Chapter 1 Assistant Prof. Raula Gaikovina Kula, Nara Institute of Science and
Technology, Japan
<EMAIL_ADDRESS>
2 Associate Prof. Gregorio Robles, Universidad Rey Juan Carlos, Spain
<EMAIL_ADDRESS>
# The Life and Death of Software Ecosystems

Raula Gaikovina Kula  Gregorio Robles

###### Abstract

Software ecosystems have gained a lot of attention in recent times. Industry
and developers gather around technologies and collaborate to their
advancement; when the boundaries of such an effort go beyond certain amount of
projects, we are witnessing the appearance of Free/Libre and Open Source
Software (FLOSS) ecosystems.

In this chapter, we explore two aspects that contribute to a healthy
ecosystem, related to the attraction (and detraction) and the death of
ecosystems. To function and survive, ecosystems need to attract people, get
them on-boarded and retain them. In Section One we explore possibilities with
provocative research questions for attracting and detracting contributors (and
users): the lifeblood of FLOSS ecosystems. Then in the Section Two, we focus
on the death of systems, exploring some presumed to be dead systems and their
state in the afterlife.

##  1 Attractors (and Detractors) to FLOSS Projects

A contributing component to the sustainability (i.e., life) of a FLOSS project
is its ability to attract new development. Although keeping current
contributors is equally important, projects risk failure if they are unable to
attract a healthy amount of new developers to provide rejuvenation and aid in
project evolution, especially in response to ever-changing external forces
(i.e., impactful events, new technologies, vulnerabilities and rivals) that
affect FLOSS projects. In this section, we discuss (1) the different forces of
attraction (and detraction) that influence contributors to participate in
specific projects, (2) the effect of these forces at the ecosystem level, and
finally present (3) three provocative research questions to further our
understanding of attracting contributors to a project.

###  1.1 Forces of Attraction (and Detraction)

We classify known forces of attraction as either motivation-related,
environmental, or a combination of the two. Internal project-driven campaigns
usually revolve around marketing strategies to attract developers. A study by
Storey et al. [8] showed that communities of FLOSS projects are shaped through
social and communication channels (sometimes referred to as social coding).
Recently Aniche et al. [3] confirmed that news channels also play an important
role in shaping and sharing knowledge among developers. Hence, owners of
projects could boost their social presence through participation on recent
topics from news aggregators such as reddit111https://www.reddit.com, Hacker
News222https://news.ycombinator.com and slashdot333https://slashdot.org. For
instance, a project may employ new or well-known or recognizable trademarks
that are trending in the news. Social media outlets and other communication
channels can be leveraged to improve project attractiveness (i.e., innovative
posts on Q&A forums such as StackOverflow444https://stackoverflow.com and
social media endorsements and collaborations through twitter or facebook).
Recently, analytical indicators of project health or fitness are aimed at
increasing the appeal of a project. In detail, the emergence of online
collaboration platforms GitHub, GitLab and BitBucket, with specific features
such as pull requests, forks, and stars depicts the fitness of a project.

Other motivations are driven by external forces. Hata et al. [6] used game
theory to identify three strategies that is likely to incite contributions.
The authors suggest that improving the code writing mechanisms (i.e., wikis,
offical webpage, contributing and coding guidelines and using multi-language
formats). Secondly, in terms of monetary incentives, sites such as
bountysource website555https://www.bountysource.com/ allow developers to be
hired as bounty hunters to fix specialized bugs in a project. Finally, the
impact of innovations such as social coding, introduced by online
collaborations of GitHub has attracted attention of developers. A lesser
explicit form of motivations is driven by a third-party with their own
interests. For instance, a company may allocate employees or provide monetary
incentives to support (i.e., keep alive) a project of interest. This is
especially in cases where a third-party is interested in stimulating further
feature development of an existing product that they are invested in.

Failing projects provide insights into some environmental forces that detract
developers from making contributions. A study by Coelho and Valente [4] found
the following reasons for failing projects: usurped by competitor, obsolete
project, lack of time and interest, outdated technologies, low
maintainability, conflicts among developers, legal problems, and acquisition.
To mitigate these detractors, the authors propose three strategies to
rejuvenate contributions in failing FLOSS projects. Firstly, projects are
encouraged to improve their stability by moving towards an organization
account instead of a personal account. Secondly, failing projects are
encouraged to transfer the project to new maintainers. This is especially
needed if the current maintainers’ activity has been deteriorating over time.
Finally, the project is encouraged to accept new core developers. This
organizational factor aims to rejuvenate and ignite fresh ideas, giving new
life to the project.

###  1.2 Forces at the Ecosystem Level

To date, existing works performed their analysis in respect to individual
projects. At a higher level of abstraction, there exists cases where the
forces of attraction (and detractions) in several projects in an ecosystem are
triggered by a common event. For instance, several studies [2, 5, 7]
investigated the eventful case of the JavaScript "left-pad" incident (see
[1]), where removal of a trivial library package caused major breakages in
thousands of projects including notable JavaScript frameworks like babel and
react.

Other examples of impactful events at the ecosystem level include responses to
wide-spreading high risk security vulnerabilities (i.e., ShellShock,
Heartbleed and Poodle), rivaling technologies (i.e., battles between competing
frameworks for specific programming domains such as PHP666A blog post for 2018
best PHP frameworks at https://coderseye.com/best-php-frameworks-for-web-
developers/ and JavaScript777A blog that shows the trend changes between rival
JavaScript frameworkshttps://stackoverflow.blog/2018/01/11/brutal-lifecycle-
javascript-frameworks/) and inadequacies in the current situation. As an
example, current inadequacies could be realized when a change in management
occurs (i.e., such as change of the middle-man in InnerSource888InnerSource
takes the lessons learned from developing FLOSS and applies them to the way
companies develop software internally. Taken from
https://paypal.github.io/InnerSourceCommons/). Changes in management (i.e.,
especially the single movement of a key contributor) may set off a chain
series of attract and distraction forces that leave behind a rippling effect
across the ecosystem. We theorize that these forces impact ecosystem
sustainability, especially if affected projects act as hubs within that
ecosystem.

###  1.3 Provocative Research Questions

To conclude this section, we formulate a set of provocative research questions
to further our understanding of attraction and detraction forces:

  * •

What are the strengths and successes of known attractor strategies to FLOSS
projects? We have identified many attraction forces. Understanding the
strength and success of these different attractors will assist us to treat
projects that may be suffering with attracting new contributors to their
projects.

  * •

How often are these attractor strategies practiced in the real-world and in
respect to different ecosystems? It is unknown to what extent and the
frequency by which these strategies are practiced by practitioners in recent
times. Furthermore, we are unclear of the environmental and ecosystem
conditions required to sustain these attraction forces.

  * •

What are the implications and impact of these forces of attraction at the
ecosystem level? We theorize that attraction forces may impact the overall
ecosystem. However, it is unclear the extent by which these forces of
attraction may affect the sustainability of the overall ecosystem itself.

##  2 On the Death of Ecosystems

Software ecosystems have gained a lot of attention in recent times. Industry
and developers gather around technologies and collaborate to their
advancement; when the boundaries of such efforts go beyond certain amount of
projects, we are witnessing the appearance of a software ecosystem. Software
ecosystems are complex in nature, as many stakeholders are involved. There are
for sure key people (e.g., Guido van Rossum in Python) and projects (such as
MySQL in the MySQL ecosystem), but activity follows a decentralized pattern,
more in the fashion of stigmergic process as known for instance from colonies
of ants.

In this section we want to focus on the death of software ecosystems. While it
is known that many FLOSS projects are discontinued, to the knowledge of the
authors we haven’t found any research on the topic on software ecosystems. We
define as the death of a project as having no activity in it for a long period
as done in other research works. So, a dead software ecosystems have would
have no activity. It should be noted that other definitions of death could be
proposed. One may think of having no users, a loss of interest in the software
industry, a decrease in developers and developer interest, etc.

On the other hand, we are not looking after projects, which are defined (i.e.,
they have a goal) and concrete software solution that has an organizational
and logistic structure (a known website, repository, mailing list, etc.).
Software ecosystems are built of many projects, which co-ordinate themselves
(or not) but that have a relationship that is in general technological
(although other types of ecosystems such as the (entire) Apache ecosystem
orchestrates around collaboration).

###  2.1 Research Questions

Current research literature has so far focused mainly in successful FLOSS
systems, to see how they are articulated and organized, in order to derive
lessons learned out of these. Our method will be exploratory and based on case
studies. Specifically, we want to address following RQs:

  * •

RQ1. What do we know of dead ecosystems?
We want to approach our study based on real cases of ecosystems that were so
in the past, but that are now inactive. So, as a first step, we performed an
unstructured search for dead ecosystems, by asking participants in the
workshop and then by looking in the web (mainly in the webpages of its
projects and in Wikipedia) for more information. The output of this research
question is a list of dead ecosystems on which the subsequent RQs will be
addressed.

  * •

RQ2. Why are these ecosystems dead?
Once we have identified dead ecosystems in RQ1, we would like to dig into the
reasons why these have become inactive. In this regard, we would like to see
if the cause of the inactivity can be technology (e.g., becoming an outdated
technology), economic (e.g., failure of funding), legal (i.e., patent or
license issues), among others. As input of information we will use Google
searches on the Internet.

  * •

RQ3. What can we learn from dead ecosystems?
Once we have identified dead ecosystems (RQ1) and have further information
into what causes are behind its death (RQ2), our goal is to see if we can
extract major insight into the topic. The final goal is, of course, to help
software ecosystems to stay "healthy".

###  2.2 Findings

Based on research questions in the prior section, this section we discuss and
present the findings of each research question.

####  2.2.1 RQ1 What do we know of dead ecosystems?

Table 1:  Summary of the studied dead ecosystems System Name | Brief Description | Discontinued Date
---|---|---
Concurrent Versioning System (CVS) | version control | May, 2008
FireFoxOS | mobile operating system | Dec, 2015
Apache Geronimo | application server | May, 2013
Maemo | mobile development platform | Feb, 2010

During the seminar in Shonan, participants were asked informally regarding
open ecosystems that have been discontinued. After much discussion, as shown
in Table 1, the following dead systems arose from the discussions.

##### Control Versioning System (CVS)

CVS is a version control system, an important component of Source
Configuration Management (SCM)999 website is at https://www.nongnu.org/cvs/.
Using it, you can record the history of sources files, and documents. The last
version of CVS was published in 2008 (see
http://savannah.nongnu.org/news/?group=cvs).

##### FirefoxOS

Firefox OS was a mobile operating system, based on HTML5 and the Linux kernel,
available for several platforms. It was developed by Mozilla Corporation under
the support of other companies and a large community of volunteers from around
the world. The operating system was designed to allow HTML5 applications to
communicate directly with device hardware using JavaScript and Open Web
APIs101010 although an official website is not found, the blog of one of the
key engineers is an example of its existence https://medium.com/@bfrancis/the-
story-of-firefox-os-cb5bf796e8fb .

In December 2015, Mozilla announced it would stop development of new Firefox
OS smartphones, and in September 2016 announced the end of development.

##### Apache Geronimo

Apache Geronimo111111website available at http://geronimo.apache.org/ is a
FLOSS application server developed by the Apache Software Foundation and
distributed under the Apache license. IBM announced May 14, 2013 that it would
withdraw and discontinue support of Apache Geronimo (see
http://www-01.ibm.com/common/ssi/rep_ca/1/897/ENUS913-081/ENUS913-081.PDF).
This was also communicated through their website and mailing lists.

##### Maemo

Maemo 121212website available at http://maemo.org/intro/ is a development
platform for handheld devices based on debian GNU / Linux. Maemo is mostly
based on open-source code and has been developed by Maemo Devices within Nokia
in collaboration with many FLOSS projects such as the Linux kernel, Debian,
and GNOME.

At the Mobile World Congress 2010, Intel and Nokia announced that they would
unite their Linux-based platforms into a single product, called MeeGo. The
Linux Foundation canceled MeeGo in September 2011 in favor of Tizen. An
emerging Finnish company, Jolla, took Mer, a successor based on the MeeGo
community, and created a new operating system: Sailfish OS, and launched a new
smartphone at the end of 2013.

####  2.2.2 RQ2 Why are these ecosystems dead?

We have investigated what happened to the projects presented in RQ1, to see if
there is any continuation. In this regard, we investigate whether or not the
original project is still alive, and if there have been any forks (i.e.,
others have taken the source code base and have evolved the software
independently). As shown in Table 2, new projects emerged in the aftermath of
the dying ecosystem.

Table 2: Emergent Projects after the death of the ecosystem System Name | Example Emergent Projects
---|---
Concurrent Versioning System (CVS) | CVSNT
FireFoxOS | Panasonic variant, H5OS, KaiOS, Jio
Apache Geronimo | Tomcat, EJB, Derby
Maemo | MeeGo, Tizan, Mer

##### CVS

Although the CVS project was discontinued, we find that due to the development
of the Microsoft Windows, Linux, Solaris, HPUX, I5os and Mac OS X ports,
evidence shows that CVS has split off into a separate project named
CVSNT131313website available at https://www.march-hare.com/cvspro/ , which is
under current, active development (i.e., the latest update as of writing was
April 2018).

