# Stitched ViTs are Flexible Vision Backbones

Zizheng Pan Jing Liu Haoyu He Jianfei Cai Bohan Zhuang
ZIP Lab, Monash University, Australia Corresponding author. E-mail: $\tt
<EMAIL_ADDRESS>
###### Abstract

Large pretrained plain vision Transformers (ViTs) have been the workhorse for
many downstream tasks. However, existing works utilizing off-the-shelf ViTs
are inefficient in terms of training and deployment, because adopting ViTs
with individual sizes requires separate training and is restricted by fixed
performance-efficiency trade-offs. In this paper, we are inspired by
stitchable neural networks, which is a new framework that cheaply produces a
single model that covers rich subnetworks by stitching pretrained model
families, supporting diverse performance-efficiency trade-offs at runtime.
Building upon this foundation, we introduce SN-Netv2, a systematically
improved model stitching framework to facilitate downstream task adaptation.
Specifically, we first propose a Two-way stitching scheme to enlarge the
stitching space. We then design a resource-constrained sampling strategy that
takes into account the underlying FLOPs distributions in the space for
improved sampling. Finally, we observe that learning stitching layers is a
low-rank update, which plays an essential role on downstream tasks to
stabilize training and ensure a good Pareto frontier. With extensive
experiments on ImageNet-1K, ADE20K, COCO-Stuff-10K, NYUv2 and COCO-2017, SN-
Netv2 demonstrates strong ability to serve as a flexible vision backbone,
achieving great advantages in both training efficiency and adaptation. Code
will be released at https://github.com/ziplab/SN-Netv2.

##  1 Introduction

General-purpose Transformer architectures [3, 15, 12, 53, 26, 5] have grown
into unprecedented scale in recent research. In computer vision, large
pretrained plain ViTs such as MAE [20], DINO [5, 32] and DeiT [44, 45] are
widely adopted as backbones for tackling downstream tasks. However, despite
the great performance, when adopting pretrained ViTs into the downstream
tasks, they all face the challenge of the huge computational and memory cost.
For example, processing a single $224\times 224$ image using DeiT3-Large [45]
requires 2.8G peak GPU memory consumption, which is 14$\times$ higher than
that of using DeiT3-Small (0.2G).

On the other hand, most existing efforts adopting Transformers as backbones
for downstream tasks can be roughly categorised into three types of
approaches: full-adoption [28, 56, 42, 16], parameter-efficient tuning [24, 7]
and adapters [9]. Specifically, full-adoption trains one scale of ViT on
downstream tasks while parameter-efficient methods additionally reduce the
number of learnable parameters. Adapter method like ViT-Adapter [9] equips
plain ViTs with pyramid feature maps for dense predictions. However, these
methods do not address the memory consumption and computational cost after
adoption.

Figure 1: The framework of the proposed Two-way Stitching, where stitching can
go Fast-to-Slow, Slow-to-Fast, or travel between anchors as a round trip. For
example, the blue line represents the forward path of a new stitched network,
which starts with blocks from the small ViT, connects a few blocks from the
large ViT, and finally propagates its activations back to the small ViT. “LoRA
SL” refers to the proposed low-rank adaptation of stitching layers.

Most recently, a promising framework has been proposed named stitchable neural
networks (SN-Netv1, Figure 2 (a)) [33], where it reassembles a new network
(stitch) by stitching pretrained model families (anchors) in a Fast-to-Slow
direction, i.e., the stitch begins with a small ViT and goes into a large ViT
via a stitching layer ($1\times 1$ convolution). At training time, SN-Net
randomly samples a stitch and train it as a normal network with gradient
descent. With a few epochs of finetuning on the same pretraining dataset
(i.e., ImageNet-1K), SN-Netv1 produces numerous networks with different
performance-efficiency trade-offs that satisfy various resource constraints.
However, many general-purpose ViTs are trained on exceptionally large datasets
(e.g., JFT [53], LAION [39]) by industry, making it impractical for
researchers to train SN-Netv1 on the same domain as the anchors. Therefore,
exploring the downstream adaptation of SN-Net becomes essential as it enables
researchers to use off-the-shelf pretrained models without significant
retraining overhead on the original dataset.

In this paper, we systematically improve SN-Netv1 for direct adaptation on
downstream dense predictions. The new method, called SN-Netv2, presents
impressive power as a strong and flexible vision backbone on downstream CV
tasks. Specifically, our improvements are in three folds: 1) Different from
SN-Netv1 that only permits the stitching direction of Fast-to-Slow, we propose
a novel Two-way stitching strategy (Figure 1), which enables stitching to go
both Fast-to-Slow and Slow-to-Fast, or travel between anchors as a round trip
(e.g., Fast-Slow-Fast). As a result, we effectively enlarge the stitching
space by $10\times$ and avoid sub-optimal stitches that reside in certain
resource constraints. However, under the default random sampling strategy,
simply enlarging the space incurs unbalanced training and hinders the overall
performance of SN-Netv1 due to the varying number of stitches residing on
different FLOPs intervals. To this end, 2) we introduce a Resource-constrained
Sampling (ROS) strategy, which draws a stitch at each iteration according to
the categorical distribution of stitches in the space [46, 30]. In this case,
ROS ensures a balanced training for stitches under different resource
constraints. 3) Last, although SN-Netv1 demonstrated the advantage of fully
finetuning the stitching layers after least-square initialization, we point
out that learning stitching layers resides on a low intrinsic dimension. On
ImageNet-1K, low-rank adaptation of stitching layers achieves competitive
performance, while it plays a key role in stabilizing training and ensuring a
smooth performance curve on downstream tasks.

With the improved techniques, we comprehensively experiment with SN-Netv2 on
ImageNet-1K, ADE20K, COCO-Stuff-10K, NYUv2 and COCO. We show SN-Netv2
demonstrates strong performance and training efficiency than typical single-
scale ViT backbone adoption on downstream dense prediction tasks. In
particular, on ImageNet-1K, SN-Netv2 obtains better performance-efficiency
trade-offs than SN-Netv1. On downstream tasks, SN-Netv2 achieves competitive
performance with anchors under equal training schedules, while eliminating the
need to train different scales of ViT backbone separately. With systematic
evaluation, we show SN-Netv2 enables flexible inference on memory and
computation-intensive dense predictions with a single network, showcasing its
potential as a next-generation backbone for a wide range of tasks.

##  2 Related Work

#### General-purpose Transformers.

Benefit from the large-scale datasets [39, 53] and powerful scaling laws [25],
recent pretrained plain ViTs [20, 2, 14, 12] have achieved strong performance
on many visual benchmarks. Based on different training objectives, ViT
pretraining can be done by either supervised learning (SL) or self-supervised
learning (SSL). In SL-based pretraining, the common practice [31, 14] is to
train ViTs with the cross-entropy loss on ImageNet-21K [38] or privately
gathered images [43]. Compared to it, SSL-based pretraining is a more
promising direction as it can leverage the vast amounts of unlabeled data.
Prevalent approaches in this area include contrastive learning [5, 8, 52, 35]
and masked image modeling (MIM) [20, 2, 58, 13, 18, 50]. In recent studies,
ViTs have been scaled up to billion parameters [53, 12, 15] to match the
prevalent large language models (LLMs) [55, 36, 3]. However, when adopting
into downstream tasks, most existing efforts [28, 16, 54, 56] fully adopt the
pretrained model in order to exploit the pretrained weights, suffering from
the huge computational cost and memory consumption. In contrast, this paper
efficiently adapts large pretrained ViTs into CV tasks as a single flexible
backbone to satisfy various resource constraints at runtime.

#### Model stitching.

Model stitching has been studied in previous works [27, 1, 11] to measure the
similarity of representations from neural networks. In particular, it implies
a sequence of well-performed networks that can be obtained by stitching the
early portion of a trained network with the last portion of another trained
network by a $1\times 1$ convolution layer (a.k.a, stitching layer). Inspired
by this observation, Yang et al. [51] proposed to dissect and reassemble a new
architecture based on the model zoo. Most recently, Pan et al. proposed SN-Net
[33] to cheaply produce numerous networks with different complexity-
performance trade-offs by stitching a family of pretrained models. However, it
only experiments with the same pretraining domain, without exploring dense
prediction performance. In this paper, we systematically improves SN-Net and
address its limitations on the stitching space, sampling strategy and
downstream training adaptation.

##  3 Rethinking Stitchable Neural Networks

Figure 2: Explanation of SN-Netv1 [33]. Figure (a): The framework of SN-Netv1
under the default Fast-to-Slow strategy, where the blue line indicates the
forward path of a stitched network by stitching a small ViT to a large ViT.
Figure (b): The naive Fast-to-Slow stitching strategy incurs some bad stitches
(highlight in red box) when stitching pretrained DeiT3-S and DeiT3-L on
ImageNet-1K.

SN-Netv1 [33] is a scalable framework built upon pretrained model families. A
simple illustration of SN-Netv1 is shown in Figure 2 (a). Specifically, given
an input $\mathbf{X}$ and two pretrained models $f_{\theta}$ and $f_{\phi}$,
$\mathcal{S}:\mathcal{A}_{\theta,l}\to\mathcal{A}_{\phi,m}$ is denoted as a
stitching layer which implements a transformation between the activation space
of the $l$-th layer of $f_{\theta}$ to the activation space of the $m$-th
layer of $f_{\phi}$. Let $Z_{\theta}$ and $G_{\phi}$ represent the function of
the first and last portion of blocks of $f_{\theta}$ and $f_{\phi}$
respectively. Next, SN-Netv1 obtains a new network architecture
$F_{\mathcal{S}}$ by

$F_{S}(\mathbf{X})=G_{\phi,m}\circ\mathcal{S}\circ Z_{\theta,l}(\mathbf{X}),$
(1)

where $\circ$ indicates the composition. With different stitching
configurations of $l$ and $m$, SN-Net cheaply produces numerous stitched
networks (i.e., stitches) that achieve good accuracy-efficiency trade-offs
between two pretrained models (i.e., anchors) after joint training. However,
despite the simple idea and its effectiveness, SN-Net still suffers from
noticeable limitations.

Stitching space. SN-Netv1 strictly follows a typical stitching direction in
the literature [1, 11], i.e., propagating activations by starting with one
anchor and ending with another. Based on whether the stitched network starts
with a small anchor (Fast) or a large anchor (Slow), Pan et al. [33] explored
two stitching directions: Fast-to-Slow and Slow-to-Fast, and demonstrated that
Fast-to-Slow generally leads to a better and more smooth performance curve.
However, simply adhering to the Fast-to-Slow direction may assemble sub-
optimal network architectures. As shown in Figure 2 (b), stitching DeiT3-S and
DeiT3-L under the default setting of SN-Netv1 produces a few stitches that
achieve worse performance than the small anchor, even with higher FLOPs.
Therefore, it is evident that the existing stitching space in SN-Netv1
requires redesign.

Sampling strategy. Training SN-Netv1 is simple but effective: at each training
iteration, SN-Netv1 randomly samples a stitch from a pre-defined configuration
set, and then train the sampled stitch as a normal network with gradient
descent. However, random sampling approach only works well when the stitches
are evenly distributed across different FLOPs constraints (e.g., 5G, 10G).
While this condition is met in the initial stitching space of SN-Netv1,
enlarging the space can result in a problematic imbalance, with some FLOPs
constraints having far fewer stitches than others. Consequently, this leads to
an unbalanced training for networks in certain FLOPs ranges, and thus
negatively impacts the performance.

