11institutetext: Primaa, Paris, France
11email<EMAIL_ADDRESS>22institutetext: Hôpital Saint Louis,
Paris, France

# Classification in Histopathology: A unique deep embeddings extractor for
multiple classification tasks

Adrien Nivaggioli 11 Nicolas Pozin 11 Rémy Peyret 11 Stéphane Sockeel 11 Marie
Sockeel 11 Nicolas Nerrienet 11 Marceau Clavel 11 Clara Simmat 11 Catherine
Miquel 22

###### Abstract

In biomedical imaging, deep learning-based methods are state-of-the-art for
every modality (virtual slides, MRI, etc.) In histopathology, these methods
can be used to detect certain biomarkers or classify lesions. However, such
techniques require large amounts of data to train high-performing models which
can be intrinsically difficult to acquire, especially when it comes to scarce
biomarkers. To address this challenge, we use a single, pre-trained, deep
embeddings extractor to convert images into deep features and train small,
dedicated classification head on these embeddings for each classification
task. This approach offers several benefits such as the ability to reuse a
single pre-trained deep network for various tasks; reducing the amount of
labeled data needed as classification heads have fewer parameters; and
accelerating training time by up to 1000 times, which allows for much more
tuning of the classification head. In this work, we perform an extensive
comparison of various open-source backbones and assess their fit to the target
histological image domain. This is achieved using a novel method based on a
proxy classification task. We demonstrate that thanks to this selection
method, an optimal feature extractor can be selected for different tasks on
the target domain. We also introduce a feature space augmentation strategy
which proves to substantially improve the final metrics computed for the
different tasks considered. To demonstrate the benefit of such backbone
selection and feature-space augmentation, our experiments are carried out on
three separate classification tasks and show a clear improvement on each of
them: microcalcifications classification (29.1% F1-score increase), lymph
nodes metastasis detection (12.5% F1-score increase), mitosis classification
(15.0% F1-score increase). We therefore show how an optimal unique feature
extractor can be used in a multi-task setting, with improved performance,
faster training and inference and optimized memory usage.

###### Keywords:

Deep Learning Unsupervised Learning Histopathology Embeddings Classification

##  1 Introduction

Digital pathology is rapidly gaining popularity as a reliable and efficient
tool for medical diagnosis [9]. It paves the way for Artificial Intelligence
(AI) based techniques usage as an aid to diagnosis, leading to significant
improvements in accuracy and efficiency [25, 21, 3, 24]. However, despite
recent advancements in AI and computer vision, their application in digital
pathology faces several challenges. For instance, training a deep learning
model requires a large volume of annotated data to avoid overfitting, which
can be difficult to acquire. Indeed, annotating medical data is a time-
consuming and expensive process, as it requires dedicated software and trained
professionals. Deep learning algorithms also require significant computational
resources, which can limit experimentation and parameter tuning. Furthermore,
some biomarkers are quite rare (vascular emboli, perineural invasion,…),
leading to small amounts of data, which makes it difficult to train accurate
deep learning models.

To address these challenges, we use a pretrained deep learning backbone to
convert images into embeddings. The embeddings are then used to train small
dedicated neural networks. In this study, we focus on histopathology
classification tasks, and refer to these networks as "classification heads".
As a consequence, we alleviate the need for labelled data and make the
training process shorter as only small specific networks need to be trained.
Furthermore, we propose a way to compare and select the best backbone to use
to extract embeddings, and we prove the importance of the augmentations in the
feature space.

In recent years, a few papers have followed similar so-called Deep Feature
Learning (DFL) approaches [14]. However, most of them use an arbitrarily
chosen backbone while other existing models may be more relevant to their
usage [14]. Most authors also focus their analysis to a single use case while
an efficient feature extractor may be used for multiple tasks [19].

In this paper, we first describe our backbone selection method and perform a
large-scale benchmark of existing open-source networks to find the most
relevant for histopathology images. Then, we detail various experiments on
histopathology classification tasks and show that a DFL-based method, using
the same backbone, outperforms traditional deep learning on every tested
classification tasks. We discuss our results and show the importance of data
augmentation in the feature space, as it greatly improves performances.

####  1.0.1 Contributions

  * •

We propose a backbone selection method. It is applied on several open source
models so as to select a backbone relevant to our tasks.

  * •

We prove the effectiveness of feature space data augmentations.

  * •

We successfully apply our Deep Feature Learning based methodology on three
histopathology classification tasks.

  * •

We prove the superiority of our approach compared to standard deep learning
approaches in digital pathology.

Figure 1: Overview of our method. A) A sample is a RGB image at zoom x20. It
is enriched with configurable variations such as: sibling patch at zoom x5,
black and white translation of the patch… Each element from the combination is
converted into embeddings through a deep network. Resulting embeddings are
concatenated to create a single deep embedding dataset. B) A small
classification head is trained on the deep embedding dataset. C) At inference
time, images are passed through the backbone and the classification head to
get a prediction.

##  2 Pretrained Backbone as Universal Histopathological Feature Extractors

###  2.1 Concept

Complex tasks may require the usage of deep networks to get satisfying
results. Such cases where little data is available and big networks are
required lead to overfitting. In digital pathology, the available datasets
often are relatively small, as it is expensive and time consuming to build
large annotated datasets that requires expert knowledge.

Thus, projecting images into a lower-dimensional feature space is a way to
address these issues as we only train small classification networks on top of
the backbone, reducing at the same time the need for labeled data, the risk of
overfitting, and the training time. Moreover, we can still benefit from deep
neural networks abstraction abilities, even if they are trained on external
data.

The crux of our method is the assumption that images, which are high-
dimensional (e.g., 256x256x3), can be projected into a smaller feature space
by using deep learning task agnostic backbones without significant loss of
information relevant for our classification tasks.

###  2.2 Backbone Selection

To ensure that deep embeddings carry sufficient information and to compare
various backbones performances, we create an evaluation dataset composed of
35334 breast histopathology images at zoom x5 (1.76 $\mu$m per pixel)
distributed amongst 23 imbalanced classes, which include both common tumor and
benign lesions. This small dataset only required few annotated data, labeled
in-house by expert pathologists. Moreover, it allows us to compare backbones
on histopathological images, which are the types of images the following
experiment are based on. The precise composition of this dataset is described
in the supplementary materials, in Table 6.

As a baseline, we train every layer of a ResNet50 [12] on this dataset,
starting with Tensorflow’s [2] ImageNet weights. We use a focal loss [15] to
fight the imbalance along with ADAM optimizer [13], and various augmentations:
scaling, shear, and flip. We achieve a maximum validation balanced accuracy of
0.6889.

For various open-source backbones, we train a classification head directly on
normalized deep embedding extracted from our image dataset. Each head is
composed of a single dense layer with a softmax activation, trained with a
focal-loss, ADAM optimizer, and Gaussian Noise as augmentation.

Table 1 provides an overview of our results. It is quite revealing in several
ways. First, most pretrained backbones outperform the baseline model, which
quickly overfits given the reduced size of the dataset. At the moment,
backbones pretrained on Medical Images perform poorly when compared to those
pretrained of ImageNet-1K or ImageNet-22K. This confirms what recent works
have found [22, 7]. One possible explanation for this is the fact that the
best performing available models have been trained in an unsupervised manner
for a long time on large GPU clusters [5], whereas those pretrained on Medical
Images had access to more limited ressources and less diverse data [7]. This
might change in the future with the increasing amount of open-source medical
datasets and research teams working with histopathological images. Vision
Transformers perform worse than regular convolution-based backbones. This is
in line with findings of previous works, which did not find clear advantages
of vision-based models in comparison with standard convolution-based ones [10,
20]. Backbones trained in an unsupervised manner [5, 4, 18] perform better
than their supervised counterparts. This is coherent with the rise of
foundation models, which are taking over the leaderboards of many Image
Processing datasets.

A visual comparison of the embeddings of the evulation dataset using the
HistoSegNet [6] and DINO ResNet-50 [5] backbones is shown in the supplementary
materials, in Figure 2.

Table 1: Comparison of various pretrained backbones Pretrained | Pretrained on | Unsupervised | Balanced | Cohen Kappa | Weighted
---|---|---|---|---|---
Backbone | Medical Images | Training | Accuracy | Score | F1-Score
HistoSegNet [6] | yes | no | 0.5676 | 0.7942 | 0.8259
ResNet50 [7] | yes | yes | 0.6519 | 0.8194 | 0.8498
ViT-S/16 [7] | yes | yes | 0.6522 | 0.8444 | 0.8706
ViT-patch16-224 [23] | no | no | 0.7098 | 0.8904 | 0.9090
DINO ViT-S/8 [5] | no | yes | 0.7287 | 0.8978 | 0.9149
Keras’s ResNet50 | no | no | 0.7404 | 0.9029 | 0.9197
ConvNeXt-B [16] | no | no | 0.7686 | 0.9135 | 0.9292
HorNet-L [18] | no | yes | 0.7894 | 0.9276 | 0.9411
SwAV ResNet-50 [4] | no | yes | 0.8196 | 0.9290 | 0.9415
DINO ResNet-50 [5] | no | yes | 0.8236 | 0.9365 | 0.9475
ResNet-50 | - | no | 0.6582 | 0.7599 | 0.7995
Fully Trained

##  3 Histopathology Classification Experiments

We apply the Deep Feature Learning method to various histopathology
classification tasks: classification of Lymph Nodes Metastasis,
Microcalcifications and Mitosis.

In histopathology, pathologists use information from various zoom levels to
perform diagnosis as both context and cellular level patterns are of
importance. Some AI based methods[8] replicate this by making prediction on
multiple images at same position but different zooms. Doing so with
traditional deep learning requires specific architectures handling multiple
input images, which increases the model dimensions and makes the method even
more prone to overfitting. Also, increasing even more the input dimension by
adding multiple input images facilitates overfitting. Using DFL, we can easily
concatenate features extracted with a same backbone from samples at various
zooms. In the following experiments, we compare various approaches:
Traditional Deep Learning at zoom x20, DFL at zoom x20, and DFL at both zoom
x20 and x5.

Based on backbones comparisons illustrated on Table 1, we use the DINO
ResNet-50 [5] as backbone. All trainings were performed on a single GeForce
RTX 2080 Ti GPU, using a Focal Loss and an ADAM optimizer. Apart from the
mitosis classification, traditional deep learning trainings used an
EfficientNet architecture and performed online image augmentation (such as
Color Jitter, Gaussian Noise, Hue/Saturation/Value random changes, random Flip
and Rotations). In the DFL setting, only Gaussian noise augmentation is
applied, directly on the extracted embeddings. The classification head is
composed of a single dense hidden layer, followed by the classification layer.
Figure 1 depicts an overview of our method.

Table 2: Dataset description Biomarker | Set | #Slides |  | #Patches
---
Positives/Negatives
Origin |  | Sample
---
Composition
Lymph Nodes Metastasis | Train | 500 | 720363 / 869493 | Camelyon17 [17] |  Zooms x20 and x5 B&W Images
Test | 50 | 57334 / 173050 | In-House
Microcalcifications | Train | 1642 | 2825 / 331842 |  In-House 3 Centers |  Zooms x20 and x5 RGB and B&W Images
Test | 284 | 632 / 57421
Mitosis | Train | 180 | 2500 / 2500 |  MIDOG21 [1] In-House |  Zoom x20 RGB and B&W Images Padding and Scaling
Test | 10 | 125 / 1108 | In-House

Test and train datasets used for every classification tasks are detailed in
Table 2. Note that samples composition for the DFL (considered zoom, color
images on which embeddings are extracted…) may be different for each task. As
explicited in Table 2, lymph nodes and microcalcification tasks are performed
on samples containing images at both zooms x20 and x5 as context matter for
theses tasks. For the mitosis classification task, high resolution details is
necessary, and only zoom x20 is used. In addition, it was found sufficient to
extract embeddings from RGB images for the lymph nodes metastasis task while
features extracted from a combination of RGB and black and white images were
more suited for mitosis and microcalcififications classification. Most images
had a size of 256*256 pixels, apart from mitosis images that had a size of
50*50 pixels. To make them fit into our backbone, we used a concatenation of
scaling and padding to 256*256 pixels. Finally, some datasets used for our
experiments are public while some were labeled on in-house data by expert
pathologists following a standard process: annotation is performed by a junior
pathologist and then reviewed by an expert.

Table 3 summarizes our results and proves the efficiency of the Deep Feature
Learning methodology over traditional Deep Learning

Table 3: Biomarker Classification Biomarker | Method | Precision | Recall | PR-AUC | Training time
---|---|---|---|---|---
Lymph Nodes Metastasis | Deep Learning | 0.848 | 0.713 | 0.848 | 7 days
DFL - Single Zoom | 0.856 | 0.769 | 0.882 | 17 minutes
DFL - Multi Zoom | 0.927 | 0.822 | 0.932 | 30 minutes
Microcalcifications | Deep Learning | 0.604 | 0.590 | 0.584 | 20 hours
DFL - Single Zoom | 0.614 | 0.603 | 0.619 | 15 minutes
DFL - Multi Zoom | 0.785 | 0.757 | 0.739 | 20 minutes
Mitosis | Deep Learning | 0.546 | 0.509 | 0.507 | 5 hours
DFL - Single Zoom | 0.574 | 0.641 | 0.584 | 4 minutes

##  4 Discussion

###  4.1 Backbone Selection Validation

We tested the backbone selection method on the microcalcification
classification task. Deep embeddings were extracted using various backbones,
and the same classification head was trained on those embeddings. Our results,
presented in Table 4, are consistent with those shown in Table 1, and support
the use of DINO ResNet-50 [5] as the backbone of choice for our experiments.

