et al., 2010).

Exceptional multiwavelength properties of the source and fast, repeating swing
between different states prompted Hjellming (1973) to call it “An astronomical
puzzle”. Interest in solving this puzzle has persisted for more than 50 years.
A new chapter in the study of the source has been opened with the launch of
the Imaging X-ray Polarimetry Explorer (IXPE), that enabled the first X-ray
polarization studies in the 2–8 keV band.

The source was previously observed by IXPE in the X-ray hard (radio quiescent)
and intermediate (minor flaring) states (Veledina et al., 2024). The first
X-ray polarimetric observations in October-November 2022 revealed high average
polarization degree (PD) of $20.6\pm 0.3$% (becoming $23$% after accounting
for the unpolarized contribution of the iron line). The polarization angle,
PA=$90\aas@@fstack{\circ}1\pm 0\aas@@fstack{\circ}4$, suggests that it is
nearly orthogonal to the extended jet emission and relativistic ejections
observed in the system. Spectropolarimetric properties suggest the X-ray
emission is produced solely by reflection, indicating the presence of
obscuring material hiding the central engine from the observer’s line of
sight. The apparent high luminosity of $10^{38}$ erg s-1 and a much higher
luminosity emitted along the funnel in the obscuring outflow ($>5.5\times
10^{39}$ erg s-1) imply that Cyg X-3 belongs to the class of ultraluminous
X-ray sources. The dramatic drop of polarization in the minor flaring state,
PD=$10.0\pm 0.5$%, was interpreted in terms of the dilution of the outflow.

In this picture, the outflow could disappear completely in the ultrasoft and
soft non-thermal states (radio quenched and major flaring states,
respectively). Here we present the X-ray polarimetric properties of Cyg X-3 in
the ultrasoft state, with the aim to verify the source accretion geometry and
to understand the physical reasons of state transitions.

##  2 Data

IXPE observed the source in the ultrasoft state on 2024 June 2–3. We follow
the standard data analysis procedures that are described in detail in Appendix
A. We use simultaneous Swift/BAT
data111https://swift.gsfc.nasa.gov/results/transients/ (Gehrels et al., 2004)
to estimate the flux in the hard X-ray band. We also monitored the source in
radio with the AMI telescope at 15 GHz. Results of radio monitoring Cyg X-3
are presented in Appendix B.

Figure 3: Time variations of the IXPE count rate in the 2–8 keV band and the
orbit-average PD and PA. Time is measured from the beginning of the
observation at MJD 60463.8124. Boundaries of the time bins (except beginning
and the end of the observation) correspond to the orbital phase 0.

##  3 Results

The source was observed in the ultrasoft X-ray spectral state (Fig. 1b), with
highly suppressed hard X-ray flux, as indicated by Swift/BAT. We note the
lower flux, with respect to its typical value (Szostek et al., 2008), around
the iron line complex (compare the black and red spectra). Radio fluxes were
likewise historically low, $4.5\pm 0.3$ mJy (Fig. 1a and Appendix B). The
average polarimetric properties are shown in Fig. 2. We find average
PD=$11.9\pm 0.5$% and PA=$94\degr\pm 1\degr$ and no significant energy
dependence, in particular, around the iron line range. The PA is shifted by
$\sim$$4\degr$ with respect to the values detected in the hard and
intermediate states.

Spectropolarimetric data suggest absence of the prominent iron line, in
contrast to hard and intermediate states (see Appendix A). The spectrum in
this state was previously well fitted either by a single blackbody component,
a multicolour disc, or a low-temperature Comptonization component (Szostek et
al., 2008; Hjalmarsdotter et al., 2009; Koljonen et al., 2018). High PD and
its independence of energy is, however, not consistent with either of the
aforementioned components.

Next, we consider variations of polarimetric properties. Prominent orbital
flux variations are detected, with the typical profile shape of a brief dip
and a more extended peak (Fig. 3a). In Fig. 3bc we show the PD and PA averaged
over different orbital cycles. In contrast to the hard state, we find
pronounced changes of PD in the ultrasoft state. The PA remains constant
within errors, yet systematically shifted with respect to the hard- and
intermediate-state data in all bins. We compared the spectropolarimetric
properties of the time bins 2 and 3 (in Fig. 3bc) to those of time bin 5 (see
Figs. 8 and 9 in Appendix A). The iron line is detected in the time bin 5,
albeit at a low significance; for the bins 2+3 its contribution is compatible
with zero.