##### FirefoxOS

After the discontinuation of Firefox OS, several variants of the OS have
emerged. Panasonic will continue to develop the operating system for use in
their Smart TVs, which runs My Home Screen, powered by the Firefox OS. Acadine
Technologies has derived their H5OS from Firefox OS as well. Li Gong, the
founder of the company, has overseen the development of Firefox OS while
serving as president of the Mozilla Corporation. Alcatel OneTouch GO FLIP uses
a fork called KaiOS141414 website at https://www.kaiostech.com/. In addition,
in July 2017, it was reported that Indian telecom operator Jio would be
launching new feature phone with OS derived from Firefox OS and the apps are
purely in HTML5 and CSS.

##### Apache Geronimo

The development of Apache Geronimo ceased around 2013, after its 3.0.1
release, when IBM and Oracle stopped to support the project in favor of their
own technologies. Geronimo is not a single technology, but is the sum of many
components, like Apache Tomcat151515website as http://tomcat.apache.org/,
Apache EJB161616website at http://tomee.apache.org/tomcat-ejb.html, Apache
Derby171717website at https://db.apache.org/derby/, among others. Many of
these components are used in the implementation components of other frameworks
as can be seen from http://arjan-tijms.omnifaces.org/2014/05/implementation-
components-used-by.html.

##### Maemo

In February 2010, the Maemo project from Nokia merged with Moblin to create
the MeeGo mobile software platform under the umbrella of the Linux Foundation.
However, the Maemo community continued to be active in Maemo. That is the
reason why Nokia transferred the Maemo ownership first to the Hildon
Foundation, and then to a German association called Maemo Community e.V. The
last general assembly of this association has been in October 2017.

MeeGo181818 A variant of MeeGo is Tizen https://www.tizen.org/ was cancelled
in September 2011, although a community-driven successor called
Mer191919website as http://www.merproject.org/ was formed that. A Finnish
start-up, Jolla, chose in 2013 Mer as the basis of the Sailfish OS operating
system for their Jolla Phone smartphones. Another Mer derivative called Nemo
Mobile is also currently developed actively.

####  2.2.3 RQ3 What can we learn from dead ecosystems?

There is little to learn from dead ecosystems, because software ecosystems, at
least those that are FLOSS, don’t die! In our quest for dead ecosystems, what
we have found are that ecosystems that have been abandoned have evolved (if
not completely, at least partially) with a given name. This means, that
organizations and names are the ones that may disappear, but the technology
can be found years later in other projects and developments. There are two
main factors that may concur to explain this situation:

  1. 1.

Forks originating from the dead ecosystem. The first one is the right to fork
that exists (and is used) in FLOSS development. Although forking (i.e.,
splitting the community by taking the technology under a new name) is
historically not welcome in the FLOSS community, it is understood in certain
contexts. One of these situations is when the project is abandoned.

  2. 2.

Technological advancements. The second one is related to the development of
technologies. This requires time, much human labor and is maintenance
intensive. A software is not only its development and its community. It is as
well the number of tests and maturity that it has achieved. Successful FLOSS
ecosystems have invested a large amount of effort in becoming mature. Even if
its key players lose their interest in the technology and the community seems
to shrink, there is always the source code, that is result of that effort. In
addition, the investment in time and learning of other technologies results in
inertia by those who are familiar with the technology. With ecosystems that
have a large community, the probabilities of even a minor part of this
community still interested in continuing with development is very high.

###  2.3 Conclusions

FLOSS ecosystems are still too young to draw conclusions from our
investigation, but as far as we have analyzed we have not found any (well-
known) FLOSS ecosystem that can be considered dead (i.e., completely
abandoned). For one or the other reason, the original software has evolved
into other systems and communities and still serves, even if the importance of
the project is not the one that used to be.

A lesson learned from our analysis is that if organizations want
sustainability of a technology or application, they should strive for the
ecosystem way. This is a lesson that could be of interest for consortia,
public bodies, and companies wanting to set a standard. The network effects of
developing a long-lasting software ecosystem is the probability that at least
a small portion of the community keeps it alive. We have seen that this is the
case from outdated technologies (like CVS) to hardware-linked software (such
as Maemo).

As there is a growing interest of corporations in FLOSS, such as the one that
can be found in OpenStack, OW2, WebKit, among others, we are sure that the
future will allow to have further examples of ecosystems and analyse how they
evolve, even when their main promoters abandon.

## References

  * Web [2018] The npm Blog — kik, left-pad, and npm; 2018.  (Accessed on 01/31/2018).  http://blog.npmjs.org/post/141577284765/kik-left-pad-and-npm.
  * Abdalkareem et al. [2017] Abdalkareem R, Nourry O, Wehaibi S, Mujahid S, Shihab E.  Why Do Developers Use Trivial Packages? An Empirical Case Study on npm.  In: Proceedings of the 11th Joint Meeting of the European Software Engineering Conference and the ACM SIGSOFT Symposium on the Foundations of Software Engineering (ESEC/FSE’17) ACM; 2017.
  * Aniche et al. [2018] Aniche M, Treude C, Steinmacher I, Wiese I, Pinto G, Storey MA, et al.  How Modern News Aggregators Help Development Communities Shape and Share Knowledge.  In: International Conference on Software Engineering (ICSE18); 2018. .
  * Coelho and Valente [2017] Coelho J, Valente MT.  Why Modern Open Source Projects Fail.  In: Proceedings of the 2017 11th Joint Meeting on Foundations of Software Engineering ESEC/FSE 2017, New York, NY, USA: ACM; 2017. p. 186–196.
  * Decan et al. [2017] Decan A, Mens T, Claes M.  An empirical comparison of dependency issues in OSS packaging ecosystems.  In: 2017 IEEE 24th International Conference on Software Analysis, Evolution and Reengineering (SANER) IEEE; 2017. p. 2–12.
  * Hata et al. [2015] Hata H, Todo T, Onoue S, Matsumoto K.  Characteristics of Sustainable OSS Projects: A Theoretical and Empirical Study.  In: 2015 IEEE/ACM 8th International Workshop on Cooperative and Human Aspects of Software Engineering; 2015. p. 15–21.
  * Kikas et al. [2017] Kikas R, Gousios G, Dumas M, Pfahl D.  Structure and Evolution of Package Dependency Networks.  In: Proceedings of the 14th International Conference on Mining Software Repositories MSR ’17, Piscataway, NJ, USA: IEEE Press; 2017. p. 102–112.
  * Storey et al. [2017] Storey MA, Zagalsky A, Filho FF, Singer L, German DM.  How Social and Communication Channels Shape and Challenge a Participatory Culture in Software Development.  IEEE Transactions on Software Engineering 2017 Feb;43(2):185–204.


# Q-ViT: Accurate and Fully Quantized Low-bit Vision Transformer

Yanjing Li1$\dagger$, Sheng Xu1$\dagger$, Baochang Zhang1,2, Xianbin Cao1$*$,
Peng Gao3, Guodong Guo4,5
1Beihang University, Beijing, P.R.China
2Zhongguancun Laboratory, Beijing, P.R.China
3Shanghai Artificial Intelligence Laboratory, Shanghai, P.R.China
4 Institute of Deep Learning, Baidu Research, Beijing, P.R.China
5 National Engineering Laboratory for Deep Learning Technology and
Application,
Beijing, P.R.China
{yanjingli, shengxu, bczhang<EMAIL_ADDRESS>
###### Abstract

The large pre-trained vision transformers (ViTs) have demonstrated remarkable
performance on various visual tasks, but suffer from expensive computational
and memory cost problems when deployed on resource-constrained devices. Among
the powerful compression approaches, quantization extremely reduces the
computation and memory consumption by low-bit parameters and bit-wise
operations. However, low-bit ViTs remain largely unexplored and usually suffer
from a significant performance drop compared with the real-valued
counterparts. In this work, through extensive empirical analysis, we first
identify the bottleneck for severe performance drop comes from the information
distortion of the low-bit quantized self-attention map. We then develop an
information rectification module (IRM) and a distribution guided distillation
(DGD) scheme for fully quantized vision transformers (Q-ViT) to effectively
eliminate such distortion, leading to a fully quantized ViTs. We evaluate our
methods on popular DeiT and Swin backbones. Extensive experimental results
show that our method achieves a much better performance than the prior arts.
For example, our Q-ViT can theoretically accelerates the ViT-S by 6.14$\times$
and achieves about 80.9% Top-1 accuracy, even surpassing the full-precision
counterpart by 1.0% on ImageNet dataset. Our codes and models are attached on
https://github.com/YanjingLi0202/Q-ViT.

††footnotetext: $\dagger$ Equal contribution. $*$ Corresponding author.

##  1 Introduction

Inspired by the success in natural language processing (NLP), transformer-
based models have shown great power in various computer vision (CV) tasks,
such as image classification [4] and object detection [2]. Pre-trained with
large-scale data, these models usually have a tremendous number of parameters.
For example, there are 632M parameters taking up 2528MB memory usage and 162G
FLOPs in the ViT-H model, which is both memory and computation expensive
during inference. This limits these models for the deployment on resource-
limited platforms. Therefore, compressed transformers are urgently needed for
real applications.

Substantial efforts have been made to compress and accelerate neural networks
for efficient online inference. Methods include compact network design [10],
network pruning [9], low-rank decomposition [3], quantization [21, 30, 32],
and knowledge distillation [24, 31]. Quantization is particularly suitable for
deployment on AI chips because it reduces the bit-width of network parameters
and activations for efficient inference. Prior post-training quantization
(PTQ) methods [18, 14] on ViTs directly compute quantized parameters based on
pre-trained full-precision models, which constrains the model performance to a
sub-optimized level without fine-tuning. Furthermore, quantizing these models
based on PTQ methods to ultra-low bits (e.g., 4 bits or lower) is ineffective
and suffers from a significant performance reduction.

Figure 1: The histogram of query values ${\bf q}$ (blue shadow) along with the
PDF curve (red line) of Gaussian distribution $N(\mu,\sigma^{2})$ [20], for 3
selected layers in DeiT-T and 4-bit fully quantized DeiT-T (baseline). $\mu$
and $\sigma^{2}$ are the statistical mean and variance of the values.

Differently, quantization-aware training (QAT) [16] methods perform
quantization during back propagation and achieve much less performance drop
with a higher compression rate generally. QAT is shown to be effective for CNN
models [17] for CV tasks. However, QAT methods remain largely unexplored for
low-bit quantization of vision transformers. Therefore, we first build a fully
quantized ViT baseline, a straightforward yet effective solution based on
common techniques. Our study discovers that the performance drop of fully
quantized ViT lies in the information distortion among the attention mechanism
in the forward process, and the ineffective optimization for eliminating the
distribution difference through distillation in the backward propagation.
First, the attention mechanism of ViT aims at modeling long-distance
dependencies [27, 4]. However, our analysis shows that a direct quantization
method leads to the information distortion, i.e., significant distribution
variation for the query module between quantized ViT and full-precision
counterpart. For example, as shown in Fig. 1, the variance difference is
0.4409 (1.2124 v.s. 1.6533) for the first block ††The Gaussian distribution
hypothesis is supported by [20]. This inevitably deteriorates the
representation capability of the attention module on capturing the global
dependency for the input. Second, the distillation for the fully quantized ViT
baseline utilizes distillation token (following [25]) to directly supervise
the classification output of the quantized ViT. However, we found that such a
simple supervision is ineffective, which is coarse-grained for the large gap
between the quantized attention scores and their full-precision counterparts.

To address the aforementioned issues, a fully quantized ViT (Q-ViT) is
developed by retaining the distribution of quantized attention modules as that
of full-precision counterparts (see the overview in Fig. 2). Accordingly, we
propose to modify the distorted distribution over quantized attention modules
through an Information Rectification Module (IRM) based on information entropy
maximization, in the forward process. While in the backward process, we
present a Distribution Guided Distillation (DGD) scheme to eliminate the
distribution variation through attention similarity loss between quantized ViT
and full-precision counterpart. The contributions of our work include:

  * •

We propose an Information Rectification Module (IRM) based on the information
theory to address the information distortion problem. IRM applies quantized
representations in the attention module with a maximized information entropy,
allowing the quantized model to restore the representation of input images.

  * •

We develope a Distribution Guided Distillation (DGD) scheme to eliminate the
distribution mismatch in distillation. DGD takes appropriate activations and
utilizes knowledge from the similarity matrices in distillation to perform
optimization accurately.

  * •

Our Q-ViT, for the first time, explores a promising way towards accurate and
low-bit ViT. Extensive experiments on the ImageNet benchmark show that our
Q-ViT outperforms the baseline by a large margin, and achieves comparable
performances with the full-precision counterparts.

