# DARTS: Double Attention Reference-based Transformer for Super-resolution

Masoomeh Aslahishahri
Department of Computer Science
University of Saskatchewan
Saskatoon, Canada
<EMAIL_ADDRESS>
&Jordan Ubbens
Department of Computer Science
University of Saskatchewan
Saskatoon, Canada
<EMAIL_ADDRESS>
Ian Stavness
Department of Computer Science
University of Saskatchewan
Saskatoon, Canada
<EMAIL_ADDRESS>

###### Abstract

We present DARTS, a transformer model for reference-based image super-
resolution. DARTS learns joint representations of two image distributions to
enhance the content of low-resolution input images through matching
correspondences learned from high-resolution reference images. Current state-
of-the-art techniques in reference-based image super-resolution are based on a
multi-network, multi-stage architecture. In this work, we adapt the double
attention block from the GAN literature, processing the two visual streams
separately and combining self-attention and cross-attention blocks through a
gating attention strategy. Our work demonstrates how the attention mechanism
can be adapted for the particular requirements of reference-based image super-
resolution, significantly simplifying the architecture and training pipeline.
We show that our transformer-based model performs competitively with state-of-
the-art models, while maintaining a simpler overall architecture and training
process. In particular, we obtain state-of-the-art on the SUN80 dataset, with
a PSNR/SSIM of 29.83 / .809. These results show that attention alone is
sufficient for the RSR task, without multiple purpose-built subnetworks,
knowledge distillation, or multi-stage training.

##  1 Introduction

Image super resolution (SR) refers to enhancing the spatial resolution of a
low-resolution (LR) image, transforming it into a high-resolution (HR) image,
while recovering crucial and realistic textures and details [51; 49]. Image
super resolution can improve the user experience of digital media content. For
example, SR can enhance the texture of video frames in computer games without
increasing the computational cost, or improve the digital zoom functionality
of smartphones. A wide range of computer vision tasks such as medical imaging
and remote sensing can also significantly benefit from image super resolution
[19]. The super resolution problem is generally divided into two sub-problems:
single image super resolution (SISR) and reference-based image super
resolution (RSR). Using classical approaches, SISR produces blurry images with
aliasing artifacts because significant degradations occur when images are
downsampled. However, recent SISR models have tried to enhance the image SR
process and produce images as close as possible to the corresponding target HR
images [4; 15; 16; 21; 3; 20].

Figure 1: Overview of DARTS. a) The proposed architecture takes two images as
input, where LR features are derived from an adapted ResNet feature extractor
(FE) and HR reference features are derived from a patch embedding (PE) module.
b) Double attention is used across the two image distributions, using both
self- and cross-attention blocks. c) Gating attention balances self- and
cross-attention matrices, modulated by a gating parameter $(\lambda)$.

The latest progress in image SR has been made by RSR, where HR reference
images are used to transfer HR textures to corresponding LR images to produce
more realistic images preserving fine-grained textures [43]. This texture
transfer is a challenging problem because it requires finding matching
correspondences between the LR input image and the HR reference image. There
are two major challenges inherent in this task: 1) the spatial
(transformation) gap between similar regions in the LR input and HR reference
images, where corresponding landmarks appear at different positions and
orientations in the two images, and 2) the resolution gap between the LR input
and HR reference images. For the transformation gap, the input image and the
reference image may be very similar in their content, but the appearance of
objects may differ due to geometrical transformations in scale, position and
shape of objects, which can lead to suboptimal texture transfer. For the
resolution gap, the amount of information between the LR input and HR
reference images is significantly imbalanced because some information is
missing in the LR input image, which can hinder the correspondence matching
process, especially in fine-grained texture regions.

Recent successes in RSR have adopted self-supervised learning [1] or a coarse-
to-fine matching module [27] to produce quality HR images. Current state-of-
the-art (SOTA) techniques are built on the general architecture which was
first presented in $C^{2}$-matching [12]. These techniques involve a
contrastive correspondence module, trained via student-teacher distillation,
which extracts correspondences from the input and reference images. These
correspondences are then transformed by a dynamic aggregation module followed
by a restoration module which outputs the super resolved image. While the
correspondences were originally extracted by a convolutional backbone,
subsequent papers have explored other options such as deformable transformers
[1]. Although the correspondence module is interchangeable, these methods tend
to maintain the same general structure as $C^{2}$-matching with multiple
subnetworks and multiple stages of training.

In this work, we present a model called DARTS, extending double attention to
jointly learn the matching correspondences of two image distributions using
self-attention and cross-attention blocks. Our key technical innovation is to
introduce separate streams across two image distributions that communicate
through an adaptive gating attention strategy [5] attending to the content of
self-attention and cross-attention blocks, regulating the attention paid to
each block by adjusting a gating parameter. This structure can accommodate
finding the matching correspondences between the two image distributions and
provide interaction between them at varying representation sizes. In our
ablation studies, we demonstrate that this structure outperforms double
attention used in a single-stream model. In contrast to current SOTA methods
such as $C^{2}$-matching [12] and DATSR [1], the entire network consists of a
single module which is trained end-to-end using gradient descent.

To the best of our knowledge, our proposed framework is a novel technique in
RSR that can achieve quantitative results which are comparable to other SOTA
techniques in the literature, while using a simplified architecture and
training strategy. The main contributions of this study are:

  * •

Using a modified double attention module to learn joint representations across
two image distributions and predict correspondences. This allows the attention
mechanism to transfer fine-grained textures from the HR reference image to the
corresponding LR image, while being robust to differences in the shape and
scale of objects.

  * •

We use a gating attention strategy to simultaneously attend to the content of
self-attention and cross-attention blocks. This technique provides the
attention heads with freedom to modulate the combination of features in each
transformer block by adjusting a gating parameter.

  * •

We empirically show that our framework achieves comparable or better results
than previous methods with more complex training strategies. On the Sun80
dataset, our approach achieves a PSNR of $29.83~{}dB$, outperforming all
current RSR methods including those based on $C^{2}$-Matching [12].

Reproducibility. An open-source implementation of our technique, pre-trained
models, and output images are available at https://github.com/bia006/DARTS.

##  2 Related Work

Single Image Super Resolution. SISR is a common problem in computer vision and
has been studied for an extended period of time. SISR aims to enhance the
spatial details within the downsampled LR image and super-resolve the LR image
to the corresponding HR target image. The first deep learning-based SISR study
used an interpolated LR image with three-layer CNN to represent the mapping
function between the LR and target HR images [6]. They later improved the SISR
process by using a deconvolutional layer to upsample the feature map to the
desired size [7]. Later, deeper networks employing residual learning and dense
skip connections were proposed for image SR [15; 16; 38; 21; 24]. The
checkerboard artifacts caused by the deconvolutional layer were reduced by
replacing it with subpixel convolutional layers to upsample the feature map
size [34]. To improve the performance of SISR models, a channel attention
block was proposed to explore inter-channel correlations [47]. Different
research adapted non-local attention to model long-range dependencies to
reproduce quality SR images [4; 23; 48; 29; 30]. Perceptual loss and MSE were
introduced to overcome the overly smoothed textures in PSNR-oriented methods
[13; 35]. Generative adversarial networks (GAN) [9] were used in image SR
models [18] and witnessed further refinement in other studies [33; 40; 46].
With the advent of vision transformers, deeper neural networks attending to
attention mechanism have been recently introduced to produce higher quality
images while preserving textures [3; 20].

Reference-based Image Super Resolution. The most significant difference
between the SISR and RSR models is that the RSR model receives additional
information for image SR in the form of a HR reference image. The additional
HR reference image along with the LR input image can improve the quality of
the reconstructed images. The texture representations significantly missing in
the LR input image can be transferred from the corresponding HR reference
image which contains the same or similar content. In [49], a multi-scale
feature transformation was proposed to fuse swapped features obtained from
local patch matching into the LR input image. A coarse-to-fine matching scheme
was introduced to reduce the computational complexity while boosting the
spatial feature matching when there is a domain shift problem between the LR
input and HR reference images [27]. Elsewhere in the literature, a contrastive
learning network was used to match relevant correspondences between the LR
input and HR reference images [12].

$C^{2}$-matching is a SOTA model for RSR [12]. The $C^{2}$-matching process
consists of a two main training stages – one to learn correspondence matching
and a second to intake correspondences and synthesize a super-resolved output
image. The first stage consists of a contrastive learning module with teacher-
student distillation, while the second stage uses a dynamic aggregation module
and a restoration network. In the original work, all of the subnetworks are
based on CNNs. Other SOTA models have been introduced, borrowing significantly
from the structure of $C^{2}$-matching while making changes to improve its
feature matching capabilities. In [1], a deformable attention block built on
UNet [32] was introduced to attend to the content of feature encoders. A
refinement module was also proposed to select/align features for better
performance [45].

StyleSwin. StyleSwin [44] is a Swin transformer-based [25] GAN model for
generating synthetic images. To enlarge the receptive field of Swin
transformer blocks, StyleSwin introduced the double attention mechanism,
applying attention across both local and shifted windows. The authors find
that this significantly improves the quality of generated images.

##  3 Approach

###  3.1 Problem Setup

In this work, we adapt double attention from image generation to RSR. While
the original application involved computing self-attention in the local and
shifted windows for a single input in order to broaden the network’s receptive
field, we adapt it to learn matching correspondences from the HR reference
image to the LR input counterpart.

We denote the input LR, the corresponding HR reference, the reconstructed HR,
and the HR target images as $I_{LR}$, $I_{ref}$, $I_{SR}$ and $I_{HR}$,
respectively. Each image distribution contains $N$ samples.

###  3.2 Model Architecture

The architecture of DARTS is illustrated in Figure 1. The network receives the
LR image and HR reference image as inputs and upsamples the LR input image by
a factor of four through a cascade of Swin transformer blocks to enhance the
spatial resolution.