Table 4: Comparing backbones for microcalcifications classification Backbone | Precision | Recall | PR-AUC
---|---|---|---
Keras’s ResNet-50 | 0.478 | 0.419 | 0.379
HorNet-L [18] | 0.593 | 0.426 | 0.435
SWaV ResNet-50 [4] | 0.610 | 0.435 | 0.460
DINO ResNet-50 [5] | 0.714 | 0.541 | 0.607

###  4.2 Deep Learning versus Deep Feature Learning

Results presented in Table 3 support our claims:

  * •

Trainings are much shorter. Instead of hours (or even sometimes days), they
only take a few minutes. This allows for more parameters finetuning, and
overall experimentation on the trainings and the dataset used.

  * •

The results are better overall when using the same zoom as in the Deep
Learning setting. This is due to the increased resistance to overfitting of
the DFL method, as the Deep Learning setting inevitably overfitted even with
extensive image augmentation,.

  * •

In a DFL setting, using multiple zooms is trivial as it simply entails
extracting several sets of embeddings during the preprocessing step.

Given a large enough dataset, traditional deep learning might outperform our
DFL-based approach, which is dependent on the quality of the extracted
embeddings. But, for our usecases and up to a couple million samples in the
training dataset, our method still outperformed traditional deep learning.

Table 5: Augmentation Influence Biomarker | Method | Gaussian Noise | Precision | Recall | PR-AUC
---|---|---|---|---|---
Lymph Nodes Metastasis | Single Zoom | No | 0.843 | 0.729 | 0.863
Single Zoom | Yes | 0.856 | 0.769 | 0.882
Multi Zoom | No | 0.919 | 0.795 | 0.915
Multi Zoom | Yes | 0.927 | 0.822 | 0.932
Microcalcifications | Single Zoom | No | 0.596 | 0.562 | 0.574
Single Zoom | Yes | 0.614 | 0.603 | 0.619
Multi Zoom | No | 0.701 | 0.619 | 0.678
Multi Zoom | Yes | 0.785 | 0.757 | 0.739

###  4.3 On Augmentations

The main drawback of our method is the difficulty of using image augmentation.
Because the image dataset is first preprocessed and transformed into a deep
embeddings dataset, images cannot be augmented on the fly as usually done in
traditionnal deep learning methods.

A partial answer would be to augment the image dataset offline, before
converting it to embeddings, but the resulting set would still be limited.
Another way is to perform augmentation on the fly directly in the feature
space. Here, we apply gaussian noise directly on the deep embeddings. This is
paramount, as the classification head otherwise overfits in a couple of
epochs. Table 5 shows the importance of this noise, as it persistently
improves performance.

We believe that on the fly augmentation in the feature space is key to push
performance further. In the future, we will explore more refined ways than
simple homogeneous gaussian noise: various statistical perturbations, various
augmentations for the different components of the feature vector, sampling in
the feature space to simulate given image augmentations (e.g color
augmentation) or image synthesis…

###  4.4 On Concatenating Deep Embeddings

We find that extracting and concatenating multiple embeddings for each image
often led to better results. When training classification heads, we tried
several combinations of deep embeddings by varying the image zoom (x5 or x20)
and the color image color space (RGB or Black and White).

An exploration path would be to extract deep embeddings with multiple
backbones. By using multiple models trained on different datasets, with
various training methodology and architectures, we may extract more
representative deep embeddings [11]. We could also further reduce the feature
space dimension through dimension reduction techniques such as PCA, LDA, or
Isomap, this would further speed up training time and reduce classification
head sizes.

##  5 Conclusion

In this paper, we proved that a Deep Feature Learning based approach
outperforms traditional deep learning on multiple histopathology
classification tasks. Faster trains, lighter networks, and smaller datasets
allows for more experimentation with limited computational ressources. We
proposed a backbone selection method that we used to compare various open-
source backbones so as to find the most relevant for histopathology
classification tasks. Our method eases the use of multiple images as inputs,
which greatly improves results. The use of gaussian noise as an on-the-fly
augmentation also proves valuable, and increases performances considerably.

Our DFL-based method is modular and, as discussed in the previous section,
many avenues are still left for us to explore. Our method could in particular
be adapted to detection and segmentation tasks, which would notably broaden
its use.

## References

  * [1] MIDOG Challenge 2021 - Grand Challenge. https://midog2021.grand-challenge.org/, [Accessed 28-Feb-2023]
  * [2] TensorFlow homepage. https://www.tensorflow.org, [Accessed 28-Feb-2023]
  * [3] Banerji, S., Mitra, S.: Deep learning in histopathology: A review. WIREs Data Mining and Knowledge Discovery 12(1) (Nov 2021). https://doi.org/10.1002/widm.1439, https://doi.org/10.1002/widm.1439
  * [4] Caron, M., Misra, I., Mairal, J., Goyal, P., Bojanowski, P., Joulin, A.: Unsupervised learning of visual features by contrasting cluster assignments (2020). https://doi.org/10.48550/ARXIV.2006.09882, https://arxiv.org/abs/2006.09882
  * [5] Caron, M., Touvron, H., Misra, I., Jégou, H., Mairal, J., Bojanowski, P., Joulin, A.: Emerging properties in self-supervised vision transformers. In: Proceedings of the International Conference on Computer Vision (ICCV) (2021)
  * [6] Chan, L., Hosseini, M.S., Rowsell, C., Plataniotis, K.N., Damaskinos, S.: Histosegnet: Semantic segmentation of histological tissue type in whole slide images. In: Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV) (October 2019)
  * [7] Chen, R.J., Krishnan, R.G.: Self-supervised vision transformers learn visual concepts in histopathology. Learning Meaningful Representations of Life, NeurIPS 2021 (2021)
  * [8] D’Amato, M., Szostak, P., Torben-Nielsen, B.: A comparison between single- and multi-scale approaches for classification of histopathology images. Frontiers in Public Health 10, 892658 (07 2022). https://doi.org/10.3389/fpubh.2022.892658
  * [9] Dawson, H.: Digital pathology – rising to the challenge. Frontiers in Medicine 9 (Jul 2022). https://doi.org/10.3389/fmed.2022.888896, https://doi.org/10.3389/fmed.2022.888896
  * [10] Deininger, L., Stimpel, B., Yuce, A., Abbasi-Sureshjani, S., Schönenberger, S., Ocampo, P., Korski, K., Gaire, F.: A comparative study between vision transformers and cnns in digital pathology (2022). https://doi.org/10.48550/ARXIV.2206.00389, https://arxiv.org/abs/2206.00389
  * [11] Gontijo-Lopes, R., Dauphin, Y., Cubuk, E.D.: No one representation to rule them all: Overlapping features of training methods. In: International Conference on Learning Representations (2022), https://openreview.net/forum?id=BK-4qbGgIE3
  * [12] He, K., Zhang, X., Ren, S., Sun, J.: Deep residual learning for image recognition. In: Proceedings of the IEEE conference on computer vision and pattern recognition. pp. 770–778 (2016)
  * [13] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization (2014), http://arxiv.org/abs/1412.6980, cite arxiv:1412.6980Comment: Published as a conference paper at the 3rd International Conference for Learning Representations, San Diego, 2015
  * [14] Kumar, A., Singh, S.K., Saxena, S., Lakshmanan, K., Sangaiah, A.K., Chauhan, H., Shrivastava, S., Singh, R.K.: Deep feature learning for histopathological image classification of canine mammary tumors and human breast cancer. Information Sciences 508, 405–421 (Jan 2020). https://doi.org/10.1016/j.ins.2019.08.072, https://doi.org/10.1016/j.ins.2019.08.072
  * [15] Lin, T.Y., Goyal, P., Girshick, R., He, K., Dollar, P.: Focal loss for dense object detection. IEEE Transactions on Pattern Analysis and Machine Intelligence 42(2), 318–327 (Feb 2020). https://doi.org/10.1109/tpami.2018.2858826, https://doi.org/10.1109/tpami.2018.2858826
  * [16] Liu, Z., Mao, H., Wu, C.Y., Feichtenhofer, C., Darrell, T., Xie, S.: A convnet for the 2020s (2022). https://doi.org/10.48550/ARXIV.2201.03545, https://arxiv.org/abs/2201.03545
  * [17] Pinchaud, N.: Camelyon17 challenge (2019)
  * [18] Rao, Y., Zhao, W., Tang, Y., Zhou, J., Lim, S.L., Lu, J.: Hornet: Efficient high-order spatial interactions with recursive gated convolutions. Advances in Neural Information Processing Systems (NeurIPS) (2022)
  * [19] Spanhol, F.A., Oliveira, L.S., Cavalin, P.R., Petitjean, C., Heutte, L.: Deep features for breast cancer histopathological image classification. In: 2017 IEEE International Conference on Systems, Man, and Cybernetics (SMC). IEEE (Oct 2017). https://doi.org/10.1109/smc.2017.8122889, https://doi.org/10.1109/smc.2017.8122889
  * [20] Springenberg, M., Frommholz, A., Wenzel, M., Weicken, E., Ma, J., Strodthoff, N.: From cnns to vision transformers – a comprehensive evaluation of deep learning models for histopathology (2022)
  * [21] Suganyadevi, S., Seethalakshmi, V., Balasamy, K.: A review on deep learning in medical image analysis. International Journal of Multimedia Information Retrieval 11(1), 19–38 (Sep 2021). https://doi.org/10.1007/s13735-021-00218-1, https://doi.org/10.1007/s13735-021-00218-1
  * [22] Tavolara, T.E., Gurcan, M.N., Niazi, M.K.K.: Contrastive multiple instance learning: An unsupervised framework for learning slide-level representations of whole slide histopathology images without labels. Cancers 14(23), 5778 (Nov 2022). https://doi.org/10.3390/cancers14235778, https://doi.org/10.3390/cancers14235778
  * [23] Wu, B., Xu, C., Dai, X., Wan, A., Zhang, P., Yan, Z., Tomizuka, M., Gonzalez, J., Keutzer, K., Vajda, P.: Visual transformers: Token-based image representation and processing for computer vision (2020)
  * [24] Wu, Y., Cheng, M., Huang, S., Pei, Z., Zuo, Y., Liu, J., Yang, K., Zhu, Q., Zhang, J., Hong, H., Zhang, D., Huang, K., Cheng, L., Shao, W.: Recent advances of deep learning for computational histopathology: Principles and applications. Cancers 14(5), 1199 (Feb 2022). https://doi.org/10.3390/cancers14051199, https://doi.org/10.3390/cancers14051199
  * [25] Zhou, S.K., Greenspan, H., Davatzikos, C., Duncan, J.S., Ginneken, B.V., Madabhushi, A., Prince, J.L., Rueckert, D., Summers, R.M.: A review of deep learning in medical imaging: Imaging traits, technology trends, case studies with progress highlights, and future promises. Proceedings of the IEEE 109(5), 820–838 (May 2021). https://doi.org/10.1109/jproc.2021.3054390, https://doi.org/10.1109/jproc.2021.3054390

##  6 Supplementary Materials

Table 6: Backbone Evaluation Dataset Composition Pathology | Training Dataset | Validation Dataset
---|---|---
Invasive Lobular Carcinoma | 9271 | 3091
Fibroadenomas | 3797 | 1266
Invasive Ductal Carcinoma | 4134 | 1378
Healthy Tissue Glands/Lobules | 2747 | 916
Simple Adenosis | 1995 | 665
Ductal carcinoma in situ | 1403 | 468
Cyst | 661 | 220
Not Interpretable | 422 | 141
Lymph Node | 397 | 132
Lobular Carcinoma in situ | 301 | 100
Microcalcifications | 263 | 88
Healthy Fibrous Tissue | 222 | 74
Apocrine Metaplasia | 164 | 55
Columnar Cell Metaplasia | 156 | 52
Inflammatory Lesion | 103 | 35
Healthy Adipose Tissue | 98 | 33
Sclerosing Adenosis | 94 | 31
Papilloma | 85 | 28
Simple Ductal Hyperplasia | 58 | 19
Atipycal Ductal Hyperplasia | 41 | 13
Blood Vessel | 38 | 12
Scar | 27 | 9
Necrosis | 23 | 8
Figure 2: Projection in 2 dimensions of the evaluation dataset embeddings
using the HistoSegNet (top) and DINO ResNet-50 (bottom) backbones (Best viewed
in color). The projection was performed using the UMAP algorithm, which had no
information regarding the class of each image. The separation of the classes
is much clearer using the DINO ResNet-50 backbone, and some classes already
lie in their own, distinct clusters, which is consistent with the performance
observed on the multiclass classification task used in the backbone selection
method. This illustrates the quality of the feature extractor for the
histopathology domain.