Stitching layers. SN-Netv1 is initially trained on the same data source of the
pretrained model families (e.g., ImageNet-1K pretrained DeiTs [44]). Under
this setting, training a stitching layer with fully finetuning has a
consistent target as the activation space $\mathcal{A}_{\theta,l}$ and
$\mathcal{A}_{\phi,m}$ may not change significantly. However, things can be
different when adopting SN-Netv1 on downstream dense prediction tasks due to
the domain gap between the pretrained data and the target data. In this case,
both $\mathcal{A}_{\theta,l}$ and $\mathcal{A}_{\phi,m}$ need to adapt to the
target domain, making it unstable and difficult to simultaneously learn many
stitches. This implies the necessity of an appropriate method for learning
stitching layers on downstream tasks.

##  4 Method

In this section, we systematically introduce our improvements over SN-Netv1,
including the redesign of stitching space, sampling strategy and stitching
layers for stable adaptation to downstream dense prediction tasks.

###  4.1 Two-way Stitching

To improve the stitching space, we first propose Two-way stitching, which
allows the stitching to travel in different ways including Fast-to-Slow (FS),
Slow-to-Fast (SF), Fast-Slow-Fast (FSF) and Slow-Fast-Slow (SFS). The design
principle of Two-way Stitching is to augment the small anchor with the blocks
from a large and strong anchor, while accelerating the large anchor by
adopting more efficient blocks from the small anchor. In this way, Two-way
stitching can leverage the strengths of different scales of anchors.

We illustrate our framework in Figure 1, where it also shows a concrete
example of an FSF stitch in the blue line of Figure 1. Specifically, its
begins with the small anchor during the forwarding pass, traverses the large
anchor along the middle route, and ultimately returns to the small anchor for
subsequent propagations. By enabling a broader stitching configuration, Two-
way stitching enlarges the stitching space by $10\times$ compared to the
initial space of SN-Netv1 (e.g., 134 vs. 13 based on DeiT3-S/L), which
facilitates the discovery of more optimal architectures, as shown in Section
5. Moreover, benefiting from the two new stitching configurations (i.e., FSF
and SFS), SN-Netv2 can produce many networks at the similar FLOPs as each
stitch can be regarded as replacing intermediate consecutive blocks at certain
positions in one anchor with the blocks from another anchor.

###  4.2 Resource-Constrained Sampling

Figure 3: Comparison on the categorical distribution of stitches between SN-
Netv1 and SN-Netv2 by stitching DeiT3-S and DeiT3-L on ImageNet-1K.

After enlarging the stitching space, we empirically find that the performance
of anchors drops significantly compared to its original performance. To
understand this, we begin with analysing the categorical distribution of
stitches in the space, i.e., $\pi(\tau)$, where $\tau$ denotes the FLOPs
constraint. In practice, we round the real FLOPs following a step $t$ to
discretize the whole FLOPs range. By default, we adopt $t=1$ in ImageNet
experiments and $t=10$ for dense prediction tasks. Benefiting from the
significantly smaller architectural space compared to neural architecture
search (i.e., NAS, $10^{2}$ vs. $10^{20}$), we can calculate $\pi(\tau)$
exactly prior to training by $\pi(\tau=\tau_{0})=\\#(\tau=\tau_{0})/E$, where
$E$ is the total number of stitches in the space and $\\#(\tau=\tau_{0})$ is
the number of stitches that yield FLOPs $\tau_{0}$.

With the exact $\pi(\tau)$, we visualize the categorical distribution of
stitches for SN-Netv2 and SN-Netv1 in Figure 3. As it shows, stitches in SN-
Netv1 are evenly distributed across different FLOPs, which ensures balanced
training for stitches at different resource constraints. However, under Two-
way Stitching, the sampling probability of anchors ($2/134$) is much lower
than stitches in other FLOPs constraints. As a result, anchors are
significantly under-trained which therefore affects the overall performance of
SN-Netv2. Inspired by recent NAS works [46, 30], we design a Resource-
constrained Sampling strategy (ROS) for SN-Netv2. Specifically, at each
training iteration, we first sample a FLOPs constraint $\tau_{0}$. Next, we
randomly sample a stitch that satisfies the constraint
$\alpha\sim\pi(\tau=\tau_{0})$. With this design, ROS effectively guarantees
balanced training for stitches at different FLOPs constraints, especially for
anchors where we increase their sampling probability by 10$\times$, e.g., from
$2/134$ to $2/13$ based on DeiT3-S/L and 13 FLOPs intervals.

###  4.3 Low-Rank Adaptation of Stitching Layers

Under Two-way stitching, the stitching layers involve with two types of
transformation matrix to control how the stitches route between anchors:
$\mathbf{M}_{1}\in\mathbb{R}^{D_{1}\times D_{2}}$ and
$\mathbf{M}_{2}\in\mathbb{R}^{D_{2}\times D_{1}}$, where $D_{1}$ and $D_{2}$
refer to model widths of the small anchor and large anchor, respectively. In
practice, they are 1$\times$1 convolutions and initialized by the least-square
(LS) solution as in SN-Netv1. Formally, let
$\mathbf{X}_{1}\in\mathbb{R}^{N\times D_{1}}$ and
$\mathbf{X}_{2}\in\mathbb{R}^{N\times D_{2}}$ be the feature maps from two
anchors at one stitching position, where $N$ denotes the length of the input
sequence. Next, the targeted transformation matrix for $\mathcal{S}_{1}$ and
$\mathcal{S}_{2}$ can be obtained respectively by

$\mathbf{M}_{1}=\mathbf{X}_{1}^{\dagger}\mathbf{X}_{2},\mathbf{M}_{2}=\mathbf{X}_{2}^{\dagger}\mathbf{X}_{1},$
(2)

where ${\dagger}$ denotes the Moore-Penrose pseudoinverse of a matrix. In SN-
Netv1, the transformation matrix $\mathbf{M}$ is fully finetuned with gradient
descent during training, which demonstrates better performance than the
initial LS solution.

However, as mentioned in Section 3, learning a good stitching layer on
downstream tasks can be difficult as anchors need to adapt to the target
domain simultaneously. Motivated by previous observations [17, 6] that sparse
finetuning via paremeter-efficient tuning (PET) can impose regularization on
the model which facilitates training stability, we propose LoRA SL, a low-rank
adaptation method for stitching layers in order to stabilize training.
Specifically, similar to LoRA [23], we freeze the LS initialized
transformation matrix $\mathbf{M}$ but constrain its update with a low-rank
decomposition. Taking $\mathcal{S}_{1}$ as a concrete example, the activation
projection of the stitching layer can be formulated by

$\mathbf{X}_{1}\mathbf{M}_{1}+\mathbf{X}_{1}\Delta\mathbf{M}_{1}=\mathbf{X}_{1}\mathbf{M}_{1}+\mathbf{X}_{1}\mathbf{B}_{1}\mathbf{A}_{1},$
(3)

where $\mathbf{B}_{1}\in\mathbb{R}^{D_{1}\times r}$,
$\mathbf{A}_{1}\in\mathbb{R}^{r\times D_{2}}$, and $r\ll min(D_{1},D_{2})$ is
the rank. In practice, $\mathbf{B}_{1}$ is initialized by Gaussian
initialization and $\mathbf{A}_{1}$ is initialized with zeros. Therefore, the
initial update $\Delta\mathbf{M}_{1}=\mathbf{B}_{1}\mathbf{A}_{1}$ is zero and
does not affect the LS solution at the beginning of training. As each
stitiching layer is responsible for multiple stitches, the low-rank update
helps to regularize the learning of $\mathbf{M}_{1}$. We show in Figure 11
that LoRA SL improves the overall performance of SN-Netv2. In Algorithm 1, we
summarize our training approach for SN-Netv2 and highlight the difference with
SN-Netv1 in bold. It is also worth noting that we do not adopt knowledge
distillation to train SN-Netv2 as it is very inefficient on downstream dense
prediction tasks.

Algorithm 1 Training Stitchable Neural Networks v2

1:Two pretrained ViTs to be stitched. The pre-computed categorical
distribution of stitches in the Two-way stitching space $\pi(\tau)$.

2:Initialize all stitching layers by least-squares matching, freeze the
weights $\mathbf{M}$ with LoRA.

3:for $i=1,...,n_{iters}$ do

4: Get next mini-batch of data $\mathbf{X}$ and label $\mathbf{Y}$.

5: Clear gradients, $optimizer.zero\\_grad()$.

6: Sample a target resource constraint $\tau_{0}$.

7: Randomly sample a stitch $\alpha$ that satisfies the constraint
$\alpha\sim\pi(\tau=\tau_{0})$.

8: Execute the current stitch, $\hat{\mathbf{Y}}=F_{\alpha}(\mathbf{X})$.

9: Compute loss, $loss=criterion(\hat{\mathbf{Y}},\mathbf{Y})$.

10: Compute gradients, $loss.backward()$.

11: Update weights, $optimizer.step()$.

12:end for

Figure 4: Performance comparison between SN-Netv1 and SN-Netv2 on ImageNet-1K
based on DeiT3. The yellow stars denote the original anchor performance. We
highlight the best performed stitches on the Pareto-frontier in SN-Netv2.
Figure 5: Semantic segmentation results of SN-Netv2 on ADE20K by stitching
DeiT3-S/L and DeiT3-B/L. We use yellow stars to denote the anchor performance,
i.e., adopt the individual anchor as a backbone for training. We highlight the
stitches at the Pareto frontier in blue line.

##  5 Experiments

In this section, we first show the advantage of SN-Netv2 over SN-Netv1 on
ImageNet-1K [38]. Next, we conduct experiments on downstream dense prediction
tasks to show the strong performance of SN-Netv2 as a promising flexible
vision backbone, including semantic segmentation on ADE20K [57], COCO-Stuff
[4], depth estimation on NYUv2 [40] and object detection on COCO-2017 [29].

###  5.1 ImageNet Classification

Implementation details. We experiment with three combinations of stitching for
DeiT3 [45], namely DeiT3-S/B, DeiT3-S/L and DeiT3-B/L. We train each setting
on ImageNet-1K by 30 epochs with a total batch size of 256 on 8 V100 GPUs. The
learning rate is 0.1$\times$ to that of default training time learning rate in
DeiT3. During training, anchors adopt the same stochastic layer drop rate as
the pretrained model family, i.e., 0.05, 0.15, 0.4 for DeiT3-S, DeiT3-B,
DeiT3-L, respectively. All other hyperparameters are the same as in DeiT3. For
all experiments, we adopt a rank of 16 for LoRA SL and 100 images for its LS
initialization. We evaluate the performance by Top-1 accuracy (%).

Figure 6: Semantic segmentation results of SN-Netv2 on COCO-Stuff-10K by
stitching DeiT3-S/L and DeiT3-B/L. We highlight the stitches at the Pareto
frontier in blue line. Table 1: Training efficiency comparison between SN-
Netv2 and individual anchors based on SETR. We measure the training cost by
A100 GPU hours. FLOPs and mIoU in SN-Netv2 are represented by a range, e.g.,
“34 - 363” means the model can cover FLOPs ranging from 34G to 363G.