To account for interaction across adjacent windows, the Swin transformer uses
shifted window partitioning in every other block. We adapt this strategy
across two image distributions for every Swin block. Given the input feature
maps $I_{LR}^{l}\in\mathbb{R}^{H\times W\times C}$ and
$I_{ref}^{l}\in\mathbb{R}^{H\times W\times C}$ of layer $l$, the following
Swin blocks operate as:

$\text{\footnotesize{Local
windows}}=\small\begin{cases}\hat{I_{LR}}^{l}=\textrm{W-MHA}(\textrm{LN}(I_{LR}^{l}))+I_{LR}^{l},\quad\quad\hat{I_{ref}}^{l}=\textrm{W-MHA}(\textrm{LN}(I_{ref}^{l}))+I_{ref}^{l}\\\
I_{LR}^{l+1}=\textrm{MLP}(\textrm{LN}(\hat{{}^{I}_{LR}}^{l}))+\hat{I_{LR}}^{l},\quad\quad\quad
I_{ref}^{l+1}=\textrm{MLP}(\textrm{LN}(\hat{I_{ref}}^{l}))+\hat{I_{ref}}^{l}\end{cases}$
(1)

and,

$\text{\footnotesize{Shifted
windows}}=\small\begin{cases}\hat{I_{LR}}^{l+1}=\textrm{SW-
MHA}(\textrm{LN}(I_{LR}^{l+1}))+I_{LR}^{l+1},\quad\quad\hat{I_{ref}}^{l+1}=\textrm{SW-
MHA}(\textrm{LN}(I_{ref}^{l+1}))+I_{ref}^{l+1}\\\
I_{LR}^{l+2}=\textrm{MLP}(\textrm{LN}(\hat{I_{LR}}^{l+1}))+\hat{I_{LR}}^{l+1},\quad\quad\quad
I_{ref}^{l+2}=\textrm{MLP}(\textrm{LN}(\hat{I_{ref}}^{l+1}))+\hat{I_{ref}}^{l+1}\end{cases}$
(2)

where W-MHA and SW-MHA represent window-based multi-head attention across the
local and shifted window partitioning respectively, and LN denotes layer
normalization. Since the computational weight of the Swin block corresponds
linearly to image size, the network is scalable to higher resolution images.
For the RSR task, we extend the Swin transformer to learn joint
representations across two image distributions, processing $I_{LR}$ and
$I_{ref}$ inputs in separate streams and computing self-attention and cross-
attention matrices. The two streams then interact through a gating attention
strategy.

A 16-block ResNet [10] with SpectralNorm [31] normalization is used to extract
image features from a $40\times 40$ LR input image and this feature map is
used as the input to the first transformer block. For the corresponding
reference image, unlike the other SOTA methods which use dense features
extracted using a pre-trained VGG network [1; 12; 49], we use random crops
from the original HR reference image without spatial loss. The random crops
are $(40\times 40),(80\times 80)$ and $(160\times 160)$ spatially, matching
the input size of each transformer block. We adapt the image patching module
from ViT [8] to extract the feature representations out of the cropped
patches. Experiments incorporating VGG features extracted from the
$relu3-1,relu2-1$ and $relu1-1$ layers as an input in each transformer block
instead of random patches yielded similar results (data not shown).

Double Attention. We use local and shifted window partitioning across the two
inputs $(I_{LR},I_{ref})$ in each transformer block, which enlarges the
receptive field by $2.5k$ across each input image, where $k\times k$ denotes
the window size (i.e. $k=8$). This strategy allows the network to attend to
the context of four windows (two local and two shifted windows) at the same
time, containing $I_{LR}$ and $I_{ref}$ representations. We compute the query
and key from $I_{LR}$ for self-attention and the query from $I_{LR}$ and key
from $I_{ref}$ for cross-attention blocks. The network alternates between the
local and shifted windows across the two distributions, first attending to the
local window on $I_{LR}$ and computing attention scores for the self-attention
block, and then attending to the context of the local window on $I_{LR}$ and
the context of shifted window on $I_{ref}$ to perform cross-attention. Next,
the network alternates between local and shifted windows across the two
distributions to compute the scores from the self- and cross-attention blocks.
In every step, we compute two attention matrices containing self- and cross-
attention scores which are potentially of different magnitudes. To avoid this
difference in magnitudes resulting in one attention matrix being overpowered
by the other, we use a gating attention block to balance their contributions.
We denote $I_{w}$ and $I_{sw}$ to represent non-overlapping patches under the
local and shifted window partitioning respectively as

$\small head_{i}=\begin{cases}\text{Self-
Attn}(I_{LR_{w}}W_{i}^{Q},I_{LR_{w}}W_{i}^{K},I_{ref_{w}}W_{i}^{V}),\\\
\text{Cross-
Attn}(I_{LR_{sw}}W_{i}^{Q},I_{ref_{sw}}W_{i}^{K},I_{ref_{sw}}W_{i}^{V})\\\
\text{Self-
Attn}(I_{LR_{sw}}W_{i}^{Q},I_{LR_{sw}}W_{i}^{K},I_{ref_{sw}}W_{i}^{V}),\\\
\text{Cross-
Attn}(I_{LR_{w}}W_{i}^{Q},I_{ref_{w}}W_{i}^{K},I_{ref_{w}}W_{i}^{V})\\\
\end{cases}$ (3)

where $W_{i}^{Q},W_{i}^{K},W_{i}^{V}\in\mathcal{R}^{C\times C}$ are the query,
key and value projection matrices for $i^{th}$ head respectively.

This technique allows the network to capture larger context more efficiently,
i.e. with a window size of $k=8\times 8$ and an input feature map of $80\times
80$, $5$ transformer blocks are required to span the entire feature map,
instead of $10$ transformer blocks.

The main architecture uses Swin transformer blocks ($i$), receiving
intermediate visual representations $I_{LR}^{i}$ and $I_{ref}^{i}$, as
illustrated in Figure 1(b). The module computes query, key and value matrices
as block inputs and we perform self-attention and cross-attention attending to
the content of $I_{LR}^{i}$ and $I_{ref}^{i}$ in each block.

For self-attention, the query $q_{i}$ and key $k_{j}$ extracted from the
$I_{LR}^{i}$ distribution are passed as inputs to a multi-head attention
block, where the attention block produces weighted feature vectors for
$I_{LR}^{i}$ conditioned on itself – in effect attending to different parts of
the same input data in a visual stream in each transformer block. For cross-
attention, the query $q_{i}$ extracted from the $I_{LR}^{i}$ distribution and
the key $k_{j}$ extracted from the $I_{ref}^{i}$ distribution are passed as
inputs to a multi-head attention block, where the attention block produces
weighted feature vectors for $I_{LR}^{i}$ conditioned on $I_{ref}^{i}$ – in
effect performing $I_{ref}^{i}$-conditioned $I_{LR}^{i}$ attention attending
to different parts of $I_{ref}^{i}$ in another visual stream in each
transformer block. This strategy simulates the attention mechanism introduced
in vision-and-language models [26].

The self-attention and cross-attention matrices are combined through a gating
attention mechanism. The value $v_{i}$ vector is extracted from the
$I_{ref}^{i}$ distribution and the rest of the attention block proceeds with
residual learning with the initial $I_{LR}^{i}$ representations.

Gating Attention. Building on the insights of [5], we use a gating attention
strategy to combine the content of self-attention and cross-attention blocks
simultaneously, as illustrated in Figure 1(c). We initialize a gating block
where each attention head $h$ maintains a learnable $\lambda_{h}$ gating
parameter regulating the attention paid to the self-attention scores versus
the cross-attention scores. The gating attention layer sums the content of
self-attention and cross-attention after softmax using the gating parameter
$\lambda_{h}$ for each attention head. A sigmoid function is used to keep the
gating parameter in distribution space. The attention matrix $A^{h}$ is given
by

$\begin{split}\text{self-
attn}&\;=\;\left[\text{softmax}\left(\frac{Q_{LR}K_{LR}^{T}}{\sqrt{D_{h}}}\right)\right]\quad,\\\
\text{cross-
attn}&\;=\;\left[\text{softmax}\left(\frac{Q_{LR}K_{ref}^{T}}{\sqrt{D_{h}}}\right)\right]\quad,\\\
A^{h}\;=\;&(1-\sigma(\lambda_{h}))\;\text{self-
attn}+\sigma(\lambda_{h})\;\text{cross-attn}\end{split}$ (4)

where $\left[\;\cdot\;\right]$ is a normalization operation and $\sigma$
denotes the sigmoid function. We initialize the gating parameter $\lambda_{h}$
to 1 as suggested in [5].

The gating strategy adjusts the importance of each attention score, in which
each attention head projects the most important representation out of the two
attention blocks. The final output is

$\text{attn}=\text{concat}(head_{0},...,head_{h})W^{O}$ (5)

where $W^{O}\in\mathcal{R}^{C\times C}$ denotes the projection matrix from
concatenated heads to output.

Positional Encoding. We use both local and global positional encodings. For
the local positional encoding, we use relative positional encoding (RPE) which
encodes the spatial information via learnable parameters interacting with
queries and keys in each attention block [25]. For the global positional
encoding we use sinusoidal position encoding (SPE), encoding positional
information as a mix of sine and cosine functions [2; 39; 42], on each
upsampling block to provide translation invariance. In practice, RPE is
applied within each transformer block, and SPE is applied on each upsampling
block informing the global position.

Architectural Details. Our framework slices input images of size $M\times M$
into $M/k\times M/k$ non-overlapping patches of $k\times k$ pixels and embeds
them into vectors of dimension $D_{emb}=96N_{h}$, where $M\times M$, $k\times
k$ and $N_{h}$ denote the input feature map size, window size, and number of
attention heads, respectively. The patches are propagated through 3 blocks
while keeping dimensionality constant. Each block contains a double attention
mechanism across $I_{LR}$ and $I_{ref}$, performing self-attention and cross-
attention. Each attention operation is followed by a 2-layer MLP with GeLU
activation and a residual connection to the output. Gating attention is
applied at the end of each transformer block.

###  3.3 Objective Functions

The objective functions of the proposed method include:

Reconstruction Loss. We use $l_{1}$ reconstruction loss with
$\text{weight}=10$.:

$L_{rec}=||I_{HR}-I_{SR}||_{1}$ (6)

Perceptual Loss. We employ perceptual loss [13] to enhance the visual quality
of the reconstructed images, given by

$L_{per}=\frac{1}{V}\sum_{i=1}^{C}||\Phi_{i}(I_{HR})-\Phi_{i}(I_{SR})||_{F}$
(7)

where $C$ and $V$ denote the channel number and volume of the feature maps.
$\Phi$ represents the $relu5-1$ features out of the VGG19 network [35]. We use
$1e-4$ as the weight for $L_{per}$.