# Sofic mean dimension of typical actions
and a comparison theorem

Lei Jin  Lei Jin: Center for Mathematical Modeling, University of Chile and
UMI 2807 - CNRS<EMAIL_ADDRESS>and  Yixiao Qiao  Yixiao Qiao (Corresponding
author): School of Mathematical Sciences, South China Normal University,
Guangzhou, Guangdong 510631, China<EMAIL_ADDRESS>
###### Abstract.

We refine two results in the paper entitled “Sofic mean dimension” by Hanfeng
Li, improving two inequalities with two equalities, respectively, for sofic
mean dimension of typical actions. On the one hand, we study sofic mean
dimension of full shifts, for which, Li provided an upper bound which however
is not optimal. We prove a more delicate estimate from above, which is optimal
for sofic mean dimension of full shifts over arbitrary alphabets (i.e. compact
metrizable spaces). Our refinement, together with the techniques (in relation
to an estimate from below) in the paper entitled “Mean dimension of full
shifts” by Masaki Tsukamoto, eventually allows us to get the exact value of
sofic mean dimension of full shifts over any finite dimensional compact
metrizable spaces. On the other hand, we investigate finite group actions. In
contrast to the case that the acting group is infinite (and amenable), Li
showed that if a finite group acts continuously on a finite dimensional
compact metrizable space, then sofic mean dimension may be different from
(strictly less than) the classical (i.e. amenable) mean dimension (an
explicitly known value in this case). We strengthen this result by proving a
sharp lower bound, which, combining with the upper bound, gives the exact
value of sofic mean dimension for all the actions of finite groups on finite
dimensional compact metrizable spaces. Furthermore, this equality leads to a
satisfactory comparison theorem for those actions, deciding when sofic mean
dimension would coincide with classical mean dimension. Moreover, our two
results, in particular, verify for a typical class of sofic group actions that
sofic mean dimension does not depend on sofic approximation sequences.

###### Key words and phrases:

Sofic mean dimension; Full shift; Finite group action.

###### 2010 Mathematics Subject Classification:

37B99; 54F45.

##  1\. Main results

###  1.1. Overview

The purpose of this paper is to obtain the exact value of sofic mean dimension
of a typical class of sofic group actions. We will state our main theorems
very quickly (i.e. with a minimum introduction) in this section (Section 1).
The definition of sofic mean dimension as well as all the other necessary
terminologies are located in Section 2. We shall prove our main results in
Section 3 and Section 4. A standard explanation in relation to two different
approaches to the definition of sofic mean dimension, for the reader’s
convenience, can be found in the appendix.

###  1.2. Convention

Throughout this paper the symbol $\mathbb{N}$ is planned to denote the set of
positive integers. All acting groups are always assumed to be countable and
discrete. If we let a sofic group $G$ act continuously on a compact metrizable
space $X$ and if we let $\Sigma$ be a sofic approximation sequence for $G$,
then we denote its sofic mean dimension by
$\operatorname{\mathrm{mdim}}_{\Sigma}(X,G)$.

###  1.3. Full shifts

Possibly the most canonical and fundamental actions are the full shifts. For a
compact metrizable space $K$ and a group $G$ we denote by $(K^{G},\sigma_{G})$
the shift action of $G$ on the product space $K^{G}$.

For $\mathbb{Z}$-actions the exact value of the mean dimension of full shifts
was obtained satisfactorily by Tsukamoto [Tsu19] (provided that the alphabet
$K$ is finite dimensional). We notice that it is straightforward to generalize
Tsukamoto’s result to actions of amenable groups (e.g. this can be done with
almost no additional effort with the help of [JQ]). But if we proceed to the
case of sofic group actions (which are a much wider class of group actions),
then the situation becomes nontrivial immediately.

Actually, Li [Li13, Theorem 7.1] showed (for any sofic approximation sequence
$\Sigma$ for $G$) that

$\operatorname{\mathrm{mdim}}_{\Sigma}(K^{G},\sigma_{G})\leq\dim(K).$

However, this upper bound for
$\operatorname{\mathrm{mdim}}_{\Sigma}(K^{G},\sigma_{G})$ is not optimal in
general. A main ingredient of this paper is to refine Li’s upper bound with a
more delicate estimate from above:

$\operatorname{\mathrm{mdim}}_{\Sigma}(K^{G},\sigma_{G})\leq\inf_{n\in\mathbb{N}}\frac{\dim(K^{n})}{n}.$

Our approach is substantially different from Li’s111Roughly speaking, Li’s
method can be regarded as (in some sense, a kind of) local analysis, i.e.
using the notion of sofic metric mean dimension which dominates sofic mean
dimension from above. On the contrary, we shall adopt a global treatment of a
sharp upper bound for
$\operatorname{\mathrm{mdim}}_{\Sigma}(K^{G},\sigma_{G})$.. We shall explain
about our difficulty shortly in Remark 1.2. As we will see in a moment, our
refinement gives an optimal upper bound for
$\operatorname{\mathrm{mdim}}_{\Sigma}(K^{G},\sigma_{G})$ which applies to all
alphabets $K$, which eventually allows us to get the exact value of sofic mean
dimension of full shifts over any finite dimensional compact metrizable
spaces. This is our first main result.

######  Theorem 1.1 (Main theorem 1).

Let $K$ be a finite dimensional compact metrizable space. Let $G$ be a sofic
group and $\Sigma$ a sofic approximation sequence for $G$. The following
assertion is true:

$\operatorname{\mathrm{mdim}}_{\Sigma}(K^{G},\sigma_{G})=\begin{cases}\dim(K),&\text{if
$K$ satisfies $\dim(K\times K)=2\dim(K)$}\\\
\dim(K)-1,&\text{otherwise}\end{cases}.$

######  Remark 1.2.

For mean dimension of $\mathbb{Z}$-actions, the above two types of
inequalities for full shifts can be deduced directly from each other. More
precisely, let us denote by
$\operatorname{\mathrm{mdim}}(K^{\mathbb{Z}},\sigma_{\mathbb{Z}})$ the
(classical) mean dimension of the $\mathbb{Z}$-shift on $K^{\mathbb{Z}}$. The
inequality $\dim(K^{n})/n\leq\dim(K)$ (for $n\in\mathbb{N}$) is generally
true. Conversely, if we assume that
$\operatorname{\mathrm{mdim}}(K^{\mathbb{Z}},\sigma_{\mathbb{Z}})\leq\dim(K)$
for any compact metrizable space $K$, then replacing the alphabet $K$ by
$K^{n}$ for an arbitrary $n\in\mathbb{N}$ and noting that
$((K^{n})^{\mathbb{Z}},\sigma_{\mathbb{Z}})$ is (dynamically) isomorphic to
$(K^{\mathbb{Z}},\sigma_{\mathbb{Z}}^{n})$ we see that
$\operatorname{\mathrm{mdim}}(K^{\mathbb{Z}},\sigma_{\mathbb{Z}})\leq\dim(K^{n})/n$
for all $n\in\mathbb{N}$. A point here is that the acting group $\mathbb{Z}$
contains a subgroup $n\mathbb{Z}$ of index $n\in\mathbb{N}$ which is possible
to be arbitrarily large, and furthermore, any subgroup action of $\mathbb{Z}$
also forms a $\mathbb{Z}$-action.222For all the actions of (infinite) amenable
groups we would like to refer to the paper [JQ] by the authors, in which we
established a satisfactory and unexpected formula for mean dimension of
product actions. But we do not state this formula here as we shall not use it
in the present paper. Unfortunately, the same trick does not apply to the
context of sofic group actions essentially, because in general we cannot
expect an arbitrary group action to have this nice property and also because
we do not have such an equality for sofic mean dimension of product actions
with respect to the same sofic approximation sequence. Thus, Li’s handling of
$\operatorname{\mathrm{mdim}}_{\Sigma}(K^{G},\sigma_{G})$ is not adequate for
the purpose of obtaining its exact value. This turns out to be one of the main
obstacles that we have to overcome. To deal with an upper bound in a more
careful way is crucial to our proof.

######  Corollary 1.3.

Let $K$ be a finite dimensional compact metrizable space and $G$ an infinite
amenable group. The classical mean dimension of $(K^{G},\sigma_{G})$ satisfies
that

$\operatorname{\mathrm{mdim}}(K^{G},\sigma_{G})=\begin{cases}\dim(K),&\text{if
$\dim(K\times K)=2\dim(K)$}\\\ \dim(K)-1,&\text{otherwise}\end{cases}.$

######  Remark 1.4.

It follows from Lemma 2.3 that if the acting group $G$ is finite then the
classical mean dimension of $(K^{G},\sigma_{G})$ satisfies the following
equality:

$\displaystyle\operatorname{\mathrm{mdim}}(K^{G},\sigma_{G})=\begin{cases}\dim(K),&\text{if
$\dim(K\times K)=2\dim(K)$},\\\
\dim(K)-1+\frac{1}{|G|},&\text{otherwise}.\end{cases}$

In contrast to the above corollary which holds for actions of infinite
amenable groups, Theorem 1.1 is more satisfactory as it applies to (and
unifies the cases for) both finite and infinite sofic groups $G$.

###  1.4. Double finite actions

The simplest examples of group actions (probably) are double finite actions,
namely, actions of finite groups on finite dimensional compact metrizable
spaces.

For an action of an amenable group $G$ on a compact metrizable space $X$ we
denote its classical mean dimension by $\operatorname{\mathrm{mdim}}(X,G)$. Li
[Li13, Section 3] showed (for any sofic approximation sequence $\Sigma$ for
$G$) that

$\operatorname{\mathrm{mdim}}_{\Sigma}(X,G)\leq\operatorname{\mathrm{mdim}}(X,G);$

if in addition, $G$ is infinite, then

$\operatorname{\mathrm{mdim}}_{\Sigma}(X,G)=\operatorname{\mathrm{mdim}}(X,G).$

However, for a finite group $G$ (which is automatically amenable and sofic)
this equality might be false. Indeed, Li [Li13, Section 3] provided a bunch of
examples of double finite actions, for which, the above inequality is strict.
So a natural and further question we may ask is to decide when (i.e. for which
kind of sofic group actions) these two values will agree.

More precisely, we let a finite group $G$ act continuously on a finite
dimensional compact metrizable space $X$. Note that in this case, by the
definition of classical mean dimension we have apparently that

$\operatorname{\mathrm{mdim}}(X,G)=\frac{1}{|G|}\cdot\dim(X).$

Nevertheless, the exact value of its sofic mean dimension (with respect to a
sofic approximation sequence $\Sigma$ for $G$) is not clear to us. In fact, Li
[Li13, Lemma 3.9] gave an upper bound for
$\operatorname{\mathrm{mdim}}_{\Sigma}(X,G)$, which is finer than
$\operatorname{\mathrm{mdim}}(X,G)$, as follows:

$\operatorname{\mathrm{mdim}}_{\Sigma}(X,G)\leq\frac{1}{|G|}\cdot\inf_{n\in\mathbb{N}}\frac{\dim(X^{n})}{n}.$

The second main result of this paper is to solve this problem completely,
strengthening Li’s estimate with an equality:

$\operatorname{\mathrm{mdim}}_{\Sigma}(X,G)=\frac{1}{|G|}\cdot\inf_{n\in\mathbb{N}}\frac{\dim(X^{n})}{n}.$

######  Theorem 1.5 (Main theorem 2).

If a finite group $G$ acts continuously on a finite dimensional compact
metrizable space $X$ and if $\Sigma$ is a sofic approximation sequence for
$G$, then

$\operatorname{\mathrm{mdim}}_{\Sigma}(X,G)=\begin{cases}\frac{1}{|G|}\cdot\dim(X),&\text{if
$\dim(X\times X)=2\dim(X)$}\\\
\frac{1}{|G|}\cdot(\dim(X)-1),&\text{otherwise}\end{cases}.$

######  Corollary 1.6 (Comparison theorem).

Suppose that a finite group $G$ acts continuously on a finite dimensional
compact metrizable space $X$ and that $\Sigma$ is a sofic approximation
sequence for $G$. Then
$\operatorname{\mathrm{mdim}}_{\Sigma}(X,G)=\operatorname{\mathrm{mdim}}(X,G)$
(resp.
$\operatorname{\mathrm{mdim}}_{\Sigma}(X,G)<\operatorname{\mathrm{mdim}}(X,G)$)
if and only if $\dim(X\times X)=2\dim(X)$ (resp. $\dim(X\times X)<2\dim(X)$).

######  Remark 1.7.

As we can see, Li’s theorem [Li13, Section 3], Theorem 1.1 and Theorem 1.5, in
particular, present a typical class of sofic group actions, for which, sofic
mean dimension does not depend on the sofic approximation sequences, while it
has not yet been confirmed if this is still true for all sofic group actions.

###  1.5. Open questions

To end this section we pose the following open problems in this direction,
which seem to be worth considering in order to have a complete understanding
of sofic mean dimension of full shifts and amenable group actions.

  * (i)

Let $K$ be an infinite dimensional compact metrizable space. Is the statement
$\operatorname{\mathrm{mdim}}_{\Sigma}(K^{G},\sigma_{G})=+\infty$ true?

  * (ii)

Let a finite group $G$ act continuously on an infinite dimensional compact
metrizable space $X$. Is the statement
$\operatorname{\mathrm{mdim}}_{\Sigma}(X,G)=+\infty$ true?