Previous IXPE observations in the hard and intermediate states revealed
significant variations of polarization at the orbital period. In Fig. 4 we
show the orbital phase-resolved polarimetric properties of the source in the
three observed states. The PD variations in the ultrasoft state are nearly
identical to those found in the intermediate state, but the PA shows a
pronounced peak around phase 0 (corresponding to the lowest X-ray fluxes).
Using the pcube analysis of the 2–5 keV and 5–8 keV energy bins, we also
checked whether the energy dependence of PD is different for these two orbital
phases. We find no difference between the two energy bins for the high X-ray
flux spectra, associated with phase 0.5, but for the low-flux bin (phase 0),
the PD decreases with energy at 90% confidence level. We also performed
orbital phase-resolved spectropolarimetry (see Figs. 6 and 7). The iron line
is detected in the spectra around phase 0; for the phase 0.5 only an upper
limit can be put (see Appendix A).

Figure 4: Orbital phase dependence of the PD and PA from the hard state (blue
triangles), the intermediate state (orange open squares), and ultrasoft state
(red circles). Green lines show the best-fit rotating vector model (Appendix
C) to the ultrasoft state data.

Figure 5: Left: sketch of the geometry associated with the hard, intermediate,
and ultrasoft states. Right: Polarization of radiation undergoing single
scattering above a funnel of half-opening angle $\alpha$ for different
observer inclinations $i$ (red lines). The filled red band depicts the
conservative PD range 10.8–12.4% of Cyg X-3 in the ultrasoft state which
translates to the inclination of $i=$26°–28°, which is very close to
$i=29\aas@@fstack{\circ}5\pm 1\aas@@fstack{\circ}2$ determined by Antokhin et
al. (2022). The blue line depicts the contour of the constant PD=23% observed
in the hard state for the model of the reflection from the inner wall of the
conical surface (from Fig. 4b in Veledina et al., 2024).

##  4 Discussion

Previous IXPE studies clearly showed that the states of Cyg X-3 do not simply
match the hard/intermediate states of other X-ray binaries, and the standard
scenarios that have been confirmed for, e.g., Swift J1727.8$-$1613 (Veledina
et al., 2023; Ingram et al., 2024; Podgorný et al., 2024) and Cyg X-1
(Krawczynski et al., 2022) cannot be applied to this source. If the soft state
of Cyg X-3 is analogous to the other soft-state sources, the PD$\sim$1–2% can
be expected (comparable to the measurements of Cyg X-1 and Swift
J1727.8$-$1613 that are thought to have similar orbital inclinations; Svoboda
et al., 2024; Steiner et al., 2024). On the contrary, we observe the average
PD$\approx$12%, that appears somewhat higher than that observed in the
intermediate state. This puts tight constraints on radiative mechanisms
relevant to this state of the system, as well as the accretion geometry.

A high PD that is independent of energy is in line with the single reflection
or scattering scenario. Presence of an obscuring medium covering the central
source along the line of sight is inevitable for the production of the high
polarization. Absence of a clear contribution of the iron line, both in
spectra and polarimetric properties, indicates the spectrum is not dominated
by the reflection off the cold and dense matter, suggested for the hard state
(Veledina et al., 2024). This conclusion is supported by the previous
spectroscopic analysis of Chandra data (Kallman et al., 2019), that find
absence of the fluorescent line from neutral iron (yet lines of highly ionised
species of Fe xxv and Fe xxvi, whose motion is associated with the compact
object, Vilhu et al. 2009, remain prominent throughout the ultrasoft state).