Model | Params (M) | FLOPs (G) | ADE20K | COCO-Stuff-10K
---|---|---|---|---
mIoU | Train Cost | mIoU | Train Cost
DeiT3-S | 23 | 32 | 45.5 | 75 | 39.6 | 40
DeiT3-B | 88 | 108 | 48.6 | 90 | 42.7 | 48
DeiT3-L | 307 | 363 | 52.3 | 174 | 47.8 | 90
SN-Netv2 + DeiT3-S/L | 338 | 34 - 363 | 45.6 - 51.4 | 120 | 40.1 - 48.2 | 60
SN-Netv2 + DeiT3-B/L | 412 | 110 - 363 | 48.5 - 52.2 | 140 | 42.7 - 48.1 | 80

Results. In Figure 4, we compare SN-Netv2 to SN-Netv1 on ImageNet-1K. With the
same training schedule, SN-Netv2 produces hundreds of stitches that satisfy a
diverse range of resource constraints. More importantly, by highlighting the
stitches on the Pareto frontier, we show SN-Netv2 can find much better
architectures, i.e., stitching configurations than SN-Netv1. This is achieved
by the enlarged stitching space from Two-way stitching, the improved ROS
sampling, as well as the effective low-rank update of stitching layers under
LoRA SL. Moreover, while SN-Netv1 results in a noticeable performance drop for
the small anchor, SN-Netv2 escapes from the sub-optimal space at the low-FLOPs
constraints and significantly improves the stitches that reside in that range.
Overall, this strongly demonstrates the advantage of SN-Netv2 over SN-Netv1.

###  5.2 Semantic Segmentation

Implementation details. To explore the power of SN-Netv2 on downstream tasks,
we first conduct comprehensive experiments on semantic segmentation, including
ADE20K [57] and COCO-Stuff-10K [4]. Our method is based on SETR [56] due to
its simple framework which well reflects the performance of plain ViT
backbones [54]. It is worth noting that while SETR is proposed along with
three different decoders: Naive, PUP and MLA, we adopt the Naive approach as
it achieves the best performance-efficiency trade-off. For all experiments, we
train with a total batch size of 16 for ADE20K and COCO-Stuff-10K. We set the
training iterations as 160K, 80K for ADE20K and COCO-Stuff-10K, respectively.
Besides, we adopt a rank of 16 for LoRA SL when stitching DeiT3-S and DeiT3-L,
and a rank of 4 for stitching DeiT3-B and DeiT3-L. Similar to ImageNet
experiments, we use 100 images for LS initialization. All other
hyperparameters are set with default choices in mmsegmentation [10]. Following
prior works [49, 54, 19], we adopt mean Intersection over Union (mIoU) as the
metric to evaluate the performance.

ADE20K results. We report our ADE20K results in Figure 5. Specifically, based
on DeiT3-S and DeiT3-L, SN-Netv2 demonstrates strong performance against
anchor performance while simultaneously supporting a diverse range of resource
constraints. By stitching DeiT3-B and DeiT3-L, SN-Netv2 achieves equal
performance with anchors at their FLOPs. Moreover, by ploting the stitches on
Pareto Frontier, we show SN-Netv2 smoothly interpolates the performance of two
solo backbone settings, which strongly demonstrates the effectiveness of our
method.

COCO-Stuff-10K results. In Figure 6, we show that stitching DeiT3-S and
DeiT3-L under SN-Netv2 even achieves better performance than anchors. More
impressively, by stitching DeiT3-B and DeiT3-L, we found some stitches that
achieve better performance than the large anchor at a lower FLOPs. It implies
that the original plain ViTs may not be the best architecture in different
domains.

Training efficiency. We demonstrate that SN-Netv2 achieves great training
advantages compared to typical backbone adoption in downstream dense
prediction tasks. As shown in Table 1, on both ADE20K and COCO-Stuff-10K,
stitching DeiT3-S/L or DeiT3-B/L can cover a wide range of performance-
efficiency trade-offs in a single network, while requiring even less GPU hours
than training the anchors separately (e.g., 140 vs. 174 + 90 on ADE20K).

Figure 7: Results of stitching DeiT3-S and DeiT3-L on NYUv2 under the
framework of DPT. Figure 8: Results of stitching DeiT3-B and DeiT3-L on NYUv2
under the framework of DPT. Figure 9: Object detection and instance
segmentation results of SN-Netv2 on COCO-2017 by stitching DeiT3-S and DeiT3-L
under Mask R-CNN [21] based ViTDet [28]. We report the results on detection
metrics in the top row and the instance segmentation metrics in the bottom
row.

###  5.3 Depth Estimation

Implementation details. Based on DPT [37], we conduct experiments on NYUv2
[40] dataset and train SN-Netv2 by stitching DeiT3-S/L and DeiT3-B/L.
Specificially, we train each model on 4 A100 GPUs with a total batch size of
16. We set the training epochs to 24. We adopt AdamW optimizer with an initial
learning rate of $2\times 10^{-5}$. For LoRA SL, we use a rank of 4 for all
experiments. We utilize common metrics for evaluting the performance on depth
estimation, including $\delta>1.25$, $\delta>1.25^{2}$, $\delta>1.25^{3}$
(where higher values indicate better performance), as well as AbsRel, RMSE,
and Log10 (where lower values indicate better performance).

Results. We report our experiment results on NYUv2 [40] in Figure 7 and Figure
8. Overall, SN-Netv2 demonstrates highly competitive performance compared to
the anchor models, while simultaneously achieving a smooth performance-
efficiency curve. In particular, we found stitching DeiT3-B and DeiT3-L is
slightly better, which we assume a smaller gap in model complexity between the
anchors can help to achieve better performance, which aligns with the
observations of SN-Netv1 [33].

###  5.4 Object Detection and Instance Segmentation

Implementation details. We experiment SN-Netv2 on COCO-2017 and adopt Mask
R-CNN based ViTDet [28]. We train all models including individual anchors on 8
A100 GPUs with a total batch size of 8 for 100 epochs. We set the rank as 16
for LoRA SL. Besides, we adopt the same layer decay rate as the baselines for
different anchors. All other hyperparameters adopt the default setting in
detectron2 [47].

Results. Based on DeiT3-S and DeiT3-L, we report the performance of SN-Netv2
on object detection and instance segmentation in Figure 9. Overall, we found
SN-Netv2 exhibits strong flexibility on the detection task as well, as
evidenced by the smooth metrics under various resource constraints. This again
demonstrates that SN-Netv2 can serve as a flexible backbone on a wide range of
CV tasks. In particular, under the similar training cost ($\sim$1500 GPU
hours), SN-Netv2 can achieve comparable performance at the FLOPs of
individually trained DeiT3-L (51.2 vs. 51.9 bbox AP), while supporting many
FLOPs-accuracy trade-offs at runtime without additional training cost.
However, we observe a performance gap between individually trained DeiT3-S
with that of SN-Netv2 at the same FLOPs (42.9 vs. 46.6 bbox AP). For this, we
hypothesize that with the heavy decoder in ViTDet ($27\times$ larger than SETR
decoder), simultaneously ensuring the performance of hundreds of stitches as
backbones can be more difficult. Nevertheless, SN-Netv2 still achieves
superior advantage in terms of the training efficiency as it only requires
training once while covering many FLOPs-accuracy trade-offs. We leave the
improvement for future work.

Figure 10: Effect of improvements over SN-Netv1 based on stitching DeiT3-S and
DeiT3-L on ImageNet-1K. From left to right, we gradually apply Two-way
stitching (TWS), resource-constrained sampling (ROS) and low-rank adaptation
of stitching layers (LoRA SL) with a rank of 16.

###  5.5 Ablation Study

Effect of improved techniques. Based on DeiT3-S and DeiT3-L, we conduct
experiments on ImageNet-1K to show the effect of Two-way stitching, resource-
constrained sampling and LoRA stitching layers. As shown in Figure 10,
benefiting form Two-way stitching, SN-Net successfully finds better stitching
configurations at the relatively low FLOPs constraints except for the small
anchor where it drops significantly. However, with the help of ROS, we improve
the performance of DeiT3-S and therefore ensure the overall performance curve.
Finally, we show LoRA SL can achieve comparable performance with the fully
finetuned baseline in Figure 10 (c).

Effect of LoRA SL on downstream tasks. We have shown that applying SN-Netv2
with or without LoRA SL on ImageNet-1K has a minor effect on the performance.
However, we emphasize that low-rank adaptation for stitching layers is
critical for ensuring good performance on downstream tasks. In Figure 11, we
report the results of SN-Netv2 with/without applying LoRA to stitching layers
on ADE20K. As it shows, without LoRA SL, there is a noticeable performance
drop for anchors when stitching DeiT3-S/L. The issue is even more pronounced
when stitching DeiT3-B/L, resulting in a highly unstable performance curve.
However, after applying LoRA to stitching layer, we achieve a more stable and
better performance curve. In this case, we speculate that low-rank update of
the stitching layer can stabilize anchor learning, thus ensuring good
performance of intermediate stitches in SN-Netv2. We show the effect of
different ranks in Section A.4.

Figure 11: Performance comparison of SN-Netv2 with (Figure (b) and (d)) and
without (Figure (a) and (c)) low-rank adaptation for stitching layers on
ADE20K. Figure 12: Figure (a): Comparing parameter-efficient tuning (PET) with
fully finetuning (Full) on SN-Netv2. Figure (b): Benchmarking different
pretrained weights of plain ViTs under SN-Netv2. Experiments are conducted on
COCO-Stuff-10K and based on Base/Large variants of ViTs.

Compared to parameter-efficient tuning on backbone. We investigate the impact
of parameter-efficient tuning (PET) on the performance of SN-Netv2 by
stithcing DeiT3-B/L. Specifically, we apply LoRA with a rank of 4 to the
backbone and train it using the same schedule as fully-finetuning on COCO-
Stuff-10K. As shown in Figure 12 (a), SN-Netv2 can produce good performance-
efficiency trade-offs under PET as well. Moreover, it even achieves better
performance than individually trained anchors with PET (e.g., 44.8 vs. 45.8
with DeiT3-L), for which we conjecture SN-Netv2 may regularize the large
anchor learning when combining with PET.

Benchmarking other pretrained ViT weights. In Figure 12 (b), we report the
results of stitching different pretrained weights based on the base and large
variants of ViTs, including MAE [20], SAM [26], AugReg [41] and BEiTv2 [34].
Overall, SN-Netv2 consistently generates good Pareto frontiers for these
weights. As pre-training objectives and datasets differ across ViTs, our
results in Figure 12 (b) demonstrate varying performance when stitching
different ViT weights, where DeiT3 achieves the best performance. Therefore,
we choose DeiT3 as the default weights.

##  6 Conclusion and Future Work

We have introduced SN-Netv2, a systematically improved training framework that
effectively employs off-the-shelf foundation ViTs to obtain a flexible and
strong vision backbone on downstream dense prediction tasks. In particular, we
have first proposed a Two-way stitching strategy to enlarge the stitching
space and effectively identified more optimal stitches in certain resource
constraints. Next, we have devised a resource-constrained sampling approach to
ensure balanced training for stitches that reside on different resource
constraints, thereby enhancing the overall performance of SN-Netv2.
Furthermore, we have demonstrated that low-rank adaptation of stitching layers
yields competitive performance on ImageNet-1K while playing a crucial role in
stabilizing training on downstream tasks. Extensive experiments across various
datasets have demonstrated that SN-Netv2 achieves great training efficiency
and inference flexibility than typical ViT backbone adoption, benefiting the
massive model deployment in real-world applications.