##  2 Related Work

Vision transformer. Motivated by the great success of Transformer in natural
language processing, researchers are trying to apply Transformer architecture
to Computer Vision tasks. Unlike mainstream CNN-based models, Transformer is
capable of capturing long-distance visual relations by its self-attention
module and provides the paradigm without image-specific inductive bias. ViT
[4] views 16 $\times$ 16 image patches as token sequence and predicts
classification via a unique class token, which shows promising results.
Subsequently, many works, such as DeiT [25] and PVT [28] achieve further
improvement on ViT, making it more efficient and applicable in downstream
tasks. CONTAINER [7] fully utilizes a hybrid ViT to aggregate dynamic and
static information, exploring a new framework for visual tasks. However, these
high-performing vision transformers are attributed to the large number of
parameters and high computational overhead, limiting their adoption.
Therefore, innovating a smaller and faster vision transformer becomes a new
trend. DynamicViT [23] presents a dynamic token sparsification framework to
prune redundant tokens progressively and dynamically, achieving competitive
complexity and accuracy trade-off. Evo-ViT [33] proposes a slow-fast updating
mechanism that guarantees information flow and spatial structure, trimming
down both the training and inference complexity. While the above works focus
on efficient model designing, this paper boosts the compression and
acceleration in the track of quantization.

Quantization. Quantizing neural networks (QNNs) often possess low-bit (1$\sim$
4-bit) weights and activations to accelerate the model inference and save the
memory usage. Specifically, ternary weights are introduced to reduce the
quantization error in TWN [13]. DoReFa-Net [35] exploits convolution kernels
with low bit-width parameters and gradients to accelerate both the training
and inference. TTQ [36] uses two full-precision scaling coefficients to
quantize the weights to ternary values. [37] presented a $2\\!\sim\\!4$-bit
quantization scheme using a two-stage approach to alternately quantize the
weights and activations, which provides an optimal trade-off among memory,
efficiency, and performance. [11] parameterizes the quantization intervals and
obtain their optimal values by directly minimizing the task loss of the
network and also the accuracy degeneration with further bit-width reduction.
[29] introduces transfer learning into network quantization to obtain an
accurate low-precision model by utilizing the Kullback-Leibler (KL)
divergence. [6] enables accurate approximation for tensor values that have
bell-shaped distributions with long tails and finds the entire range by
minimizing the quantization error. In our Q-ViT, we aim to implement an
accurate, fully quantized vision transformer under the QAT paradigm.

Figure 2: Overview of Q-ViT, applying Information Rectification Module (IRM)
for maximizing representation information and Distribution Guided Distillation
(DGD) for accurate optimization.

##  3 Baseline of Fully Quantized ViT

First of all, we build a baseline to study the fully quantized ViT since it
has never been proposed in previous works. A straightforward solution is to
quantize the representations (weights and activations) in ViT architecture in
the forward propagation and apply distillation to the optimization in the
backward propagation.

Quantized ViT architecture.  We briefly introduce the technology of neural
network quantization. We first introduce a general asymmetric activation
quantization and symmetric weight quantization scheme as

$\displaystyle Q_{a}(x)$
$\displaystyle=\lfloor\operatorname{clip}\\{(x-z)/\alpha_{x},-Q_{n}^{x},Q_{p}^{x}\\}\rceil\;\;\;\;Q_{w}({\bf
w})=\lfloor\operatorname{clip}\\{{\bf w}/\alpha_{\bf w},-Q_{n}^{\bf
w},Q_{p}^{\bf w}\\}\rceil$ (1) $\displaystyle\hat{x}$
$\displaystyle=Q_{a}(x)\times\alpha_{x}+z,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\hat{\bf
w}=Q_{w}({\bf w})\times\alpha_{\bf w}.$

Here, $\operatorname{clip}\\{y,r_{1},r_{2}\\}$ returns $y$ with values below
$r_{1}$ set as $r_{1}$ and values above $r_{2}$ set as $r_{2}$, and $\lfloor
y\rceil$ rounds $y$ to the nearest integer. With quantizing activations to
signed $a$ bits and weights to signed $b$ bits,
$Q_{n}^{x}=2^{a-1},Q_{p}^{x}=2^{a-1}-1$ and $Q_{n}^{\bf w}=2^{b-1},Q_{p}^{\bf
w}=2^{b-1}-1$. In general, the forward and backward propagation of
quantization function in quantized network is formulated as

$\displaystyle\operatorname{Forward:}\;\;\operatorname{Q-Linear}(x)=\hat{x}\cdot\hat{\bf
w}=\alpha_{x}\alpha_{\bf w}((Q_{a}(x)+z/\alpha_{x})\otimes Q_{w}({\bf w})),$
(2)
$\displaystyle\operatorname{Backward:}\;\frac{\partial\mathcal{J}}{\partial
x}=\frac{\partial\mathcal{J}}{\partial\hat{x}}\frac{\partial\hat{x}}{\partial
x}=\left\\{\begin{aligned}
&\frac{\partial\mathcal{J}}{\partial\hat{x}}&\operatorname{if}x\in[-Q_{n}^{x},Q_{p}^{x}]\\\
&0&\operatorname{otherwise}\;\;\;\;\;\;\;\;\;\;\;\\\ \end{aligned}\right.,$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{\partial\mathcal{J}}{\partial{\bf
w}}=\frac{\partial\mathcal{J}}{\partial x}\frac{\partial x}{\partial\hat{\bf
w}}\frac{\partial\hat{\bf w}}{\partial{\bf w}}=\left\\{\begin{aligned}
&\frac{\partial\mathcal{J}}{\partial x}\frac{\partial x}{\partial\hat{\bf
w}}&\operatorname{if}{\bf w}\in[-Q_{n}^{\bf w},Q_{p}^{\bf w}]\\\
&0&\operatorname{otherwise}\;\;\;\;\;\;\;\;\;\;\;\;\;\\\
\end{aligned}\right.,$

where $\mathcal{J}$ is loss function, $Q(\cdot)$ is applied in the forward
propagation while the straight-through estimator (STE) [1] is used to retain
the derivation of gradient in backward propagation. $\otimes$ denotes the
matrix multiplication with efficient bit-wise operations.

The input images are first encoded as patches and passes through several
transformer blocks. Such transformer block consists of two components: Multi-
Head Self-Attention (MHSA) and Multi-Layer Perceptron (MLP). The computation
of attention weight depends on the corresponding query ${\bf q}$, key ${\bf
k}$ and value ${\bf v}$, and the quantized computation in one attention head
is

${\bf q}=\operatorname{Q-Linear}_{q}(x),{\bf
k}=\operatorname{Q-Linear}_{k}(x),{\bf v}=\operatorname{Q-Linear}_{v}(x),$ (3)

where
$\operatorname{Q-Linear}_{q},\operatorname{Q-Linear}_{k},\operatorname{Q-Linear}_{v}$
denote the three quantized linear layers for ${\bf q},{\bf k},{\bf v}$,
respectively. Thus, the attention weight is formulated as

$\displaystyle{\bf A}$ $\displaystyle=\frac{1}{\sqrt{d}}(Q_{a}({\bf q})\otimes
Q_{a}({\bf k})^{\top}),$ (4) $\displaystyle Q_{\bf A}$
$\displaystyle=Q_{a}(\operatorname{softmax}({\bf A})).$

Training for Quantized ViT.  Knowledge distillation is an essential
supervision approach for training quantized neural networks, which bridges the
performance gap between quantized models and full-precision counterparts. The
usual practice is using distillation through attention as described in [25]

$\displaystyle\mathcal{L}_{\operatorname{dist}}$
$\displaystyle=\frac{1}{2}\mathcal{L}_{\operatorname{CE}}(\psi(Z_{q}),y)+\frac{1}{2}\mathcal{L}_{\operatorname{CE}}(\psi(Z_{q}),y_{t}),$
(5) $\displaystyle y_{t}$ $\displaystyle=\mathop{\arg\max}_{c}Z_{t}(c).$

##  4 Proposed Q-ViT

Figure 3: Analysis of bottleneck from architecture perspective. We report the
accuracy of 2-bit quantized DeiT-S and DeiT-B on ImageNet dataset about
replacing full-precision structure.

With aforementioned quantizing and training pipeline, a fully quantized ViT
baseline is built, which however has a large performance gap with full-
precision counterparts. Our study shows that the baseline suffers an severe
information distortion among quantized attention scores in the forward
propagation and distillation direction misleading in the backward propagation.

###  4.1 Performance degeneration of Fully Quantized ViT Baseline

Intuitively, in the fully quantized ViT baseline, the information
representation capability largely depends on the transformer-based
architecture, such as the attention weight in MHSA module. However, the
performance improvement brought by such architecture is severely limited by
the quantized parameters, while the rounded and discrete quantization also
significantly affect the optimization. The phenomenon identifies the
bottleneck of the fully quantized ViT baseline comes from architecture and
optimization for the forward and backward propagation, respectively.

Architecture bottleneck.  We replace each module with the full-precision
counterpart respectively and compare the accuracy drop as shown in Fig. 3.
These ablation experiments are conducted the same as experiments in Tab. 2. We
find that quantizing query, key, value and attention weight (i.e.,
$\operatorname{softmax}({\bf A})$ in Eq. (4) to 2 bits brings the most
significant drop of accuracy among all parts of the ViT, up to 10.03%. While
quantized MLP layers and quantized weights of linear layers in MHSA brings
only 1.78% and 4.26% drop, respectively. And once query, key, value and
attention weight are quantized, even keep all weights of linear layers in MHSA
module full-precision, the performance drops (10.57%) are still significant.
Thus, improving the attention structure is pivotal to solve the performance
drop problem of quantized ViT.

Optimization bottleneck.  We calculate l2-norm distances between each
attention weight among different blocks in DeiT-S architecture as shown in
Fig. 4. The MHSA modules in full-precision ViT with different depth learn
different representations from images. As mentioned in [22], lower ViT layers
attend both locally and globally while higher ViT layers pay most attention to
global representations. However, the fully quantized ViT (blue lines in Fig.
4) fails to learn accurate attention map distances. Thus, it requires a new
design that could utilize the full-precision teacher’s information better.

###  4.2 Information Rectification in Q-Attention

To address the information distortion of quantized representations in the
forward propagation, we propose an efficient Q-Attention structure based on
information theory, which statistically maximizes the entropy of
representation and revives the attention mechanism in the fully quantized ViT.
Since the representations with extremely compressed bit-width in fully
quantized ViT have limited capabilities, the ideal quantized representation
should preserve the given full-precision counterparts as much as possible,
which means the mutual information between quantized and full-precision
representations should be maximized as mentioned in [20].

We further show the statistical results that the distribution of query and key
values in ViT architectures intending to follow Gaussian distributions under
the distilling supervision, whose histograms are in bell-shape [20]. For
example, in Fig. 1 and Fig. 5, we have shown the query and key distributions
and their corresponding Probability Density Function (PDF) using the
calculated mean and standard deviation for each MHSA layer. Therefore, the
distributions of query and key in the MHSA modules of full-precision
counterparts are formulated as

$\displaystyle{\bf q}\sim\mathcal{N}(\mu({\bf q}),\sigma({\bf
q})),\;\;\;\;{\bf k}\sim\mathcal{N}(\mu({\bf k}),\sigma({\bf k})).$ (6)

Since the weight and the activation with extremely compressed bit-width in
fully quantized ViT have limited capabilities, the ideal quantization process
should preserve the corresponding full-precision counterparts as much as
possible, thus the mutual information between quantized and full-precision
representations should be maximized [20]. As shown in [19], for Gaussian
distribution, the quantizers with maximum output entropy (MOE) and minimum
average error (MAE) are approximately the same within a multiplicative
constant. Thus the process of minimizing the error between full-precision
values and quantized values is equivalent to maximizing the information
entropy of the quantized values. Thus, when the deterministic quantization
function is applied to quantized ViT, such objective is equivalent to
maximizing the information entropy $\mathcal{H}(Q_{\bf x})$ of quantized
representation $Q_{\bf x}$ [19] in Eq. (4), which is defined as

$\displaystyle\mathcal{H}(Q_{a}({\bf x}))=-\sum_{q_{\bf x}}p(q_{\bf x})\log
p(q_{\bf x})=\frac{1}{2}\log 2\pi e\sigma^{2}_{\bf x},$ (7)
$\displaystyle\mathop{\max}\mathcal{H}(Q_{a}({\bf x}))=\frac{n\ln
2}{2^{n}},\;\;\operatorname{when}\;p(q_{\bf x})=\frac{1}{2^{n}},$

where $q_{\bf x}$ is the random quantized variables in $Q_{a}({\bf x})$ (which
is $Q_{a}({\bf q})$ or $Q_{a}({\bf k})$ in different conditions) with
probability mass function $p(\cdot)$. For better retaining the information
contained in the MHSA modules from the full-precision counterparts, the
information entropy in the quantization process should be maximized.