Adversarial Loss. We perform adversarial training with the discriminator
architecture used in StyleSwin [44]. We use hinge [22] with $R_{1}$ gradient
penalty as the adversarial loss function [14]. We employ balanced consistency
regularization (bCR) [50] in training with equal weights on the real and fake
images. The weight for the adversarial loss is $1e-4$. We also directly adopt
the Wavelet discriminator from StyleSwin [44] to combat blocking artifacts.

| Method | CUFED5 | SUN80 | Urban100 | Manga109 | WR-SR
---|---|---|---|---|---|---
| SRCNN [6] | 25.33 / .745 | 28.26 / .781 | 24.41 / .738 | 27.2 / .850 | 27.27 / .767
| EDSR [21] | 25.93 / .777 | 28.52 / .792 | 25.51 / .783 | 28.93 / .891 | 28.07 / .793
| RCAN [47] | 26.06 / .769 | 29.86 / .810 | 25.42 / .768 | 29.38 / .895 | 28.25 / .799
SISR | SwinIR [20] | 26.62 / .790 | 30.11 / .817 | 26.26 / .797 | 30.05 / .910 | 28.06 / .797
|  ESRGAN [40] | 21.90 / .633 | 24.18 / .651 | 20.91 / .620 | 23.53 / .797 | 26.07 / .726
| ENet [33] | 24.24 / .695 | 26.24 / .702 | 23.63 / .711 | 25.25 / .802 | 25.47 / .699
|  RankSRGAN [46] | 22.31 / .635 | 25.60 / .667 | 21.47 / .624 | 25.04 / .803 | 26.15 / .719
| CrossNet [51] | 25.48 / .764 | 28.52 / .793 | 25.11 / .764 | 23.36 / .741 | -
|  SRNTT [49] | 25.61 / .764 | 27.59 / .756 | 25.09 / .774 | 27.54 / .862 | 26.53 / .745
| SRNTT-rec [49] | 26.24 / .784 | 28.54 / .793 | 25.50 / .783 | 28.95 / .885 | 27.59 / .780
|  MASA [27] | 24.92 / .729 | 27.12 / .708 | 23.78 / .712 | 27.44 / .849 | 25.76 / .717
| MASA-rec [27] | 27.54 / .814 | 30.15 / .815 | 26.09 / .786 | 30.28 / .909 | 28.19 / .796
|  TTSR [43] | 25.53 / .765 | 28.59 / .774 | 24.62 / .747 | 28.70 / .886 | 26.83 / .762
| TTSR-rec [43] | 27.09 / .804 | 30.02 / .814 | 25.87 / .784 | 30.09 / .907 | 27.97 / .792
RSR |  $C^{2}$-Matching [12] | 27.16 / .805 | 29.75 / .799 | 25.52 / .764 | 29.73 / .893 | 27.80 / .780
|  $C^{2}$-Matching-rec [12] | 28.24 / .841 | 30.18 / .817 | 26.03 / .785 | 30.47 / .911 | 28.32 / .801
|  DATSR [1] | 27.95 / .835 | 29.77 / .800 | 25.92 / .775 | 29.75 / .893 | 27.87 / .787
| DATSR-rec [1] | 28.72 / .856 | 30.20 / .818 | 26.52 / .798 | 30.49 / .912 | 28.34 / .805
|  RRSR [45] | 28.09 / .835 | 29.57 / .793 | 25.68 / .767 | 29.82 / .893 | 27.89 / .784
| RRSR-rec [45] | 28.83 / .856 | 30.13 / .816 | 26.21 / .790 | 30.91 / .913 | 28.41 / .804
| DARTS (ours) | 26.6 / .781 | 29.83 / .809 | 25.6 / .772 | 29.8 / .898 | 27.78 / .787
| DARTS-rec (ours) | 26.4 / .781 | 29.9 / .81 | 25.51 / .770 | 29.8 / .901 | 27.8 / .786
Table 1: Quantitative comparisons using PSNR / SSIM metrics. The SISR and RSR
methods are grouped accordingly. The ‘-rec’ denotes only reconstruction
($l_{1}$) loss. The methods shaded in grey use a GAN loss. The highest values
for networks trained with multiple loss functions are shown in red and the
highest values for networks trained with only reconstruction ($l_{1}$) loss
are shown in blue.

##  4 Experiments

###  4.1 Experimental Settings

Datasets. Performance was evaluated using CUFED5 [49], SUN80 [37], Urban100
[11], and Manga109 [28]. For CUFED5, the training set contains $11,871$ pairs
of input and reference images. The CUFED5 test set contains $126$ images, in
which every input image is paired with five reference images from different
levels of similarity. The input image and its respective five references have
been selected from an album of the same event. The Webly-Referenced SR dataset
[12] consists of $80$ image pairs where each pair contains an input image and
reference image. The SUN80 dataset contains $80$ images in which each input
image has $20$ reference images. For the Urban100 and Manga109 datasets, which
are SISR datasts, we adopt the same evaluation strategy as [43; 49]. These
datasets consist of $100$ and $109$ images respectively and a random image
from the same dataset is used as a reference image. The LR input images are
created by downsampling the HR target images $4\times$ using bicubic
interpolation.

Evaluation Metrics. Peak-Signal-to-Noise Ratio (PSNR) and Structural
Similarity Index (SSIM) [41] are used as evaluation metrics. The Y channel of
YCrCb colorspace is used for measuring the PSNR and SSIM values on each
dataset.

###  4.2 Implementation Details

For training, we use the Adam solver [17] with $\beta_{1}=0.0,\beta_{2}=0.99$.
We use a single-cycle learning rate schedule [36], with a maximum learning
rate of $1e-4$. During training, we use a batch size of 4 with one sample per
GPU. At inference time we use $(192\times 192)$, $(384\times 384)$, and
$(768\times 768)$ patches from the reference image.

###  4.3 Results and Analysis

We compare the proposed technique with recent CNN- and transformer-based
networks from the literature. For SISR methods, we include SRCNN [6], EDSR
[21], RCAN [47], SwinIR [20], ESRGAN [40], ENet [33] and RankSRGAN [46]. For
RSR methods, we include CrossNet [51], SRNTT [49], MASA [27], TTSR [43],
$C^{2}$-Matching [12], DATSR [1] and RRSR [45].

Quantitative results. Table 1 shows a quantitative comparison of DARTS against
existing SOTA methods. The proposed method achieves results which are
competitive with SOTA on many datasets, and achieves SOTA in the SUN80
benchmark. Methods using the general $C^{2}$-matching architecture [1; 45]
still achieve the best performance in the majority of datasets – however,
these models require a more complex architecture and training procedure to
achieve these results.

Qualitative results. Fig 2 shows qualitative comparisons with SOTA methods. We
compare DARTS with ESRGAN [40], RankSRGAN [46], SRNTT [49] and DATSR [1]. The
outputs of DARTS show smooth images with fewer artifacts. As shown in the top
left example, our model has reconstructed the face without mixing neighboring
features. As shown in the second row of examples, DARTS can reconstruct the
word "AIDA" more accurately than competing models. It can also reconstruct the
fine features of the buildings without mismatching features and their relative
positions. Overall, DARTS is able to preserve the integrity of global geometry
while also performing fine-grained texture reconstruction.

input image | LR | ESRGAN | RankSRGAN | input image | LR | ESRGAN | RankSRGAN
---|---|---|---|---|---|---|---
reference image | SRNTT | DATSR | DARTS (ours) | reference image | SRNTT | DATSR | DARTS (ours)
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|  |  |  |  |  |  |
|  |  |  |  |  |  |
Figure 2: Qualitative comparisons of our results with ESRGAN [40], RankSRGAN [46], SRNTT [49], and DATSR [1]. Methods | #params | FLOPs
---|---|---
SRNTT-rec [49] | 5.75M | 4.1G
TTSR-rec [43] | 6.2M | 32.93G
$C2$-Matching-rec [12] | 8.9M | -
DATSR-rec [1] | 18.0M | -
DARTS-rec | 22.29M | 24.8G
Table 2: Comparison of the number of network parameters and FLOPs. ‘-rec’ denotes only the reconstruction ($l_{1}$-norm) loss. Compute information not available for $C^{2}$-matching and DATSR due to an inability to compile required libraries on the available hardware. | Self-attn | Cross-attn | Gating-attn | PSNR/SSIM
---|---|---|---|---
a) | ✓ | ✓ | ✓ |  26.4 / .781
b) | ✓ | ✓ | ✗ | 26.1 / .771
c) | ✓ | ✗ | ✗ | 26.08 / .77
d) | ✗ | ✓ | ✗ | 25.96 / .768
Table 3: Gating and two-stream attention work well together. We run ablation
studies on DARTS-rec trained for 50 epochs on the CUFED5 dataset. a) shows
results using the full DARTS architecture, b) shows the performance when the
gating parameter is frozen, c) uses only self-attention, and d) uses only
cross-attention.

Ablation studies. We perform three ablative studies as shown in Table 3. These
ablations show the individual effects of gating attention, self-attention and
cross-attention. Gating attention consists of both self-attention and cross-
attention blocks that attend to the queries and keys from $I_{LR}$, as well as
the queries from $I_{LR}$ and keys from $I_{ref}$ in each transformer block.
The value vector is extracted from the $I_{ref}$ distribution. When gating
attention is removed for ablation b), the $\lambda$ gating parameter is frozen
at 0, paying equal attention to both self- and cross-attention scores. Self-
attention consists of a single self-attention block that only attends to the
query, key and value feature vectors from the $I_{LR}$ distribution in each
transformer block. This attention structure is very similar to the double
attention introduced in [44]. Cross-attention consists of a single cross-
attention block that only attends to the queries from $I_{LR}$, and keys and
values from $I_{ref}$ in each transformer block. This attention structure is
similar to vision and linguistic tasks proposed in [26].

|  |  |
---|---|---|---
a) Robustness to scale transformation | b) Robustness to rotation transformation
Figure 3: DARTS shows more robustness than SRNTT [49] to scale (a) and
rotation (b) across three levels of augmentation (small, medium, large), as
measured by AEE (lower is better) and PSNR (higher is better).