We have tested various scenarios where the observed polarization arises from
the reflection at the inner walls of the funnel (we use the same setup as
described in Podgorný et al., 2022, 2023; Veledina et al., 2024). We
considered the orbital inclination in the range $20\degr<i<40\degr$, in line
with previous estimates (Vilhu et al., 2009; Antokhin et al., 2022), and
varied density and ionisation of the medium. When decreasing the density of
the funnel walls (with respect to the hard-state $N_{\rm H}=10^{24}$ cm-2),
the PD becomes a decreasing function of energy in the IXPE range, because the
matter becomes effectively transparent to high-energy radiation. We also
considered cases of higher ionisation. With increasing ionisation, we
generally observe the decrease of the overall PD owing to the enhanced role of
multiple scatterings. We find that in order to reproduce the lack of the
neutral/moderately ionized iron lines, the ionisation parameter (giving the
ratio of ionising flux to the number density of matter) $\xi\gtrsim 5000$ is
required. This value is somewhat high, yet still within the boundaries of the
allowed range $2\lesssim\log\xi\lesssim 4$ coming from the line spectroscopy
(Kallman et al., 2019). However, at these high values the multiple scatterings
occur within a large fraction of the envelope, akin to a spherical shell,
hence the PD drops down to 1%, becoming inconsistent with the observed levels.

The observed blackbody-like spectrum and PD=10–15% naturally appear in the
scenario where the intrinsic radiation undergoes single Thomson scattering in
the medium along the axis of the optically thick funnel and the observer sees
only this scattered component (see Fig. 5). In this case, the observed average
PD gives the direct estimate on the inclination. For single Thomson scattering
regime we have

$P=\frac{1-\mu^{2}}{1+\mu^{2}},$ (1)

where $\mu$ is the cosine of the scattering angle. Thus PD=$11.9\pm 0.5$%
translates to inclination
$i=\arccos\left[(1-P)/(1+P)\right]^{1/2}=27\aas@@fstack{\circ}5\pm
0\aas@@fstack{\circ}5$. This constraint is relevant to a geometry of an
infinitely narrow funnel. Calculations that include averaging over the non-
zero opening angle $\alpha$ of a conical funnel, for an isotropic central
source and equal scattering probability within the funnel, show that PD
dependence on $\alpha$ is weak (see nearly vertical contours of constant PD in
Fig. 5).

Orbital variability can be used as an independent tracer of accretion
geometry. Orbital profiles of gamma-ray fluxes were previously explained by a
jet-orbital axis misalignment (Dubus et al., 2010; Zdziarski et al., 2018) or
bending of the jet/funnel (Dmytriiev et al., 2024). Radio fluxes were likewise
varying due to changing wind absorption (Zdziarski et al., 2018).
Interestingly though, we find absence of the orbital modulation of radio
fluxes during IXPE observations (Fig. 11 in Appendix B). Polarization
properties of the precessing funnel can be tested using the rotating vector
model (see Appendix C; Radhakrishnan & Cooke, 1969; Poutanen, 2020). We note
that this model does not necessarily assume that the radiation beam collimated
by the funnel is precessing, but there could be an asymmetry in the
distribution of scattering material, e.g., formed by the bow shock (Antokhin
et al., 2022) with the geometry fixed in the corotating frame of the binary.
In Fig. 4, we show the best-fit model with green line. The fit gives
$\chi^{2}$/d.o.f=25.7/8. The model fails in reproducing the data because the
amplitude of PA variations in excess of $20\degr$ translates to the changes of
PD, predicted by this model for $i=27\degr$, by about 9%, much larger than the
observed PD amplitude. While the model can reproduce orbital variations of PA,
precessing funnel scenario cannot be responsible for the orbital changes of
PD. This may be related to a more complex dependence of PD on the angle
between the funnel axis and the line of sight.

We note that the orbital-average ultrasoft-state PD shows pronounced
variations from one orbital cycle to another (Fig. 3). The changes could
indicate secular variations of funnel parameters, but might as well be related
to the switching between the dominating mechanisms responsible for the
polarization production: reflection at the funnel walls and scattering
within/on top of the funnel itself. The latter possibility can be verified by
comparing the spectra corresponding to different polarization levels: those
coming from the scattering are expected to have no pronounced neutral iron
line. In our case, however, the normalisation of the line in time bin 5 and an
upper limit in time bins 2+3 are comparable.