However, direct application of quantization function converting the values
into finite fixed points brings irreversible disturbance to the distributions
and the information entropy $\mathcal{H}(Q_{a}({\bf q}))$ and
$\mathcal{H}(Q_{a}({\bf k}))$ degenerates to a much lower level than the full-
precision counterparts. To mitigate the information degradation from the
quantization process in the attention mechanism, a Information Rectification
Module (IRM) is proposed for effectively maximizing the information entropy of
quantized attention weights

$\displaystyle Q_{a}(\tilde{\bf q})=Q_{a}(\frac{{\bf q}-\mu({\bf
q})+\beta_{\bf q}}{\gamma_{\bf q}\sqrt{\sigma^{2}({\bf q})+\epsilon_{\bf
q}}}),\;\;\;\;Q_{a}(\tilde{\bf k})=Q_{a}(\frac{{\bf k}-\mu({\bf k})+\beta_{\bf
k}}{\gamma_{\bf k}\sqrt{\sigma^{2}({\bf k})+\epsilon_{\bf k}}}),$ (8)

where $\gamma_{\bf q},\beta_{\bf q}$ and $\gamma_{\bf k},\beta_{\bf k}$ are
learnble parameters for modifying the distribution of $\tilde{\bf q}$, while
$\epsilon_{\bf q}$ and $\epsilon_{\bf k}$ are constants preventing the
denominator being 0. The learning rates of learnable $\gamma_{\bf
q},\beta_{\bf q}$ and $\gamma_{\bf k},\beta_{\bf k}$ are same as the whole
network. Thus after IRM, the information entropy
$\mathcal{H}(Q_{a}({\tilde{\bf q}}))$ and $\mathcal{H}(Q_{a}({\tilde{\bf
k}}))$ are formulated as

$\displaystyle\mathcal{H}(Q({\tilde{\bf q}}))=\frac{1}{2}\log 2\pi
e[\gamma^{2}_{\bf q}(\sigma^{2}_{\bf q}+\epsilon_{\bf
q})],\;\;{\color[rgb]{0,0,0}\mathcal{H}(Q({\tilde{\bf k}}))=\frac{1}{2}\log
2\pi e[\gamma^{2}_{\bf k}(\sigma^{2}_{\bf k}+\epsilon_{\bf k})].}$ (9)

Then to revive the attention mechanism to capture critic elements by
information entropy maximization, the learnable parameters $\gamma_{\bf
q},\beta_{\bf q}$ and $\gamma_{\bf k},\beta_{\bf k}$ reshape the distributions
of the query and key values to achieve the state of information maximization.
In a nutshell, in our IRM-Attention structure, the information entropy of
quantized attention weight is maximized to alleviate its severe information
distortion and revive the attention mechanism.

Figure 4: Attention-distance comparison for full-precision DeiT-Small (green
lines) , fully quantized DeiT-Small baseline (blue lines), and Q-ViT (yellow
lines) for same input. Q-ViT shows similar behavior with the full- precision
model, while baseline suffers indistinguishable attention distance for
information degradation.

###  4.3 Distribution Guided Distillation through Attention

To address the attention distribution mismatch occurred in fully quantized ViT
baseline in the backward propagation, we further propose a Distribution Guided
Distillation (DGD) scheme with apposite distilled activations and the well-
designed similarity matrices to effectively utilize knowledge from the
teacher, which optimizes the fully quantized ViT more accurately.

Figure 5: The histogram of query and key values ${\bf q},{\bf k}$ (blue
shadow) along with the PDF curve (red line) of Gaussian distribution
$N(\mu,\sigma^{2})$ [20], for 3 selected layers in DeiT-T and 4-bit Q-ViT.
$\mu$ and $\sigma^{2}$ are the statistical mean and variance of the values.

As an optimization technique based on element-level comparison of activation,
distillation allows the quantized ViT to mimic the full-precision teacher
model about output logits. However, we find that the distillation procedure
used in previous ViT and fully quantized ViT baseline (Sec. 3) unable to
deliver meticulous supervision to attention weights (shown in Fig. 4), leading
to insufficient optimization. To solve the optimization insufficiency in the
distillation of the fully quantized ViT, we propose the Distribution-Guided
Distillation (DGD) method in Q-ViT. We first build patch-based similarity
pattern matrices for distilling the upstream query and key instead of
attention following [26], which is formulated as

$\displaystyle\tilde{G}^{l}_{{\bf q}_{h}}=\tilde{\bf
q}^{l}_{h}\cdot({\tilde{\bf q}^{l}_{h}})^{\top},\;\;G^{(l)}_{{\bf
q}_{h}}=\tilde{G}^{l}_{{\bf q}_{h}}/\|\tilde{G}^{l}_{{\bf q}_{h}}\|_{2},$ (10)
$\displaystyle\tilde{G}^{l}_{{\bf k}_{h}}=\tilde{\bf
k}^{l}_{h}\cdot({\tilde{\bf k}^{l}_{h}})^{\top},\;\;G^{(l)}_{{\bf
k}_{h}}=\tilde{G}^{l}_{{\bf k}_{h}}/\|\tilde{G}^{l}_{{\bf k}_{h}}\|_{2},$

where $\|\cdot\|_{2}$ denotes $\ell_{2}$ normalization, and $l,h$ are layer
index and head index. Previous work shows that matrices constructed in this
way are regarded as the specific patterns reflecting the semantic
comprehension of network [26]. And the patches encoded from input images
containing high-level understanding of parts, objects, and scenes [8]. Thus
such semantic-level distillation target provide guided and meticulous
supervision to the quantized ViT. The corresponding $\tilde{G}^{l}_{{\bf
q}_{h};T}$ and $\tilde{G}^{l}_{{\bf k}_{h};T}$ are constructed in the same way
by the teacher’s activation. Thus combining the original distillation loss in
Eq. (5), the final distillation loss is then formulated as

$\displaystyle\mathcal{L}_{\operatorname{DGD}}=\sum_{l\in[1,L]}\sum_{h\in[1,H]}\|\tilde{G}^{l}_{{\bf
q}_{h};T}-\tilde{G}^{l}_{{\bf q}_{h}}\|_{2}+\|\tilde{G}^{l}_{{\bf
k}_{h};T}-\tilde{G}^{l}_{{\bf k}_{h}}\|_{2},$ (11)
$\displaystyle\mathcal{L}_{\operatorname{distillation}}=\mathcal{L}_{\operatorname{dist}}+\mathcal{L}_{\operatorname{DGD}},$

where $L$ and $H$ denote the number of ViT layers and heads. With the proposed
Distribution Guided Distillation, the Q-ViT retains the distribution over
query and key from the full-precision counterparts (as shown in Fig. 5).

Our DGD scheme first provides the distribution-aware optimization direction
together with processing appropriate distilled parameters and then constructs
similarity matrices to eliminate scale differences and numerical instability,
thereby improves fully quantized ViT by accurate optimization.

##  5 Experiments

In this section, we evaluate the performance of the proposed Q-ViT model for
image classification task using popular DeiT [25] and Swin [15] backbones. To
the best of our knowledge, there is no publicly available source code on
quantization-aware training of vision transformer at this point, so we
implement the baseline and LSQ [5] methods by ourselves.

###  5.1 Datasets and Implementation Details

Datasets. The experiments are carried out on the ILSVRC12 ImageNet
classification dataset [12]. The ImageNet dataset is more challenging due to
its large scale and greater diversity. There are 1000 classes and 1.2 million
training images, and 50k validation images in it. In our experiments, we use
the classic data augmentation method described in [25].

Experimental settings. In our experiments, we initialize the weights of
quantized model with the corresponding pretrained full-precision model. The
quantized model is trained for 300 epochs with batch-size 512 and the base
learning rate $2e\\!-\\!4$. We do not use warm-up scheme. For all the
experiments, we apply LAMB [34] optimizer with weight decay set as 0. Other
training settings follow DeiT [25] or Swin Transformer [15]. Note that we use
8-bit for the patch embedding (first) layer and the classification (last)
layer following [5].

Bakcbone. We evaluate our quantization method on two popular vision
transformer implementation: DeiT [25] and Swin Transformer [15]. The DeiT-S,
DeiT-B, Swin-T and Swin-S are adopted as the backbone models, whose Top-1
accuracy on ImageNet dataset are 79.9%, 81.8%, 81.2%, and 83.2% respectively.
For a fair comparison, we utilize the official implementation of DeiT and Swin
Transformer.

###  5.2 Ablation Study

We give quantitative results of the proposed IRM and DGD in Tab. 1 As shown in
Tab. 1, the fully quantized ViT baseline suffers a severe performance drop on
classification task (0.2%, 2.1% and 11.7% with 2/3/4-bit, respectively). IRM
and DGD improve the performance when used alone, and the two techniques
further boost the performance considerably when combined together. For
example, the IRM improve the 2-bit Baseline by 1.7% and the DGD achieves 2.3%
performance improvement. While combining the IRM and DGD together, the
performance improvement achieves 3.8%.

Table 1: Evaluating the components of Q-ViT based on ViT-S backbone. Method | #Bits | Top-1 | #Bits | Top-1 | #Bits | Top-1
---|---|---|---|---|---|---
Full-precision | 32-32 | 79.9 | - | - | - | -
Baseline | 4-4 | 79.7 | 3-3 | 77.8 | 2-2 | 68.2
+IRM | 4-4 | 80.2 | 3-3 | 78.2 | 2-2 | 69.9
+DGD | 4-4 | 80.4 | 3-3 | 78.5 | 2-2 | 70.5
1-7 +IRM+DGD (Q-ViT) | 4-4 | 80.9 | 3-3 | 79.0 | 2-2 | 72.0

To conclude, the two techniques can promote each other to improve Q-ViT and
close the performance gap between fully quantized ViT and full-precision
counterpart.

Table 2: Quantization results on ImageNet dataset. “#Bits” (W-A) is the bit width for weights and activation. Network | Method | #Bits | Size(MB) | FLOPs(G) | Top-1 | Top-5
---|---|---|---|---|---|---
DeiT-S | Full-precision | 32-32 | 88.2 | 4.3 | 79.9 | 95.0
VT-PTQ | 8${}_{\operatorname{MP}}$-8${}_{\operatorname{MP}}$ | 22.2 | - | 78.1 | -
LSQ | 4-4 | 11.4 | 0.7 | 79.6 | 94.6
Baseline | 4-4 | 11.4 | 0.7 | 79.7 | 94.5
Q-ViT | 4-4 | 11.4 | 0.7 | 80.9 | 94.9
2-7 | LSQ | 3-3 | 8.7 | 0.4 | 77.3 | 93.0
| Baseline | 3-3 | 8.7 | 0.4 | 77.5 | 93.3
| Q-ViT | 3-3 | 8.7 | 0.4 | 79.0 | 94.2
2-7 | LSQ | 2-2 | 6.0 | 0.2 | 68.0 | 86.4
| Baseline | 2-2 | 6.0 | 0.2 | 68.2 | 86.5
| Q-ViT | 2-2 | 6.0 | 0.2 | 72.1 | 90.3
DeiT-B | Full-precision | 32-32 | 346.2 | 16.8 | 81.8 | 95.6
VT-PTQ | 8${}_{\operatorname{MP}}$-8${}_{\operatorname{MP}}$ | 86.8 | - | 81.3 | -
LSQ | 4-4 | 44.1 | 2.2 | 80.9 | 95.1
Baseline | 4-4 | 44.1 | 2.2 | 81.1 | 95.3
Q-ViT | 4-4 | 44.1 | 2.2 | 83.0 | 96.1
2-7 | LSQ | 3-3 | 33.4 | 1.4 | 79.0 | 94.5
| Baseline | 3-3 | 33.4 | 1.4 | 79.3 | 94.9
| Q-ViT | 3-3 | 33.4 | 1.4 | 81.0 | 95.1
2-7 | LSQ | 2-2 | 22.7 | 0.8 | 70.3 | 88.6
| Baseline | 2-2 | 22.7 | 0.8 | 70.4 | 88.8
| Q-ViT | 2-2 | 22.7 | 0.8 | 74.2 | 92.2
Swin-T | Full-precision | 32-32 | 114.2 | 4.5 | 81.2 | 95.5
LSQ | 4-4 | 14.6 | 0.6 | 80.2 | 95.2
Baseline | 4-4 | 14.6 | 0.6 | 80.5 | 95.4
Q-ViT | 4-4 | 14.6 | 0.6 | 82.5 | 97.3
2-7 | LSQ | 3-3 | 11.2 | 0.3 | 79.7 | 94.9
| Baseline | 3-3 | 11.2 | 0.3 | 79.8 | 95.1
| Q-ViT | 3-3 | 11.2 | 0.3 | 80.9 | 96.1
2-7 | LSQ | 2-2 | 7.7 | 0.2 | 70.4 | 88.8
| Baseline | 2-2 | 7.7 | 0.2 | 70.6 | 89.0
| Q-ViT | 2-2 | 7.7 | 0.2 | 74.7 | 92.5
Swin-S | Full-precision | 32-32 | 199.8 | 8.7 | 83.2 | 96.2
LSQ | 4-4 | 7.0 | 1.1 | 82.5 | 97.1
Baseline | 4-4 | 7.0 | 1.1 | 82.9 | 97.3
Q-ViT | 4-4 | 7.0 | 1.1 | 84.4 | 98.3
2-7 | LSQ | 3-3 | 5.5 | 0.6 | 80.6 | 95.7
| Baseline | 3-3 | 5.5. | 0.6 | 80.9 | 95.9
| Q-ViT | 3-3 | 5.5 | 0.6 | 82.7 | 97.5
2-7 | LSQ | 2-2 | 3.9 | 0.3 | 72.4 | 90.2
| Baseline | 2-2 | 3.9 | 0.3 | 72.7 | 90.6
| Q-ViT | 2-2 | 3.9. | 0.3 | 76.9 | 94.9

###  5.3 Main Results

The experimental results are shown in Tab. 2. We compare our method with
2/3/4-bit baseline and LSQ [5] based on the same frameworks for the task of
image classification with the ImageNet dataset. We also report the
classification performance of the 8-bit post-training quantization networks
percentile VT-PTQ [18]. We firstly evaluate the proposed method on DeiT-S and
DeiT-B models.

