# FAIR Enough: Develop and Assess a FAIR-Compliant Dataset for Large Language
Model Training?

Shaina Raza
Vector Institute for Artificial Intelligence
Toronto, ON, Canada
<EMAIL_ADDRESS>&Shardul Ghuge
Vector Institute for Artificial Intelligence
Toronto, ON, Canada
<EMAIL_ADDRESS>&Chen Ding
Toronto Metropolitan University
Toronto, ON, Canada
<EMAIL_ADDRESS>&Elham Dolatabadi
Vector Institute for Artificial Intelligence
York University Toronto, ON, Canada
<EMAIL_ADDRESS>&Deval Pandya
Vector Institute for Artificial Intelligence
Toronto, ON, Canada
<EMAIL_ADDRESS>Corresponding author. Email:
<EMAIL_ADDRESS>
###### Abstract

The rapid evolution of Large Language Models (LLMs) underscores the critical
importance of ethical considerations and data integrity in AI development,
emphasizing the role of FAIR (Findable, Accessible, Interoperable, Reusable)
data principles. While these principles have long been a cornerstone of
ethical data stewardship, their application in LLM training data is less
prevalent, an issue our research aims to address. Our study begins with a
review of existing literature, highlighting the significance of FAIR
principles in data management for model training. Building on this foundation,
we introduce a novel framework that incorporates FAIR principles into the LLM
training process. A key aspect of this approach is a comprehensive checklist,
designed to assist researchers and developers in consistently applying FAIR
data principles throughout the model development lifecycle. The practicality
and effectiveness of our framework are demonstrated through a case study that
involves creating a FAIR-compliant dataset to detect and reduce biases. This
case study not only validates the usefulness of our framework but also
establishes new benchmarks for more equitable, transparent, and ethical
practices in LLM training. We offer this framework to the community as a means
to promote technologically advanced, ethically sound, and socially responsible
AI models.

_Keywords_ Large Language Models, FAIR Principles, Data Management, AI Ethics,
Interoperability, Dataset Curation, Bias

##  1 Introduction

In the dynamic realm of artificial intelligence (AI), Language Models (LMs)
play a pivotal role in advancing diverse AI applications Jiang et al. (2020).
From virtual assistants to content generation, LMs have become indispensable
in both shaping the future trajectory of academic research and in their
widespread practical applications. The impact of this transformation is
further amplified by the advancements of Large Language Models (LLMs) Zhao et
al. (2023b), such as OpenAI’s GPT series, Gemini, LLaMa, and Falcon. As of
January 2024, LLM development has garnered $18.2 billion in funding and $2.1
billion in revenue, showcasing its substantial economic impact TrendFeedr
(2024). The rapid success of LLMs underscores the importance of diverse data
for broadening their applicability across different domains. This idea is also
well-supported in research Bender et al. (2021); Wang et al. (2023), which
affirms the importance of high-quality data in refining these models. The
training of these big models on data from diverse sources highlights the
complex ethical and responsible data practice challenges in their
implementation Chang et al. (2023); Wang et al. (2023).

The FAIR data principles, which stand for Findable, Accessible, Interoperable,
and Reusable Dunning et al. (1970); Boeckhout et al. (2018), were initially
established to improve the stewardship of scientific data. These principles
are highly beneficial for any model development lifecycle Wise et al. (2019)
and have become increasingly recognized in responsible AI development
Deshpande and Sharp (2022). Recently, the relevance of FAIR principles has
been particularly underscored in generative AI due to ethical challenges such
as bias, privacy concerns, and the potential misuse of AI-generated content of
Ethics (2024); Partescano et al. (2021). This highlights the growing
importance of ensuring data used in AI is findable, accessible, interoperable,
and reusable, adhering to ethical standards. For LMs, including advanced LLMs
111’LLMs’ here includes both ’LMs’ and ’LLMs’, with ’LLMs’ representing more
advanced versions, adherence to FAIR data principles is becoming crucial. The
manner in which data is curated and managed significantly impacts the
reliability, ethical integrity, and overall effectiveness of these models.