Scale and Rotation Invariance. To demonstrate the robustness of our model to
scale and rotation, we perform an analysis similar to the analyses conducted
in [12]. We rebuild the CUFED5 dataset with small, medium and large degrees of
scale and rotation transformations. We use the scaled and rotated input images
as reference images only at inference. We use Average End-to-point Error (AEE)
and PSNR to assess the accuracy of matching correspondences and image
restoration performance, respectively. Figure 3 demonstrates the invariance of
our model to the scale and rotation of objects during inference versus SRNTT
[49]. As the degree of transformation increases, the AEE increases as well
indicating more mismatched reconstructed features. Based on AEE, our model
exhibits performance which is superior to SRNTT, even though we did not use
large transformations during training. Based on PSNR, the restoration
performance of DARTS is more robust than that of SRNTT as well.

Limitations. The main limitation of DARTS is the memory footprint of the model
compared to previous architectures. Table 2 shows that DARTS has more
parameters than SRNTT, TTSR, $C^{2}$-matching, and DATSR. However, the large
memory requirement is largely due to the fact that attention is computed
across four windows at once. This limits our experiments on consumer hardware
to small batch sizes, as each GPU is only able to perform forward and backward
passes on one sample at a time. It also limits the size of HR reference
images, as mentioned in Section 4.2. This could potentially impact the
performance of the model as it is sometimes necessary to crop the reference
image into smaller patches.

##  5 Conclusion

In this work we proposed DARTS, a transformer model for reference-based image
super-resolution. This proposed model consists of a two-stream architecture
which simultaneously attends to the low-resolution input image and the high-
resolution reference image, combining self-attention and cross-attention using
a gating-attention strategy. The architecture is conceptually simple and
consists of a single module which can be trained end-to-end in a single
training stage. Quantitative and qualitative evaluations show that the
proposed method is competitive with prior state-of-the-art methods which use a
complex, multi-stage process with multiple submodules. It is also shown to be
less sensitive to scale and rotation transformations of the reference images.
DARTS represents a new direction for reference-based image super-resolution
which is consolidated in the attention mechanism, as an alternative to more
complicated SOTA methods based on $C^{2}$-matching.

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# Gromov–Hausdorff convergence of state spaces for spectral truncations

Walter D. van Suijlekom  Institute for Mathematics, Astrophysics and Particle
Physics, Radboud University Nijmegen, Heyendaalseweg 135, 6525 AJ Nijmegen,
The Netherlands<EMAIL_ADDRESS>
(Date: January 12, 2021)

###### Abstract.

We study the convergence aspects of the metric on spectral truncations of
geometry. We find general conditions on sequences of operator system spectral
triples that allows one to prove a result on Gromov–Hausdorff convergence of
the corresponding state spaces when equipped with Connes’ distance formula. We
exemplify this result for spectral truncations of the circle, Fourier series
on the circle with a finite number of Fourier modes, and matrix algebras that
converge to the sphere.

##  1\. Introduction

We continue our study of spectral truncations of (noncommutative) geometry
that we started in [10] and here focus on the metric convergence aspect of so-
called operator system spectral triples. This is part of a program that tries
to extend the spectral approach to geometry to cases where (possibly) only
part of the spectral data is available, very much in line with [11]. And even
though the mathematical motivation should be sufficient, there is a clear
physical motivation for this. Indeed, from experiments we will only have
access to part of the spectrum since we are limited by the power and
resolution of our detectors: we typically study physical phenomena up to a
certain energy scale and with finite resolution.

The usual spectral approach to geometry [9] in terms of a $*$-algebra
$\mathcal{A}$ of operators on $\mathcal{H}$ and a self-adjoint operator $D$ on
$\mathcal{H}$ has been adapted in [11, 10] to deal with such spectral
truncations. The $*$-algebra is replaced by an operator system $E$ (dating
back to [7]), which is by definition a $*$-closed subspace of
$\mathcal{B}(\mathcal{H})$ containing the identity.

More precisely, we have the following definition.

######  Definition 1.

An operator system spectral triple is a triple $(\mathcal{E},\mathcal{H},D)$
where $\mathcal{E}$ is a dense subspace of an operator system $E$ in
$\mathcal{B}(\mathcal{H})$, $\mathcal{H}$ is a Hilbert space and $D$ is a
self-adjoint operator in $\mathcal{H}$ with compact resolvent and such that
$[D,T]$ is a bounded operator for all $T\in\mathcal{E}$.

An operator system comes with an ordering, namely, one can speak of positive
operators in $E\subseteq\mathcal{B}(\mathcal{H})$. As a consequence states on
$E$ can be defined as positive linear functionals of norm 1. The above triple
then induces a (generalized) distance function on the state space
$\mathcal{S}(E)$ by setting

(1)
$d(\varphi,\psi)=\sup_{x\in\mathcal{E}}\left\\{|\varphi(x)-\psi(x)|:\|x\|_{1}\leq
1\right\\}$

where $\|\cdot\|_{1}$ denote the Lipschitz semi-norm:

$\|x\|_{1}=\|[D,x]\|;\qquad(x\in\mathcal{E}).$

If $\mathcal{E}=\mathcal{A}$ is a $*$-algebra then this reduces to the usual
distance function [8, 9] on the state space of the $C^{*}$-algebra
$A=\overline{\mathcal{A}}$ . It also agrees with the definition of quantum
metric spaces based on order-unit spaces given in [27, 28, 17, 18, 21, 22,
23]. Note, however, that in the present work we restrict our attention to the
metric structure on the state spaces, that is to say, as an ordinary metric
space. In contrast, in loc.cit. the authors develop the notion of quantum
metric space and quantum Gromov–Hausdorff distance which are formulated in the
dual category (of $C^{*}$-algebras, order-unit spaces, et cetera) and with a
more general version of the above distance function.

So, we will study the properties of this metric distance function and the
notions of Gromov–Hausdorff convergence it gives rise to. We consider
sequences of spectral triples on operator systems and formulate general
conditions under which we prove the state spaces equipped with the above
distance functions to converge to a limiting state space. The latter is also
described by an operator system spectral triple. One of the novelties of our
work is that we use the ideas of correspondences between compact metric spaces
and their relation to Gromov–Hausdorff convergence as described for instance
in [6, Section 7.3.3].

We exemplify our main result on Gromov–Hausdorff convergence by considering:

  * •

spectral truncations on the circle;

  * •

Fourier series with only a finite number of non-zero Fourier coefficients;

  * •

matrix algebras converging to the sphere.

Previous results in the literature on the distance function for spectral
truncations have been reported in [11, 14, 13]. However, in these works the
distance function on states of the truncated system was only computed after
pulling back these states to the original metric geometry. Extensions of the
results contained in the present paper to tori are contained in the master’s
thesis [4].

The convergence of matrix algebras to the sphere was studied by Rieffel in
[29] while computer simulations were performed in [3]. Using the general
approach below we re-establish part of this convergence result, namely, the
Gromov–Hausdorff convergence of the corresponding (classical) metric spaces.

We note that other convergence results on the distance function on quantum
spaces are obtained for quantum tori in [20], for coherent states on the Moyal
plane in [12]. More generally, in [13] certain sets of states have been
identified for which the Connes’ distance formula has good convergence
properties with respect to a given metric on a Riemannian manifold.

#### Acknowledgements

I would like to thank IHÉS for their hospitality and support during a visit in
February 2020. I thank Alain Connes, Jens Kaad and Marc Rieffel for fruitful
discussions. I am grateful to an anonymous referee for useful remarks and
suggestions.

##  2\. Gromov–Hausdorff convergence for operator systems

Given a sequence of operator system spectral triples
$(\mathcal{E}_{n},\mathcal{H}_{n},D_{n})$ we want to understand when and how
this approximates an operator system spectral triple
$(\mathcal{E},\mathcal{H},D)$. We will adopt the point of view of [28] and
consider the convergence (in Gromov–Hausdorff distance) of the corresponding
state spaces $\mathcal{S}(E_{n})\to\mathcal{S}(E)$ equipped with the distance
formula (1). Since this notion of convergence is most suited to deal with
compact metric spaces, we will assume below (cf. Theorem 5) that the topology
defined by the metric $d$ coincides with the weak-$*$ topology (with respect
to which we know the state spaces to be compact). For the examples that follow
this assumption is indeed satisfied, see also Remark 3 below.

######  Definition 2.

Let $\\{(\mathcal{E}_{n},\mathcal{H}_{n},D_{n})\\}_{n}$ be a sequence of
operator system spectral triples and let $(\mathcal{E},\mathcal{H},D)$ be an
operator system spectral triple. An approximate order isomorphism for this set
of data is given by linear maps $R_{n}:E\to E_{n}$ and $S_{n}:E_{n}\to E$ for
any $n$ such that the following three condition hold:

  1. (1)

the maps $R_{n},S_{n}$ are positive, unital maps

  2. (2)

there exist sequences $\gamma_{n},\gamma_{n}^{\prime}$ both converging to zero
such that

$\displaystyle\|S_{n}\circ R_{n}(a)-a\|$
$\displaystyle\leq\gamma_{n}\|a\|_{1},$ $\displaystyle\|R_{n}\circ
S_{n}(h)-h\|$ $\displaystyle\leq\gamma_{n}^{\prime}\|h\|_{1}.$

In other words, we use the Lipschitz semi-norms to quantify how close the
positive maps $R_{n}$ and $S_{n}$ are to being each other’s inverse (i.e. form
an order isomorphism) as $n\to\infty$.

We will call a map between operator systems $C^{1}$-contractive if it is
contractive with respect to both the operator norms and the Lipschitz semi-
norms (thus assuming that we are given two operator system spectral triples
for them). Finally, we say that the pair of maps $(R_{n},S_{n})$ is a
$C^{1}$-approximate order isomorphism if $(R_{n},S_{n})$ is an approximate
order isomorphism in the above sense and for which all maps $R_{n}$ and
$S_{n}$ are $C^{1}$-contractive.