For DeiT-S backbone, compared with 8-bit VT-PTQ method, our 4bit Q-ViT
achieves a much larger compression ratio than 8-bit VT-PTQ, but with
significant performance improvement (78.1% vs. 80.9%). And it is worth noting
that the proposed 2-bit model significantly compresses the DeiT-S by
21.5$\times$ on FLOPs. The proposed method boosts the performance of 2/3/4-bit
Baseline by 3.9%, 1.5% and 1.2% with the same architecture and bit-width,
which is significant on the ImageNet dataset. For larger DeiT-B, as shown in
Tab. 2, the performance of the proposed method outperforms the 2/3/4-bit
Baseline by 3.8%, 1.7% and 1.9%, a large margin. Also note that the proposed
2/3/4-bit model significantly compresses the DeiT-B by 21$\times$, 12$\times$
and 7.6$\times$ on FLOPs. Compared with 8-bit post-training quantization
methods, our method achieves significantly higher compression rate, and the
performance improvement is significant.

Also, our method generates convincing results on Swin-transformers. As shown
in Tab. 2, the performance of the proposed method with Swin-T outperforms the
2/3/4-bit Baseline method by 4.1% , 2.1% and 2.0%, a large margin. Compared
with 8-bit post-training quantization methods, our method achieves
significantly higher compression rate, and comparable performance. Note that
our 4-bit Q-ViT surpasses the full-precision by 1.3% counterpart using Swin-T,
which demonstrates the significance of our Q-ViT. For larger Swin-S, the
performance of the proposed method outperforms the 2/3/4-bit Baseline by 4.3%,
1.8% and 1.5%. Also note that our 4-bit Q-ViT surpasses the full-precision by
1.1% counterpart using Swin-S and significantly compresses the Swin-S by
7.9$\times$ , which demonstrates the effectiveness and efficiency of our
Q-ViT.

##  6 Conclusion

In this paper, we introduce Q-ViT to improve the fully quantized ViTs with
high compression ratio and competitive performance. We first build a
theoretical framework of fully quantized ViT and analysis the bottlenecks of
the fully quantized ViT baseline. Then we introduce Information Rectification
Module and Distribution Guided Distillation to Q-ViT for performance
improvement. Our proposed Q-ViTs achieve comparable performance with full-
precision counterparts with ultra-low bit weights and activations. Our work
gives an insightful analysis and effective solution about the crucial issues
in ViT full quantization, which blazes a promising path for the extreme
compression of ViT.

##  7 Acknowledgement

This work was supported in part by the National Natural Science Foundation of
China under Grant 62076016, under Grant 62206272 and 61827901, Beijing Natural
Science Foundation-Xiaomi Innovation Joint Fund L223024, Foundation of China
Energy Project GJNY-19-90.

## References

  * [1] Yoshua Bengio, Nicholas Léonard, and Aaron Courville.  Estimating or propagating gradients through stochastic neurons for conditional computation.  arXiv preprint arXiv:1308.3432, 2013.
  * [2] Nicolas Carion, Francisco Massa, Gabriel Synnaeve, Nicolas Usunier, Alexander Kirillov, and Sergey Zagoruyko.  End-to-end object detection with transformers.  In Proc. of ECCV, pages 213–229, 2020.
  * [3] Misha Denil, Babak Shakibi, Laurent Dinh, Marc’Aurelio Ranzato, and Nando De Freitas.  Predicting parameters in deep learning.  In Proc. of NeurIPS, pages 2148–2156, 2013.
  * [4] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, et al.  An image is worth 16x16 words: Transformers for image recognition at scale.  arXiv preprint arXiv:2010.11929, 2020.
  * [5] Steven K Esser, Jeffrey L McKinstry, Deepika Bablani, Rathinakumar Appuswamy, and Dharmendra S Modha.  Learned step size quantization.  arXiv preprint arXiv:1902.08153, 2019.
  * [6] Jun Fang, Ali Shafiee, Hamzah Abdel-Aziz, David Thorsley, Georgios Georgiadis, and Joseph H Hassoun.  Post-training piecewise linear quantization for deep neural networks.  In Proc. of ECCV, pages 69–86, 2020.
  * [7] Peng Gao, Jiasen Lu, Hongsheng Li, Roozbeh Mottaghi, and Aniruddha Kembhavi.  Container: Context aggregation network.  In Proc. of NeurIPS, pages 19160–19171, 2021.
  * [8] Kaiming He, Xinlei Chen, Saining Xie, Yanghao Li, Piotr Dollár, and Ross Girshick.  Masked autoencoders are scalable vision learners.  arXiv preprint arXiv:2111.06377, 2021.
  * [9] Yang He, Guoliang Kang, Xuanyi Dong, Yanwei Fu, and Yi Yang.  Soft filter pruning for accelerating deep convolutional neural networks.  In Proc. of IJCAI, pages 2234–2240, 2018.
  * [10] Andrew G Howard, Menglong Zhu, Bo Chen, Dmitry Kalenichenko, Weijun Wang, Tobias Weyand, Marco Andreetto, and Hartwig Adam.  Mobilenets: Efficient convolutional neural networks for mobile vision applications.  arXiv preprint arXiv:1704.04861, 2017.
  * [11] Sangil Jung, Changyong Son, Seohyung Lee, Jinwoo Son, Jae-Joon Han, Youngjun Kwak, Sung Ju Hwang, and Changkyu Choi.  Learning to quantize deep networks by optimizing quantization intervals with task loss.  In Proc. of CVPR, pages 4350–4359, 2019.
  * [12] Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton.  Imagenet classification with deep convolutional neural networks.  In Proc. of NeurIPS, pages 1097–1105, 2012.
  * [13] Fengfu Li, Bo Zhang, and Bin Liu.  Ternary weight networks.  arXiv preprint arXiv:1605.04711, 2016.
  * [14] Yang Lin, Tianyu Zhang, Peiqin Sun, Zheng Li, and Shuchang Zhou.  Fq-vit: Fully quantized vision transformer without retraining.  arXiv preprint arXiv:2111.13824, 2021.
  * [15] Ze Liu, Yutong Lin, Yue Cao, Han Hu, Yixuan Wei, Zheng Zhang, Stephen Lin, and Baining Guo.  Swin transformer: Hierarchical vision transformer using shifted windows.  In Proc. of ICCV, pages 10012–10022, 2021.
  * [16] Zechun Liu, Zhiqiang Shen, Marios Savvides, and Kwang-Ting Cheng.  Reactnet: Towards precise binary neural network with generalized activation functions.  In Proc. of ECCV, pages 143–159, 2020.
  * [17] Zechun Liu, Baoyuan Wu, Wenhan Luo, Xin Yang, Wei Liu, and Kwang-Ting Cheng.  Bi-real net: Enhancing the performance of 1-bit cnns with improved representational capability and advanced training algorithm.  In Proc. of ECCV, 2018.
  * [18] Zhenhua Liu, Yunhe Wang, Kai Han, Wei Zhang, Siwei Ma, and Wen Gao.  Post-training quantization for vision transformer.  Proc. of NeurIPS, 2021.
  * [19] D Messerschmitt.  Quantizing for maximum output entropy (corresp.).  IEEE Transactions on Information Theory, 17(5):612–612, 1971.
  * [20] Haotong Qin, Yifu Ding, Mingyuan Zhang, Qinghua Yan, Aishan Liu, Qingqing Dang, Ziwei Liu, and Xianglong Liu.  Bibert: Accurate fully binarized bert.  In Proc. of ICLR, pages 1–24, 2022.
  * [21] Haotong Qin, Ruihao Gong, Xianglong Liu, Mingzhu Shen, Ziran Wei, Fengwei Yu, and Jingkuan Song.  Forward and backward information retention for accurate binary neural networks.  In Proc. of CVPR, pages 2250–2259, 2020.
  * [22] Maithra Raghu, Thomas Unterthiner, Simon Kornblith, Chiyuan Zhang, and Alexey Dosovitskiy.  Do vision transformers see like convolutional neural networks?  Proc. of NeurIPS, pages 1–13, 2021.
  * [23] Yongming Rao, Wenliang Zhao, Benlin Liu, Jiwen Lu, Jie Zhou, and Cho-Jui Hsieh.  Dynamicvit: Efficient vision transformers with dynamic token sparsification.  Proc. of NeurIPS, pages 1–14, 2021.
  * [24] Adriana Romero, Nicolas Ballas, Samira Ebrahimi Kahou, Antoine Chassang, Carlo Gatta, and Yoshua Bengio.  Fitnets: Hints for thin deep nets.  In Proc. of ICLR, pages 1–13, 2015.
  * [25] Hugo Touvron, Matthieu Cord, Matthijs Douze, Francisco Massa, Alexandre Sablayrolles, and Hervé Jégou.  Training data-efficient image transformers & distillation through attention.  In Proc. of ICML, pages 10347–10357, 2021.
  * [26] Frederick Tung and Greg Mori.  Similarity-preserving knowledge distillation.  In Proc. of ICCV, pages 1365–1374, 2019.
  * [27] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin.  Attention is all you need.  Proc. of NeurIPS, 30, 2017.
  * [28] Wenhai Wang, Enze Xie, Xiang Li, Deng-Ping Fan, Kaitao Song, Ding Liang, Tong Lu, Ping Luo, and Ling Shao.  Pyramid vision transformer: A versatile backbone for dense prediction without convolutions.  In Proc. of ICCV, pages 568–578, 2021.
  * [29] Zheng Xie, Zhiquan Wen, Jing Liu, Zhiqiang Liu, Xixian Wu, and Mingkui Tan.  Deep transferring quantization.  In Proc. of ECCV, pages 625–642, 2020.
  * [30] Sheng Xu, Yanjing Li, Tiancheng Wang, Teli Ma, Baochang Zhang, Peng Gao, Yu Qiao, Jinhu Lu, and Guodong Guo.  Recurrent bilinear optimization for binary neural networks.  arXiv preprint arXiv:2209.01542, 2022.
  * [31] Sheng Xu, Yanjing Li, Bohan Zeng, Teli Ma, Baochang Zhang, Xianbin Cao, Peng Gao, and Jinhu Lu.  Ida-det: An information discrepancy-aware distillation for 1-bit detectors.  arXiv preprint arXiv:2210.03477, 2022.
  * [32] Sheng Xu, Junhe Zhao, Jinhu Lu, Baochang Zhang, Shumin Han, and David Doermann.  Layer-wise searching for 1-bit detectors.  In Proc. of CVPR, pages 5682–5691, 2021.
  * [33] Yifan Xu, Zhijie Zhang, Mengdan Zhang, Kekai Sheng, Ke Li, Weiming Dong, Liqing Zhang, Changsheng Xu, and Xing Sun.  Evo-vit: Slow-fast token evolution for dynamic vision transformer.  arXiv preprint arXiv:2108.01390, 2021.
  * [34] Yang You, Jing Li, Sashank Reddi, Jonathan Hseu, Sanjiv Kumar, Srinadh Bhojanapalli, Xiaodan Song, James Demmel, Kurt Keutzer, and Cho-Jui Hsieh.  Large batch optimization for deep learning: Training bert in 76 minutes.  Proc. of ICLR, pages 1–37, 2020.
  * [35] Shuchang Zhou, Yuxin Wu, Zekun Ni, Xinyu Zhou, He Wen, and Yuheng Zou.  Dorefa-net: Training low bitwidth convolutional neural networks with low bitwidth gradients.  arXiv preprint arXiv:1606.06160, 2016.
  * [36] Chenzhuo Zhu, Song Han, Huizi Mao, and William J Dally.  Trained ternary quantization.  In Proc. of ICLR, pages 1–10, 2017.
  * [37] Bohan Zhuang, Chunhua Shen, Mingkui Tan, Lingqiao Liu, and Ian Reid.  Towards effective low-bitwidth convolutional neural networks.  In Proc. of CVPR, pages 7920–7928, 2018.


#  ${\cal N}=1$ supersymmetry in $4D$ gravity with torsion and non-metricity

Cecilia Bejarano<EMAIL_ADDRESS>Instituto de Astronomía y Física del
Espacio (IAFE-UBA-CONICET), Ciudad Universitaria, Pabellón IAFE, Ciudad de
Buenos Aires (1428), Argentina  Eric Lescano<EMAIL_ADDRESS>Division of
Theoretical Physics, Rudjer Bošković Institute, Bijenička 54, 10000 Zagreb,
Croatia

###### Abstract

We study the inclusion of fermionic degrees of freedom and ${\cal N}=1$
supersymmetry in 4-dimensional manifolds with arbitrary torsion and non-
metricity tensors deforming the connection. We inspect the closure of local
diffeomorphism, Lorentz and supersymmetric transformations on the vielbein and
gravitino at leading order in fermions. The closure of a pair of
supersymmetric transformations acting on the vielbein restricts the form of
the spin connection. The dependency of the latter on the fundamental fields is
exclusively given by the vielbein, up to Lorentz and/or diffeomorphisms
brackets, independently of the torsion or non-metricity deformations of the
affine connection, which is not restricted by supersymmetry at this order. We
have then studied the couple of the Rarita-Schwinger theory to a generic
gravitational Lagrangian; the supersymmetric variation of the former restricts
the form of the bosonic Lagrangian in order to accomplish invariance. Special
cases where $R_{\mu\nu}{}^{ab}=0$, $Q_{\mu\nu\rho}=0$ or
$T_{\mu\nu}{}^{\rho}=0$ are provided.