Limitations and societal impact. This paper mainly aims to improve SN-Netv1
for flexible ViT adoption on downstream tasks, which however leaves parameter
efficient approaches [24, 23, 7] under-explored. Future work may also consider
a better training strategy to improve the performance of stitches on the
Pareto frontier. Besides, SN-Netv2 requires multi-GPU training, which
inevitably results in a substantial electricity consumption and carbon
emission.

##  Appendix A Appendix

We organize our supplementary material as follows.

  * •

In Section A.1, we compare different stitching types at initialization based
on ImageNet-1K.

  * •

In Section A.2, we compare different stitching types after training based on
ImageNet-1K.

  * •

In Section A.3, we compare SN-Netv2 with SN-Netv1 on semantic segmentation.

  * •

In Section A.4, we explore the effect of different ranks in LoRA SL.

  * •

In Section A.5, we describe the details of the benchmarked pretrained ViTs.

Figure 13: Performance comparison between SF and FS stitches on ImageNet-1K at
initialization. Figure 14: Performance comparison between FSF and SFS stitches
on ImageNet-1K at initialization. Figure 15: Ranking visualization of
different types of stitches on image classification, semantic segmentation and
object detection.

###  A.1 Performance Comparisons of Different Stitching Types at
Initialization

By default, we randomly sample 100 images from the train set and solve the
least-square problem to initialize the stitching layers. After initialization,
we directly compare the performance of different types of stitches based on
DeiT3 models and ImageNet-1K. Overall, the observations depicted in Figure 13
indicate that Fast-to-Slow (FS) outperforms at high FLOPs, while Slow-to-Fast
(SF) excels at low FLOPs. This highlights the effectiveness of Two-way
stitching, as no single stitching direction emerges as the superior choice
across all FLOPs intervals. Similarly, in Figure 14, we find that Fast-Slow-
Fast (FSF) generally outperforms Slow-Fast-Slow (SFS) at low FLOPs, whereas
SFS stitching prevails at high FLOPs. This implies that benefiting from the
enlarged stitching space, SN-Netv2 can find a large number of more optimal
architectures than SN-Netv1 (which uses FS only), enabling them to achieve
favorable performance right from the initialization phase.

Figure 16: Performance comparison between SN-Netv1 and SN-Netv2 on ADE20K and
COCO-Stuff-10K by stitching DeiT3-S and DeiT3-L.

###  A.2 Performance Comparison of Different Stitching Types after Training

We show in Figure 15 that different stitching types present varying
performance in different resource constraints after training. Similar to the
phenomenon in Section A.1, our findings reveal that SF outperforms FS at low
FLOPs, while FS becomes superior at high FLOPs, particularly for ImageNet-1K
and ADE20K. We attribute this to the fact that networks adopting a few early
blocks from the other anchor tend to perform better, indicating that stitching
at the early stages of ViTs is more effective when stitched only once. For
this, we assume early blocks involve with general low level features [22, 48],
which makes them more amenable to stitch. Besides, FSF stitches are generally
better than that of SFS after training. These observations can serve as
valuable guidelines for future stitching designs.

###  A.3 Performance Comparison of SN-Netv1 and SN-Netv2 on Semantic
Segmentation

Based on DeiT3-S and DeiT3-L, we compare the performance of SN-Netv2 to SN-
Netv1 on ADE20K and COCO-Stuff-10K. By default, we adopt a rank of 16 for LoRA
SL and train each model on ADE20K with a 160K schedule and COCO-Stuff-10K with
a 80K schedule. The total batch size is 16 for all experiments. As shown in
Figure 16, it is evident that SN-Netv2 exhibits significant improvement for
stitches at low FLOPs, while also achieving competitive performance at high
FLOPs. Notably, the highlighted Pareto frontier demonstrates that SN-Netv2
surpasses SN-Netv1 in terms of the overall performance-efficiency trade-offs.

###  A.4 Effect of Different Ranks in LoRA SL

Figure 17: Effect of different ranks in LoRA SL based on stitching DeiT3-S and
DeiT3-L on COCO-Stuff-10K. “Full” refers to fully finetune the stitching
layers.

In Figure 17, we explore the effect of different ranks in LoRA SL by stitching
DeiT3-S and DeiT3-L on COCO-Stuff-10K. In general, we observe that different
low ranks perform similar, where they can effectively produce smoothly
increasing Pareto frontiers. However, without the low-rank update, the
performance of the stitches at the higher FLOPs drops (>250G FLOPs).
Therefore, it indicates that low-rank update may regularize stitches learning
during the training time.

###  A.5 Details of Benchmarked Pretrained ViTs

We have shown in Section 5.5 that different pretrained ViTs under SN-Netv2
performs differently due to the various training objectives and datasets. In
this section, we provide the details of the benchmarked ViT weights.

  * •

DeiT3 [45]. ImageNet-21K pretrained
DeiT3-B111https://dl.fbaipublicfiles.com/deit/deit_3_base_224_21k.pth and
DeiT3-L222https://dl.fbaipublicfiles.com/deit/deit_3_large_224_21k.pth.

  * •

MAE [20]. ImageNet-1K finetuned
MAE-B333https://dl.fbaipublicfiles.com/mae/finetune/mae_finetuned_vit_base.pth
and
MAE-L444https://dl.fbaipublicfiles.com/mae/finetune/mae_finetuned_vit_large.pth.

  * •

SAM [26]. SA-1B [26] pretrained
SAM-B555https://dl.fbaipublicfiles.com/segment_anything/sam_vit_b_01ec64.pth
and
SAM-L666https://dl.fbaipublicfiles.com/segment_anything/sam_vit_l_0b3195.pth.

  * •

AugReg [41]. ImageNet-21K pretrained and ImageNet-1K finetuned
ViT-B777https://huggingface.co/timm/vit_base_patch16_384.augreg_in21k_ft_in1k/blob/main/pytorch_model.bin
and
ViT-L888https://huggingface.co/timm/vit_large_patch16_384.augreg_in21k_ft_in1k/blob/main/pytorch_model.bin.

  * •

BEiTv2 [34]. ImageNet-1K pretrained, then ImageNet-21K finetuned, and finally
ImageNet-1K finetuned
BEiTv2-B999https://conversationhub.blob.core.windows.net/beit-share-
public/beitv2/beitv2_base_patch16_224_pt1k_ft21kto1k.pth and
BEiTv2-L101010https://conversationhub.blob.core.windows.net/beit-share-
public/beitv2/beitv2_large_patch16_224_pt1k_ft21kto1k.pth.

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# RSMA for Dual-Polarized Massive MIMO Networks: A SIC-Free Approach

Arthur S. de Sena12, Pedro H. J. Nardelli2, Daniel B. da Costa1, Petar
Popovski3,
Constantinos B. Papadias4, Mérouane Debbah1 Emails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>1 Technology Innovation Institute,
United Arab Emirates 2 Lappeenranta-Lahti University of Technology, Finland
3 Aalborg University, Denmark 4 The American College of Greece, Greece

###### Abstract

Aiming at overcoming practical issues of successive interference cancellation
(SIC), this paper proposes a dual-polarized rate-splitting multiple access
(RSMA) technique for a downlink massive multiple-input multiple-output (MIMO)
network. By modeling the effects of polarization interference, an in-depth
theoretical analysis is carried out, in which we derive tight closed-form
approximations for the outage probabilities and ergodic sum-rates. Simulation
results validate the accuracy of the theoretical analysis and confirm the
effectiveness of the proposed approach. For instance, under low to moderate
cross-polar interference, our results show that the proposed dual-polarized
MIMO-RSMA strategy outperforms the single-polarized MIMO-RSMA counterpart for
all considered levels of residual SIC error.

###### Index Terms:

Massive MIMO, dual-polarized antenna arrays, rate-splitting multiple access.

##  I Introduction

Massive multiple-input multiple-output (MIMO) has become an indispensable
technology for fifth-generation (5G) wireless communications systems and
beyond. Nevertheless, the installation of tens to hundreds of antennas in a
tight physical space can create a strong channel correlation. This issue
limits the size of practical antenna arrays and can hamper the performance of
massive MIMO [1]. Fortunately, the polarization domain provides an efficient
way to mitigate this limitation. Specifically, since electromagnetic waves
with orthogonal polarizations propagate with a low correlation, it is possible
to implement orthogonal dual-polarized antennas and build an array with twice
the number of antennas of a single-polarized array using identical physical
dimensions [2]. Moreover, the polarization domain offers a new degree of
freedom (DoF) to MIMO systems, which can be exploited for generating
multiplexing and diversity gains [2, 1, 3].

Efficient multiple access (MA) techniques are also essential for supporting
the stringent requirements of beyond-5G systems. In particular, rate-splitting
multiple access (RSMA) has arisen as a robust next-generation MA technique for
MIMO systems [4, 5]. At the base station (BS), RSMA divides each users’
message into two parts. The first part of each message is encoded into a
single super symbol, called the common message, and mapped to the BS antennas
through a common precoder, which is intended for all uses. The second part of
each message, called private message, is transmitted via a private precoder
that should be decoded only at the intended user. Upon reception, users rely
on successive interference cancellation (SIC) to recover the transmitted
messages. Thanks to these features, RSMA can deliver high spectral and energy
efficiencies, optimality in terms of DoF, and robustness even in scenarios
with imperfect channel state information (CSI) [4]. Moreover, RSMA can
outperform all conventional MA techniques, including non-orthogonal multiple
access (NOMA) and orthogonal multiple access (OMA) techniques [5].

Despite the above benefits, there are still issues that need to be studied and
tackled. First, due to the SIC protocol in RSMA, the common message is always
detected with interference from private messages, which has degrading effects
on the system data rates. Moreover, existing RSMA-related works make the
idealistic assumption that SIC can be carried out perfectly. However, due to
hardware limitations, degraded CSI, and other issues, SIC errors are likely to
happen in practice. As demonstrated in [6], the residual interference left by
imperfect SIC can strongly harm the performance of SIC-based schemes. Thus,
strategies for combating the effects of imperfect SIC in RSMA are necessary.
In particular, the performance superiority and additional DoF of dual-
polarized MIMO systems can be exploited to alleviate interference issues of
SIC [3]. Nevertheless, to the best of our knowledge, the study of dual-
polarized MIMO-RSMA systems is still missing in the literature, and there is
no reported investigation of the harmful effects of imperfect SIC in RSMA
schemes.

This major gap in the literature motivates the development of this work.
Specifically, by modeling the practical issues of depolarization phenomena, we
propose a low-complexity dual-polarized RSMA strategy for multiplexing common
and private messages via the polarization domain in a massive MIMO network.
This approach removes the need to execute SIC in the receivers and,
consequently, frees the system from the detrimental effects of imperfect SIC.
An in-depth theoretical study is carried out on the proposed transmission
approach, where, first, we investigate the statistical properties of the
achieved highly correlated channel gains. For overcoming the challenging
statistical characterization, we assume that the channel gains are independent
and determine their approximate distributions. Closed-form expressions for the
outage probability and ergodic sum-rate are derived based on the obtained
distributions. Simulation results supported by insightful discussions validate
the theoretical analysis and demonstrate the potential performance
improvements enabled by the dual-polarized RSMA scheme.