Note that the positivity and unitality condition on $R_{n},S_{n}$ in
particular implies that we may pull back states as follows:

$\displaystyle
R_{n}^{*}:\mathcal{S}(E_{n})\to\mathcal{S}(E);\qquad\varphi_{n}\mapsto\varphi_{n}\circ
R_{n},$ $\displaystyle
S_{n}^{*}:\mathcal{S}(E)\to\mathcal{S}(E_{n});\qquad\varphi\mapsto\varphi\circ
S_{n}.$

######  Remark 3.

Even though it would be more natural to consider completely positive maps
$R_{n},S_{n}$ between the operators systems $E_{n}$ and $E$, this turns out
not to be necessary for the proof of our main result as in fact we restrict
our attention to the states space and the metric thereon. However, in all
examples discussed below we find that $E$ is a commutative $C^{*}$-algebra so
that these maps are in fact completely positive (cf. [25, Theorems 3.9 and
3.11]).

Let us denote the distance functions (1) for
$(\mathcal{E}_{n},\mathcal{H}_{n},D_{n})$ and $(\mathcal{E},\mathcal{H},D)$ by
$d_{E_{n}}$ and $d_{E}$, respectively.

######  Proposition 4.

If $(R_{n},S_{n})$ is a $C^{1}$-approximate order isomorphism for
$(\mathcal{E}_{n},\mathcal{H}_{n},D_{n})$ and $(\mathcal{E},\mathcal{H},D)$,
then

  1. (1)

For all $\varphi_{n},\psi_{n}\in\mathcal{S}(E_{n})$ we have

$d_{E}(\varphi_{n}\circ R_{n},\psi_{n}\circ R_{n})\leq
d_{E_{n}}(\varphi_{n},\psi_{n})\leq d_{E}(\varphi_{n}\circ R_{n},\psi_{n}\circ
R_{n})+2\gamma_{n}^{\prime}.$

  2. (2)

For all $\varphi,\psi\in\mathcal{S}(E)$ we have

$d_{E_{n}}(\varphi\circ S_{n},\psi\circ S_{n})\leq d_{E}(\varphi,\psi)\leq
d_{E_{n}}(\varphi\circ S_{n},\psi\circ S_{n})+2\gamma_{n}.$

###### Proof.

Since $R_{n}$ is Lipschitz contractive it follows that if $\|a\|_{1}\leq 1$
then also $\|R_{n}(a)\|_{1}\leq 1$. Hence

$\sup_{a\in\mathcal{E}}\left\\{|\varphi\circ R_{n}(a)-\psi\circ
R_{n}(a)|:\|a\|_{1}\leq
1\right\\}\leq\sup_{h\in\mathcal{E}_{n}}\left\\{|\varphi(h)-\psi(h)|:\|h\|_{1}\leq
1\right\\}.$

This establishes the first inequality (also proven in [11, Proposition 3.6]).

For the second, note that for all $h\in\mathcal{E}_{n}$ with $\|h\|_{1}\leq 1$
we have

$\displaystyle|\varphi_{n}(h)-\psi_{n}(h)|$
$\displaystyle\leq|\varphi_{n}(R_{n}(S_{n}(h)))-\psi_{n}(R_{n}(S_{n}(h)))|$
$\displaystyle\quad+|\varphi_{n}(h)-\varphi_{n}(R_{n}(S_{n}(h)))|+|\psi_{n}(h)-\psi_{n}(R_{n}(S_{n}(h)))|$
$\displaystyle\leq d_{E}(\varphi_{n}\circ R_{n},\psi_{n}\circ
R_{n})+2\gamma_{n}^{\prime}.$

since $\|\varphi_{n}\|=\|\psi_{n}\|=1$ and $\|S_{n}(h)\|_{1}\leq\|h\|_{1}\leq
1$. The second claim follows similarly. ∎

The final justification for the above definition of $C^{1}$-approximate order
isomorphism is our following, main result.

######  Theorem 5.

Suppose $(\mathcal{E}_{n},\mathcal{H}_{n},D_{n})$ ($n=1,2,\cdots$) and
$(\mathcal{E},\mathcal{H},D)$ are operator system spectral triples such that
the topologies on $\mathcal{S}(E_{n})$ and $\mathcal{S}(E)$ defined by the
metrics $d_{E_{n}}$ and $d_{E}$, respectively, agree with the weak-$*$
topologies on them.

If $(R_{n},S_{n})$ is a $C^{1}$-approximate order isomorphism for
$(\mathcal{E}_{n},\mathcal{H}_{n},D_{n})$ and $(\mathcal{E},\mathcal{H},D)$,
then the state spaces $(\mathcal{S}(E_{n}),d_{E_{n}})$ converge to
$(\mathcal{S}(E),d_{E})$ in Gromov–Hausdorff distance.

###### Proof.

This follows by applying techniques for correspondences between metric spaces
and their relation to Gromov–Hausdorff convergence via the notion of
distortion (cf. [6, Theorem 7.3.25]). In the case at hand one has for the
Gromov–Hausdorff distance that

$d_{\textup{GH}}\left((\mathcal{S}(E_{n}),d_{E_{n}}),(\mathcal{S}(E),d_{E})\right)\leq\frac{1}{2}\textup{dis}(\mathfrak{R}_{n})$

where the correspondence $\mathfrak{R}_{n}\subset\mathcal{S}(E_{n})\times
S(E)$ is defined by

$\mathfrak{R}_{n}=\left\\{(\varphi_{n},R_{n}^{*}(\varphi_{n})):\varphi_{n}\in\mathcal{S}(E_{n})\right\\}\cap\left\\{(S_{n}(\varphi),\varphi):\varphi\in\mathcal{S}(E)\right\\}$

with distortion (cf. [6, Defn 7.3.21])

$\textup{dis}(\mathfrak{R}_{n})=\sup\left\\{\left|d_{E_{n}}(\varphi_{n},\varphi_{n}^{\prime})-d_{E}(\varphi,\varphi^{\prime})\right|:(\varphi_{n},\varphi),(\varphi_{n}^{\prime},\varphi^{\prime})\in\mathfrak{R}_{n}\right\\}$

But from Proposition 4 it follows that this can be bounded by
$2\gamma_{n}+2\gamma_{n}^{\prime}$, which converges to 0 as $n\to\infty$. ∎

######  Remark 6.

Given an operator system spectral triple $(E,\mathcal{H},D)$, it is an
interesting question to see when the metric topology on $\mathcal{S}(E)$
defined by $d$ coincides with the weak-$*$ topology. For the commutative case,
this was already established in [8] while more generally it is also shown in
that paper that if the set $\\{h\in E:\|[D,h]\|\leq 1\\}/\mathbb{C}1$ is
bounded in $E$, then $d$ is a metric. Rieffel then established [26, Theorem
1.8] the most general result (for the compact case) stating that when the
above set is totally bounded, the $d$-topology agrees with the weak-$*$
topology.

Below we will consider only finite-dimensional operator system spectral
triples for which also $\ker[D,\cdot]=\mathbb{C}$, so that the above set is
totally bounded. Indeed, in this case $\|[D,\cdot]\|$ induces a norm on the
quotient $E/\mathbb{C}1$ for which the unit ball is compact. Hence, under
these assumptions the $d$-topology coincides with the weak$-*$ topology; a
fact that we will tacitly use throughout the rest of the paper.

######  Remark 7.

Following up on the previous remark, there is a close relation between our
notion of $C^{1}$-approximate order isomorphism and the
$(\varepsilon,C)$-approximations of operator systems $E$ equipped with a
Lipschitz semi-norm $L$ that were considered recently in [16]. Such
$(\varepsilon,C)$-approximations are defined by two maps $\imath,\varphi:E\to
F$ into some other operator system $F$ such that (1)
$C^{-1}\|a\|\leq\|\imath(a)\|\leq C\|a\|$, $\|\varphi(a)\|\leq C\|a\|$ for all
$a\in E$; (2) $\text{ran}(\varphi)\subseteq F$ is finite dimensional; and (3)
$\|\imath(a)-\varphi(a)\|\leq\varepsilon L(a)$.

Thus, if $(R_{n},S_{n})$ is a $C^{1}$-approximate order isomorphism in the
sense we just defined, then, with the additional assumption that
$\text{ran}(R_{n})$ or $\text{ran}(S_{n})$ is finite-dimensional, the pair
$(\imath=\text{id}_{E},S_{n}\circ R_{n})$ gives a
$(\gamma_{n},1)$-approximation of the pair $(E,\|\cdot\|_{1})$. Note that in
all examples discussed below this assumption of finite-dimensionality will be
met.

##  3\. Examples of Gromov–Hausdorff convergence

###  3.1. Spectral truncations of the circle converge

We will analyze a spectral truncation of the distance function on the circle,
the latter being described by the spectral triple

(2)
$\left(\mathcal{A}=C^{\infty}(S^{1}),\mathcal{H}=L^{2}(S^{1}),D=-i\frac{d}{dx}\right).$

We will consider a spectral truncation defined by the orthogonal projection
$P=P_{n}$ of rank $n$ onto span${}_{\mathbb{C}}\\{e_{1},e_{2},\ldots,e_{n}\\}$
for some fixed $n\geq 1$, where $e_{k}(x)=e^{ikx}$ ($k\in\mathbb{Z}$) is the
orthonormal (Fourier) eigenbasis of $D$. In the following we will suppress the
representation of $C(S^{1})$ on $L^{2}(S^{1})$ by pointwise multiplication and
simply write $f$ for the corresponding bounded operator. An arbitrary element
$T=PfP$ in $PC(S^{1})P$ can be written as an $n\times n$ Toeplitz matrix
$T_{kl}=\langle e_{k},fe_{l}\rangle=a_{k-l}$ in terms of the Fourier
coefficients $a_{j}$ of $f$. In matrix form we thus have

(3) $T=\begin{pmatrix}a_{0}&a_{-1}&\cdots&a_{-n+2}&a_{-n+1}\\\
a_{1}&a_{0}&a_{-1}&&a_{-n+2}\\\ \vdots&a_{1}&a_{0}&\ddots&\vdots\\\
a_{n-2}&&\ddots&\ddots&a_{-1}\\\ a_{n-1}&a_{n-2}&\cdots&a_{1}&a_{0}\\\
\end{pmatrix}.$

The corresponding operator system $PC(S^{1})P=PC^{\infty}(S^{1})P$ is called
the Toeplitz operator system and is denoted by $C(S^{1})^{(n)}$; it has been
analyzed at length in [10].