##  I Introduction

In Ref. F , Fock initiated the inclusion of fermionic degrees of freedom in
General Relativity (GR). The standard procedure requires, on the one hand,
fields with Grassmanian statistics and, on the other, a Clifford algebra for
the gamma matrices. The fermionic fields are usually defined on the tangent
space of a Riemannian manifold so the fundamental gravitational field is given
by a tetrad (or vielbein). While the GR Lagrangian for the gravity sector is
given by the Ricci scalar, fermionic contributions can be easily coupled to
the Einstein-Hilbert Lagrangian depending on the spin of the Grassmanian
field: a free spin-$\frac{1}{2}$ field is coupled considering the Dirac
Lagrangian while a free spin-$\frac{3}{2}$ field is coupled using the Rarita-
Schwinger (RS) Lagrangian RandS . The latter is supersymmetric invariant
meaning that if we couple the RS Lagrangian to a GR framework and we include
supersymmetric transformations, the supersymmetric transformation of the RS
Lagrangian compensates the transformation of the Ricci scalar. Furthermore in
this scenario the Majorana spinor can be interpreted as a gravitino field
WandZ .

In the last years, several proposals were considered to deform the
gravitational sector considering arbitrary torsion or non-metricity tensors.
When the connection is given by the Levi-Civita connection (and the torsion
and the non-metricity tensors vanish), the resulting theory is GR. For the
curvature-less Weitzenbock connection the framework is known as Teleparallel
Equivalent of GR, while for a gravitational theory defined by a torsion-free,
flat connection we have Symmetric Teleparallel Equivalent of GR. All these
theories share the same dynamics (known as geometrical trinity of gravity),
but this is not true for the modified versions of the latter, i.e.,
$f(T)$-teleparallel theory f(T)1 -f(T)10  and $f(Q)$-symmetric teleparallel
theory f(Q)1 -f(Q)4 . In presence of spinors, these kind of proposals where
studied in RecentSpin1 -RecentSpin3 . In vacuum solutions, both the torsion
and the non-metricity are useful geometrical pieces to deform a connection
with covariant extra terms, known as contortion and distortion. Particularly,
when one includes fermionic matter, the commutator of a pair of covariant
derivatives acting on the spinors contains new contributions. Moreover, in
presence of a non-trivial non-metricity the covariant derivative of the gamma
matrix is not null.

The main goal of this work is to study how to couple the RS dynamics for
backgrounds with arbitrary torsion and non-metricity, and to inspect the
restrictions that ${\cal N}=1$ supersymmetry forces on the gravitational
sector. Our main results are: On the one hand we study the closure of the
symmetry transformations by considering the leading order terms in fermions
Tanii . Interestingly enough, by taking into account arbitrary torsion and
non-metricity tensors, the commutator of a pair of supersymmetric
transformations on the vielbein forbids deformations on the spin-connection up
to Lorentz and/or diffeomorphims bracket. This result is independent of the
fermionic Lagrangian so it applies for spin-$\frac{1}{2}$ fields as well as
$\frac{3}{2}$ fields. On the other hand, we study the restrictions on the
gravitational theory when the RS Lagrangian is coupled. We performed the
supersymmetric transformation of this Lagrangian in presence of arbitrary
torsion and non-metricity tensors, and then we give the specific constraints
for geometries with $R_{\mu\nu}{}^{ab}=0$, $Q_{\mu\nu\rho}=0$ or
$T_{\mu\nu}{}^{\rho}=0$.

The results of this work show agreement between two large fields of knowledge:
on the one hand, the field of GR and its several modifications related to the
inclusion of torsion and non-metricity as fundamental degrees of freedom and,
on the other hand, the inclusion of fermionic matter related to the vacuum
fields through supersymmetry. The current state of the art indicates several
extension to this work, as we discussed in Section V.

##  II Field content, symmetries and covariant derivatives

We consider a fermionic matter field coupled to an arbitrary geometry. The
fundamental fields are a four-dimensional Majorana spinor field, $\psi^{a}$,
and a four-dimensional vielbein $e_{\mu}{}^{a}$ 111We give our conventions in
appendix A.

Majorana spinors satisfy a reality condition,

$\displaystyle\psi_{a}^{c}=C\bar{\psi}^{T}=\psi_{a}\,,$ (1)

where $C$ is a charged conjugation matrix which satisfies,

$\displaystyle C^{-1}\gamma^{a}C=-\gamma^{aT}\,,\quad C^{T}=-C\,.$ (2)

The gamma matrices fulfill a Clifford algebra

$\displaystyle\Big{\\{}\gamma^{a},\gamma^{b}\Big{\\}}=2\eta^{ab}\,,$ (3)

with $\eta^{ab}$ the inverse of the Minwkowski metric $\eta_{ab}$. In turn,
the metric tensor is obtained from the orthonormality condition as

$\displaystyle e_{\mu}{}^{a}\eta_{ab}e_{\nu}{}^{b}=g_{\mu\nu}\,.$ (4)

The symmetry rules of the fundamental fields are given by

$\displaystyle\delta e_{\mu}{}^{a}$ $\displaystyle=$
$\displaystyle\xi^{\nu}\partial_{\nu}e_{\mu}{}^{a}+\partial_{\mu}\xi^{\nu}e_{\nu}{}^{a}+e_{\mu}{}^{b}\Lambda_{b}{}^{a}+\frac{1}{2}\bar{\epsilon}\gamma^{a}\psi_{\mu}$
(5) $\displaystyle\delta\psi^{a}$ $\displaystyle=$
$\displaystyle\xi^{\nu}\partial_{\nu}\psi^{a}+\psi^{b}\Lambda_{b}{}^{a}-\frac{1}{4}\Lambda_{bc}\gamma^{bc}\psi^{a}+e^{\mu}{}_{a}\nabla_{\mu}\epsilon\,,$
(6)

where $\xi^{\mu}$, $\Lambda_{a}{}^{b}$ and $\epsilon$ are arbitrary parameters
for local diffeomorphisms, Lorentz and supersymmetry transformations,
respectively. From Eq. (5) one can observe that the vielbein transforms as a
vector with respect to infinitesimal diffeomorphims and local Lorentz
transformations, while the spinor field $\psi^{a}$ is a scalar under
infinitesimal diffeomorphims but transforms as a vector and a spinor with
respect to local Lorentz transformations.

The covariant derivative on these fields is expressed in terms of the affine
connection, $\Gamma_{\mu\nu}^{\rho}$, and the spin connection,
$\omega_{\mu}{}^{a}{}_{b}$, i.e.,

$\displaystyle\nabla_{\mu}e_{\nu}{}^{a}$ $\displaystyle=$
$\displaystyle\partial_{\mu}e_{\nu}{}^{a}-\Gamma_{\mu\nu}^{\sigma}e_{\sigma}{}^{a}+\omega_{\mu}{}^{a}{}_{b}e_{\nu}{}^{b}\,,$
(7) $\displaystyle\nabla_{\mu}\psi_{a}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\psi_{a}+\omega_{\mu
a}{}^{b}\psi_{b}+\frac{1}{4}\omega_{\mu bc}\gamma^{bc}\psi_{a}\,.$ (8)

We decompose the affine connection in the usual way,

$\displaystyle\Gamma_{\mu\nu}{}^{\rho}=\tilde{\Gamma}_{\mu\nu}{}^{\rho}-K_{\mu\nu}{}^{\rho}-N_{\mu\nu}{}^{\rho}\,$
(9)

where $\tilde{\Gamma}_{\mu\nu}{}^{\rho}$ is the Levi-Civita connection and

$\displaystyle K_{\mu\nu\rho}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(-T_{\mu\nu\rho}+T_{\nu\rho\mu}-T_{\rho\mu\nu})$
$\displaystyle N_{\mu\nu\rho}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(Q_{\mu\nu\rho}+Q_{\nu\mu\rho}-Q_{\rho\mu\nu})$ (10)

the contorsion and the distorsion, respectively, with the non-metricity and
the torsion defined as follows

$\displaystyle\nabla_{\mu}g_{\nu\rho}$ $\displaystyle=$ $\displaystyle
Q_{\mu\nu\rho}\,,$ (11) $\displaystyle T_{\mu\nu}{}^{\rho}$ $\displaystyle=$
$\displaystyle\Gamma_{\mu\nu}^{\rho}-\Gamma_{\nu\mu}^{\rho}\,.$ (12)

In this work we will focus on the leading order in fermionic contributions, so
$\Gamma$ is a generic affine connection and we do not need to impose a
particular fermionic torsion, which is a requirement in the full picture Tanii
. Using the previous definitions, we impose the following vielbein
compatibility in presence of torsion and non-metricity,

$\displaystyle\nabla_{\mu}e_{\nu}{}^{a}=N_{\mu\nu}{}^{\rho}e_{\rho}{}^{a}+K_{\mu\nu}{}^{\rho}e_{\rho}{}^{a}\,,$
(13)

or, equivalently,

$\displaystyle\tilde{\nabla}_{\mu}e_{\nu}{}^{a}\equiv\partial_{\mu}e_{\nu}{}^{a}-\tilde{\Gamma}_{\mu\nu}^{\sigma}e_{\sigma}{}^{a}+\omega_{\mu}{}^{a}{}_{b}e_{\nu}{}^{b}=0\,.$
(14)

The previous expression implies that the spin connection is defined as in GR
or, in other words, it does not contain extra tensorial deformations related
to the torsion and/or the non-metricity. Moreover, from Eq. (13) it is
straightforward to obtain

$\displaystyle
2\nabla_{\rho}e_{(\mu}{}^{a}\eta_{ab}e_{\nu)}{}^{b}=Q_{\rho\mu\nu}\,,$ (15)

in agreement with Eq. (11).

We can now define the curved version of the Majorana spinor field,
$\psi^{\mu}=e^{\mu}{}_{a}\psi^{a}$ and the curved version of the gamma matrix,
$\gamma^{\mu}=e^{\mu}{}_{a}\gamma^{a}$ which satisfies

$\displaystyle\Big{\\{}\gamma^{\mu},\gamma^{\nu}\Big{\\}}=2g^{\mu\nu}\,.$ (16)

The covariant derivative acting on a curved gamma matrix is given by

$\displaystyle\nabla_{\mu}\gamma^{\nu}=\frac{1}{2}Q_{\mu}{}^{\nu}{}_{\rho}\gamma^{\rho}\,,$
(17)

and the commutator of a pair of covariant derivatives acting on a generic
spinor reads

$\displaystyle\Big{[}\nabla_{\mu},\nabla_{\nu}\Big{]}\epsilon=$
$\displaystyle\frac{1}{4}R_{\mu\nu}{}^{ab}(\omega)\gamma_{ab}\epsilon-2\Gamma_{[\mu\nu]}^{\rho}\nabla_{\rho}\epsilon$
(18)
$\displaystyle-\frac{1}{2}\omega_{\mu}{}^{ab}\nabla_{\nu}\gamma_{ab}\epsilon$

while the same communtator acting on a the spinor field gives

$\displaystyle\Big{[}\nabla_{\mu},\nabla_{\nu}\Big{]}\psi_{\rho}=$
$\displaystyle\frac{1}{4}R_{\mu\nu}{}^{ab}(\omega)\gamma_{ab}\psi_{\rho}-R_{\mu\nu}{}^{\sigma}{}_{\rho}(\Gamma)\psi_{\sigma}$
(19)
$\displaystyle-2\Gamma_{[\mu\nu]}^{\sigma}\nabla_{\sigma}\psi_{\rho}-\frac{1}{2}\omega_{\mu}{}^{ab}\nabla_{\nu}\gamma_{ab}\psi_{\rho}\,.$

In the next part of the work we compute the closure of the different
transformations and, particularly, we analyze the role of Eq. (14) in the
closure of the supersymmetric transformations.