Notation and Special Functions: Bold-faced lower-case letters denote vectors
and upper-case represent matrices. The transpose and the Hermitian transpose
of $\mathbf{A}$ are represented, respectively, by $\mathbf{A}^{T}$ and
$\mathbf{A}^{H}$, the operator tr$\\{\mathbf{A}\\}$ computes the trace of
$\mathbf{A}$, and $[\mathbf{A}]_{i:j}$ returns a sub-matrix of $\mathbf{A}$
containing its columns from $i$ to $j$. The symbol $\otimes$ represents the
Kronecker product, $\mathbf{I}_{M}$ represents the identity matrix of
dimension $M\times M$, and $\mathbf{0}_{M,N}$ denotes the $M\times N$ matrix
with all zero entries. In addition, $\mathrm{E}(\cdot)$ denotes expectation,
$\Gamma(\cdot)$ is the Gamma function [7, eq. (8.310.1)],
$\gamma(\cdot,\cdot)$ is the lower incomplete Gamma function [7, eq.
(8.350.1)], $\bm{e}_{n}(\cdot)$ denotes the truncated Taylor series of the
exponential function with $n$ terms [7, eq. (1.211.1)], and
$\mathrm{Ei}(\cdot)$ corresponds to the exponential integral [7, eq.
(8.211.1)].

##  II System Model

Figure 1: Proposed system model. A dual-polarized massive MIMO-RSMA base
station serves dual-polarized users distributed into different spatial groups.

We consider a massive MIMO network in which a single BS communicates in
downlink mode with $L$ users. The BS is equipped with $M/2$ pairs of co-
located dual-polarized transmit antennas, and each user employs one pair of
dual-polarized receive antennas, such that each antenna pair contains one
horizontally and one vertically polarized antenna element. Due to the closely
spaced antennas, the wireless channels of users located in similar angular
sectors become correlated. The BS exploits this characteristic and clusters
the users into $G$ groups based on the likeness of their channel covariance
matrices. For simplicity, we assume that each group contains $U$ users, i.e.,
$L=GU$, and that users within each group share a common covariance matrix
$\mathbf{\bar{R}}_{g}=\mathbf{I}_{2}\otimes\mathbf{R}_{g}$, in which
$\mathbf{R}_{g}$ is the covariance matrix corresponding to each polarization
with rank denoted by $r_{g}$. Under such assumptions, we can represent the
wireless channel for the $u$th user in the $g$th group by [2]

$\displaystyle\mathbf{H}_{gu}$
$\displaystyle=\sqrt{\zeta_{gu}}\begin{bmatrix}\mathbf{h}^{vv}_{gu}&\sqrt{\chi}\mathbf{h}^{vh}_{gu}\\\
\sqrt{\chi}\mathbf{h}^{hv}_{gu}&\mathbf{h}^{hh}_{gu}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}\sqrt{\zeta_{gu}}\mathbf{U}_{g}\bm{\Lambda}^{\frac{1}{2}}_{g}\mathbf{g}^{vv}_{gu}&\sqrt{\zeta_{gu}\chi}\mathbf{U}_{g}\bm{\Lambda}^{\frac{1}{2}}_{g}\mathbf{g}^{vh}_{gu}\\\
\sqrt{\zeta_{gu}\chi}\mathbf{U}_{g}\bm{\Lambda}^{\frac{1}{2}}_{g}\mathbf{g}^{hv}_{gu}&\sqrt{\zeta_{gu}}\mathbf{U}_{g}\bm{\Lambda}^{\frac{1}{2}}_{g}\mathbf{g}^{hh}_{gu}\end{bmatrix}\in\mathbb{C}^{M\times
2},$ (1)

where $\bm{\Lambda}_{g}\in\mathbb{R}^{\bar{r}_{g}\times\bar{r}_{g}}_{>0}$ is a
diagonal matrix formed by $\bar{r}_{g}$ nonzero eigenvalues of
$\mathbf{R}_{g}$ sorted in descending order,
$\mathbf{U}_{g}\in\mathbb{C}^{\frac{M}{2}\times\bar{r}_{g}}$ comprises the
corresponding $\bar{r}_{g}$ left eigenvectors of $\mathbf{R}_{g}$ obtained
from the singular value decomposition (SVD),
$\mathbf{g}^{ij}_{gu}\in\mathbb{C}^{\bar{r}_{g}\times 1}$ is a vector that
models the reduced-dimension fast-fading channels from polarization $i$ to
polarization $j$, in which $i,j\in\\{v,h\\}$, with $v$ and $h$ denoting,
respectively, the vertical and horizontal polarizations, whose entries follow
the complex Gaussian distribution with zero-mean and unit-variance,
$\zeta_{gu}$ denotes the large-scale fading coefficient, and $\chi$ represents
the inverse of the cross-polar discrimination (iXPD) parameter that models the
level of cross-polar interference.

###  II-A RSMA for Dual-Polarized Massive MIMO

Inspired by the recent works [2, 6, 1], we implement a two-stage transmission
approach where, in the first stage, spatial multiplexing is performed for
separating the multiple groups of users and, in the second stage, the RSMA
technique is employed to serve the users within each group. As anticipated in
Section I, the BS splits each data message into a common and a private part.
The common parts are encoded into a common symbol111RSMA schemes with multiple
common symbols and multiple layers of SIC also exist [4]. However, the study
of these more complicated strategies goes beyond the scope of this work.,
$c_{g}$, which should be decoded by all users within the $g$th group, while
the private parts are encoded into private symbols, $p_{gu}$, each one
intended to a particular user. Lastly, the private and common symbols are
multiplied by linear precoders and then superimposed in the power domain for
transmission, resulting in the following data stream:

$\displaystyle\mathbf{x}$
$\displaystyle=\sum_{m=1}^{G}\mathbf{K}_{m}\left(\mathbf{c}_{m}\sqrt{\alpha_{m}}c_{m}+\sum_{n=1}^{U}\mathbf{p}_{mn}\sqrt{\beta_{mn}}p_{mn}\right),$
(2)

where $\alpha_{m}$ and $\beta_{mn}$ denote, respectively, the power allocation
coefficients for the common and private messages,
$\mathbf{K}_{m}=\mathbf{I}_{2}\otimes\mathbf{F}_{m}\in\mathbb{C}^{M\times\bar{M}}$
is the precoding matrix responsable for performing spatial multiplexing of
user groups, in which
$\mathbf{F}_{m}\in\mathbb{C}^{\frac{M}{2}\times\frac{\bar{M}}{2}}$ represents
the precoding matrix for each polarization, $\bar{M}$ is a parameter that
controls the dimension of the projected channel, and
$\mathbf{c}_{m}\in\mathbb{C}^{\bar{M}\times 1}$ and
$\mathbf{p}_{mn}\in\mathbb{C}^{\bar{M}\times 1}$ are the precoding vectors for
the common and private messages, respectively, satisfying
$\|\mathbf{c}_{m}\|^{2}=1$ and $\|\mathbf{p}_{mn}\|^{2}=1$.

At the users’ side, the common message is detected first while treating the
private message as interference. Then, each user executes SIC to subtract the
common message from the composite signal. After that, the private message is
finally recovered. This conventional approach is effective for retrieving the
transmitted messages and can deliver remarkable performance gains. However, as
discussed before, there are limitations. In this paper, we exploit the
polarization domain to cope with the SIC-related issues, as explained next.

Instead of transmitting a superimposed stream, the private and common messages
are transmitted through independent polarized data streams, i.e., each message
is assigned to one polarization. Such a strategy will enable users to decode
common and private messages without relying on SIC. In addition to freeing the
system from errors of imperfect SIC, users should be able to recover the
common message without interference of the private messages, i.e., in ideal
conditions. In practice, however, users will experience cross-polar
interference, an issue that will be investigated in our analysis. For
simplicity, the common message is assigned to the vertical polarization and
the private messages to the horizontal polarization. With this strategy, the
precoding vector for the common message can be written as
$\mathbf{c}_{g}=[(\mathbf{c}^{v}_{g})^{H},\mathbf{0}_{1,\frac{\bar{M}}{2}}]^{H}$,
and for the private message as
$\mathbf{p}_{gu}=[\mathbf{0}_{1,\frac{\bar{M}}{2}},(\mathbf{p}^{h}_{gu})^{H}]^{H}$,
where $\mathbf{c}^{v}_{m}\in\mathbb{C}^{\frac{\bar{M}}{2}\times 1}$ and
$\mathbf{p}^{h}_{mn}\in\mathbb{C}^{\frac{\bar{M}}{2}\times 1}$ are the
precoding vectors for the common and private messages corresponding to
polarizations $v$ and $h$, respectively. As a result, the $u$th user in the
$g$th group will receive the following signal

$\displaystyle\mathbf{y}_{gu}$
$\displaystyle=\mathbf{H}^{H}_{gu}\sum_{m=1}^{G}\mathbf{K}_{m}\begin{bmatrix}\mathbf{c}^{v}_{m}\sqrt{\alpha_{m}}c_{m}\\\
\sum_{n=1}^{U}\mathbf{p}^{h}_{mn}\sqrt{\beta_{mn}}p_{mn}\end{bmatrix}+\begin{bmatrix}n^{v}_{gu}\\\
n^{h}_{gu}\end{bmatrix},$ (3)

where $n^{i}_{gu}\in\mathbb{C}$ is the additive white noise observed in
polarization $i\in\\{v,h\\}$, which follows the complex Gaussian distribution
with zero mean and variance $\sigma^{2}$.

##  III Precoder Design

In order to employ RSMA to each group separately, the precoding matrix
$\mathbf{K}_{g}$ should be designed to eliminate the inter-group interference.
Such a goal can be accomplished by exploiting the null space spanned by the
left eigenvector matrices of interfering groups, i.e., by defining
$\mathbf{U}^{*}_{g}=[\mathbf{U}_{1},\cdots,\mathbf{U}_{g-1},\mathbf{U}_{g+1},\cdots,\mathbf{U}_{G}]\in\mathbb{C}^{\frac{M}{2}\times\sum_{g^{\prime}\neq
g}\bar{r}_{g^{\prime}}}$, $\mathbf{F}_{g}$ can be constructed from the
orthonormal basis of $\mathrm{null}\\{\mathbf{U}^{*}_{g}\\}$, which can be
obtained from the eigenvectors corresponding to the zero eigenvalues of
$\mathbf{U}^{*}_{g}$. More specifically, let
$\mathbf{E}^{0}_{g}\in\mathbb{C}^{\frac{M}{2}\times\frac{M}{2}-\sum_{g^{\prime}\neq
g}\bar{r}_{g^{\prime}}}$ denote the matrix comprising the last
$\frac{M}{2}-\sum_{g^{\prime}\neq g}\bar{r}_{g^{\prime}}$ left eigenvectors of
$\mathbf{U}^{*}_{g}$ computed via SVD. Then, the desired precoding matrix can
be given by
$\mathbf{K}_{g}=\mathbf{I}_{2}\otimes\mathbf{F}_{g}=\mathbf{I}_{2}\otimes\big{[}\mathbf{E}^{0}_{g}\big{]}_{1:\frac{\bar{M}}{2}}$,
in which, due to the dimension of $\mathbf{E}^{0}_{g}$, it is required that
$\frac{\bar{M}}{2}\leq\frac{M}{2}-\sum_{g^{\prime}\neq
g}\bar{r}_{g^{\prime}}$, $\frac{M}{2}>\sum_{g^{\prime}\neq
g}\bar{r}_{g^{\prime}}$, and $\frac{\bar{M}}{2}\leq\bar{r}_{g}\leq r_{g}$. To
satisfy these constraints, the parameter $\bar{r}_{g}$ is configured as
$\mathrm{min}\\{r_{g},\lfloor(\frac{M}{2}-\frac{\bar{M}}{2})/(G-1)\rfloor\\}$.