An operator system spectral triple for the Toeplitz operator system is given
by $(C(S^{1})^{(n)},PL^{2}(S^{1}),PDP)$.

####  3.1.1. Fejér kernel

Clearly, the compression $f\mapsto PfP$ by $P=P_{n}$ defines a positive map
$R_{n}:C(S^{1})\to C(S^{1})^{(n)}$. In order to find an approximate inverse to
this map we take inspiration from [29, Section 2]. In fact, we define
$S_{n}:C(S^{1})^{(n)}\to C(S^{1})$ to be its (formal) adjoint when we equip
$C(S^{1})$ with the $L^{2}$-norm and $C(S^{1})^{(n)}$ with the (normalized)
Hilbert–Schmidt norm. Let $\alpha_{x}$ denote the natural action of $S^{1}$ on
$C(S^{1})^{(n)}$, and define a norm 1 vector $|\psi\rangle$ in $PL^{2}(S^{1})$
by

$|\psi\rangle=\frac{1}{\sqrt{n}}\left(e_{1}+\cdots+e_{n}\right).$

Figure 1. The Fejér kernel
$F_{N}=\frac{1}{N}\frac{\sin^{2}(Nx/2)}{\sin^{2}(x/2)}$ for $N=10$.

######  Proposition 8.

The map $S_{n}:C(S^{1})^{(n)}\to C(S^{1})$ defined for any $T\in
C(S^{1})^{(n)}$ by
$S_{n}(T)(x)=\operatorname{Tr}\left(|\psi\rangle\langle\psi|\alpha_{x}(T)\right)$
satisfies

$\langle
f,S_{n}(T)\rangle_{L^{2}(S^{1})}=\frac{1}{n}\operatorname{Tr}\left((R_{n}(f))^{*}T\right).$

Moreover, we may write

$S_{n}(R_{n}(f))(x)=\sum_{k=-n+1}^{n-1}\left(1-\frac{|k|}{n}\right)a_{k}e^{ikx}=(F_{n}*f)(x)$

in terms of the Fejér kernel $F_{n}$ and the Fourier coefficients $a_{k}$ of
$f$.

###### Proof.

Let us first check the formula for $S_{n}(T)$ by computing that

$\operatorname{Tr}\left(|\psi\rangle\langle\psi|\alpha_{x}(T)\right)=\frac{1}{n}\sum_{k,l}T_{kl}e^{i(k-l)x}=\frac{1}{n}\sum_{k=-n+1}^{n-1}(n-|k|)a_{k}e^{ikx}$

Thus, $S_{n}(T)=F_{n}*f$ when $T=PfP$ —again understanding $f$ as an operator
acting on $L^{2}(S^{1})$— and we may use elementary Fourier theory (cf. [30,
Proposition 3.1(vi)])to derive

$\displaystyle\langle g,S_{n}(T)\rangle$ $\displaystyle=\langle
g,F_{n}*f\rangle=\sum_{k=-n+1}^{n-1}\overline{b_{k}}a_{k}\left(1-\frac{|k|}{n}\right),$
where $b_{k}$ are the Fourier coefficients of $g$. On the other hand, we have
$\displaystyle\frac{1}{n}\operatorname{Tr}\left((R_{n}(g))^{*}T\right)$
$\displaystyle=\frac{1}{n}\sum_{k,l=-n+1}^{n-1}\overline{b_{k-l}}a_{k-l}=\frac{1}{n}\sum_{k=-n+1}^{n-1}\overline{b_{k}}a_{k}\left(n-|k|\right).\qed$

####  3.1.2. The circle as a limit of its spectral truncations

Let us now show in a series of Lemma’s that the conditions of Definition 2 are
satisfied.

######  Lemma 9.

For any $f\in C^{\infty}(S^{1})$ we have $\|R_{n}(f)\|\leq\|f\|$ and
$\|[D,R_{n}(f)]\|\leq\|[D,f]\|$.

###### Proof.

Since $R_{n}(f)=PfP$ and $P$ commutes with $D$ this follows directly since $P$
is a projection. ∎

######  Lemma 10.

There exists a sequence $\\{\gamma_{n}\\}$ converging to 0 such that

$\|f-S_{n}(R_{n}(f))\|\leq\gamma_{n}\|[D,f]\|$

for all $f\in C^{\infty}(S^{1})$.

###### Proof.

Since $\|[D,f]\|$ is equal to the Lipschitz constant for $f$ [8, Proposition
1] we have that $|f(x)-f(z)|\leq|x-z|\|[D,f]\|$. We use this in the following
estimate

$\displaystyle|f(x)-S_{n}(R_{n}(f))(x)|$
$\displaystyle\leq\frac{1}{2\pi}\int_{-\pi}^{\pi}F_{n}(y)|f(x)-f(y-x)|dy$
$\displaystyle\leq\frac{1}{2\pi}\int_{-\pi}^{\pi}F_{n}(y)|y|dy\cdot\|[D,f]\|=:\gamma_{n}\|[D,f]\|.$

Becasue the Fejér kernels form an approximate delta-function at $0$ (cf. [30,
Theorem 4.1 and Lemma 5.1]) imply that $\gamma_{n}\to 0$. ∎

######  Lemma 11.

For any $T\in C(S^{1})^{(n)}$ we have $\|S_{n}(T)\|\leq\|T\|$ and
$\|[D,S_{n}(T)]\|\leq\|[D,T]\|$.

###### Proof.

First note that $[D,S_{n}(T)]$ is a function on $S^{1}$ (it is $i$ times the
derivative of $S_{n}(T)$). Moreover, we have

$\displaystyle|[D,S_{n}(T)](x)|$
$\displaystyle=|\operatorname{Tr}\left(|\psi\rangle\langle\psi|\alpha_{x}([D,T])\right)|\leq\||\psi\rangle\langle\psi|\|_{1}\|\alpha_{x}([D,T])\|\leq\|[D,T]\|.$

Since this holds for any $x$, we may take the supremum to arrive at the
desired inequality. The other inequality is even easier. ∎

######  Lemma 12.

There exists a sequence $\\{\gamma_{n}^{\prime}\\}$ converging to 0 such that

$\|T-R_{n}(S_{n}(T))\|\leq\gamma^{\prime}_{n}\|[D,T]\|$

for all $T\in C(S^{1})^{(n)}$.

###### Proof.

Write $T=PgP$ for $g=\sum_{k}b_{k}e^{ikx}$. Then the matrix coefficients of
the Toeplitz matrix $T-R_{n}(S_{n}(T))$ are given by

$\displaystyle(T_{kl}-R_{n}(S_{n}(T))_{kl})$
$\displaystyle=\begin{pmatrix}b_{k-l}\end{pmatrix}-\begin{pmatrix}1-\frac{|k-l|}{n}b_{k-l}\end{pmatrix}=\begin{pmatrix}\frac{|k-l|}{n}b_{k-l}\end{pmatrix}$
$\displaystyle=(T_{n}-T_{n}^{*})\odot\begin{pmatrix}\frac{k-l}{n}b_{k-l}\end{pmatrix}$
$\displaystyle=\frac{1}{n}(T_{n}-T_{n}^{*})\odot\begin{pmatrix}[D,T]\end{pmatrix}$

in terms of the Schur product $\odot$ with $T_{n}$ and $T_{n}^{*}$ where

$T_{n}=\begin{pmatrix}1&0&\cdots&0\\\ 1&1&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots\\\ 1&1&\cdots&1\end{pmatrix}.$

Now the norm of the map $A\mapsto T_{n}\odot A$ for $A\in M_{n}(\mathbb{C})$
coincides with $\|T_{n}\|_{cb}$ (cf. [25, Chapter 8]). In [1, Theorem 1] the
following estimate for this norm was derived:

$\|T_{n}\|_{cb}\leq\left(1+\frac{1}{\pi}(1+\log(n))\right).$

Hence we have

$\displaystyle\|T-R_{n}(S_{n}(T))\|$
$\displaystyle\leq\frac{2}{n}\|T_{n}\|_{cb}\|[D,T]\|\leq\gamma_{n}^{\prime}\|[D,T]\|$

where
$\gamma_{n}^{\prime}:=\frac{2}{n}\left(1+\frac{1}{\pi}(1+\log(n))\right)$. It
is clear that $\gamma_{n}^{\prime}\to 0$ as $n\to\infty$. ∎

Thus we find that the pair of maps $(R_{n},S_{n})$ for
$\\{(C(S^{1})^{(n)},P_{n}L^{2}(S^{1}),P_{n}DP_{n})\\}_{n}$ and
$(C^{\infty}(S^{1}),L^{2}(S^{1}),D)$ forms a $C^{1}$-approximate order
isomorphism. We may conclude from Theorem 5 that

######  Proposition 13.

The sequence of state spaces $\\{(\mathcal{S}(C(S^{1})^{(n)}),d_{n})\\}_{n}$
converges to $(\mathcal{S}(C(S^{1})),d)$ in Gromov–Hausdorff distance.

Using a simple Python script we have computed the distance function for states
on $C(S^{1})^{(n)}$ of the form $S_{n}^{*}({\rm ev}_{x})$ for $n=3,5,9$, where
${\rm ev}_{x}$ is the pure state on $C(S^{1})$ given by evaluation at $x$. The
optimization problem for computing the distance has been solved numerically
using the standard sequential least squares programming (SLSQP) method and we
claim absolutely no originality or proficiency here. We have illustrated the
numerical results in Figure 2.

Figure 2.  The distance function $d_{n}(0,x)\equiv d_{n}(0,S_{n}^{*}({\rm
ev}_{x}))$ on the Toeplitz operator system (Proposition 13) for $n=3,5,9$. The
blue band corresponds to the lower bounds $d(0,x)-2\gamma_{n}$ given in
Proposition 4 with the constants $\gamma_{n}$ given in Lemma 10.

###  3.2. Fejér–Riesz operator systems converge to the circle

In [10] we found the dual operator system of $C(S^{1})^{(n)}$ to be equal to
the operator system of functions on $S^{1}$ with only a finite number of non-
zero Fourier coefficients. It gives a different type of truncation, this time
taking place at the level of the function algebra, as opposed to a spectral
truncation in Hilbert space.