Time variations of PD may potentially be linked to the changes of the optical
depth of the scattering material: higher Thomson optical depth $\tau$ leads to
higher scattered fluxes. Higher $\tau$ also leads to the reduction of PD, as
per the increased role of (unpolarized) contribution of higher-order
scatterings. Changes of $\tau$ may be the reason for the visible anti-
correlation between the orbital fluence and measured PD (see Fig. 3). The PD
drop by a factor of 1.5 between the time bins 2+3 and 5 corresponds to the
transition from an optically thin medium to the medium of Thomson optical
depth $\tau\approx 0.35$. We can then obtain the average bolometric luminosity
of the system using the ultrasoft state flux estimate from Hjalmarsdotter et
al. (2009) (with updated distance estimate and when accounting for the
scattered fraction corresponding to $\tau=0.35$). We obtain the intrinsic
luminosity estimate $L_{\rm int}\approx 1.4\times 10^{39}$ erg s-1. This
estimate is close to the Eddington limit for He-rich material if the compact
object has mass $\lesssim 5$ solar masses, in line with previous results
(Zdziarski et al., 2013; Koljonen & Maccarone, 2017).

At the same time, the spectra corresponding to the high flux phases (orbital
phase 0.5) show strong suppression of the iron line, as compared to the low
fluxes (phase 0, see Appendix A). This may indicate systematic changes between
scattering (phase 0.5) and reflection (phase 0) mechanisms at different
orbital phases, attributed to the changing viewing angle of precessing funnel.
The observed high-flux PD=$11.2\pm 0.4$% corresponds to the scattering angle
(inclination) in this phase $i=26\aas@@fstack{\circ}7\pm
0\aas@@fstack{\circ}5$.

The polarimetric properties discussed above imply that during the ultrasoft
state, when the jet is quenched, the funnel interior and the channel above it
become filled with matter, coming from the WR wind and perhaps from the inner
walls of the outflow (this picture is similar to the jet cocoon scenario,
Koljonen et al., 2018). The amount of material along the funnel axis in this
state is much larger than in the hard state, when the powerful jet cleared the
material within the funnel beam (and further away above it) and the observer
saw only the light reflected from the inner walls. The drop of polarization
during the intermediate (minor flaring) state might then be explained by the
accumulation of matter close at the inner boundaries of the funnel and
replacement of the reflection at the sharp inner boundary by the reflection
within the volume close to this boundary (see Fig. 5). The effects of matter
ionisation may also play a role in the PD reduction: in this case, the funnel
becomes partially transparent to the reflected radiation (but not the incident
one).

##  5 Summary

We have studied the X-ray polarimetric properties of the X-ray binary Cyg X-3
during the ultrasoft state. The high PD=$11.9\pm 0.5$% and PA=$94\degr\pm
1\degr$ indicate that the central source is covered by a thick envelope,
similar to the previously observed hard and intermediate states. The
polarization of the continuum radiation can arise from a single scattering off
the optically thin medium located along the funnel axis. This is different
from the scenario for the other states, where the X-ray continuum is dominated
by the reflection from funnel walls.

We show that the PD in this case can serve as a direct probe of the orbital
inclination, $i=27\aas@@fstack{\circ}5\pm 0\aas@@fstack{\circ}5$. We discuss
the changes of the orbit-average and phase-dependent PDs and find that this
scenario can be realised in the phases with highest flux, with PD=$11.2\pm
0.4$%, leading to the conservative inclination estimate $26\degr<i<28\degr$.

Changes of the orbital-average PD may be related to the optical depth
variations of scattering matter. The transition between the hard and ultrasoft
states may in turn be caused by the filling of the funnel by the matter
(coming either from the funnel walls or from the enhanced WR wind), linked to
the jet quenching. The major flaring state can then be related to the matter
ejection from the funnel, its eventual emptying and transition of the source
to the hard state. This closes the loop of the correlated radio and X-ray
activity in Cyg X-3.