##  III Closure of the transformations

The closure of a pair of symmetry transformations is satisfied when

$\displaystyle\Big{[}\delta_{1},\delta_{2}\Big{]}\\{e_{\mu}{}^{a},\psi^{a}\\}=\delta_{3}\\{e_{\mu}{}^{a},\psi^{a}\\}\,.$
(20)

The closures are given by the following brackets,

$\displaystyle\Big{[}\delta_{\xi_{1}},\delta_{\xi_{2}}\Big{]}$
$\displaystyle=$
$\displaystyle\delta_{\xi_{3}}\,,\quad\xi^{\mu}_{3}=2\xi^{\nu}_{[2}\partial_{\nu}\xi_{1]}^{\mu}\,.$
(21) $\displaystyle\Big{[}\delta_{\Lambda},\delta_{\xi}\Big{]}$
$\displaystyle=$
$\displaystyle\delta_{\tilde{\Lambda}}\,,\quad\tilde{\Lambda}_{ab}=\xi^{\mu}\partial_{\mu}\Lambda_{ab}\,.$
(22) $\displaystyle\Big{[}\delta_{\epsilon},\delta_{\xi}\Big{]}$
$\displaystyle=$
$\displaystyle\delta_{\tilde{\epsilon}}\,,\quad\tilde{\epsilon}=\xi^{\mu}\partial_{\mu}\tilde{\epsilon}\,.$
(23) $\displaystyle\Big{[}\delta_{\Lambda_{1}},\delta_{\Lambda_{2}}\Big{]}$
$\displaystyle=$
$\displaystyle\delta_{\Lambda_{3}}\,,\quad\Lambda_{3c}{}^{a}=2\Lambda_{[1c}{}^{b}\Lambda_{2]b}{}^{a}\,.$
(24) $\displaystyle\Big{[}\delta_{\Lambda},\delta_{\epsilon}\Big{]}$
$\displaystyle=$
$\displaystyle\delta_{\tilde{\epsilon}}\,,\quad\tilde{\epsilon}=\frac{1}{4}\Lambda_{ab}\gamma^{ab}\epsilon\,,$
(25)

where $\xi,\epsilon,\Lambda$ are the infinitesimal parameters of the
supersymmetric, diffeomorphism, Lorentz transformations, respectively. It is
worth to note that the previous closures are satisfied in any gravity theory.
However, the closure $[\delta_{\epsilon_{1}},\delta_{\epsilon_{2}}]$ forces
Eq. (14) up to a Lorentz/diffeomorphism bracket. Let us illustrate this point
performing the explicit computation,

$\displaystyle\Big{[}\delta_{\epsilon_{1}},\delta_{\epsilon_{2}}\Big{]}e_{\mu}{}^{a}=\frac{1}{2}\bar{\epsilon}_{2}\gamma^{a}\nabla_{\mu}\epsilon_{1}-(1\leftrightarrow
2)\,.$ (26)

Using the explicit form of the covariant derivative on the spinor
$\epsilon_{1}$, the Clifford algebra and the following properties

$\displaystyle\bar{\epsilon}_{2}\gamma^{a}\partial_{\mu}\epsilon_{1}$
$\displaystyle=$
$\displaystyle-\partial_{\mu}\bar{\epsilon}_{1}\gamma^{a}\epsilon_{2}\,,$
$\displaystyle\bar{\epsilon}_{2}\gamma^{abc}\epsilon_{2}$ $\displaystyle=$
$\displaystyle\bar{\epsilon}_{1}\gamma^{abc}\bar{\epsilon}_{2}\,,$

the expression (26) can be written as

$\displaystyle\Big{[}\delta_{\epsilon_{1}},\delta_{\epsilon_{2}}\Big{]}e_{\mu}{}^{a}=\frac{1}{2}\partial_{\mu}(\bar{\epsilon}_{2}\gamma^{a}\epsilon_{1})+\frac{1}{2}\bar{\epsilon}_{2}\gamma^{c}\omega_{\mu}{}^{a}{}_{c}\epsilon_{1}\,.$
(27)

The previous terms can be written as an infinitesimal diffeomorphism
transformation with
$\tilde{\xi}^{\nu}=\frac{1}{2}e^{\nu}{}_{a}\bar{\epsilon}_{2}\gamma^{a}\epsilon_{1}$,

$\displaystyle\delta_{\tilde{\xi}}e_{\mu}{}^{a}$ $\displaystyle=$
$\displaystyle
e^{\nu}{}_{b}\bar{\epsilon}_{2}\gamma^{b}\epsilon_{1}\partial_{[\nu}e_{\mu]}{}^{a}+\frac{1}{2}\partial_{\mu}(\bar{\epsilon}_{2}\gamma^{a}\epsilon_{1})\,$
(28) $\displaystyle=$
$\displaystyle-\frac{1}{2}e^{\nu}{}_{b}\bar{\epsilon}_{2}\gamma^{b}\epsilon_{1}\omega_{\nu}{}^{a}{}_{c}e_{\mu}{}^{c}$
$\displaystyle+\frac{1}{2}\bar{\epsilon}_{2}\gamma^{c}\epsilon_{1}\omega_{\mu}{}^{a}{}_{c}e_{\mu}{}^{c}+\frac{1}{2}\partial_{\mu}(\bar{\epsilon}_{2}\gamma^{a}\epsilon_{1})\,,$

where in the last step we use the vielbein compatibility given by Eq. (14).
Finally,

$\displaystyle\Big{[}\delta_{\epsilon_{1}},\delta_{\epsilon_{2}}\Big{]}=\delta_{\tilde{\xi}}+\delta_{\tilde{\Lambda}}\,,\quad\tilde{\Lambda}^{a}{}_{c}=\frac{1}{2}e^{\nu}{}_{b}\bar{\epsilon}_{2}\gamma^{b}\epsilon_{1}\omega_{\nu}{}^{a}{}_{c}\,.$
(29)

Then, the closure of the supersymmetry algebra forces the use of the standard
spin connection before introducing dynamics. This means that the affine
connection can be generic, but the vielbein compatibility cannot contain terms
with torsion or non-metricity.

In the next part of the work we explore the RS Lagrangian. We explicitly show
the conditions which arise from supersymmetric invariance in presence of a
non-trivial torsion and non-metricity. We will focus on leading order terms in
fermions, and that is why we omit the study of
$\Big{[}\delta_{\epsilon_{1}},\delta_{\epsilon_{2}}\Big{]}\psi^{a}$ in the
present section.

##  IV Coupling the Rarita-Schwinger Lagrangian

The minimal matter Lagrangian that can be coupled to a gravity theory to turn
the latter into a supergravity theory (with ${\cal N}=1$ supersymmetric
invariance) is the RS Lagrangian which can be understood as the generalization
of the Dirac Lagrangian for spin-$\frac{3}{2}$ fields. We then consider the
following action principle,

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int d^{4}x({\cal
L}_{\textrm{gravity}}+{\cal L}_{RS})\,,$ (30)

with ${\cal L}_{\textrm{gravity}}$ a generic gravitational vacuum Lagrangian
for gravity and

$\displaystyle{\cal L}_{RS}$ $\displaystyle=$
$\displaystyle-e\bar{\psi}_{\mu}\gamma^{\mu\nu\rho}\nabla_{\nu}\psi_{\rho}+\mathcal{O}(\psi^{4})\,.$
(31)

Our goal now is to compute the supersymmetric variation of the matter
contributions in order to find restrictions on the gravity sector. For this
reason we apply the last transformations of equations (5) and (6) on Eq. (31).
After a lengthy but straightforward computation the supersymmetric variation
of the RS Lagrangian reads

$\displaystyle\delta_{\epsilon}{\cal L}_{RS}=$
$\displaystyle-\frac{1}{2}e\bar{\epsilon}Q_{\mu}{}^{\sigma}{}_{\sigma}\gamma^{\mu\nu\rho}\nabla_{\nu}\psi_{\rho}+e\bar{\epsilon}\nabla_{\mu}\gamma^{\mu\nu\rho}\nabla_{\nu}\psi_{\rho}$
(32)
$\displaystyle+e\bar{\epsilon}\gamma^{\mu\nu\rho}\nabla_{\mu}\nabla_{\nu}\psi_{\rho}-e\bar{\psi}_{\mu}\gamma^{\mu\nu\rho}\nabla_{\nu}\nabla_{\rho}\epsilon\,.$

We provide in Appendix B the products between gamma matrices relations
required for this computation. The new terms in this expression are related to
non-metricity effects (first line) and to torsion contributions (commutators
of the covariant derivatives in second line) through equations (18) and (19).

From the supersymmetric variation of Eq. (32) one learns lessons for different
proposals of theories of gravity when $R_{\mu\nu}{}^{ab}$, $Q_{\mu\nu\rho}$ or
$T_{\mu\nu}{}^{\rho}$ vanish. We summarize our findings in Table 1.

$ef(Q,T)$ | $R_{\mu\nu}{}^{ab}=0$ | $\begin{aligned} \delta_{\epsilon}f(Q,T)&=\frac{1}{2}e\bar{\epsilon}Q_{\mu}{}^{\sigma}{}_{\sigma}\gamma^{\mu\nu\rho}\nabla_{\nu}\psi_{\rho}-e\bar{\epsilon}\nabla_{\mu}\gamma^{\mu\nu\rho}\nabla_{\nu}\psi_{\rho}\\\ &+\frac{1}{2}e\bar{\epsilon}\gamma^{\mu\nu\rho}(2\Gamma_{[\mu\nu]}^{\sigma}\nabla_{\sigma}\psi_{\rho}+\frac{1}{2}\omega_{\mu ab}\nabla_{\nu}\gamma^{ab}\psi_{\rho})\\\ &-\frac{1}{2}e\bar{\psi}_{\mu}\gamma^{\mu\nu\rho}(2\Gamma_{[\nu\rho]}^{\sigma}\nabla_{\sigma}\epsilon+\frac{1}{2}\omega_{\nu ab}\nabla_{\rho}\gamma^{ab}\epsilon)\end{aligned}$
---|---|---
$ee^{\mu}{}_{a}e^{\nu}{}_{b}R_{\mu\nu}{}^{ab}+f(T)$ | $Q_{\mu\nu\rho}=0$ | $\begin{aligned} \delta_{\epsilon}f(T)=-\frac{1}{2}e\bar{\epsilon}\gamma^{\mu\nu\rho}T_{\nu\rho}{}^{\sigma}\nabla_{\sigma}\psi_{\rho}+\frac{1}{2}e\bar{\psi}_{\mu}\gamma^{\mu\nu\rho}T_{\nu\rho}{}^{\sigma}\nabla_{\sigma}\epsilon\end{aligned}$
$ee^{\mu}{}_{a}e^{\nu}{}_{b}R_{\mu\nu}{}^{ab}+f(Q)$ | $T_{\mu\nu}{}^{\rho}=0$ | $\begin{aligned} \delta_{\epsilon}f(Q)&=\frac{1}{2}e\bar{\epsilon}Q_{\mu}{}^{\sigma}{}_{\sigma}\gamma^{\mu\nu\rho}\nabla_{\nu}\psi_{\rho}-e\bar{\epsilon}\nabla_{\mu}\gamma^{\mu\nu\rho}\nabla_{\nu}\psi_{\rho}\\\ &-\frac{1}{2}e\bar{\epsilon}\gamma^{\mu\nu\rho}(Q_{\mu\nu}{}^{\sigma}\nabla_{\sigma}\psi_{\rho}-\frac{1}{2}\omega_{\mu ab}\nabla_{\nu}\gamma^{ab}\psi_{\rho})\\\ &+\frac{1}{2}e\bar{\psi}_{\mu}\gamma^{\mu\nu\rho}(Q_{\nu\rho}{}^{\sigma}\nabla_{\sigma}\epsilon-\frac{1}{2}\omega_{\nu ab}\nabla_{\rho}\gamma^{ab}\epsilon)\end{aligned}$
Table 1:  The first two columns correspond to particular bosonic Lagrangians
which the Rarita-Schwinger Lagrangian can be coupled to. In the third column
we show the constraints imposed by ${\cal N}=1$ supersymmetric invariance.

The last two rows of the table reduce to the well known Einstein-Hilbert
Lagrangian when $T_{\mu\nu}{}^{\rho}=0$ or $Q_{\mu\nu\rho}=0$, respectively.
In that case (i.e., GR), the full action is ${\cal N}=1$ supersymmetric
invariant due to

$\displaystyle\delta_{\epsilon}(ee^{\mu}{}_{a}e^{\nu}{}_{b}R_{\mu\nu}{}^{ab})=$
$\displaystyle\frac{1}{2}e\bar{\epsilon}\gamma_{\rho}\psi^{\rho}e_{a}{}^{\mu}e_{b}{}^{\nu}R_{\mu\nu}{}^{ab}(\omega)$
(33)
$\displaystyle-e\bar{\epsilon}\gamma^{\mu}\psi_{a}e_{b}{}^{\nu}R_{\mu\nu}{}^{ab}(\omega)\,.$

The previous terms cancel the terms of Eq. (32) which are independent of
$T_{\mu\nu}{}^{\rho}$ and $Q_{\mu\nu\rho}$.

##  V Conclusions and future directions

We have analyzed the inclusion of ${\cal N}=1$ supersymmetric transformations
in a general gravity framework, where the torsion and the non-metricity
tensors are arbitrary. We have studied the closure of local Lorentz,
diffeomorphisms and ${\cal N}=1$ supersymmetry transformations on the
fundamental fields at leading order in fermions (bilineal contributions at the
level of the action). We have found that the closure of a pair of
supersymmetric transformations acting on the vielbein reproduce a
diffeomorphism and a Lorentz transformation when the spin connection satisfies
Eq. (14), and all the possible deformations of the spin connection are
restricted by the closure of the supersymmetric transformations, although they
do not restrict the deformations of the affine connection (Eq. (9)). The
inspection of the closure of the symmetry transformations is independent to
the dynamics.