Now, we concentrate on the design of the precoding vector for the private
massages. The role of $\mathbf{p}^{h}_{gu}$ is to ensure that each private
message reaches only its intended user in the assigned polarization. This
implies that $\mathbf{p}^{h}_{gu}$ must be orthogonal to the subspace spanned
by the effective channel $(\mathbf{h}^{hh}_{gu^{\prime}})^{H}\mathbf{F}_{g}$
of users $u^{\prime}\neq u$, which can be achieved by

$\displaystyle\mathbf{p}^{h}_{gu}=\mathrm{null}\\{\mathbf{F}^{H}_{g}[\mathbf{h}^{hh}_{g1},\cdots,\mathbf{h}^{hh}_{g(u-1)},\mathbf{h}^{hh}_{g(u+1)},\cdots,\mathbf{h}^{hh}_{gU}]\\},$
(4)

in which we must have $\bar{M}>U-1$ to ensure the existence of a non-trivial
null space.

In turn, $\mathbf{c}^{v}_{g}$ should be designed to broadcast the common
messages to all users. Different strategies and optimization procedures have
been proposed in the literature, including matched filter precoding (MFP),
weighted MFP, precoding for max-min fairness, among others [4]. However, such
sophisticated precoding designs can lead to an intractable analysis. As an
alternative, we construct $\mathbf{c}^{v}_{g}$ as a random precoder with
independent and identically distributed entries following the standard complex
Gaussian distribution, as in [5].

##  IV Performance Analysis

In this section, we carry out a statistical characterization of the effective
channel gains observed in the signal-to-interference-plus-noise ratios
(SINRs), based on which we derive closed-form approximations for the outage
probabilities. Tight approximations for ergodic sum-rates are also derived.

###  IV-A Signal Detection and SINR Analysis

With $\mathbf{K}_{g}$ and $\mathbf{p}^{h}_{gu}$ given in Section III, the
signal in (3) can be simplified as

$\displaystyle\mathbf{y}_{gu}=\begin{bmatrix}\sqrt{\zeta_{gu}}(\mathbf{h}^{vv}_{gu})^{H}\mathbf{F}_{g}\mathbf{c}^{v}_{g}\sqrt{\alpha_{g}}c_{g}\\\
\sqrt{\zeta_{gu}}(\mathbf{h}^{hh}_{gu})^{H}\mathbf{F}_{g}\mathbf{p}^{h}_{gu}\sqrt{\beta_{gu}}p_{gu}\end{bmatrix}$
$\displaystyle+\begin{bmatrix}\sqrt{\zeta_{gu}\chi}(\mathbf{h}^{hv}_{gu})^{H}\mathbf{F}_{g}\sum_{n=1}^{U}\mathbf{p}^{h}_{gn}\sqrt{\beta_{gn}}p_{gn}\\\
\sqrt{\zeta_{gu}\chi}(\mathbf{h}^{vh}_{gu})^{H}\mathbf{F}_{g}\mathbf{c}^{v}_{g}\sqrt{\alpha_{g}}c_{g}\end{bmatrix}+\begin{bmatrix}n^{v}_{gu}\\\
n^{h}_{gu}\end{bmatrix}.$ (5)

As can be observed in (IV-A), upon reception, users become able to recover the
common and private messages directly from the assigned polarization,
effortless without SIC. However, both messages are corrupted by cross-polar
interference that $\mathbf{p}^{h}_{gu}$ is unable to cancel. Consequently, the
SINR experienced when detecting the common message can be given by

$\displaystyle\gamma^{c}_{gu}$
$\displaystyle=\frac{\zeta_{gu}|(\mathbf{h}^{vv}_{gu})^{H}\mathbf{F}_{g}\mathbf{c}^{v}_{g}|^{2}\alpha_{g}}{\zeta_{gu}\chi\sum_{n=1}^{U}|(\mathbf{h}^{hv}_{gu})^{H}\mathbf{F}_{g}\mathbf{p}^{h}_{gn}|^{2}\beta_{gn}+\sigma^{2}}.$
(6)

Analogously, the SINR observed by the $u$th user when decoding its private
message can be written as

$\displaystyle\gamma^{p}_{gu}=\frac{\zeta_{gu}|(\mathbf{h}^{hh}_{gu})^{H}\mathbf{F}_{g}\mathbf{p}^{h}_{gu}|^{2}\beta_{gu}}{\zeta_{gu}\chi|(\mathbf{h}^{vh}_{gu})^{H}\mathbf{F}_{g}\mathbf{c}^{v}_{g}|^{2}\alpha_{g}+\sigma^{2}}.$
(7)

###  IV-B Statistical Characterization of Channel Gains

Let the gain in the numerator of (6) be denoted by
$\varsigma^{c}_{gu}=\zeta_{gu}|(\mathbf{h}^{vv}_{gu})^{H}\mathbf{F}_{g}\mathbf{c}^{v}_{g}|^{2}\alpha_{g}$
and the interference term by
$\omega^{c}_{gu}=\zeta_{gu}\chi\sum_{n=1}^{U}|(\mathbf{h}^{hv}_{gu})^{H}\mathbf{F}_{g}\mathbf{p}^{h}_{gn}|^{2}\beta_{gn}$.
Then, the squared norm in $\varsigma^{c}_{gu}$ can be expanded as
$|(\mathbf{h}^{vv}_{gu})^{H}\mathbf{F}_{g}\mathbf{c}^{v}_{g}|^{2}=(\mathbf{c}^{v}_{g})^{H}\mathbf{F}^{H}_{g}\mathbf{h}^{vv}_{gu}(\mathbf{h}^{vv}_{gu})^{H}\mathbf{F}_{g}\mathbf{c}^{v}_{g}=\mathrm{tr}\\{\mathbf{F}^{H}_{g}\mathbf{h}^{vv}_{gu}(\mathbf{h}^{vv}_{gu})^{H}\mathbf{F}_{g}\mathbf{c}^{v}_{g}(\mathbf{c}^{v}_{g})^{H}\\}$.
Given that $\mathbf{c}^{v}_{g}$ is an isotropic unit vector independent of
$(\mathbf{h}^{vv}_{gu})^{H}\mathbf{F}_{g}$ and that
$\mathrm{E}\\{\mathbf{c}^{v}_{g}(\mathbf{c}^{v}_{g})^{H}\\}=\frac{1}{\bar{M}}\mathbf{I}_{\bar{M}}$,
we have that
$\mathrm{E}\\{|(\mathbf{h}^{vv}_{gu})^{H}\mathbf{F}_{g}\mathbf{c}^{v}_{g}|^{2}\\}=\frac{1}{\bar{M}}\mathrm{tr}\\{\mathbf{F}^{H}_{g}\mathbf{h}^{vv}_{gu}(\mathbf{h}^{vv}_{gu})^{H}\mathbf{F}_{g}\\}$.
Since $\mathbf{g}^{vv}_{gu}$ in (II) is a complex Gaussian distributed vector
and $\mathbf{F}_{g}$ is a semi-unitary matrix (with orthonormal columns), the
vector $\mathbf{F}^{H}_{g}\mathbf{h}^{vv}_{gu}$ is still complex Gaussian
distributed. Consequently, the gain
$\varsigma^{c}_{gu}=\zeta_{gu}|(\mathbf{h}^{vv}_{gu})^{H}\mathbf{F}_{g}\mathbf{c}^{v}_{g}|^{2}\alpha_{g}$
follows a Gamma distribution with shape parameter $1$ and rate parameter
$\phi/\zeta_{gu}\alpha_{g}$, with
$\phi=\bar{M}/\mathrm{tr}\\{\mathbf{F}^{H}_{g}\mathbf{R}_{g}\mathbf{F}_{g}\\}$,
whose probability distribution function (PDF) can be found in [8, eq.
(15.1.1)].

Now, we focus on $\omega^{c}_{gu}$. Note that $\omega^{c}_{gu}$ consists of a
sum of $U$ correlated random variables (RVs). Consequently, determining the
exact distribution of $\omega^{c}_{gu}$ becomes challenging. For overcoming
this issue, we approximate the PDF of $\omega^{c}_{gu}$ by assuming that the
sum terms are independent. Then, we can expand the squared norm in
$\omega^{c}_{gu}$ as
$\mathrm{tr}\\{\mathbf{F}^{H}_{g}\mathbf{h}_{gu}^{hv}(\mathbf{h}_{gu}^{hv})^{H}\mathbf{F}_{g}\mathbf{p}^{h}_{gn}(\mathbf{p}^{h}_{gn})^{H}\\}$.
Since $\mathbf{p}^{h}_{gn}$ is uniformly distributed with the Haar measure
$O(\bar{M})$, we have that
$\mathrm{E}\\{\mathbf{p}^{h}_{gn}(\mathbf{p}^{h}_{gn})^{H}\\}=\frac{1}{\bar{M}}\mathbf{I}_{\bar{M}}$.
As a result, the distribution of each term of the summation in
$\omega^{c}_{gu}$ follows a Gamma distribution with shape parameter $1$ and
rate parameter ${\phi}/{\zeta_{gu}\chi\beta_{gn}}$, for $n=1,\cdots,U$.
However, given that we employ a uniform power allocation among the private
messages222As stated in [5], uniform power allocation for multi-user MIMO is
widely employed in the literature and used in practical 4G and 5G systems., we
have that $\beta_{g1}=\beta_{g2}=\cdots=\beta_{gU}$. This implies that the
terms of the sum in $\omega^{c}_{gu}$ have equal rate parameters.
Consequently, the PDF of $\omega^{c}_{gu}$ can be approximated by a Gamma PDF
with shape parameter $U$ and rate parameter
${\phi}/{\zeta_{gu}\chi\beta_{gu}}$.

We characterize now the gains in the SINR for the private messages in (7). Let
$\varsigma^{p}_{gu}=\zeta_{gu}|(\mathbf{h}^{hh}_{gu})^{H}\mathbf{F}_{g}\mathbf{p}^{h}_{gu}|^{2}\beta_{gu}$
and
$\omega^{p}_{gu}=\zeta_{gu}\chi|(\mathbf{h}^{vh}_{gu})^{H}\mathbf{F}_{g}\mathbf{c}^{v}_{g}|^{2}\alpha_{g}$.
As can be noticed, $\varsigma^{p}_{gu}$ and $\omega^{p}_{gu}$ have a similar
form of $\varsigma^{c}_{gu}$, which implies that such gains can be
characterized similarly as we have characterized $\varsigma^{c}_{gu}$.
Therefore, the full details for this statistical characterization are omitted.
In short, $\varsigma^{p}_{gu}$ and $\omega^{p}_{gu}$ follow Gamma
distributions with shape parameters $1$ and rate parameters given by
$\phi/\zeta_{gu}\beta_{gu}$ and $\phi/\zeta_{gu}\chi\alpha_{g}$, respectively.