###### Acknowledgements.

The Imaging X-ray Polarimetry Explorer (IXPE) is a joint US and Italian
mission. The US contribution is supported by the National Aeronautics and
Space Administration (NASA) and led and managed by its Marshall Space Flight
Center (MSFC), with industry partner Ball Aerospace (contract NNM15AA18C). The
Italian contribution is supported by the Italian Space Agency (Agenzia
Spaziale Italiana, ASI) through contract ASI-OHBI-2022-13-I.0, agreements ASI-
INAF-2022-19-HH.0 and ASI-INFN-2017.13-H0, and its Space Science Data Center
(SSDC) with agreements ASI-INAF-2022-14-HH.0 and ASI-INFN 2021-43-HH.0, and by
the Istituto Nazionale di Astrofisica (INAF) and the Istituto Nazionale di
Fisica Nucleare (INFN) in Italy. This research used data products provided by
the IXPE Team (MSFC, SSDC, INAF, and INFN) and distributed with additional
software tools by the High-Energy Astrophysics Science Archive Research Center
(HEASARC), at NASA Goddard Space Flight Center (GSFC). This research has been
supported by the Academy of Finland grant 355672 (AV, VA) and the Vilho, Yrjö,
and Kalle Väisälä foundation (SVF). ADM, FLM, EC, FM and PS are partially
supported by MAECI with grant CN24GR08 “GRBAXP: Guangxi-Rome Bilateral
Agreement for X-ray Polarimetry in Astrophysics”. AAZ acknowledges support
from the Polish National Science Center grants 2019/35/B/ST9/03944,
2023/48/Q/ST9/00138, and from the Copernicus Academy grant CBMK/01/24. JPod
and MD acknowledge the support from the Czech Science Foundation project GACR
21–06825X and the institutional support from the Astronomical Institute
RVO:6798581. DG and LR thank the staff of Lord’s Bridge, Cambridge, for the
their support in making the AMI observations.

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##  Appendix A IXPE data

IXPE is the imaging polarimetry NASA/ASI mission launched in December 2021
(Weisskopf et al., 2022). It contains three grazing-incidence telescopes, each
consisting of a mirror module assembly, which focuses X-rays onto a focal-
plane polarization-sensitive gas pixel detector unit (DU; Soffitta et al.,
2021; Baldini et al., 2021). IXPE observed Cyg X-3 on 2024 June 2–3 (ObsID
03250301) with the net exposure time of 60 ks.

The data were processed with the ixpeobssim package (Baldini et al., 2022)
version 31.0.1 using the CalDB released on 2024 February 28. Source photons
were extracted from a circular region centred on the source, with the radius
of 80″. Due to the brightness of the source, background subtraction was not
applied and the unweighted approach was used (Di Marco et al., 2022, 2023).

We first perform a model-independent analysis of the orbital-averaged IXPE
polarimetric data using the pcube algorithm included in the ixpeobssim
package, based on the formalism by Kislat et al. (2015). We compute the
PD=$\sqrt{q^{2}+u^{2}}$ and the PA=$\frac{1}{2}\arctan(u/q)$, using the
normalised Stokes parameters $q=Q/I$ and $u=U/I$.

To investigate the orbital dependence of the polarimetric data, we folded the
observation with the quadratic ephemeris (model 2 from Table 2) of Antokhin &
Cherepashchuk (2019) using the xpphase method from the ixpeobssim package. The
data were grouped in 9 bins and then the pcube algorithm was applied for each
phase bin in the energy range 2–8 keV.

Next, we utilised spectral information. The energy spectra were fitted
simultaneously using the xspec package version 12.14.0 (Arnaud, 1996) using
$\chi^{2}$ statistics and the version 20240101_v013 of the instrument response
functions. The reported uncertainties are at the 68.3% confidence level
(1$\sigma$) unless stated otherwise. The spectropolarimetric analysis was
performed using the weighted approach (Di Marco et al., 2022).