More precisely, we will consider the so-called Fejér–Riesz operator system:

(4)
$C^{*}(\mathbb{Z})_{(n)}=\left\\{a=(a_{k})_{k\in\mathbb{Z}}:\text{supp}(a)\subset(-n,n)\right\\}.$

The elements in $C^{*}(\mathbb{Z})_{(n)}$ are thus given by sequences with
finite support of the form

$a=(\ldots,0,a_{-n+1},a_{-n+2},\ldots,a_{-1},a_{0},a_{1},\ldots,a_{n-2},a_{n-1},0,\ldots)$

and this allows to view $C^{*}(\mathbb{Z})_{(n)}$ as an operator subsystem of
$C^{*}(\mathbb{Z})\cong C(S^{1})$.

The adjoint $a\mapsto a^{*}$ is given by $a^{*}_{k}=\overline{a}_{-k}$ and an
element $a\in C^{*}(\mathbb{Z})_{(n)}$ is positive iff $\sum_{k}a_{k}e^{ikx}$
defines a positive function on $S^{1}$.

Since this naturally is an operator subsystem of $C(S^{1})$ it is natural to
consider the following spectral triple:

(5)
$\left(C^{*}(\mathbb{Z})_{(n)},\mathcal{H}=L^{2}(S^{1}),D=-i\frac{d}{dx}\right).$

We will be looking for positive and contractive maps $K_{n}:C(S^{1})\to
C^{*}(\mathbb{Z})_{(n)}$ and $L_{n}:C^{*}(\mathbb{Z})_{(n)}\to C(S^{1})$
satisfying the conditions of Definition 2 so that we can apply Theorem 5 to
conclude Gromov–Hausdorff convergence of the corresponding state spaces.

We introduce

$\displaystyle K_{n}:C(S^{1})$ $\displaystyle\to C^{*}(\mathbb{Z})_{(n)}$
$\displaystyle f$ $\displaystyle\mapsto F_{n}\ast f$ where we recall that
$F_{n}=\sum_{|k|\leq n-1}(1-|k|/n)e^{ikx}$ is the Fejér kernel so that $K_{n}$
indeed maps to $C^{*}(\mathbb{Z})_{(n)}$ considered as an operator subsystem
of $C(S^{1})$. The map $L_{n}$ is simply the linear embedding of
$C^{*}(\mathbb{Z})_{(n)}$ as an operator subsystem of $C^{*}(\mathbb{Z})\cong
C(S^{1})$: $\displaystyle L_{n}:C^{*}(\mathbb{Z})_{(n)}$ $\displaystyle\to
C(S^{1})$ $\displaystyle(a_{k})$
$\displaystyle\mapsto\left(x\mapsto\sum_{k}a_{k}e^{ikx}\right).$

Positivity and contractiveness of $K_{n}$ for the norm and Lipschitz norm is
an easy consequence of the good kernel properties of $F_{n}$ while for $L_{n}$
they are trivially satisfied.

######  Lemma 14.

There exists a sequence $\gamma_{n}$ converging to 0 such that

$\|L_{n}\circ K_{n}(f)-f\|\leq\gamma_{n}\|[D,f]\|$

for all $f\in C^{\infty}(S^{1})$.

###### Proof.

Since $L_{n}\circ K_{n}(f)=F_{n}\ast f$ the proof is analogous to that of
Lemma 10. ∎

######  Lemma 15.

There exists a sequence $\gamma_{n}^{\prime}$ converging to 0 such that

$\|K_{n}\circ L_{n}(a)-a\|\leq\gamma_{n}^{\prime}\|[D,a]\|$

for all $a\in C^{*}(\mathbb{Z})_{(n)}$.

###### Proof.

From the Fourier coefficients of the Fejér kernel we find that

$K_{n}\circ L_{n}(a)-a=\left(-\frac{|k|}{n}a_{k}\right)_{k}.$

We will estimate the sup-norm of the function
$f(x)=\frac{1}{n}\sum_{k}|k|a_{k}e^{ikx}$ by the Lipschitz norm of $a$. First
of all, we may write $f$ as a convolution product $f=g\ast h$ where
$g=\sum_{k=-n+1}^{n-1}\text{sgn}(k)e^{ikx}$ and
$h=\frac{1}{n}\sum_{k=-n+1}^{n-1}ka_{k}e^{ikx}=\frac{1}{n}[D,a]$. Then
$\|f\|_{\infty}\leq\|g\|_{1}\|h\|_{\infty}$ where

$\|g\|_{1}\leq\|g\|_{2}=\sqrt{2n-1}.$

We conclude that $\|g\ast
h\|_{\infty}\leq\gamma^{\prime}_{n}\|[D,a]\|_{\infty}$ with
$\gamma_{n}^{\prime}=\frac{\sqrt{2n-1}}{n}\to 0$ as $n\to\infty$. ∎

We conclude that the pair of maps $(K_{n},L_{n})$ for
$\\{(C^{*}(\mathbb{Z})_{(n)},L^{2}(S^{1}),D)\\}_{n}$ and
$(C^{\infty}(S^{1}),L^{2}(S^{1}),D)$ forms a $C^{1}$-approximate order
isomorphism and we have

######  Proposition 16.

The sequence of state spaces
$\\{(\mathcal{S}(C^{*}(\mathbb{Z})_{(n)}),d_{n})\\}_{n}$ converges to
$(\mathcal{S}(C(S^{1})),d)$ in Gromov–Hausdorff distance.

We again illustrate the numerical results for the first few cases in Figure 3.
As compared to the Toeplitz operator system (Figure 2) the optimization is
much more cumbersome. This is essentially due to the fact that it involves the
computation of a supremum norm of a trigonometric polynomial.

######  Remark 17.

If we recall the duality between $C(S^{1})^{(n)}$ and
$C^{*}(\mathbb{Z})_{(n)}$ as operator systems from [10] it is quite surprising
that both operator system spectral triples converge to the circle as
$n\to\infty$.

Figure 3.  The distance function $d_{n}(0,x)\equiv d_{n}(0,L_{n}^{*}{\rm
ev}_{x})$ on the Fejér–Riesz operator system (Proposition 16) for $n=2,3$. The
blue band corresponds to the lower bounds $d(0,x)-2\gamma_{n}$ given in
Proposition 4 with the constants $\gamma_{N}$ given in Lemma 14.

###  3.3. Matrix algebras converge to the sphere

In [28, 29] Rieffel analyzed quantum Gromov–Hausdorff convergence for so-
called quantum metric spaces. Such a space is given by a pair $(A,L)$ of an
order-unit space $A$ and a so-called Lipschitz norm $L$ on $A$. At first
sight, such spaces appear to be more general than (operator system) spectral
triples and the distance function they give rise to. However, as Rieffel shows
in [28, Appendix 2] Dirac operators are universal in the sense that the
Lipschitz semi-norms can always be realized as norms of commutators with a
self-adjoint operator $D$. Note that it remains an open question, however, for
what Lipschitz semi-norms one can find an operator $D$ with compact resolvent
implementing that semi-norm.

We show below that the corresponding state spaces with Connes’ distance
formula converge in Gromov–Hausdorff distance to the state space on the round
two-sphere. This is closely connected —certainly at the technical level— to
the results of [29] which is that the matrix algebras that describe the fuzzy
two-sphere converge in quantum Gromov–Hausdorff distance to the round two-
sphere. Even though for much of the analysis we may refer to [28, 29] we do
formulate the main results in our framework of operator system spectral
triples and, as said, restrict our attention to the classical metric spaces.

We will describe the round two-sphere by the following spectral triple:

(6) $(C^{\infty}(S^{2}),\mathbb{C}^{2}\otimes L^{2}(S^{2}),D_{S^{2}})$

We write
$S^{2}=\\{(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}:x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=1\\}$
so that the following vector fields

$X_{jk}=x_{j}\partial_{k}-x_{k}\partial_{j};\qquad(j<k).$

are tangent to $S^{2}$. Of course, these vector fields are fundamental vector
fields and generate the Lie algebra $su(2)$. Note that the normal vector field
is given by $\vec{x}$ itself.

In terms of the three Pauli matrices we may then write the Dirac operator as
[32]

(7) $D_{S^{2}}=(\vec{x}\cdot\vec{\sigma})\sum_{j<k}\sigma^{j}\sigma^{k}\otimes
X_{jk}.$

Note that $(\vec{x}\cdot\vec{\sigma}):=\sum_{k=1}^{3}x_{k}\sigma_{k}$ acts as
the chirality operator and makes sure that the spinor bundle on $S^{2}$ is
actually non-trivial (as it should).

The fuzzy sphere [24] is obtained when one considers spherical harmonics on
the sphere only up to some maximum total spin. More precisely, it is described
by a matrix algebra $L(V_{n})$ where $V_{n}$ is the $n$-dimensional
irreducible representation of $SU(2)$. A Dirac operator on the fuzzy sphere
was introduced in [15] (see also [2]):

(8) $D_{n}:=\sum_{j<k}\sigma^{j}\sigma^{k}\otimes[L_{jk},\cdot]$

where $L_{jk}$ are standard generators of $su(2)$ in the $n$-dimensional
representation, satisfying

$[L_{jk},L_{lm}]=\delta_{kl}L_{jm}-\delta_{km}L_{jl}-\delta_{jl}L_{km}+\delta_{jm}L_{kl}.$

This gives rise to the following spectral triple

(9) $(L(V_{n}),\mathbb{C}^{2}\otimes L(V_{n}),D_{n})$

The comparison between (7) and (8) is convincing, except for the absence of
the chirality operator in the case of the fuzzy sphere. However, as shown in
[2] this can be repaired for by a doubling of the representation space and a
corresponding doubling constructing for the Dirac operator. Note that this
does not alter the corrresponding Lipschitz norms, so we may just as well work
with the Dirac operator defined in (8).

######  Remark 18.

The paper [2] also contains a detailed discussion on the nature of the
spectral truncation that applies to the case at hand (see [2, Section 6.3]).
It depends on the decomposition of the Hilbert space of spinors into
irreducible representations of $\text{Spin}(3)$. However, since we will not
need the specific form of the truncation here, we refrain from including it
here.