  * (iii)

Let an infinite sofic group $G$ act continuously on a finite dimensional
compact metrizable space $X$. Is the statement
$\operatorname{\mathrm{mdim}}_{\Sigma}(X,G)\leq 0$ true?

We give some remarks here around this topic. The first problem is concerning
sofic mean dimension of full shifts over infinite dimensional alphabets. For
the case $G=\mathbb{Z}$ this problem was initially posed by Tsukamoto [Tsu19,
Problem 1.3]. The second and third problems are closely related to sofic mean
dimension of double finite actions, as they remove one of the finiteness
restrictions, respectively. Moreover, regarding Question (iii), it is clear
that $\operatorname{\mathrm{mdim}}_{\Sigma}(X,G)<+\infty$ (by definition); and
if $G$ is an infinite amenable group then the solution to Question (iii) is
obviously affirmative (by the definition of classical mean dimension and by
Li’s theorem [Li13, Section 3] stating that sofic mean dimension coincides
with classical mean dimension provided the acting group is infinite and
amenable). Finally, we conjecture that the answers to Questions (i) and (ii)
are positive, while the answer to Question (iii) is negative. We hope to
return to further research in relation to all the above questions in the
future.

### Acknowledgements

L. Jin was supported by Basal Funding AFB 170001 and Fondecyt Grant No.
3190127. Y. Qiao was supported by NNSF of China No. 11901206.

##  2\. A brief review of sofic mean dimension

###  2.1. Historical background

Both the notions of sofic groups and mean dimension originate with Misha
Gromov [Gro99a, Gro99b] in 1999. Mean dimension is a fairly good numerical
topological invariant of dynamical systems. It was investigated systematically
within the framework of amenable group actions by Lindenstrauss and Weiss
[LW00] (including wonderful applications to dynamical systems) around 2000. As
an analogue of topological dimension (i.e. the Lebesgue covering dimension),
its advantage has now been shown excellently in the study of dynamical systems
whose topological entropy is equal to infinity and whose phase space has
infinite topological dimension. In 2013, Li [Li13] successfully extended the
definition of mean dimension to the context of sofic group actions. Strictly
speaking, Li [Li13] introduced a new concept with a fully novel view, called
sofic mean dimension, which applies to all sofic group actions; and further,
Li [Li13] established its connection with classical mean dimension as well,
which is highly satisfactory and which we have mentioned in the previous
section. It is worth pointing out that Li’s generalization to the soficity
setting is totally nontrivial. This section is dedicated to fundamental
material on terminologies and notations. The writing of this section is mostly
borrowed, for consistency, from the paper [JQ] by the authors, with necessary
modifications adapting to the present paper.

###  2.2. Group actions

Throughout this subsection we let $G$ be a topological group. The terminology
“$G$ acts continuously on (or simply, acts on) a compact metrizable space $X$”
is planned to be understood a continuous mapping

$\Phi:G\times X\to X,\quad(g,x)\mapsto gx$

satisfying the following conditions:

$\Phi(e,x)=x,\quad\Phi(gh,x)=\Phi(g,\Phi(h,x)),\quad\forall x\in X,\;\forall
g,h\in G,$

where $e$ is the identity element of the group $G$. We generally omit the
mapping $\Phi$ provided we have already gotten the action of $G$ on $X$ clear.

Full shifts are a typical class of group actions. This notion has appeared in
Theorem 1.1. Let $K$ be a compact metrizable space. The shift action of $G$ on
the product space $K^{G}$ is defined as follows:

$\sigma_{G}:G\times K^{G}\to K^{G},\quad(g,(x_{h})_{h\in
G})\mapsto(x_{hg})_{h\in G}.$

Here $K$ is usually called an alphabet.

Let $G$ act continuously on compact metrizable spaces $X_{n}$, respectively,
where $n$ ranges over $R$ which is some subset (finite or infinite) of
$\mathbb{N}$. We define the product action of $G$ on the product space
$\prod_{n\in R}X_{n}$ as follows:

$g(x_{n})_{n\in R}=(gx_{n})_{n\in R},\quad\forall g\in
G,\;\forall(x_{n})_{n\in R}\in\prod_{n\in R}X_{n}.$

Product actions will be considered in the sequel (e.g. the proofs of the main
theorems).

###  2.3. Sofic groups

We denote by $|F|$ the cardinality of a set $F$. For every $d\in\mathbb{N}$ we
write $[d]$ for the set $\\{k\in\mathbb{N}:1\leq k\leq d\\}$ and
$\operatorname{\mathrm{Sym}}(d)$ for the group of permutations of $[d]$. A
group $G$ is sofic if there is a sequence

$\Sigma=\\{\sigma_{i}:G\to\operatorname{\mathrm{Sym}}(d_{i})\\}_{i\in\mathbb{N}}$

together with a sequence $\\{d_{i}\\}_{i\in\mathbb{N}}\subset\mathbb{N}$ such
that the following three conditions are satisfied:

$\displaystyle\bullet\quad\quad\lim_{i\to\infty}\frac{1}{d_{i}}|\\{k\in[d_{i}]:\sigma_{i}(st)(k)=\sigma_{i}(s)\sigma_{i}(t)(k)\\}|=1\quad\text{for
all $s,t\in G$;}$
$\displaystyle\bullet\quad\quad\lim_{i\to\infty}\frac{1}{d_{i}}|\\{k\in[d_{i}]:\sigma_{i}(s)(k)\neq\sigma_{i}(t)(k)\\}|=1\quad\text{for
all distinct $s,t\in G$;}$
$\displaystyle\bullet\quad\quad\lim_{i\to\infty}d_{i}=+\infty.$

Such a sequence $\Sigma$ is called a sofic approximation sequence for $G$.

######  Remark 2.1.

Note that the third condition will be fulfilled automatically if we
additionally assume the group $G$ to be infinite.

######  Remark 2.2.

The sofic groups form a fairly extensive class, which contain in particular
all amenable groups and all residually finite groups. We notice that the
existence of a non-sofic group is still unknown.

###  2.4. A dichotomy in dimension theory

We denote by $\dim(K)$ the topological dimension (i.e. the Lebesgue covering
dimension) of a compact metrizable space $K$. If the space $K$ is empty, then
we set $\dim(K)=-\infty$. For a finite dimensional (nonempty) compact
metrizable space $K$, since it was classically known [Dra01] that

$2\dim(K)-1\leq\dim(K\times K)\leq 2\dim(K)$

and since $\dim(K)$ must be a nonnegative integer, we have either

$\dim(K\times K)=2\dim(K)$

or

$\dim(K\times K)=2\dim(K)-1.$

For a friendly treatment of the following result [Dra01] in dimension theory
we refer to [Tsu19, Theorem 2.5].

######  Lemma 2.3.

Let $K$ be a finite dimensional compact metrizable space. Then for every
$n\in\mathbb{N}$

$\dim(K^{n})=\begin{cases}n\dim(K),&\text{if $K$ satisfies $\dim(K\times
K)=2\dim(K)$},\\\ n\dim(K)-n+1,&\text{otherwise}.\end{cases}$

###  2.5. $\epsilon$-embeddings, $\rho_{2}$ and $\rho_{\infty}$

Let $X$ and $P$ be two compact metrizable spaces, and let $\rho$ be a
compatible metric on $X$. For $\epsilon>0$ a continuous mapping $f:X\to P$ is
called an $\epsilon$-embedding with respect to $\rho$ if $f(x)=f(x^{\prime})$
implies $\rho(x,x^{\prime})<\epsilon$, for all $x,x^{\prime}\in X$. Let
$\operatorname{\mathrm{Widim}}_{\epsilon}(X,\rho)$ be the minimum
(topological) dimension $\dim(P)$ of a compact metrizable space $P$ which
admits an $\epsilon$-embedding $f:X\to P$ with respect to $\rho$.

######  Remark 2.4.

We may verify that $\dim(X)=\lim_{\epsilon\to
0}\operatorname{\mathrm{Widim}}_{\epsilon}(X,\rho)$.

Let $K$ be a compact metrizable space with a compatible metric $\rho$ on it.
For every $n\in\mathbb{N}$ we define on the product space $K^{n}$ two
compatible metrics $\rho_{2}$ and $\rho_{\infty}$ as follows:

$\rho_{2}\left((x_{i})_{i\in[n]},(y_{i})_{i\in[n]}\right)=\sqrt{\frac{1}{n}\sum_{i\in[n]}(\rho(x_{i},y_{i}))^{2}},$
$\rho_{\infty}\left((x_{i})_{i\in[n]},(y_{i})_{i\in[n]}\right)=\max_{i\in[n]}\rho(x_{i},y_{i}).$

We do not include $n\in\mathbb{N}$ in the notations $\rho_{2}$ and
$\rho_{\infty}$ because this does not cause any ambiguity.

The following result due to Tsukamoto [Tsu19, Lemma 3.1] is very practical. We
shall employ it in our argument.

######  Theorem 2.5.

Let $K$ be a finite dimensional compact metrizable space. Let $\rho$ be a
compatible metric on $K$. Then there is some $\delta>0$ such that

$\operatorname{\mathrm{Widim}}_{\epsilon}(K^{n},\rho_{\infty})\geq
n\cdot(\dim(K)-1)$

for all $n\in\mathbb{N}$ and all $0<\epsilon<\delta$.

###  2.6. Amenable group actions and classical mean dimension

A group $G$ is said to be amenable if there exists a sequence
$\\{F_{n}\\}_{n\in\mathbb{N}}$ of nonempty finite subsets of $G$ such that for
any $g\in G$

$\lim_{n\to\infty}\frac{|F_{n}\triangle gF_{n}|}{|F_{n}|}=0.$

Such a sequence $\\{F_{n}\\}_{n\in\mathbb{N}}$ is called a Følner sequence of
the group $G$.

Let an amenable group $G$ act continuously on a compact metrizable space $X$.
Take a Følner sequence $\\{F_{n}\\}_{n\in\mathbb{N}}$ of $G$ and a compatible
metric $\rho$ on $X$. For a nonempty finite subset $F$ of $G$ we set

$\rho_{F}(x,x^{\prime})=\rho_{\infty}\left((gx)_{g\in F},(gx^{\prime})_{g\in
F}\right),\quad\forall\,x,x^{\prime}\in X.$

It is clear that $\rho_{F}$ is also a compatible metric on $X$. The mean
dimension of $(X,G)$ is defined by

$\operatorname{\mathrm{mdim}}(X,G)=\lim_{\epsilon\to
0}\lim_{n\to\infty}\frac{\operatorname{\mathrm{Widim}}_{\epsilon}(X,\rho_{F_{n}})}{|F_{n}|}.$

It is well known that the limits in the above definition always exist. The
value $\operatorname{\mathrm{mdim}}(X,G)$ is independent of the choices of a
Følner sequence $\\{F_{n}\\}_{n\in\mathbb{N}}$ of $G$ and a compatible metric
$\rho$ on $X$.

###  2.7. Sofic mean dimension and its relation to classical mean dimension

Suppose that a sofic group $G$ acts continuously on a compact metrizable space
$X$. Let

$\Sigma=\\{\sigma_{i}:G\to\operatorname{\mathrm{Sym}}(d_{i})\\}_{i\in\mathbb{N}}$

be a sofic approximation sequence for $G$. We equip $X$ with a compatible
metric $\rho$. For a finite subset $F$ of $G$, $\delta>0$ and a map
$\sigma:G\to\operatorname{\mathrm{Sym}}(d)$ (where $d\in\mathbb{N}$) we define

$\operatorname{\mathrm{Map}}(\rho,F,\delta,\sigma)=\\{\phi:[d]\to
X:\rho_{2}(\phi\circ\sigma(s),s\phi)\leq\delta,\,\forall s\in F\\}.$

We consider the set $\operatorname{\mathrm{Map}}(\rho,F,\delta,\sigma)$ as a
compact subspace of the product space $X^{d}$. In our context we usually write
$\phi=(\phi_{l})_{l\in[d]}\in X^{d}$ for $\phi:[d]\to X$ (i.e. for $l\in[d]$
we write $\phi_{l}$ for $\phi(l)$). The sofic mean dimension of $(X,G)$ is
defined by

$\operatorname{\mathrm{mdim}}_{\Sigma}(X,G)=\sup_{\epsilon>0}\inf_{F\subset
G\text{ finite,
}\,\delta>0}\limsup_{i\to\infty}\frac{\operatorname{\mathrm{Widim}}_{\epsilon}\left(\operatorname{\mathrm{Map}}(\rho,F,\delta,\sigma_{i}),\rho_{\infty}\right)}{d_{i}}.$

Similar to the classical mean dimension, the definition of
$\operatorname{\mathrm{mdim}}_{\Sigma}(X,G)$ does not depend on the compatible
metrics $\rho$ on $X$. Nevertheless, as we mentioned previously, it is not
clear yet if there is an example of a sofic approximation sequence
$\Sigma^{\prime}$ different from $\Sigma$, which leads to a different value
$\operatorname{\mathrm{mdim}}_{\Sigma^{\prime}}(X,G)$.

Li [Li13, Section 3] built a satisfying bridge between sofic mean dimension
and classical mean dimension as follows.