We performed spectropolarimetric analysis of Cyg X-3 at high and low flux
levels corresponding to the time intervals when the IXPE count rate was higher
or lower than 32 cnt s-1 (see the light curve in top panel of Fig. 3). These
levels roughly give the fluxes around orbital phase 0.5 and 0, respectively.
The spectral model we used as the basis of the spectropolarimetric analysis,
given the soft-state of the source, was tbabs*(polconst*diskbb+gauss+gauss),
where the two unpolarised gaussian lines are fixed at energies 2.4 and 6.6 keV
with the width $\sigma$ of 0.15 and 0.25 keV, respectively. The gaussians are
responsible for the contribution of the unpolarized lines of Si , S , Fe  and
other species identified by high-resolution spectroscopy (Kallman et al.,
2019). We also used a constant factor const to account for uncertainties in
the effective area of different DUs. The best-fit results for the high and low
fluxes are reported in Table 1. The deconvolved spectrum of Stokes parameters
$I$, $Q$, and $U$ in $EF_{E}$ representation are shown in Figs. 6 and 7.

Then, we performed the same spectropolarimetric analysis checking variation of
the orbital-average polarization with time. In particular, we performed the
analysis putting together the data in the time bins 2 and 3 and in the time
bin 5 (see Fig. 3), corresponding to the highest and lowest PD. The best-fit
results are given in Table 2. The deconvolved spectrum of Stokes parameters
$I$, $Q$, and $U$ in $EF_{E}$ representation are shown in Figs. 8 and 9.

Table 1: Model parameters for the spectropolarimetric fit for the high and low fluxes. Component | Parameter | Low flux | High flux
---|---|---|---
tbabs |  $N_{\rm H}$ ($10^{22}$ cm-2) | $7.5\pm 0.2$ | $7.0\pm 0.1$
diskbb |  $kT_{\rm in}$ (keV) | $1.8\pm 0.4$ | $1.64\pm 0.01$
| norm |  $32\pm$3 | $117\pm 4$
polconst | PD (%) | $15.2\pm 0.9$ | $11.2\pm 0.4$
| PA (deg) | $104.8\pm 1.6$ | $91.7\pm 0.9$
gauss1 |  $E$ (keV) | [2.4] | [2.4]
|  $\sigma$ (keV) | [0.15] | [0.15]
| norm ($\times 10^{-2}$) | $2.6\pm 0.6$ | $4.2\pm 0.5$
gauss2 |  $E$ (keV) | [6.6] | [6.6]
|  $\sigma$ (keV) | [0.25] | [0.25]
| norm ($\times 10^{-3}$) | $6.0\pm 1.6$ | $<2.5$
const | DU1 | [1] | [1]
| DU2 | $1.020\pm 0.006$ | $1.016\pm 0.002$
| DU3 | $1.006\pm 0.006$ | $0.998\pm 0.002$
|  $\chi^{2}$/dof | 475/432 | 521/432
Table 2: Model parameters for the spectropolarimetric fit for the time bins 2+3 and 5 of Fig. 3. Component | Parameter | Time bins 2+3 | Time bin 5
---|---|---|---
tbabs |  $N_{\rm H}$ ($10^{22}$ cm-2) | $6.40\pm 0.12$ | $7.02\pm 0.16$
diskbb |  $kT_{\rm in}$ (keV) | $1.74\pm 0.19$ | $1.64\pm 0.02$
| norm | $76\pm 4$ | $103\pm 6$
polconst | PD (%) | $13.9\pm 0.5$ | $8.5\pm 0.7$
| PA (deg) | $97.0\pm 1.1$ | $95\pm 2$
gauss1 |  $E$ (keV) | [2.4] | [2.4]
|  $\sigma$ (keV) | [0.15] | [0.15]
| norm ($\times 10^{-2}$) | $3.1\pm 0.6$ | $4.9\pm 0.9$
gauss2 |  $E$ (keV) | [6.6] | [6.6]
|  $\sigma$ (keV) | [0.25] | [0.25]
| norm ($\times 10^{-3}$) | $<3.7$ | $3\pm 2$
const | DU1 | [1] | [1]
| DU2 | $1.007\pm 0.004$ | $1.024\pm 0.005$
| DU3 | $0.991\pm 0.003$ | $1.003\pm 0.005$
|  $\chi^{2}$/dof | 479/432 | 491/432
Figure 6: Spectropolarimetric properties of Cyg X-3 corresponding to the high
fluxes with the IXPE count rate above 32 cnt s-1 (see Fig. 3) around orbital
phase 0.5. Figure 7: Same as Fig. 6 but for the low fluxes with the IXPE count
rate below 32 cnt s-1 (around orbital phase 0.0). Figure 8: Same as Fig. 6 but
averaged over the orbital phases, during the time bins 2 and 3 (see Fig. 3).
Figure 9: Same as Fig. 6 but averaged over the orbital phases, during the time
bin 5 (see Fig. 3).