###  IV-C Outage Probability

In RSMA, users need to successfully decode both common and private messages
for them to be able to reconstruct the intended original information.
Therefore, an outage event will occur either if the rate of the common message
or the rate of the private message drops below the corresponding target data
rate. Mathematically, the outage probability for the $u$th user in the $g$th
group can be obtained by

$\displaystyle P^{\text{out}}_{gu}$ $\displaystyle=P^{c}_{gu}\cup
P^{p}_{gu}=P^{c}_{gu}+P^{p}_{gu}-P^{c}_{gu}P^{p}_{gu},$ (8)

where $P^{c}_{gu}=\mathrm{Pr}\\{\log_{2}(1+\gamma^{c}_{gu})<R^{c}_{g}\\}$ and
$P^{p}_{gu}=\mathrm{Pr}\\{\log_{2}(1+\gamma^{p}_{gu})<R^{p}_{gu}\\}$, with
$R^{c}_{g}$ and $R^{p}_{gu}$ representing, respectively, the target data rates
for the common and private messages, which are measured in bits per channel
use (bpcu). A tight closed-form approximation for the outage probability of
the proposed strategy is derived next.

#### Proposition I

When the BS employs the proposed dual-polarized MIMO-RSMA scheme, the outage
probability for the common message experienced by the $u$th user in the $g$th
group can be approximated by

$\displaystyle P^{c}_{gu}$
$\displaystyle=1-\left(\frac{\alpha_{g}}{\alpha_{g}+\chi\beta_{gu}\tau_{g}^{c}}\right)^{U}e^{-\frac{\phi\tau_{g}^{c}}{\rho\zeta_{gu}\alpha_{g}}},$
(9)

where $\tau^{c}_{g}=2^{R^{c}_{g}}-1$ and $\rho=1/\sigma^{2}$ is the signal-to-
noise ratio (SNR).

Proof: Please, see Appendix A. $\blacksquare$

#### Proposition II

When the BS employs the proposed dual-polarized MIMO-RSMA scheme, the outage
probability for the private message experienced by the $u$th user in the $g$th
group can be approximated by

$\displaystyle P^{p}_{gu}$
$\displaystyle=1-\frac{\beta_{gu}}{\chi\alpha_{g}\tau_{gu}^{p}+\beta_{gu}}e^{-\frac{\phi\tau_{gu}^{p}}{\rho\zeta_{gu}\beta_{gu}}},$
(10)

where $\tau_{gu}^{p}=2^{R_{gu}^{p}}-1$.

Proof: Please, see Appendix B. $\blacksquare$

The general outage probability, $P^{\text{out}}_{gu}$, for the dual-polarized
MIMO-RSMA scheme can be easily obtained by replacing (9) and (10) into (8).

###  IV-D Ergodic Sum-Rates

For ensuring a successful decoding at all users, the ergodic rate for the
common message is computed by averaging the minimum of the common message’s
instantaneous rates achieved by the users [4]. Therefore, the ergodic sum-rate
for the $g$th group can be computed by

$\displaystyle C_{g}$ $\displaystyle=C^{p}_{g}+C^{c}_{g},$ (11)

where $C^{p}_{g}=\mathrm{E}\\{\sum_{u=1}{}^{U}\hskip
11.38109pt\log_{2}(1+\gamma^{p}_{gu})\\}$ and
$C^{c}_{g}=\mathrm{E}\\{\sum_{u=1}{}^{U}\hskip
11.38109pt\min_{l\in\\{1,\cdots,U\\}}\\{\log_{2}(1+\gamma^{c}_{gl})\\}\\}$ are
the ergodic sum-rates for the private and common messages, respectively. The
following propositions provide a tight approximation for the ergodic sum-rate.

#### Proposition III

When the BS employs the proposed dual-polarized MIMO-RSMA scheme, the ergodic
sum-rate for the common message experienced in the $g$th group can be
approximated by

$\displaystyle
C^{c}_{g}=\text{\small$\sum_{u=1}^{U}\frac{(-1)^{U^{2}-1}}{\mathrm{ln}(2)}\left(\frac{\alpha_{g}}{\chi\beta_{gu}-\alpha_{g}}\right)^{U^{2}}e^{\frac{\phi\sum_{l=1}^{U}\frac{1}{\zeta_{gl}}}{\rho\alpha_{g}}}$}$
$\times\left[\mathrm{Ei}\left(-\frac{\phi\sum_{l=1}^{U}\frac{1}{\zeta_{gl}}}{\rho\alpha_{g}}\right)-\bm{e}_{U^{2}-1}\left(\frac{\phi(\chi\beta_{gu}-\alpha_{g})\sum_{l=1}^{U}\frac{1}{\zeta_{gl}}}{\chi\rho\beta_{gu}\alpha_{g}}\right)\right.$
$\times
e^{-\frac{\phi(\chi\beta_{gu}-\alpha_{g})\sum_{l=1}^{U}\frac{1}{\zeta_{gl}}}{\chi\rho\beta_{gu}\alpha_{g}}}\mathrm{Ei}\left(-\frac{\phi\sum_{l=1}^{U}\frac{1}{\zeta_{gl}}}{\rho\chi\beta_{gu}}\right)+e^{-\frac{\phi\sum_{l=1}^{U}\frac{1}{\zeta_{gl}}}{\rho\alpha_{g}}}\sum_{m=1}^{U^{2}-1}\frac{1}{m!}$
$\left.\times\left(-\frac{\chi\beta_{gu}-\alpha_{g}}{\alpha_{g}}\right)^{m}\sum_{k=0}^{m-1}(m-k-1)!\left(-\frac{\phi\sum_{l=1}^{U}\frac{1}{\zeta_{gl}}}{\rho\chi\beta_{gu}}\right)^{K}\right].$
(12)

Proof: Please, see Appendix C. $\blacksquare$

#### Proposition IV

When the BS employs the proposed dual-polarized MIMO-RSMA scheme, the ergodic
sum-rate for the private message experienced in the $g$th group can be
approximated by

$\displaystyle C^{p}_{g}$
$\displaystyle=\sum_{u=1}^{U}\frac{\beta_{gu}}{\mathrm{ln}(2)(\beta_{gu}-\chi\alpha_{g})}\left[e^{\frac{\phi}{\rho\zeta_{gu}\chi\alpha_{g}}}\mathrm{Ei}\left(-\frac{\phi}{\rho\zeta_{gu}\chi\alpha_{g}}\right)\right.$
$\displaystyle\left.-e^{\frac{\phi}{\rho\zeta_{gu}\beta_{gu}}}\mathrm{Ei}\left(-\frac{\phi}{\rho\zeta_{gu}\beta_{gu}}\right)\right].$
(13)

Proof: Please, see Appendix D. $\blacksquare$

##  V Numerical and Simulation Results

This section validates the theoretical analysis and demonstrates the
performance superiority of the dual-polarized MIMO-RSMA strategy over
conventional single-polarized and dual-polarized baseline systems, including
MIMO-OMA, which implements time-division multiple access, single-polarized
MIMO-RSMA, single-polarized MIMO-NOMA, and the dual-polarized MIMO-NOMA scheme
from [2].

The BS of the dual-polarized schemes employs $\frac{M}{2}=50$ co-located pairs
of dual-polarized antennas, i.e., a total of $M=100$ transmit antennas. For
fair performance comparisons, the single-polarized systems implement the same
number of antennas, and the covariance matrices for all systems are generated
using the one-ring model [1]. We assume that users are distributed among $G=4$
spatial groups, with the $g$th group positioned at the azimuth angle given by
$\theta_{g}=30^{\circ}+(g-1)160^{\circ}$ and a distance of $170$ m from the BS
to its center. The presented results are based on the first group, which is
located at the azimuth angle of $30^{\circ}$. Within each group there are
$U=3$ users, in which users $1$, $2$, and $3$ are located, respectively, at
$d_{1}=200$ m, $d_{2}=170$ m, and $d_{3}=140$ m from the BS. Under this setup,
the large-scale fading coefficient for the $u$th user is modeled by
$\zeta_{u}=\delta d_{u}^{-\eta}$, where $\delta$ is a BS array gain parameter
that is adjusted based on the desired users’ performance, and $\eta$ is the
path-loss exponent. These parameters are configured as $\delta=4\times 10^{4}$
and $\eta=2.5$. Moreover, we set $\bar{M}=6$ and adopt a fixed power
allocation, where, for the RSMA schemes, we adjust $\alpha=0.7$ and
$\beta_{u}=(1-\alpha)/U=0.1$, for $u=1,\cdots,3$. For the NOMA schemes, the
power coefficients of users $1,2$, and $3$ are set to $5/8,2/8$, and $1/8$,
respectively, whereas for OMA, the BS transmits at each time-slot using its
full power.

Figs. 2(a) and 2(b) compare the analytical and simulated outage probabilities
achieved with the proposed dual-polarized MIMO-RSMA scheme. Fig. 2(a),
specifically, shows outage probability curves for different sets of target
rates considering a scenario with negligible cross-polar interference, i.e.,
for $\chi=0$. It can be seen in this scenario that, independently of the
specified target rates, the approximate analytical curves generated with (9)
and (10) can follow the simulated ones with high accuracy. In Fig. 2(b), the
performance of the dual-polarized MIMO-RSMA is tested for different levels of
cross-polar interference, and an accurate agreement between simulated and
analytical results can also be observed.

Figure 2: Analytical and simulated outage probabilities: (a) for different
values of target rates with $\chi=0$, and (b) for different values of $\chi$
with $R^{c}=0.5$ bpcu, $R^{p}_{1}=0.1$ bpcu, $R^{p}_{2}=0.5$ bpcu, and
$R^{p}_{3}=1.2$ bpcu.

Figs. 3(a) and 3(b) present the outage sum-rates achieved by the proposed
dual-polarized MIMO-RSMA strategy and by other conventional MA systems. For a
fair comparison, the target rates for schemes other than RSMA are adjusted as
$R^{c}+R^{p}_{u}$. Fig. 3(a) reveals the performance superiority of the
proposed dual-polarized MIMO-RSMA scheme over the baseline systems in
scenarios with and without residual SIC errors. The effects of increasing the
SIC error factor on the outage sum-rates can be observed in Fig. 3(b). As can
be seen, the MIMO-NOMA systems rapidly become less spectrally efficient than
MIMO-OMA with the increase of $\xi$. On the other hand, the dual-polarized
MIMO-RSMA scheme show robustness to imperfect SIC.

The analytical ergodic sum-rate is validated in Fig. 4(a). As can be observed,
a near-perfect agreement between analytical and simulated curves can be
obtained for all values of $\chi$. Last, Fig. 4(b) shows simulated ergodic
sum-rates achieved with different schemes. As we can see, the proposed dual-
polarized MIMO-RSMA can outperform all baseline schemes. For instance, for an
SNR value of $26$ dB, in the scenario with perfect SIC, the dual-polarized
MIMO-RSMA achieves a remarkable sum-rate of $19.14$ bpcu, against only $7.29$
bpcu observed for the dual-polarized MIMO-NOMA.

Figure 3: Outage sum-rate curves: (a) versus SNR with different values of
$\xi$, and (b) versus $\xi$ for a fixed SNR value of $24$ dB.
($R^{c}=0.5,R^{p}_{1}=0.1,R^{p}_{2}=1,R^{p}_{3}=2$ bpcu, $\chi=0.001$). Figure
4: (a) analytical and simulated ergodic sum-rates of dual-polarized MIMO-RSMA
for various values of $\chi$, and (b) simulated ergodic sum-rates of various
MA schemes for different values of $\xi$ with $\chi=0.001$.