######  Theorem 2.6.

If an infinite amenable group $G$ acts continuously on a compact metrizable
space $X$ then
$\operatorname{\mathrm{mdim}}_{\Sigma}(X,G)=\operatorname{\mathrm{mdim}}(X,G)$.

For product actions of sofic groups Li [Li13, Section 2] showed a useful
inequality for sofic mean dimension:

######  Lemma 2.7.

Let a sofic group $G$ act continuously on compact metrizable spaces $X_{n}$,
respectively, where $n$ runs over some $R\subset\mathbb{N}$. Let $\Sigma$ be a
sofic approximation sequence for $G$. For the product action of $G$ on the
product space $\prod_{n\in R}X_{n}$ we have the following subadditivity
$\operatorname{\mathrm{mdim}}_{\Sigma}(\prod_{n\in R}X_{n},G)\leq\sum_{n\in
R}\operatorname{\mathrm{mdim}}_{\Sigma}(X_{n},G)$.

##  3\. Proof of Theorem 1.1

Theorem 1.1 will follow directly from Lemma 2.3, Lemma 3.1 and Lemma 3.2. Let
us start with settings. We let $K$ be a finite dimensional compact metrizable
space, $G$ a sofic group and

$\Sigma=\\{\sigma_{i}:G\to\operatorname{\mathrm{Sym}}(d_{i})\\}_{i\in\mathbb{N}}$

a sofic approximation sequence for $G$. What we are dealing with is the shift
action $\sigma_{G}$ of $G$ on the product space $K^{G}$.

We denote by $e$ the identity element of the group $G$. We fix a (countable)
family $\\{\alpha_{g}\\}_{g\in G}$ of positive real numbers such that

$\alpha_{e}=1,\quad\quad\sum_{g\in G}\alpha_{g}<2.$

We fix a compatible metric $D$ on the alphabet $K$ and take a metric $\rho$ on
$K^{G}$ compatible with the product topology as follows:

$\rho(x,y)=\sum_{g\in G}\alpha_{g}D(x_{g},y_{g}),\quad(x=(x_{g})_{g\in
G},y=(y_{g})_{g\in G}\in K^{G}).$

For simplicity we write $\operatorname{\mathrm{diam}}(K^{G})$ for the diameter
of the product space $K^{G}$ with respect to the metric $\rho$. Without loss
of generality we may assume $\operatorname{\mathrm{diam}}(K^{G})=1$.

######  Lemma 3.1 (Estimate from above).

$\operatorname{\mathrm{mdim}}_{\Sigma}(K^{G},\sigma_{G})\leq\inf_{n\in\mathbb{N}}\frac{\dim(K^{n})}{n}.$

###### Proof.

First of all, we note that

$\inf_{n\in\mathbb{N}}\frac{\dim(K^{n})}{n}=\lim_{i\to\infty}\frac{\dim(K^{d_{i}})}{d_{i}},$

because $\dim(\cdot\times\cdot)$ (and in particular, $\dim(K^{n})$ with
respect to $n\in\mathbb{N}$) is subadditive and because $d_{i}\to+\infty$ as
$i\to\infty$.

We take $\epsilon>0$ and $\eta>0$ arbitrarily and fix them temporarily. Since
both of them are arbitrary, it suffices (by definition) to prove that there
exist a finite (and nonempty) subset $F_{0}$ of $G$ and some $\delta_{0}>0$
satisfying that

$\limsup_{i\to\infty}\frac{\operatorname{\mathrm{Widim}}_{\epsilon}\left(\operatorname{\mathrm{Map}}(\rho,F_{0},\delta_{0},\sigma_{i}),\rho_{\infty}\right)}{d_{i}}\leq\eta+\lim_{i\to\infty}\frac{\dim(K^{d_{i}})}{d_{i}}.$

We choose a finite subset $F_{0}$ of $G$, containing the identity element $e$,
such that if $\xi=(\xi_{g})_{g\in G}$ and
$\xi^{\prime}=(\xi^{\prime}_{g})_{g\in G}$ coming from $K^{G}$ satisfy
$D(\xi_{s},\xi^{\prime}_{s})\leq\epsilon/2$ for all $s\in F_{0}$, then they
satisfy $\rho(\xi,\xi^{\prime})<\epsilon$. We take $\delta_{0}>0$ sufficiently
small such that

$\delta_{0}<\min\left\\{\frac{\eta}{(\dim(K)+1)\cdot|F_{0}|^{2}},\;\frac{\epsilon^{2}}{16}\right\\}.$

So now the main task is to estimate the term
$\operatorname{\mathrm{Widim}}_{\epsilon}(\operatorname{\mathrm{Map}}(\rho,F_{0},\delta_{0},\sigma_{i}),\rho_{\infty})$,
for all sufficiently large $i\in\mathbb{N}$, from above.

We fix $i\in\mathbb{N}$ for the moment.

For every $j\in[d_{i}]$ we define a mapping

$J_{j}:\operatorname{\mathrm{Map}}(\rho,F_{0},\delta_{0},\sigma_{i})\to[0,1]$

by sending

$\phi=(\phi_{j})_{j\in[d_{i}]}\in\operatorname{\mathrm{Map}}(\rho,F_{0},\delta_{0},\sigma_{i})\subset(K^{G})^{d_{i}}$

to

$J_{j}(\phi)=\max\left\\{\max_{s\in
F_{0}}\left(\rho(s\phi_{j},\phi_{\sigma_{i}(s)(j)})-\sqrt{\delta_{0}}\right),0\right\\},$

where $\phi_{j}=(\phi_{j}(g))_{g\in G}\in K^{G}$. Clearly, each mapping
$J_{j}:\operatorname{\mathrm{Map}}(\rho,F_{0},\delta_{0},\sigma_{i})\to[0,1]$
is continuous.

Let $CK=([0,1]\times K)/\sim$ be the cone generated by $K$, where
$(0,x)\sim(0,x^{\prime})$ for all $x,x^{\prime}\in K$. We denote by $\lambda
x$ the equivalence class of $(\lambda,x)\in[0,1]\times K$. We set $\ast=0x$
and call the symbol $\ast$ the vertex of the cone. We notice that the
following inequality is clear:

$\dim(K)\leq\dim(CK)\leq\dim(K)+1.$

We construct a mapping $H_{i}$ as follows:

$H_{i}:\operatorname{\mathrm{Map}}(\rho,F_{0},\delta_{0},\sigma_{i})\to
K^{d_{i}}\times(CK)^{|F_{0}|\cdot d_{i}},$
$\phi=(\phi_{j})_{j\in[d_{i}]}\,\mapsto\,\left(\left(\phi_{j}(e)\right)_{j\in[d_{i}]},\;\left(J_{j}(\phi)\phi_{j}|_{F_{0}}\right)_{j\in[d_{i}]}\right),$

where we write $\phi_{j}|_{F_{0}}$ for $(\phi_{j}(s))_{s\in F_{0}}\in
K^{F_{0}}$, which is the restriction of $\phi_{j}\in K^{G}$ to $K^{F_{0}}$,
and where we write $J_{j}(\phi)\phi_{j}|_{F_{0}}$ for
$(J_{j}(\phi)\phi_{j}(s))_{s\in F_{0}}\in(CK)^{|F_{0}|}$. Obviously,
$H_{i}:\operatorname{\mathrm{Map}}(\rho,F_{0},\delta_{0},\sigma_{i})\to
K^{d_{i}}\times(CK)^{|F_{0}|\cdot d_{i}}$ is a continuous mapping (since every
mapping
$J_{j}:\operatorname{\mathrm{Map}}(\rho,F_{0},\delta_{0},\sigma_{i})\to[0,1]$
($j\in[d_{i}]$) is continuous).

We now claim that
$H_{i}:\operatorname{\mathrm{Map}}(\rho,F_{0},\delta_{0},\sigma_{i})\to
K^{d_{i}}\times(CK)^{|F_{0}|\cdot d_{i}}$ is an $\epsilon$-embedding with
respect to $\rho_{\infty}$.

To verify this claim, we take $\varphi=(\varphi_{j})_{j\in[d_{i}]}$ and
$\psi=(\psi_{j})_{j\in[d_{i}]}$ in
$\operatorname{\mathrm{Map}}(\rho,F_{0},\delta_{0},\sigma_{i})$ and we assume
that $H_{i}(\varphi)=H_{i}(\psi)$. We then need to show that
$\rho_{\infty}(\varphi,\psi)<\epsilon$. In fact, it follows from
$H_{i}(\varphi)=H_{i}(\psi)$ that for all $j\in[d_{i}]$

$\varphi_{j}(e)=\psi_{j}(e),\quad\quad
J_{j}(\varphi)\varphi_{j}|_{F_{0}}=J_{j}(\psi)\psi_{j}|_{F_{0}}.$

For each $j\in[d_{i}]$ we have two cases:

  * •

either
$J_{j}(\varphi)\varphi_{j}|_{F_{0}}=J_{j}(\psi)\psi_{j}|_{F_{0}}=\ast^{|F_{0}|}$,
i.e. $J_{j}(\varphi)=J_{j}(\psi)=0$;

  * •

or
$J_{j}(\varphi)\varphi_{j}|_{F_{0}}=J_{j}(\psi)\psi_{j}|_{F_{0}}\in(CK\setminus\\{\ast\\})^{|F_{0}|}$,
i.e. $J_{j}(\varphi)=J_{j}(\psi)>0$;

where we denote by the notation $\ast^{|F_{0}|}$ the unique element
$(\ast,\ast,\dots,\ast)$ in the set $\\{\ast\\}^{|F_{0}|}$. If some
$j\in[d_{i}]$ encounters the former case, then

$\rho(s\varphi_{j},\varphi_{\sigma_{i}(s)(j)})\leq\sqrt{\delta_{0}},\quad\rho(s\psi_{j},\psi_{\sigma_{i}(s)(j)})\leq\sqrt{\delta_{0}},\quad\quad\forall
s\in F_{0}.$

Since $s\varphi_{j}(e)=\varphi_{j}(s)$ and $s\psi_{j}(e)=\psi_{j}(s)$, this
implies that

$D(\varphi_{j}(s),\varphi_{\sigma_{i}(s)(j)}(e))\leq\sqrt{\delta_{0}},\quad
D(\psi_{j}(s),\psi_{\sigma_{i}(s)(j)}(e))\leq\sqrt{\delta_{0}},\quad\quad\forall
s\in F_{0}.$

We note that

$\varphi_{l}(e)=\psi_{l}(e),\quad\quad\forall l\in[d_{i}].$

Hence we have

$D(\varphi_{j}(s),\psi_{j}(s))\leq 2\sqrt{\delta_{0}}<\epsilon/2,\quad\forall
s\in F_{0}.$

Thus, we deduce from the choice of $F_{0}$ that in this case

$\rho(\varphi_{j},\psi_{j})<\epsilon.$

If some $j\in[d_{i}]$ encounters the latter case, then it follows directly
from the definition of $CK$ that

$\varphi_{j}|_{F_{0}}=\psi_{j}|_{F_{0}}.$

By the choice of $F_{0}$, this implies that in this case we also have

$\rho(\varphi_{j},\psi_{j})<\epsilon.$

Therefore we conclude that

$\rho_{\infty}(\varphi,\psi)=\max_{j\in[d_{i}]}\rho(\varphi_{j},\psi_{j})<\epsilon.$

This proves the claim.

By this claim we get an upper bound for
$\operatorname{\mathrm{Widim}}_{\epsilon}(\operatorname{\mathrm{Map}}(\rho,F_{0},\delta_{0},\sigma_{i}),\rho_{\infty})$:

$\operatorname{\mathrm{Widim}}_{\epsilon}(\operatorname{\mathrm{Map}}(\rho,F_{0},\delta_{0},\sigma_{i}),\rho_{\infty})\leq\dim(H_{i}(\operatorname{\mathrm{Map}}(\rho,F_{0},\delta_{0},\sigma_{i}))).$

So it remains to deal with the term
$\dim(H_{i}(\operatorname{\mathrm{Map}}(\rho,F_{0},\delta_{0},\sigma_{i})))$.

Here let us put a slightly more detailed explanation in relation to the
construction of $H_{i}$. We take an arbitrary $j\in[d_{i}]$. Intuitively
speaking, for any
$\varphi\in\operatorname{\mathrm{Map}}(\rho,F_{0},\delta_{0},\sigma_{i})$ if
$J_{j}(\varphi)=0$ then we say that $\varphi_{j}$ is good, because the family
$\\{\varphi_{k}(e):k\in[d_{i}]\\}$ is able to tell almost everything fairly
well about $\varphi_{j}\in K^{G}$ (i.e. it covers information sufficiently
close to $\\{\varphi_{j}(s):s\in F_{0}\\}$), while if $J_{j}(\varphi)>0$ then
we say that $\varphi_{j}$ is bad, because in this case the needed information
indicated by the family $\\{\varphi_{k}(e):k\in[d_{i}]\\}$ is far from
necessary for admitting an $\epsilon$-embedding unless we additionally record
almost the whole $\varphi_{j}$ (namely, $\\{\varphi_{j}(s):s\in F_{0}\\}$
itself). But when we look at any fixed $j\in[d_{i}]$ it is always possible for
$\varphi_{j}$ to be good or bad as
$\varphi\in\operatorname{\mathrm{Map}}(\rho,F_{0},\delta_{0},\sigma_{i})$
changes. So seemingly we have to record a quite large rectangle
$\\{\varphi_{k}(s):s\in F_{0},k\in[d_{i}]\\}$. However, if we simply record
nearly everything then the dimension will be absolutely out of control. The
point here is that for a given $j\in[d_{i}]$ we cannot expect $\varphi_{j}$ to
move in a continuous way between good and bad regions as $\varphi$ moves
within $\operatorname{\mathrm{Map}}(\rho,F_{0},\delta_{0},\sigma_{i})$. This
is why we need consider the cone $CK$ when building $H_{i}$, as to recording
every $\varphi_{j}$ continuously in a more flexible and productive approach.