##  Appendix B Radio behaviour at 15 GHz

Cygnus X-3 is monitored at 15 GHz, usually daily, with the Large Array of
Arcminute MicroKelvin Imager (AMI; Zwart et al. 2008; Hickish et al. 2018).
AMI consists of eight 12.8-m antennas sited at the Mullard Radio Astronomy
Observatory near Cambridge, UK. The AMI receivers cover the band from 13 to 18
GHz, and are of a single linear polarisation, Stokes $I+Q$. Observations are
usually made daily, although some are missed due to adverse weather conditions
(high winds or heavy rain), or technical issues. Also the number of antennas
available varies due to technical issues. Analysis is done using custom
software, reduce_dc (Perrott et al., 2013). Each observation consists of
multiple 10-min scans of Cyg X-3, interleaved with short ($\sim 2$-min)
observations of a nearby compact source, which were used for phase
calibration. The flux density scale is set using nearby observations of 3C
286, which are usually made daily. The day-to-day flux density uncertainty is
estimated at $\approx 5$%. Usually short observations, with two 10 min scans
are made daily, but sometimes longer observations are made. Figure 10 shows
the AMI observations of Cyg X-3 for several months before and weeks after the
IXPE observations. These show that at 15 GHz, Cyg X-3 was in a faint state
after a larger outburst in early April. These AMI observations include a
longer, $\approx 4$ hour, observation of Cyg X-3 made on 2024 June 3, during
the IXPE observations, which is shown in Fig. 11. The source does not show any
pronounced orbital variability in flux density.

Figure 10: AMI observations of Cyg X-3 at 15 GHz from early March to mid June
of 2024. Each point is the average flux density in a 10-min bin, with an error
bar giving the statistical error. The dashed blue line indicates the IXPE
observation date. Figure 11: Same as Fig. 10 but for the $\approx 4$ h (i.e.
approximately one orbital period) long observation on 2024 June 3,
simultaneous with IXPE.

##  Appendix C Orbital variability

To describe the orbital variability using the rotating vector model, we
consider a funnel inclined at angle $\theta$ relative to the orbital axis and
having azimuthal angle $\phi_{0}$ measured in the orbital plane from the line
connecting the black hole to the WR star. The orbital axis make angle $i$ to
the line of sight. The PA of the scattered radiation can be written as
$\chi=\chi_{0}+\chi_{\rm a}+\pi/2$, where $\chi_{\rm a}$ is the position angle
on the sky of the orbital angular momentum (measured from north to east) and
$\chi_{0}$ is given by the rotating vector model (equation (30) in Poutanen
2020):

$\tan\chi_{0}=\frac{-\sin\theta\ \sin(\phi+\phi_{0})}{\sin i\cos\theta-\cos
i\sin\theta\cos(\phi+\phi_{0})}.$ (2)

The PD is determined by Eq. (1) where the cosine of the scattering angle
varies with the orbital phase as

$\mu=\cos i\cos\theta+\sin i\sin\theta\cos(\phi+\phi_{0}).$ (3)

The model thus has four parameters: $i$, $\theta$, $\chi_{\rm a}$, and
$\phi_{0}$. We fitted the ultrasoft state polarization data with this model
obtaining $\chi^{2}=25.7$ for 8 dof. The best-fit parameters are
$i=27\aas@@fstack{\circ}7$, $\theta=3\degr$, $\chi_{\rm
a}=6\aas@@fstack{\circ}5$, and $\phi_{0}=-120\degr$ (i.e. nearly in the
direction of motion). This direction is not far from the position of the bow
shock (Antokhin et al., 2022) where an excess of the optical depth is
expected. We did not try to get the confidence level on the parameters,
because the fit is not acceptable.