##  VI Conclusions

We have proposed a novel MA strategy for overcoming interference issues of SIC
via the polarization domain. Simulation results validated the theoretical
analysis and revealed the performance improvements enabled by that the
proposed dual-polarized MIMO-RSMA. It has also been demonstrated that the
MIMO-RSMA schemes can impressively outperform conventional single and dual-
polarized baseline systems.

## Acknowledgments

This work was partly supported by the Academy of Finland via: (a) FIREMAN
consortium n.326270 as part of CHIST-ERA grant CHIST-ERA-17-BDSI-003, (b)
EnergyNet Fellowship n.321265/n.328869/n.352654, and (c) X-SDEN project
n.349965.

##  Appendix A Proof of Proposition I

The outage probability for the common message can be calculated by

$\displaystyle P^{c}_{gu}$
$\displaystyle=\int_{0}^{\infty}\int_{0}^{\tau_{g}^{c}(y+\sigma^{2})}f_{\varsigma^{c}_{gu}\omega^{c}_{gu}}(x,y)dxdy,$
(A-1)

where $\tau_{g}^{c}=2^{R^{c}_{g}}-1$, and
$f_{\varsigma^{c}_{gu}\omega^{c}_{gu}}(x,y)$ denotes the joint PDF of
$\varsigma^{c}_{gu}$ and $\omega^{c}_{gu}$. Given that $\varsigma^{c}_{gu}$
and $\omega^{c}_{gu}$ are correlated, deriving their exact joint PDF leads to
an intractable mathematical analysis. As an alternative, we assume that
$\varsigma^{c}_{gu}$ and $\omega^{c}_{gu}$ are independent RVs. From Section
IV-B, we know that $\varsigma^{c}_{gu}$ follows a Gamma distribution with
shape parameter $1$ and rate parameter $\phi/\zeta_{gu}\alpha_{g}$, and
$\omega^{c}_{gu}$ follows a Gamma distribution with shape parameter $U$ and
rate parameter $\phi/\zeta_{gu}\chi\beta_{gu}$. Therefore, by recalling [8,
eq. (15.1.1)], the following is achieved

$\displaystyle P^{c}_{gu}$
$\displaystyle=\text{\small$\frac{\phi^{U}}{\chi^{U}\Gamma(U)\zeta_{gu}^{U}\beta_{gu}^{U}}\left(\int_{0}^{\infty}y^{U-1}e^{-y\frac{\phi}{\zeta_{gu}\chi\beta_{gu}}}dy\right.$}$
$\displaystyle\text{\small$-\left.e^{-\frac{\phi\tau_{g}^{c}\sigma^{2}}{\zeta_{gu}\alpha_{g}}}\int_{0}^{\infty}y^{U-1}e^{-y\left(\frac{\phi}{\zeta_{gu}\chi\beta_{gu}}+\frac{\phi\tau_{g}^{c}}{\zeta_{gu}\alpha_{g}}\right)}dy\right)$}.$
(A-2)

The two integrals in (A) are of the form $\int_{0}^{\infty}y^{a-1}e^{-yb}dy$,
which has a solution given by $b^{-a}\Gamma(a)$ [7, eq. (3.381.4)]. Therefore,
by defining $\rho=\sigma_{n}^{-2}$, (A) can be solved as

$\displaystyle P^{c}_{gu}$
$\displaystyle=1-\left(\frac{\alpha_{g}}{\alpha_{g}+\chi\beta_{gu}\tau_{g}^{c}}\right)^{U}e^{-\frac{\phi\tau_{g}^{c}}{\rho\zeta_{gu}\alpha_{g}}},$
(A-3)

which completes the proof. $\blacksquare$

##  Appendix B Proof of Proposition II

Recall from Section IV-B that $\varsigma^{p}_{gu}$ and $\omega^{p}_{gu}$
follow Gamma distributions with shape parameters 1 and rate parameters
$\phi/\zeta_{gu}\beta_{gu}$ and $\phi/\zeta_{gu}\chi\alpha_{g}$, respectively.
Given this, and assuming that $\varsigma^{p}_{gu}$ and $\omega^{p}_{gu}$ are
independent, their joint PDF can be obtained by

$\displaystyle f_{\varsigma^{p}_{gu}\omega^{p}_{gu}}(x,y)$
$\displaystyle=\frac{\phi^{2}}{\zeta_{gu}^{2}\chi\alpha_{g}\beta_{gu}}e^{-y\frac{\phi}{\zeta_{gu}\chi\alpha_{g}}}e^{-x\frac{\phi}{\zeta_{gu}\beta_{gu}}}.$
(B-1)

Then, with the SINR in (7), and defining $\tau_{gu}^{p}=2^{R^{p}_{gu}}-1$, the
outage probability for the private massages can be derived as follows

$\displaystyle P^{p}_{gu}$
$\displaystyle=\frac{\phi}{\zeta_{gu}\chi\alpha_{g}}\left(\int_{0}^{\infty}e^{-y\frac{\phi}{\zeta_{gu}\chi\alpha_{g}}}dy\right.$
$\displaystyle-\left.e^{-\frac{\phi}{\zeta_{gu}\beta_{gu}}\tau_{gu}^{p}\sigma^{2}}\int_{0}^{\infty}e^{-y\left(\frac{\phi}{\zeta_{gu}\beta_{gu}}\tau_{gu}^{p}+\frac{\phi}{\zeta_{gu}\chi\alpha_{g}}\right)}dy\right)$
$\displaystyle=1-\frac{\beta_{gu}}{\chi\alpha_{g}\tau_{gu}^{p}+\beta_{gu}}e^{-\frac{\phi\tau_{gu}^{p}}{\rho\zeta_{gu}\beta_{gu}}},$
(B-2)

which completes the proof. $\blacksquare$

##  Appendix C Proof of Proposition III

First, we need to obtain the CDF of $\min\\{\gamma^{c}_{gu}\\}$, for
$u=1,\cdots,U$, denoted by $F_{\text{\tiny min$\gamma^{c}_{gu}$}}(z)$. From
Appendix A, we can easily achieve the CDF of $\gamma^{c}_{gu}$ by replacing
$\tau^{c}_{gu}$ by $z$ in the outage probability expression in (A-3). In view
of this, we can exploit the theory of Order Statistics to calculate
$F_{\text{\tiny min$\gamma^{c}_{gu}$}}(z)$. Specifically, by knowing that
$\gamma^{c}_{g1},\cdots,\gamma^{c}_{gU}$ are not identically distributed, we
can recall [9, eq. (5.4.11)] to obtain the desired CDF as follows

$\displaystyle F_{\text{\tiny min$\gamma^{c}_{gu}$}}(z)$
$\displaystyle=1-\prod_{l=1}^{U}\left(\frac{\alpha_{g}}{\alpha_{g}+\chi\beta_{gu}z}\right)^{U}e^{-z\frac{\phi}{\rho\zeta_{gl}\alpha_{g}}}$
$\displaystyle=1-\alpha_{g}^{U^{2}}\left({\alpha_{g}+\chi\beta_{gu}z}\right)^{-U^{2}}e^{-z\frac{\phi\sum_{l=1}^{U}\frac{1}{\zeta_{gl}}}{\rho\alpha_{g}}}.$
(C-1)

Now that we know $F_{\text{\tiny min$\gamma^{c}_{gu}$}}(z)$, the desired sum-
rate can be obtained by a Riemann-Stieltjes integral, as follows

$\displaystyle C^{c}_{g}$
$\displaystyle=\sum_{u=1}^{U}\mathrm{E}\left\\{\min_{l\in\\{1,\cdots,U\\}}\left\\{\log_{2}\left(1+\gamma^{c}_{gl}\right)\right\\}\right\\}$
$\displaystyle=\sum_{u=1}^{U}\int_{-\infty}^{\infty}\log_{2}\left(1+z\right)dF_{\text{\tiny
min$\gamma^{c}_{gu}$}}(z).$ (C-2)

Next, by integrating by parts and applying the transformation
$\alpha_{g}+\chi\beta_{gu}z=t$, we achieve

$\displaystyle C^{c}_{g}$
$\displaystyle=\sum_{u=1}^{U}\frac{\alpha^{U^{2}}_{g}}{\mathrm{ln}(2)}\left[-\alpha_{g}^{-U^{2}}e^{\frac{\phi\sum_{l=1}^{U}\frac{1}{\zeta_{gl}}}{\rho\alpha_{g}}}\mathrm{Ei}\left(-\frac{\phi\sum_{l=1}^{U}\frac{1}{\zeta_{gl}}}{\rho\alpha_{g}}\right)\right.$
$\displaystyle+U^{2}e^{\frac{\phi\sum_{l=1}^{U}\frac{1}{\zeta_{gl}}}{\rho\alpha_{g}}}\int_{\alpha_{g}}^{\infty}t^{-U^{2}-1}\mathrm{Ei}\left(-\left(\frac{\chi\beta_{gu}-\alpha_{g}}{\chi\beta_{gu}}\right.\right.$
$\displaystyle\left.+\left.\left.\frac{t}{\chi\beta_{gu}}\right)\frac{\phi\sum_{l=1}^{U}\frac{1}{\zeta_{gl}}}{\rho\alpha_{g}}\right)dt\right].$
(C-3)

For solving the integral in (C), we make use of [10, eq. (4.1.24)], in which,
after some simplifications, we can finally obtain the desired ergodic sum-rate
expression in (IV-D), which completes the proof. $\blacksquare$

##  Appendix D Proof of Proposition IV

For deriving the ergodic sum-rate of the private messages, we obtain the CDF
of $\gamma^{p}_{gu}$ from the outage probability in (B). Then, similarly as in
Appendix C, the desired sum-rate is calculated by a Riemann-Stieltjes
integral, as follows

$\displaystyle C^{p}_{g}$
$\displaystyle=\sum_{u=1}^{U}\frac{\beta_{gu}}{\mathrm{ln}(2)}\int_{0}^{\infty}(1+z)^{-1}(\chi\alpha_{g}z+\beta_{gu})^{-1}e^{-z\frac{\phi}{\rho\zeta_{gu}\beta_{gu}}}dz.$
(D-1)

By integrating by parts and applying the transformation
$\chi\alpha_{g}z+\beta_{gu}=t$, we get

$\displaystyle C^{p}_{g}$
$\displaystyle=\sum_{u=1}^{U}\frac{\beta_{gu}}{\mathrm{ln}(2)}\left[\beta_{gu}^{-1}e^{\frac{\phi}{\rho\zeta_{gu}\beta_{gu}}}\mathrm{Ei}\left(-\frac{\phi}{\rho\zeta_{gu}\beta_{gu}}\right)\right.$
$\displaystyle+e^{\frac{\phi}{\rho\zeta_{gu}\beta_{gu}}}\int_{\beta_{gu}}^{\infty}t^{-2}\mathrm{Ei}\left(-\left(\frac{t}{\chi\alpha_{g}}\right.\right.$
$\displaystyle+\left.\left.\left.\frac{\chi\alpha_{g}-\beta_{gu}}{\chi\alpha_{g}}\right)\frac{\phi}{\rho\zeta_{gu}\beta_{gu}}\right)dt\right].$
(D-2)

Finally, by recalling [10, eq. (4.1.21)], and performing some algebraic
manipulations, the ergodic sum-rate of the private messages for the $g$th
group can be obtained as in (IV-D), which completes the proof. $\blacksquare$

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