Next we estimate
$\dim(H_{i}(\operatorname{\mathrm{Map}}(\rho,F_{0},\delta_{0},\sigma_{i})))$
from above. For any $\phi=(\phi_{j})_{j\in[d_{i}]}$ coming from
$\operatorname{\mathrm{Map}}(\rho,F_{0},\delta_{0},\sigma_{i})$ we set

$\Omega_{\phi}(\rho,F_{0},\delta_{0},\sigma_{i})=[d_{i}]\setminus\\{j\in[d_{i}]:\rho(s\phi_{j},\phi_{\sigma_{i}(s)(j)})\leq\sqrt{\delta_{0}},\;\,\forall
s\in F_{0}\\}.$

This is a subset of $[d_{i}]$. On the one hand, since
$\phi=(\phi_{j})_{j\in[d_{i}]}\in\operatorname{\mathrm{Map}}(\rho,F_{0},\delta_{0},\sigma_{i})$,
we have

$\sum_{j\in[d_{i}]}\rho(s\phi_{j},\phi_{\sigma_{i}(s)(j)})^{2}\leq\delta_{0}^{2}\cdot
d_{i},\quad\forall\,s\in F_{0}$

and therefore

$\delta_{0}\cdot|\Omega_{\phi}(\rho,F_{0},\delta_{0},\sigma_{i})|\leq\sum_{j\in[d_{i}]}\sum_{s\in
F_{0}}\rho(s\phi_{j},\phi_{\sigma_{i}(s)(j)})^{2}\leq|F_{0}|\cdot\delta_{0}^{2}\cdot
d_{i}.$

It follows that

$0\leq|\Omega_{\phi}(\rho,F_{0},\delta_{0},\sigma_{i})|\leq|F_{0}|\cdot\delta_{0}\cdot
d_{i}.$

On the other hand, we notice that for all
$j\in[d_{i}]\setminus\Omega_{\phi}(\rho,F_{0},\delta_{0},\sigma_{i})$ we have
$J_{j}(\phi)=0$. This implies that for any given
$\phi\in\operatorname{\mathrm{Map}}(\rho,F_{0},\delta_{0},\sigma_{i})$ there
are $(d_{i}-|\Omega_{\phi}(\rho,F_{0},\delta_{0},\sigma_{i})|)\cdot|F_{0}|$
entries of $H_{i}(\phi)$, which will take the value $\ast$. More precisely,
the compact metrizable space
$H_{i}(\operatorname{\mathrm{Map}}(\rho,F_{0},\delta_{0},\sigma_{i}))$ is
contained in the compact metrizable space

$\bigcup_{l\in\mathbb{Z},\;\,0\leq l\leq|F_{0}|\cdot\delta_{0}\cdot
d_{i}}\,\;\bigcup_{k_{1},\dots,k_{d_{i}}\in\\{0,1\\},\;\,k_{1}+\cdots+k_{d_{i}}=l}\,\;K^{d_{i}}\times\prod_{j=1}^{d_{i}}(\\{\ast\\}^{|F_{0}|})^{1-k_{j}}\times((CK)^{|F_{0}|})^{k_{j}},$

where $\\{\ast\\}^{|F_{0}|}\times((CK)^{|F_{0}|})^{0}=\\{\ast\\}^{|F_{0}|}$
and $(\\{\ast\\}^{|F_{0}|})^{0}\times(CK)^{|F_{0}|}=(CK)^{|F_{0}|}$. Therefore
we deduce that

$\displaystyle\dim(H_{i}(\operatorname{\mathrm{Map}}(\rho,F_{0},\delta_{0},\sigma_{i})))$
$\displaystyle\leq\max_{0\leq l\leq|F_{0}|\cdot\delta_{0}\cdot
d_{i}}\left(\dim(K^{d_{i}})+l\cdot|F_{0}|\cdot\dim(CK)\right)$
$\displaystyle\leq\dim(K^{d_{i}})+|F_{0}|^{2}\cdot\delta_{0}\cdot
d_{i}\cdot(\dim(K)+1).$

We note that here we have used the following classically known equality and
inequality: For any compact metrizable spaces $A$ and $A^{\prime}$

$\dim(A\cup A^{\prime})=\max\\{\dim(A),\dim(A^{\prime})\\},\quad\dim(A\times
A^{\prime})\leq\dim(A)+\dim(A^{\prime}).$

Thus, we obtain that

$\operatorname{\mathrm{Widim}}_{\epsilon}(\operatorname{\mathrm{Map}}(\rho,F_{0},\delta_{0},\sigma_{i}),\rho_{\infty})\leq\dim(K^{d_{i}})+|F_{0}|^{2}\cdot\delta_{0}\cdot
d_{i}\cdot(\dim(K)+1).$

Since $i\in\mathbb{N}$ is arbitrary, we conclude (from the choice of
$\delta_{0}$) that

$\limsup_{i\to\infty}\frac{\operatorname{\mathrm{Widim}}_{\epsilon}\left(\operatorname{\mathrm{Map}}(\rho,F_{0},\delta_{0},\sigma_{i}),\rho_{\infty}\right)}{d_{i}}\leq\eta+\lim_{i\to\infty}\frac{\dim(K^{d_{i}})}{d_{i}}.$

This is as desired. ∎

######  Lemma 3.2 (Estimate from below).

$\operatorname{\mathrm{mdim}}_{\Sigma}(K^{G},\sigma_{G})\geq\inf_{n\in\mathbb{N}}\frac{\dim(K^{n})}{n}.$

###### Proof.

First of all, let us make a reduction. In order to show the statement it
suffices to prove

$\operatorname{\mathrm{mdim}}_{\Sigma}(K^{G},\sigma_{G})\geq\dim(K)-1.$

In fact, such an inequality will apply to all finite dimensional compact
metrizable spaces $K$, in particular to $K^{n}$, for any $n\in\mathbb{N}$,
which will indicate that

$\operatorname{\mathrm{mdim}}_{\Sigma}((K^{n})^{G},\sigma_{G})\geq\dim(K^{n})-1,\quad\forall
n\in\mathbb{N}.$

Since the action $((K^{n})^{G},\sigma_{G})$ is (dynamically) isomorphic to the
product action of $\sigma_{G}$ on $(K^{G})^{n}$, we will have, by Lemma 2.7,
that

$\operatorname{\mathrm{mdim}}_{\Sigma}(K^{G},\sigma_{G})\geq\frac{\dim(K^{n})-1}{n},\quad\forall
n\in\mathbb{N}$

which will imply that

$\operatorname{\mathrm{mdim}}_{\Sigma}(K^{G},\sigma_{G})\geq\lim_{n\to\infty}\frac{\dim(K^{n})}{n}=\inf_{n\in\mathbb{N}}\frac{\dim(K^{n})}{n}.$

This is as required.

In the remaining part of the proof we prove

$\operatorname{\mathrm{mdim}}_{\Sigma}(K^{G},\sigma_{G})\geq\dim(K)-1.$

By Theorem 2.5 we can find some $\epsilon>0$ sufficiently small, which depends
only on the finite dimensional compact metric space $(K,D)$, such that for all
$n\in\mathbb{N}$

$\operatorname{\mathrm{Widim}}_{\epsilon}(K^{n},D_{\infty})\geq
n\cdot(\dim(K)-1).$

We fix it in the proof. We take $\delta>0$ and a (nonempty) finite subset $F$
of $G$ arbitrarily, and fix them in the proof. By definition it suffices to
show that

$\limsup_{i\to\infty}\frac{\operatorname{\mathrm{Widim}}_{\epsilon}(\operatorname{\mathrm{Map}}(\rho,F,\delta,\sigma_{i}),\rho_{\infty})}{d_{i}}\geq\dim(K)-1.$

For every $i\in\mathbb{N}$ we define a mapping $P_{i}$ as follows:

$P_{i}:K^{d_{i}}\to(K^{G})^{d_{i}},\quad
p=(p_{j})_{j\in[d_{i}]}\to\phi=(\phi_{j})_{j\in[d_{i}]},$
$\phi_{j}=(\phi_{j}(g))_{g\in G}\in K^{G},\quad\forall j\in[d_{i}],$
$\phi_{j}(g)=\begin{cases}p_{\sigma_{i}(g)(j)},&\;\text{ if }\,g\in
G\setminus\\{e\\}\\\ p_{j},&\;\text{ if }\,g=e\end{cases}.$

Clearly, for any $i\in\mathbb{N}$ the mapping
$P_{i}:K^{d_{i}}\to(K^{G})^{d_{i}}$ is continuous and it is distance-
increasing with respect to the metric $D_{\infty}$ on $K^{d_{i}}$ and the
metric $\rho_{\infty}$ on $(K^{G})^{d_{i}}$, i.e.

$D_{\infty}(p,p^{\prime})\leq\rho_{\infty}(P_{i}(p),P_{i}(p^{\prime})),\quad\forall
p,p^{\prime}\in K^{d_{i}}.$

We observe that for all sufficiently large $i\in\mathbb{N}$ it will be true
that $P_{i}(K^{d_{i}})$ is contained in
$\operatorname{\mathrm{Map}}(\rho,F,\delta,\sigma_{i})$. (We will verify this
observation in a moment.) It follows that

$\displaystyle\limsup_{i\to\infty}\frac{\operatorname{\mathrm{Widim}}_{\epsilon}(\operatorname{\mathrm{Map}}(\rho,F,\delta,\sigma_{i}),\rho_{\infty})}{d_{i}}$
$\displaystyle\geq\limsup_{i\to\infty}\frac{\operatorname{\mathrm{Widim}}_{\epsilon}(P_{i}(K^{d_{i}}),\rho_{\infty})}{d_{i}}$
$\displaystyle\geq\limsup_{i\to\infty}\frac{\operatorname{\mathrm{Widim}}_{\epsilon}(K^{d_{i}},D_{\infty})}{d_{i}}$
$\displaystyle\geq\limsup_{i\to\infty}\frac{d_{i}\cdot(\dim(K)-1)}{d_{i}}$
$\displaystyle=\dim(K)-1.$

This will end the proof.

Finally, it remains to verify that for all sufficiently large $i\in\mathbb{N}$
we have that $P_{i}(K^{d_{i}})$ is a subset of
$\operatorname{\mathrm{Map}}(\rho,F,\delta,\sigma_{i})$. We choose a
(nonempty) finite subset $E$ of $G$, containing the identity element $e\in G$,
such that if $\xi=(\xi_{g})_{g\in G}$ and
$\xi^{\prime}=(\xi^{\prime}_{g})_{g\in G}$ in $K^{G}$ satisfy
$\xi_{g}=\xi^{\prime}_{g}$ for all $g\in E$, then they satisfy
$\rho(\xi,\xi^{\prime})<\delta/2$. For every $i\in\mathbb{N}$ we let

$Q_{i}=\left\\{j\in[d_{i}]:\sigma_{i}(s)\circ\sigma_{i}(t)(j)=\sigma_{i}(st)(j),\,\;\forall
s\in E,\,\forall t\in F\cup\\{e\\}\right\\}.$

Since both $E$ and $F$ are finite, there is some $k\in\mathbb{N}$ sufficiently
large such that for any integer $i>k$

$\frac{|Q_{i}|}{d_{i}}>1-\frac{3\delta^{2}}{8}.$

Now we show that

$P_{i}(K^{d_{i}})\subset\operatorname{\mathrm{Map}}(\rho,F,\delta,\sigma_{i}),\quad\;\forall
i>k.$

We fix an integer $i>k$ and take an arbitrary $p=(p_{j})_{j\in[d_{i}]}\in
K^{d_{i}}$. Let $\phi=P_{i}(p)$ and write $\phi=((\phi_{j}(g))_{g\in
G})_{j\in[d_{i}]}\in(K^{G})^{d_{i}}$. We note that for any $j\in Q_{i}$ we
have $\sigma_{i}(e)(j)=j$, and therefore it is clear that

$\phi_{j}(g)=p_{\sigma_{i}(g)(j)},\quad\quad\forall g\in G,\;\,\forall j\in
Q_{i}.$

It follows that for any $j\in Q_{i}\cap(\sigma_{i}(t))^{-1}(Q_{i})$ and any
$t\in F\cup\\{e\\}$

$t\phi_{j}(s)=\phi_{j}(st)=p_{\sigma_{i}(st)(j)}=p_{\sigma_{i}(s)\circ\sigma_{i}(t)(j)}=\phi_{\sigma_{i}(t)(j)}(s),\quad\forall
s\in E.$

By the choice of $E$ we have

$\rho(t\phi_{j},\phi_{\sigma_{i}(t)(j)})<\delta/2,\quad\quad\forall t\in
F,\;\;\forall j\in Q_{i}\cap(\sigma_{i}(t))^{-1}(Q_{i}).$

Thus, we conclude that for all $t\in F$

$\displaystyle\rho_{2}(\phi\circ\sigma_{i}(t),t\phi)$
$\displaystyle=\sqrt{\frac{1}{d_{i}}\cdot\sum_{j\in[d_{i}]}\left(\rho(\phi_{\sigma_{i}(t)(j)},t\phi_{j})\right)^{2}}$
$\displaystyle\leq\sqrt{\frac{\delta^{2}}{4}+\frac{\left|[d_{i}]\setminus(Q_{i}\cap(\sigma_{i}(t))^{-1}(Q_{i}))\right|}{d_{i}}}$
$\displaystyle\leq\sqrt{\frac{\delta^{2}}{4}+2\cdot\left(1-\frac{|Q_{i}|}{d_{i}}\right)}$
$\displaystyle\leq\delta.$

This implies that
$\phi\in\operatorname{\mathrm{Map}}(\rho,F,\delta,\sigma_{i})$. The statement
follows. ∎

##  4\. Proof of Theorem 1.5

The proof of Theorem 1.5 will be fulfilled by Lemma 2.3, Lemma 4.1 and Lemma
4.2. To begin with, we put some settings. Let $G$ be a finite group. Suppose
that $G$ acts continuously on a finite dimensional compact metrizable space
$X$. We fix a compatible metric $\rho$ on $X$ and a sofic approximation
sequence
$\Sigma=\\{\sigma_{i}:G\to\operatorname{\mathrm{Sym}}(d_{i})\\}_{i\in\mathbb{N}}$
for $G$. Without loss of generality, we assume
$\operatorname{\mathrm{diam}}(X)=1$.