On the other hand, we have obtained the form of the supersymmetric
transformation in Eq. (32) by taking into account arbitrary non-metricity and
torsion tensors. In this way, the gravitational Lagrangian results constrained
in order to accomplish supersymmetric invariance. In the present work, we have
shown the explicit variations of ${\cal L_{\textrm{gravity}}}$ for three
particular cases where $R_{\mu\nu}{}^{ab}$, $Q_{\mu\nu\rho}$ or
$T_{\mu\nu}{}^{\rho}$ vanish.

We close this section with some possible follow-up projects:

  * •

The fermionic contributions at the level of the action principle might be used
to establish new equivalences between gravity theories. The equation of motion
for the fermionic field(s) might be useful to cure gravitational
inequivalences considering different bilinears contributions, even for
Lagrangians that are not supersymmetric.

  * •

Different constructions of $f(Q)$ cosmologies CosmoQ1 -CosmoQ6  requiere non-
linear $f(Q)$ Lagrangians to include dark components of the Universe from a
variational principle. The inclusion of ${\cal N}=1$ supersymmetry in $f(Q)$
cosmological scenarios, which can be easily related to string cosmology
scenarios Gasperini , allows the possibility of condensates in these models.
This, in turn, could be useful to explain the statistically-significant
tension in the realistic cosmological scenarios CosmoR .

  * •

It would be useful to extend the notion of supersymmetric modified gravities
into the language of T-duality invariant cosmological frameworks T-duality1
-T-duality5 , where the notion of generalized geometry arises. The agreement
between Riemannian geometries and the generalized geometry required to
manifest T-duality invariance was studied in Victor  and more recently in
Cederwall . Following these lines it would be nice to extend the construction
of black/worm holes Palatini1 -Palatini2  but considering T-duality invariant
approaches as in BlairArv .

  * •

The results of this work can be extended to the next leading order in
fermions. It is well known that when 4-fermion contributions are considered in
the RS Lagrangian, the torsion requieres a specific fermionic contribution
Tanii . It would be interesting to study if it is possible to replace these
contributions with fermionic extra terms deforming the non-metricity tensor.

### Acknowledgements

C.B acknowledges CONICET for support. The work of E.L is supported by the
Croatian Science Foundation project IP-2019-04-4168. E.L also thanks CONICET
for the support during the initial stages of this project.

##  Appendix A Conventions

We use Greek letters as space-time indices, $\mu=0,\dots,3$, and Latin letters
as flat indices, $a=0,\dots,3$. We use the notation
$T_{[ab]}=\frac{1}{2}(T_{ab}-T_{ab})$ and its generalization to further
indices.

The fundamental fields are a 4-dimensional vielbein, $e_{\mu}{}^{a}$ and a
4-dimensional Majorana gravitino, $\psi_{a}$ (when we write fermionic
quantities, we do not explicitly write spinorial indices). We use the
notation:
$\gamma^{ab}=\gamma^{[a}\gamma^{b]}=\frac{1}{2}\gamma^{a}\gamma^{b}-\frac{1}{2}\gamma^{b}\gamma^{a}$
and its generalization for more indices.

The covariant derivative is given by $\nabla_{\mu}$ and our convention is that
$\nabla_{\mu}$ covariantizes the derivative of an object with respect to all
the symmetries that the object transforms under.

##  Appendix B Gamma matrix relations

The Clifford algebra (3) determines the following identities for the gamma
matrices,

$\displaystyle\gamma_{{ab}}\gamma_{c}$
$\displaystyle=\gamma_{{abc}}+2\gamma_{[a}\eta_{b]c}\;,$ (34a)
$\displaystyle\gamma_{a}\gamma_{{bc}}$
$\displaystyle=\gamma_{{abc}}+2\eta_{a[b}\gamma_{c]}\;,$ (34b)
$\displaystyle\gamma_{{ab}}\gamma^{{cd}}$
$\displaystyle=\gamma_{{ab}}{}^{{cd}}+4\gamma_{[a}{}^{[d}\eta_{b]}{}^{c]}+2\eta_{[b}{}^{[c}\,\eta_{a]}{}^{d]}\;,$
(34c) $\displaystyle\gamma_{{ab}}\gamma^{{cde}}$
$\displaystyle=\gamma_{{ab}}{}^{{cde}}+6\gamma_{[a}{}^{[{de}}\eta_{b]}{}^{c]}+6\gamma^{[e}\eta_{[b}{}^{c}\,\eta_{a]}{}^{d]}\;,$
(34d) $\displaystyle\gamma_{{abc}}\gamma^{{de}}$
$\displaystyle=\gamma_{{abc}}{}^{{de}}+6\gamma_{[{ab}}{}^{[e}\eta_{c]}{}^{d]}+6\gamma_{[a}\eta_{c}{}^{[d}\,\eta_{b]}{}^{e]}\;.$
(34e)

## References

  * (1) V. Fock, ’Geometrization of Dirac’s theory of the electron’, Z.Phys. 57, 261-277 (1929).
  * (2) W. Rarita and J. Schwinger, ’On a theory of particles with half-integral spin’, Phys. Rev. 60, 61 (1941).
  * (3) J. Wess, B. Zumino, ’Supergauge Transformations in Four-Dimensions’, Nucl.Phys.B 70 (1974) 39-50 (1974).
  * (4) G. R. Bengochea, Rafael Ferraro, ’Dark torsion as the cosmic speed-up’, Phys.Rev.D 79 (2009) 124019
  * (5) Li, B.; Sotiriou, T.P.; Barrow, J.D. f(T) gravity and local Lorentz invariance. Phys. Rev. D 2011, 83, 064035, [gr-qc/1010.1041].
  * (6) Li, M.; Miao, R.X.; Miao, Y.G. Degrees of freedom of f(T) gravity. JHEP 2011, 07, 108, [hep-th/1105.5934].
  * (7) Ferraro, R.; Guzmán, M.J. Quest for the extra degree of freedom in f(T) gravity. Phys. Rev. D 2018, 98, 124037, [gr-qc/1810.07171].
  * (8) Ferraro, R.; Guzmán, M.J. Hamiltonian formalism for f(T) gravity. Phys. Rev. D 2018, 97, 104028, [gr-qc/1802.02130].
  * (9) Järv, L.; Hohmann, M.; Krššák, M.; Pfeifer, C. Flat connection for rotating spacetimes in extended teleparallel gravity theories. Universe 2019, 5, 142, [gr-qc/1905.03305].
  * (10) Hohmann, M.; Järv, L.; Krššák, M.; Pfeifer, C. Modified teleparallel theories of gravity in symmetric spacetimes. Phys. Rev. D 2019, 100, 084002, [gr-qc/1901.05472].
  * (11) Hashim, M.; El Hanafy, W.; Golovnev, A.; El-Zant, A. Toward a concordance teleparallel Cosmology I: Background Dynamics 2020. [astro-ph.CO/2010.14964].
  * (12) Golovnev, A.; Guzman, M.J. Non-trivial Minkowski backgrounds in f(T) gravity. Phys. Rev. D 2021, 103, 044009, [gr-qc/2012.00696].
  * (13) Beltrán Jiménez, J.; Golovnev, A.; Koivisto, T.; Veermäe, H. Minkowski space in f(T) gravity. Phys. Rev. D 2021, 103, 024054, [gr-qc/2004.07536].
  * (14) Lazkoz, R.; Lobo, F.S.N.; Ortiz-Baños, M.; Salzano, V. Observational constraints of f(Q) gravity. Phys. Rev. D 2019, 100, 104027, [gr-qc/1907.13219].
  * (15) Mandal, S.; Wang, D.; Sahoo, P.K. Cosmography in f(Q) gravity. Phys. Rev. D 2020, 102, 124029, [gr-qc/2011.00420].
  * (16) Ayuso, I.; Lazkoz, R.; Salzano, V. Observational constraints on cosmological solutions of f(Q) theories. Phys. Rev. D 2021, 103, 063505, [astro-ph.CO/2012.00046].
  * (17) Yang, J.Z.; Shahidi, S.; Harko, T.; Liang, S.D. Geodesic deviation, Raychaudhuri equation, Newtonian limit, and tidal forces in Weyl-type f(Q, T) gravity. Eur. Phys. J. C 2021, 81, 111, [gr-qc/2101.09956].
  * (18) M. Adak, O. Sert, M. Kalay and M. Sari, ’Symmetric Teleparallel Gravity: Some exact solutions and spinor couplings’, Int.J.Mod.Phys.A 28 (2013) 1350167, [gr-qc/0810.2388].
  * (19) J. B. Jiménez, L. Heisenberg, T. Koivisto,’The coupling of matter and spacetime geometry’, Class.Quant.Grav. 37 (2020) 19, 195013, [hep-th/2004.04606].
  * (20) S. Vignolo, S. Carloni, R. Cianci, F. Esposito, L. Fabbri, ’Spinor fields in f(Q)-gravity’, Class.Quant.Grav. 39 (2022) 1, 015009, Class.Quant.Grav. 39 (2022) 015009, [gr-qc/2112.03628].
  * (21) Y. Tanii, ’Introduction to supergravities in diverse dimensions’, contribution to: YITP Workshop on Supersymmetry, [hep-th/9802138].
  * (22) R. Lazkoz, F.S.N. Lobo, M. Ortiz-Baños, V. Salzano, ’Observational constraints of $f(Q)$ gravity’, Phys.Rev.D 100 (2019) 10, 104027, [gr-qc/1907.13219].
  * (23) J. Lu, X. Zhao, G. Chee, ’Cosmology in symmetric teleparallel gravity and its dynamical system’, Eur.Phys.J.C 79 (2019) 6, 530, [gr-qc/1906.08920].
  * (24) J. B. Jiménez, L. Heisenberg, T. S. Koivisto, S. Pekar, ’Cosmology in $f(Q)$ geometry’, Phys.Rev.D 101 (2020) 10, 103507, [gr-qc/1906.10027].
  * (25) B. J. Barros, T. Barreiro, T. Koivisto, N. J. Nunes, ’Testing $F(Q)$ gravity with redshift space distortions’, Phys.Dark Univ. 30 (2020) 100616, [gr-qc/2004.07867].
  * (26) N. Frusciante, ’Signatures of $f(Q)$-gravity in cosmology’, Phys.Rev.D 103 (2021) 4, 044021, e-Print: 2101.09242 [astro-ph.CO].
  * (27) Khyllep, W.; Paliathanasis, A.; Dutta, J. Cosmological solutions and growth index of matter perturbations in f(Q) gravity 2021. [gr-qc/2103.08372].
  * (28) M. Gasperini, ‘Elements of string cosmology’, Cambridge University Press, 2007, ISBN:978-0-521-18798-5.
  * (29) E. Di Valentino, L. Anchordoqui, Ö. Akarsu, Y. Ali-Haimoud, L. Amendola et al., ‘Cosmology intertwined III: $f\sigma_{8}$ and $S_{8}$’, Astropart.Phys. 131 (2021) 102604, e-Print: 2008.11285 [astro-ph.CO].
  * (30) S. Angus, K. Cho and J. H. Park, ‘Einstein Double Field Equations’, Eur. Phys. J. C 78 (2018) 500, [hep-th/1804.00964].
  * (31) O. Hohm and B. Zwiebach, ‘Duality invariant cosmology to all orders in $\alpha^{\prime}$’, Phys. Rev. D 100, 126011 (2019), [hep-th/1905.06963].
  * (32) E. Lescano and N. Mirón–Granese, ‘On the phase space in Double Field Theory’, JHEP 07 (2020) 239, JHEP 20 (2020) 239, [hep-th/2003.09588].
  * (33) E. Lescano and N. Mirón–Granese, ‘Double Field Theory with matter and its cosmological application’, [hep-th/2111.03682].
  * (34) E. Lescano and N. Mirón–Granese, ‘Double Field Theory with matter and the generalized Bergshoeff-de Roo identification’, [hep-th/2207.04041].
  * (35) V. A. Penas, ‘Deformed Weitzenböck Connections and Double Field Theory’, Fortsch.Phys. 67 (2019) 3, 1800077, [hep-th/1807.01144]
  * (36) M. Cederwall, J. Palmkvist, ‘Teleparallelism in the algebraic approach to extended geometry’, JHEP 04 (2022) 164, [hep-th/2112.08403].
  * (37) C. Bejarano, G. J. Olmo, D. Rubiera-Garcia, ‘What is a singular black hole beyond General Relativity?’, Phys. Rev. D 95, 064043 (2017) [gr-qc/1702.01292].
  * (38) C. Bejarano, F. S. N. Lobo, G. J. Olmo, D. Rubiera-Garcia, ‘Palatini wormholes and energy conditions from the prism of General Relativity’, Eur. Phys. J. C 77, (2017) 776 [gr-qc/1607.01259].
  * (39) Alex S. Arvanitakis and C. D. A. Blair, ‘Black hole thermodynamics, stringy dualities and double field theory’, Class.Quant.Grav. 34 (2017) 5, 055001, [hep-th/1608.04734].