Let us now proceed to show that there is a $C^{1}$-approximate order
isomorphism $(\breve{\sigma},\sigma)$ for the sequence of spectral triples
defined in (9) and the spectral triple of (6). As a consequence, we thus re-
establish part of the conclusion of [29, Theorem 3.2] that the fuzzy sphere
converges to the two-sphere in Gromov–Hausdorff distance as $n\to\infty$. Note
that loc.cit. goes further in establishing that this limit is the unique limit
of the sequence of fuzzy spheres (and also extends to more general coadjoint
orbits). The reason we have included this example here is that it is
formulated entirely in terms of spectral triples, and fits the general
framework set up in Section 2.

####  3.3.1. Berezin symbol and Berezin quantization

Following [29] we start by defining maps $\sigma:L(V_{n})\to C(S^{2})$ and
$\breve{\sigma}:C(S^{2})\to L(V_{n})$. Given a projection $P\in L(V_{n})$,
say, on the highest-weight vector of $V_{n}$, we define the Berezin symbol
$\sigma:L(V_{n})\to C(S^{2})$ by [5]

(10) $\sigma(T)(g)\equiv\sigma_{T}(g):=\operatorname{Tr}(T\alpha_{g}(P))$

where $\alpha_{g}$ is the action of $g\in SU(2)$ induced by conjugation on
$L(V)$. Since $\alpha_{u}(P)=P$ for all $u\in U(1)$, it follows that
$\sigma(T)$ is $U(1)$-invariant and thus descends to a function on
$SU(2)/U(1)=S^{2}$. Moreover, we readily see that $\sigma$ is an
$SU(2)$-equivariant map which will turn out to be useful later.

We let $\breve{\sigma}:C(S^{2})\to L(V_{n})$ be the adjoint of the map
$\sigma$ when $C(S^{2})$ comes equipped with the $L^{2}$-inner product and
$L(V_{n})$ with the Hilbert–Schmidt inner product. There is also the following
explicit expression (cf. [29, Sect.2]).

######  Proposition 19.

The map $\breve{\sigma}$ defined by
$\breve{\sigma}(f)\equiv\breve{\sigma}_{f}=n\int f(g)\alpha_{g}(P)dg$
satisfies

$\langle
f,\sigma_{T}\rangle=\frac{1}{n}\operatorname{Tr}(\breve{\sigma}_{f}T).$

Moreover, we may write the so-called Berezin transform as a convolution
product

$\sigma(\breve{\sigma}_{f})(g)=(f\ast H_{P})(g)\equiv\int
f(gh^{-1})H_{P}(h)dh$

where $H_{P}$ is a probability measure defined by

$H_{P}(g)=n\operatorname{Tr}(P\alpha_{g}(P)).$

###### Proof.

As in [29, Sect.2] we check the formula for $\breve{\sigma}(f)$ by computing
that

$\langle f,\sigma_{T}\rangle=\int
f(g)\operatorname{Tr}(T\alpha_{g}(P))dg=\operatorname{Tr}\left(\int
f(g)T\alpha_{g}(P)dg\right)$

so that the result follows.

For the Berezin transform we then indeed have that

$\displaystyle\sigma(\breve{\sigma}_{f})(g)$
$\displaystyle=\operatorname{Tr}\left(\breve{\sigma}_{f}\alpha_{x}(P)\right)=\operatorname{Tr}\left(n\int
f(h)\alpha_{h}(P)dh\alpha_{g}(P)\right)$ $\displaystyle=n\int
f(h)\operatorname{Tr}(P\alpha_{h^{-1}g}(P))dh=n\int f(gh^{-1})H_{P}(h)dh$

using also that $H_{P}(h^{-1})=H_{P}(h)$. ∎

Again, one readily observes that $\breve{\sigma}$ is an $SU(2)$-equivariant
map.

####  3.3.2. The sphere as a limit of matrix algebras

We now show in a series of Lemma’s that the conditions of Definition 2 hold
for $R_{n}=\breve{\sigma}$ and $S_{n}=\sigma$.

######  Lemma 20.

For any $f\in C^{\infty}(S^{2})$ we have $\|\breve{\sigma}_{f}\|\leq\|f\|$ and
$\|[D_{n},\breve{\sigma}_{f}]\|\leq\|[D_{S^{2}},f]\|$.

###### Proof.

The contractive property of $\breve{\sigma}$ is proved for instance in [19,
Theorem 1.3.5] where $\breve{\sigma}$ is the Berezin quantization map. Then,
by $SU(2)$-equivariance of $\breve{\sigma}$ we have

$[D_{n},\breve{\sigma}_{f}]=\sum_{j<k}\sigma^{j}\sigma^{k}\otimes[L_{jk},\breve{\sigma}_{f}]=\left(\sum_{j<k}\sigma^{j}\sigma^{k}\otimes\breve{\sigma}([X_{jk},f])\right)=(1\otimes\breve{\sigma})[D_{S^{2}},f].$

Since $\breve{\sigma}$ is a positive map from a commutative domain to a
$C^{*}$-algebra, it follows by a Theorem by Stinespring [31] (cf. [25, Theorem
3.11]) that $\breve{\sigma}$ is completely positive. But then, it follows from
[25, Proposition 3.6] that $\breve{\sigma}$ is completely bounded with
$\|\breve{\sigma}\|_{cb}=\|\breve{\sigma}\|$. In particular,
$\|1\otimes\breve{\sigma}\|\leq\|\breve{\sigma}\|\leq 1$ so that it follows
that

$\|[D_{n},\breve{\sigma}_{f}]\|\leq\|(1\otimes\breve{\sigma})\|\|[D_{S^{2}},f]\|\leq\|[D_{S^{2}},f]\|.\qed$

######  Lemma 21.

There exists a sequence $\\{\gamma_{n}\\}$ converging to 0 such that

$\|f-\sigma(\breve{\sigma}_{f})\|\leq\gamma_{n}\|[D_{S^{2}},f]\|$

for all $f\in C^{\infty}(S^{2})$.

###### Proof.

We exploit the expression for $\sigma(\breve{\sigma}_{f})$ as a convolution
product from Proposition 19. Indeed,

$\displaystyle|f(g)-\sigma(\breve{\sigma}_{f})(g)|$
$\displaystyle=\left|\int(f(g)-f(h))H_{P}(h^{-1}g)dh\right|$
$\displaystyle\leq\|f\|_{\text{Lip}}\int
d(g,h)H_{P}(h^{-1}g)dh=\|f\|_{\text{Lip}}\int d(e,h)H_{P}(h)dh$

where $d$ is the $SU(2)$-invariant (round) distance on $SU(2)/U(1)$ and
$\|f\|_{\text{Lip}}$ is the corresponding Lipschitz seminorm of $f$. Since
$\|f\|_{\text{Lip}}=\|[D_{S^{2}},f]\|$ by standard arguments [9, Sect. VI.1]
and $\int d(e,h)H_{P}(h)dh\to 0$ as $n\to\infty$, the result follows. ∎

######  Lemma 22.

For any $T\in L(V_{n})$ we have $\|\sigma_{T}\|\leq\|T\|$ and
$\|[D_{S^{2}},\sigma_{T}]\|\leq\|[D_{n},T]\|$.

###### Proof.

The map $\sigma$ is a contraction:

$\|\sigma_{T}\|=\sup_{g}|\operatorname{Tr}T\alpha_{g}(P)|\leq\|T\|\sup_{g}\operatorname{Tr}|\alpha_{g}(P)|=\|T\|.$

Since $\sigma$ is also $SU(2)$-equivariant we again find that

$[D_{S^{2}},\sigma_{T}]=\sum_{j<k}\sigma^{j}\sigma^{k}\otimes[X_{jk},\sigma_{T}]=\left(\sum_{j<k}\sigma^{j}\sigma^{k}\otimes\sigma([L_{jk},T])\right)=(1\otimes\sigma)[D_{n},T].$

Since the range of $\sigma$ is a commutative $C^{*}$-algebra it follows from
[25, Theorem 3.9] that $\|\sigma\|_{cb}=\|\sigma\|$. Hence

$\|[D_{S^{2}},\sigma_{T}]\|\leq\|(1\otimes\sigma)\|\|[D_{n},T]\|\leq\|[D_{n},T]\|$

since $\sigma$ is a contraction. ∎

######  Lemma 23.

There exists a sequence $\\{\gamma_{n}^{\prime}\\}$ converging to 0 such that

$\|T-\breve{\sigma}(\sigma_{T})\|\leq\gamma^{\prime}_{n}\|[D_{n},T]\|$

for all $T\in L(V_{n})$.

###### Proof.

This is based on a highly non-trivial result [29, Theorem 6.1] which states
that there exists a sequence $\\{\gamma_{n}^{\prime}\\}$ converging to 0 such
that

$\|T-\breve{\sigma}(\sigma_{T})\|\leq\gamma^{\prime}_{n}L_{n}(T)$

for all $T\in L(V_{n})$, where $L_{n}$ is the Lipschitz norm on $L(V_{n})$
defined by

$L_{n}(T)=\sup_{g\neq e}\frac{\|\alpha_{g}(T)-T\|}{l(g)}$

for a length function $g$ on $SU(2)$ that induces the round metric on $S^{2}$.
However, as in the proof of [26, Theorem 3.1] we may estimate

$L_{n}(T)\leq\sup_{X\in su(2)}\\{\|[X,T]\|:\|X\|\leq 1\\}$

while the right-hand side can be bounded from above by $k\|[D,T]\|$ for some
constant $k$ independent of $n$ (as in the display preceding [26, Theorem
4.2]. ∎

We have thus verified that the maps $(\breve{\sigma},\sigma)$ between
$\\{L(V_{n}),\mathbb{C}^{2}\otimes L(V_{n}),D_{n})$ and
$(C^{\infty}(S^{2}),\mathbb{C}^{2}\otimes L^{2}(S^{2}),D_{S^{2}})$ forms a
$C^{1}$-approximate order isomorphism and we may conclude from Theorem 5 that

######  Proposition 24.

The sequence of state spaces $\\{(\mathcal{S}(L(V_{n})),d_{n})\\}_{n}$
converges in Gromov–Hausdorff distance to $(\mathcal{S}(C(S^{2})),d)$.

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