######  Lemma 4.1 (Estimate from above).

$\operatorname{\mathrm{mdim}}_{\Sigma}(X,G)\leq\frac{1}{|G|}\cdot\inf_{n\in\mathbb{N}}\frac{\dim(X^{n})}{n}.$

###### Proof.

This was initially shown in [Li13]. We reproduce it here, with a slightly
simplified proof (in terms of $\operatorname{\mathrm{Widim}}$), for
completeness. The difference between Li’s proof and our proof is that we shall
consider $\epsilon$-embeddings into a cone.

We fix $\epsilon>0$ and $\theta>0$ arbitrarily. It suffices to show that there
is some $\delta_{0}>0$ such that

$\limsup_{i\to\infty}\frac{\operatorname{\mathrm{Widim}}_{\epsilon}(\operatorname{\mathrm{Map}}(\rho,G,\delta_{0},\sigma_{i}),\rho_{\infty})}{d_{i}}\leq\frac{1}{|G|}\cdot\inf_{n\in\mathbb{N}}\frac{\dim(X^{n})}{n}+\theta.$

For every $i\in\mathbb{N}$ we let $Q_{i}\subset[d_{i}]$ be the intersection of
the following two sets:

$\left\\{j\in[d_{i}]:\sigma_{i}(g)(j)\neq\sigma_{i}(h)(j),\;\forall g\neq h\in
G\right\\},$
$\left\\{j\in[d_{i}]:\sigma_{i}(g)(\sigma_{i}(h)(j))=\sigma_{i}(gh)(j),\;\forall
g,h\in G\right\\}.$

For any $j\in[d_{i}]$ and $A\subset[d_{i}]$ we set

$\sigma_{i}(G)(j)=\\{\sigma_{i}(g)(j):g\in
G\\}\subset[d_{i}],\quad\sigma_{i}(G)(A)=\\{\sigma_{i}(g)(a):g\in G,a\in
A\\}\subset[d_{i}].$

We note that by the definition of $Q_{i}$ we have

$|\sigma_{i}(G)(j)|=|G|,\quad\quad\sigma_{i}(G)(\sigma_{i}(G)(j))=\sigma_{i}(G)(j),\quad\quad\forall
j\in Q_{i}.$

It follows that $\sigma_{i}(G)(j)=\sigma_{i}(G)(l)$, for all
$l\in\sigma_{i}(G)(j)$. We put a $j\in Q_{i}$ into a set $C_{i}$ and remove
all the elements of $\sigma_{i}(G)(j)$ from $Q_{i}$. By dealing with the
resulting sets finitely many times (with the same method) we can find333We
notice that the situation here is very simple because $G$ is a finite group.
In a much more complicated case, i.e. where the acting group is amenable but
not necessarily finite, it is also possible to have such a one-to-one
correspondence. But we do not need this in our proof. For the general
statement we refer to [Li13, Section 3]. some $C_{i}\subset Q_{i}$ of the
maximum cardinality satisfying that the mapping

$G\times C_{i}\to[d_{i}],\quad(g,j)\mapsto\sigma_{i}(g)(j)$

is injective. We take $\tau>0$ with $\tau\cdot\dim(X)<\theta/2$. Since $G$ is
finite, there exists some $i_{0}\in\mathbb{N}$ such that

$|\sigma_{i}(G)(C_{i})|>(1-\tau)\cdot d_{i},\quad\forall i\geq i_{0}.$

We take $\delta_{0}>0$ satisfying that
$\delta_{0}\cdot|G|\cdot(\dim(X)+1)<\theta/2$ and that if two points
$x,x^{\prime}\in X$ satisfy $\rho(x,x^{\prime})\leq 2\sqrt{\delta_{0}}$ then
they satisfy $\max_{g\in G}\rho(gx,gx^{\prime})<\epsilon$.

We fix $i\in\mathbb{N}$ for the moment. For each $j\in[d_{i}]$ we define a
continuous mapping
$J_{j}:\operatorname{\mathrm{Map}}(\rho,G,\delta_{0},\sigma_{i})\to[0,1]$ by
sending
$\phi=(\phi_{j})_{j\in[d_{i}]}\in\operatorname{\mathrm{Map}}(\rho,G,\delta_{0},\sigma_{i})$
to

$J_{j}(\phi)=\max\left\\{\max_{g\in
G}\left(\rho(g\phi_{j},\phi_{\sigma_{i}(g)(j)})-\sqrt{\delta_{0}}\right),0\right\\}.$

Now we define a continuous mapping

$H_{i}:\operatorname{\mathrm{Map}}(\rho,G,\delta_{0},\sigma_{i})\to
X^{C_{i}}\times(CX)^{d_{i}}\times X^{[d_{i}]\setminus\sigma_{i}(G)(C_{i})},$
$\phi=(\phi_{j})_{j\in[d_{i}]}\mapsto\left((\phi_{j})_{j\in
C_{i}},(J_{j}(\phi)\phi_{j})_{j\in[d_{i}]},(\phi_{j})_{j\in[d_{i}]\setminus\sigma_{i}(G)(C_{i})}\right).$

We recall here that $CX=([0,1]\times X)/\sim$ denotes the cone generated by
$X$, where $(0,x)\sim(0,x^{\prime})$ for any $x,x^{\prime}\in X$, and that
$\lambda x$ is the equivalence class of $(\lambda,x)\in[0,1]\times X$ (in
particular, the symbol $\ast=0x$ is to denote the vertex of $CX$).

We claim that $H_{i}$ is an $\epsilon$-embedding with respect to
$\rho_{\infty}$. To verify this, we suppose that $H_{i}(\varphi)=H_{i}(\psi)$
for
$\varphi,\psi\in\operatorname{\mathrm{Map}}(\rho,G,\delta_{0},\sigma_{i})$. We
need to show $\rho(\varphi_{j},\psi_{j})<\epsilon$ for all $j\in[d_{i}]$. If
$j\in[d_{i}]$ satisfies $j\in[d_{i}]\setminus\sigma_{i}(G)(C_{i})$ or
$J_{j}(\varphi)=J_{j}(\psi)>0$, then it is clear that $\varphi_{j}=\psi_{j}$.
So we assume $j\in\sigma_{i}(G)(C_{i})$ and $J_{j}(\varphi)=J_{j}(\psi)=0$. We
write $j=\sigma_{i}(s)(c)$ for some $s\in G$ and $c\in C_{i}\subset Q_{i}$.
Since
$\rho(s^{-1}\varphi_{j},\varphi_{\sigma_{i}(s^{-1})(j)})\leq\sqrt{\delta_{0}}$,
$\rho(s^{-1}\psi_{j},\psi_{\sigma_{i}(s^{-1})(j)})\leq\sqrt{\delta_{0}}$ and
since
$\varphi_{\sigma_{i}(s^{-1})(j)}=\varphi_{c}=\psi_{c}=\psi_{\sigma_{i}(s^{-1})(j)}$,
we have $\rho(s^{-1}\varphi_{j},s^{-1}\psi_{j})\leq 2\sqrt{\delta_{0}}$. Thus,
$\rho(\varphi_{j},\psi_{j})<\epsilon$. This proves the claim.

It follows from this claim that
$\operatorname{\mathrm{Widim}}_{\epsilon}(\operatorname{\mathrm{Map}}(\rho,G,\delta_{0},\sigma_{i}),\rho_{\infty})$
is bounded from above by
$\dim(H_{i}(\operatorname{\mathrm{Map}}(\rho,G,\delta_{0},\sigma_{i})))$. To
estimate the latter term, we consider, for any
$\phi\in\operatorname{\mathrm{Map}}(\rho,G,\delta_{0},\sigma_{i})$, the set
$P_{i}(\phi)=\\{j\in[d_{i}]:J_{j}(\phi)>0\\}$. Since
$\phi\in\operatorname{\mathrm{Map}}(\rho,G,\delta_{0},\sigma_{i})$, we have
$\delta_{0}\cdot|P_{i}(\phi)|\leq\sum_{g\in
G,j\in[d_{i}]}\rho(g\phi_{j},\phi_{\sigma_{i}(g)(j)})^{2}\leq|G|\cdot\delta_{0}^{2}\cdot
d_{i}$ and hence we deduce that $|P_{i}(\phi)|\leq|G|\cdot\delta_{0}\cdot
d_{i}$. This implies that any element in
$H_{i}(\operatorname{\mathrm{Map}}(\rho,G,\delta_{0},\sigma_{i}))$ has entries
at most $\left(|C_{i}|+|G|\cdot\delta_{0}\cdot
d_{i}+|[d_{i}]\setminus\sigma_{i}(G)(C_{i})|\right)$, which do not take the
value $\ast$. Thus,
$\dim(H_{i}(\operatorname{\mathrm{Map}}(\rho,G,\delta_{0},\sigma_{i})))$ is
bounded from above by

$\dim(X^{|C_{i}|})+\dim(CX)\cdot|G|\cdot\delta_{0}\cdot
d_{i}+\dim(X)\cdot|[d_{i}]\setminus\sigma_{i}(G)(C_{i})|$

which, for all $i\geq i_{0}$, does not exceed

$\dim(X^{|C_{i}|})+(\dim(X)+1)\cdot|G|\cdot\delta_{0}\cdot
d_{i}+\dim(X)\cdot\tau\cdot d_{i}\leq\dim(X^{|C_{i}|})+\theta\cdot d_{i}.$

Finally, we note that $|C_{i}|\to+\infty$ (because $d_{i}\to+\infty$) as
$i\to\infty$ and that $|G|\cdot|C_{i}|\leq d_{i}$ (for all $i\in\mathbb{N}$).
Thus, we conclude with

$\limsup_{i\to\infty}\frac{\dim(X^{|C_{i}|})}{d_{i}}\leq\frac{1}{|G|}\cdot\lim_{i\to\infty}\frac{\dim(X^{|C_{i}|})}{|C_{i}|}=\frac{1}{|G|}\cdot\inf_{n\in\mathbb{N}}\frac{\dim(X^{n})}{n}.$

∎

######  Lemma 4.2 (Estimate from below).

$\operatorname{\mathrm{mdim}}_{\Sigma}(X,G)\geq\frac{1}{|G|}\cdot\inf_{n\in\mathbb{N}}\frac{\dim(X^{n})}{n}.$

###### Proof.

First of all, we note that Lemma 2.7 allows us to reduce the statement to the
following issue:

$\operatorname{\mathrm{mdim}}_{\Sigma}(X^{|G|},G)\geq\inf_{n\in\mathbb{N}}\frac{\dim(X^{n})}{n}.$

We fix a compatible metric $D=\rho_{\infty}$ on the product space $X^{|G|}$.
We would like to remind the reader to keep in mind that the remaining part of
this proof is concerned only with the product action $(X^{|G|},G)$.

To show the reduced statement it suffices to prove

$\operatorname{\mathrm{mdim}}_{\Sigma}(X^{|G|},G)\geq\dim(X)-1.$

Indeed, because this inequality will apply to all those actions $(X^{|G|},G)$,
in particular, to $((X^{n})^{|G|},G)$ which is (dynamically) isomorphic to the
product action $((X^{|G|})^{n},G)$, for any $n\in\mathbb{N}$, we shall deduce
from Lemma 2.7 that

$\operatorname{\mathrm{mdim}}_{\Sigma}(X^{|G|},G)\geq\frac{\operatorname{\mathrm{mdim}}_{\Sigma}((X^{|G|})^{n},G)}{n}=\frac{\operatorname{\mathrm{mdim}}_{\Sigma}((X^{n})^{|G|},G)}{n}\geq\frac{\dim(X^{n})-1}{n}.$

Since $n\in\mathbb{N}$ is arbitrary, this will imply that

$\operatorname{\mathrm{mdim}}_{\Sigma}(X^{|G|},G)\geq\lim_{n\to\infty}\frac{\dim(X^{n})-1}{n}=\inf_{n\in\mathbb{N}}\frac{\dim(X^{n})}{n}.$

This is as desired.

Next we prove

$\operatorname{\mathrm{mdim}}_{\Sigma}(X^{|G|},G)\geq\dim(X)-1.$

By Theorem 2.5 there exists some $\epsilon>0$ (sufficiently small) which
depends only on the finite dimensional compact metric space $(X,\rho)$, such
that

$\operatorname{\mathrm{Widim}}_{\epsilon}(X^{n},\rho_{\infty})\geq
n\cdot(\dim(X)-1),\quad\quad\forall\;n\in\mathbb{N}.$

We take $\delta>0$ arbitrarily. We fix $\epsilon$ and $\delta$ in the proof.
By definition we need to show that

$\limsup_{i\to\infty}\frac{\operatorname{\mathrm{Widim}}_{\epsilon}(\operatorname{\mathrm{Map}}(D,G,\delta,\sigma_{i}),D_{\infty})}{d_{i}}\geq\dim(X)-1.$

For every $i\in\mathbb{N}$ we let $Q_{i}\subset[d_{i}]$ be the intersection of
the following two sets:

$\left\\{j\in[d_{i}]:\sigma_{i}(g)(j)\neq\sigma_{i}(h)(j),\;\forall g\neq h\in
G\right\\},$
$\left\\{j\in[d_{i}]:\sigma_{i}(g)(\sigma_{i}(h)(j))=\sigma_{i}(gh)(j),\;\forall
g,h\in G\right\\}.$

For any $j\in[d_{i}]$ and $A\subset[d_{i}]$ we set

$\sigma_{i}(G)(j)=\\{\sigma_{i}(g)(j):g\in
G\\}\subset[d_{i}],\quad\sigma_{i}(G)(A)=\\{\sigma_{i}(g)(a):g\in G,a\in
A\\}\subset[d_{i}].$

We note that by the definition of $Q_{i}$ we have

$|\sigma_{i}(G)(j)|=|G|,\quad\quad\sigma_{i}(G)(\sigma_{i}(G)(j))=\sigma_{i}(G)(j),\quad\quad\forall
j\in Q_{i}.$

It follows that $\sigma_{i}(G)(j)=\sigma_{i}(G)(l)$, for all
$l\in\sigma_{i}(G)(j)$. As what we have explained in the proof of Lemma 4.1,
we can find some $C_{i}\subset Q_{i}$ satisfying that the mapping

$G\times C_{i}\to[d_{i}],\quad(g,j)\mapsto\sigma_{i}(g)(j)$

is injective. Since $G$ is finite, there is some $i_{0}\in\mathbb{N}$
(sufficiently large) such that

$|\sigma_{i}(G)(C_{i})|>(1-\delta^{2})\cdot d_{i},\quad\quad\forall\;i\geq
i_{0}.$

For every $i\in\mathbb{N}$ we define a mapping $P_{i}$ as follows:

$P_{i}:X^{d_{i}}\to(X^{|G|})^{d_{i}},\quad
p=(p_{j})_{j\in[d_{i}]}\to\phi=(\phi_{j})_{j\in[d_{i}]}=((\phi_{j}(g))_{g\in
G})_{j\in[d_{i}]},$
$\phi_{j}(g)=\begin{cases}hg^{-1}p_{\sigma_{i}(g)(k)},&\quad\text{ if
}\;j=\sigma_{i}(h)(k),\;h\in G,\,k\in C_{i}\\\ p_{j},&\quad\text{ if
}\;j\in[d_{i}]\setminus\sigma_{i}(G)(C_{i})\end{cases}.$

The mapping $P_{i}:X^{d_{i}}\to(X^{|G|})^{d_{i}}$ is well defined because the
mapping

$G\times C_{i}\to[d_{i}],\quad(g,j)\mapsto\sigma_{i}(g)(j)$

is injective. Clearly, $P_{i}:X^{d_{i}}\to(X^{|G|})^{d_{i}}$ is continuous.
Since for any $j\in[d_{i}]$ there is some $g\in G$ such that
$\phi_{j}(g)=p_{j}$, it is distance-increasing with respect to the metric
$\rho_{\infty}$ on $X^{d_{i}}$ and the metric $D_{\infty}$ on
$(X^{|G|})^{d_{i}}$, i.e.

$\rho_{\infty}(p,p^{\prime})\leq
D_{\infty}(P_{i}(p),P_{i}(p^{\prime})),\quad\forall p,p^{\prime}\in
X^{d_{i}}.$

We claim that for all sufficiently large $i\in\mathbb{N}$ it will be true that
$P_{i}(X^{d_{i}})$ is contained in
$\operatorname{\mathrm{Map}}(D,G,\delta,\sigma_{i})$. We will verify the claim
in a moment. By this claim and by the choice of $\epsilon>0$ we will obtain
that

$\displaystyle\limsup_{i\to\infty}\frac{\operatorname{\mathrm{Widim}}_{\epsilon}(\operatorname{\mathrm{Map}}(D,G,\delta,\sigma_{i}),D_{\infty})}{d_{i}}$
$\displaystyle\geq\limsup_{i\to\infty}\frac{\operatorname{\mathrm{Widim}}_{\epsilon}(P_{i}(X^{d_{i}}),D_{\infty})}{d_{i}}$
$\displaystyle\geq\limsup_{i\to\infty}\frac{\operatorname{\mathrm{Widim}}_{\epsilon}(X^{d_{i}},\rho_{\infty})}{d_{i}}$
$\displaystyle\geq\limsup_{i\to\infty}\frac{d_{i}\cdot(\dim(X)-1)}{d_{i}}$
$\displaystyle=\dim(X)-1.$

This will end the proof.

Now we prove that $P_{i}(X^{d_{i}})$ is contained in
$\operatorname{\mathrm{Map}}(D,G,\delta,\sigma_{i})$ for all integers $i\geq
i_{0}$. We fix an integer $i\geq i_{0}$. We take an arbitrary
$p=(p_{j})_{j\in[d_{i}]}\in X^{d_{i}}$. Let $\phi=P_{i}(p)$ and write
$\phi=(\phi_{j})_{j\in[d_{i}]}=((\phi_{j}(g))_{g\in
G})_{j\in[d_{i}]}\in(X^{|G|})^{d_{i}}$. We note that $C_{i}\subset Q_{i}$. By
the construction of the mapping $P_{i}:X^{d_{i}}\to(X^{|G|})^{d_{i}}$ we have
obviously that if $j=\sigma_{i}(h)(c)$ for some $h\in G$ and $c\in C_{i}$ then
for any $s\in G$ and any $g\in G$

$s\phi_{j}(g)=shg^{-1}p_{\sigma_{i}(g)(c)}=\phi_{\sigma_{i}(sh)(c)}(g)=\phi_{\sigma_{i}(s)(\sigma_{i}(h)(c))}(g)=\phi_{\sigma_{i}(s)(j)}(g).$

Thus, for any $j\in\sigma_{i}(G)(C_{i})$

$s\phi_{j}=\phi_{\sigma_{i}(s)(j)},\quad\quad\forall\,s\in G.$

It follows that for all $s\in G$

$\displaystyle D_{2}(s\phi,\phi\circ\sigma_{i}(s))$
$\displaystyle=\sqrt{\frac{1}{d_{i}}\cdot\sum_{j\in[d_{i}]}\left(D(\phi_{\sigma_{i}(s)(j)},s\phi_{j})\right)^{2}}$
$\displaystyle=\sqrt{\frac{1}{d_{i}}\cdot\sum_{j\in[d_{i}]\setminus\sigma_{i}(G)(C_{i})}\left(D(\phi_{\sigma_{i}(s)(j)},s\phi_{j})\right)^{2}}$
$\displaystyle\leq\sqrt{\frac{d_{i}-|\sigma_{i}(G)(C_{i})|}{d_{i}}}$
$\displaystyle\leq\delta.$

This implies that $\phi\in\operatorname{\mathrm{Map}}(D,G,\delta,\sigma_{i})$.
Thus, we conclude. ∎

## Appendix

The definition of sofic mean dimension was originally introduced by Li [Li13]
(using open covers). However, we notice that the definition which we stated in
the body of this paper (in terms of $\operatorname{\mathrm{Widim}}$) differs
from Li’s. The appendix is devoted to clarifying that these two definitions
are equivalent for sofic mean dimension.

Let $Y$ be a compact space and $\mathcal{U}$ a finite open cover of $Y$. We
denote

$\mathrm{ord}(\mathcal{U})=\max_{y\in
Y}\sum_{U\in\mathcal{U}}1_{U}(y)-1,\quad\quad\mathcal{D}(\mathcal{U})=\min_{\mathcal{V}\succ\mathcal{U}}\mathrm{ord}(\mathcal{V}),$

where $1_{U}(y)$ takes the value $1$ (resp. $0$) if $y\in U$ (resp. $y\notin
U$) and where $\mathcal{V}\succ\mathcal{U}$ means that $\mathcal{V}$ ranges
over finite open covers of $Y$ refining $\mathcal{U}$ (i.e. every element of
$\mathcal{V}$ is contained in some element of $\mathcal{U}$). For
$d\in\mathbb{N}$ we denote by $\mathcal{U}^{d}$ the finite open cover of
$Y^{d}$ consisting of all those open sets in $Y^{d}$ of the form
$U_{1}\times\cdots\times U_{d}$ (where $U_{1},\dots,U_{d}\in\mathcal{U}$).

Let a sofic group $G$ act continuously on a compact metrizable space $X$ and
let
$\Sigma=\\{\sigma_{i}:G\to\operatorname{\mathrm{Sym}}(d_{i})\\}_{i\in\mathbb{N}}$
be a sofic approximation sequence for $G$. We fix a compatible metric $\rho$
on $X$. For a finite open cover $\mathcal{U}$ of $X$, a nonempty finite subset
$F$ of $G$, $\delta>0$, and $i\in\mathbb{N}$ we denote
$\mathcal{D}(\mathcal{U}^{d_{i}}\cap\operatorname{\mathrm{Map}}(\rho,F,\delta,\sigma_{i}))$
by $\mathcal{D}(\mathcal{U},\rho,F,\delta,\sigma_{i})$. Recall that we have
already given the definition of $\operatorname{\mathrm{mdim}}_{\Sigma}(X,G)$
in Section 2. Now what we want to see is

$\operatorname{\mathrm{mdim}}_{\Sigma}(X,G)=\sup_{\mathcal{U}}\inf_{F\subset
G\text{ finite,
}\,\delta>0}\limsup_{i\to\infty}\frac{\mathcal{D}(\mathcal{U},\rho,F,\delta,\sigma_{i})}{d_{i}},$

where $\mathcal{U}$ runs over finite open covers of $X$.

In fact, this can be shown with a simple and standard argument. The key point
is to consider a Lebesgue number of a finite open cover $\mathcal{U}$ with
respect to $\rho$, and to use the following fact [LW00]: For a finite open
cover $\mathcal{U}$ of a compact metrizable space $X$,
$\mathcal{D}(\mathcal{U})\leq n$ (where $n$ is a nonnegative integer) exactly
when there is a compact metrizable space $P$ of topological dimension $n$,
which admits a continuous mapping $f:X\to P$ satisfying that there exists a
finite open cover $\mathcal{V}$ of $P$ such that $f^{-1}(\mathcal{V})$ refines
$\mathcal{U}$. Lastly, we note that although a compatible metric $\rho$ on the
space $X$ is also involved in the right-hand side of the above equality, it is
independent of the choice of a compatible metric $\rho$ on $X$ (for details
see [Li13, Section 2]).

## References

  * [Dra01] A. N. Dranishnikov. Cohomological dimension theory of compact metric spaces. Topology Atlas Invited Contributions 6 (2001), 61 pp. http://at.yorku.ca/t/a/i/c/43.htm
  * [Gro99a] Misha Gromov. Endomorphisms of symbolic algebraic varieties. Journal of the European Mathematical Society 1 (1999), 109–197.
  * [Gro99b] Misha Gromov. Topological invariants of dynamical systems and spaces of holomorphic maps: I. Math. Phys. Anal. Geom. 2 (1999), 323–415.
  * [JQ] Lei Jin, Yixiao Qiao. Mean dimension of product spaces: a fundamental formula. Submitted. Preprint on arXiv:2102.10358.
  * [Li13] Hanfeng Li. Sofic mean dimension. Advances in Mathematics 244 (2013), 570–604.
  * [LW00] Elon Lindenstrauss, Benjamin Weiss. Mean topological dimension. Israel Journal of Mathematics 115 (2000), 1–24.
  * [Tsu19] Masaki Tsukamoto. Mean dimension of full shifts. Israel Journal of Mathematics 230 (2019), 183–193.