Seminal works in data science and management Dunning et al. (1970); Wilkinson
et al. (2016); Hasnain and Rebholz-Schuhmann (2018); Jacobsen et al. (2020);
Shmueli et al. (2021) have explored FAIR principles in various domains. Their
application in the training and development of LLMs, however, is a developing
area of research. While recent advances in LLM studies Chang et al. (2023);
Wang et al. (2023) have focused on aligning LLMs with ethical standards, the
direct incorporation of FAIR principles in LLM training is less explicit.
Challenges specific to LLMs, such as addressing data biases Ntoutsi et al.
(2020) and enhancing model explainability and interpretability Singh et al.
(2023), further highlight the need for ethically and balanced training
approaches that implicitly align with FAIR principles. This emphasis on data
ethics is in line with broader trends in AI ethics and data science Raji et
al. (2021); Jobin et al. (2019) stress the necessity of integrating these
principles in LLM development.

In this work, we shed some light on FAIR data principles and propose a model
development lifecycle for LLM training that meticulously incorporates FAIR
principles at each stage. At the heart of our approach is the creation of a
FAIR-compliant dataset, designed to include a wide array of narratives from
diverse sources. The central contribution of our work, however, extends beyond
simple data collection and preparation for LLM training. We have placed a
strong emphasis on rigorously aligning the dataset with FAIR principles,
significantly boosting the training efficiency of LLMs. We acknowledge that
adherence to FAIR principles (even at the strict-most level) may not equate to
absolute ethical compliance, however, it represents a crucial step in that
direction. Our research establishes a foundational framework for further
advanced studies in this field. The primary contributions of our work are:

###  1.1 Contributions

  1. 1.

An exploration of FAIR data principles within the realm of AI research,
including the provision of a comprehensive checklist for researchers and
developers.

  2. 2.

The introduction of an innovative framework that integrates FAIR data
principles throughout the LLM training lifecycle, ensuring ethical and
effective application in various AI contexts.

  3. 3.

The demonstration of the practical benefits of a FAIR-compliant dataset
through a case study. This study specifically focuses on identifying and
mitigating biases 222The topic of bias is broad-ranging, our case study
specifically focuses on addressing linguistic biases targeting protected
groups. Fletcher et al. (2021); Garrido-Muñoz et al. (2021) prior to training
LLMs. This approach utilizes FAIR principles in building the dataset and in
training various LLMs.

The remainder of the paper is structured as follows: Section 2 presents “FAIR
Data Principles: Theoretical Background and Significance". In Section 3, we
delve into “Data Management Challenges in Large Language Models". Section 4 is
dedicated to “Framework for Integrating FAIR Data Principles in the
Development of Large Language Models". The paper then moves to Section 5 for a
comprehensive “Discussion", and concludes with the Section 6 for “Conclusion".

###  1.2 Literature Selection Criteria

This study examines publications from the last five years, in English-language
articles from prominent AI and ML journals and conferences. We explored
several databases and proceedings from high-quality journals from Elsevier,
Springer, Nature portfolio, IEEE Transactions, and key conferences like
NeurIPS, ICML, ACL and such. Recognizing the fast-paced evolution of LLM
research, we included relevant preprints and seminal works for a comprehensive
perspective. This search resulted in 135 papers, out of which we carefully
chose about 75 that directly pertained to FAIR data principles in the context
of training models. We noted a scarcity of work combining LLMs with FAIR data
principles, a gap that our study aims to address.

The search query is :

⬇

("Findability" OR "data discovery" OR "metadata standards" OR "persistent
identifiers") AND ("Accessibility" OR "data access" OR "data sharing policies"
OR "authentication and authorization mechanisms") AND ("Interoperability" OR
"data integration" OR "standardized data formats" OR "cross-domain data
exchange") AND ("Reusability" OR "data documentation" OR "data quality
assurance" OR "long-term data preservation") AND ("large language models"

OR "LLMs" OR "AI models" OR "machine learning models") AND ("training data
management" OR "ethical data sourcing" OR "bias in AI datasets" OR
"responsible data use in AI") AND ("ethical considerations in AI" OR "AI
ethics" OR "responsible AI" OR "ethical AI development") PUBLISHED FROM 2018
TO 2023 IN English IN Journals and Conferences (e.g., JAIR, JMLR, IEEE
Transactions, NeurIPS, ICLR, ACL, Springer, Nature portfolio, Elsevier).

###  1.3 Comparative Analysis of Related Work

We have conducted a meta-review of key articles on FAIR data principles,
categorizing them by domain and their relevance to LLMs. We also highlight the
difference of our work compared to the previous works. The findings are
summarized in Table 1.

Table 1: Summary of Review Papers on FAIR Data Principles Reference |  Summary |  Domain Focus |  LLM Relevance
---|---|---|---
Alvarez-Romero et al. (2023) |  Discusses data governance for FAIR principles in health data management. |  Health Data |  Not Applicable
Inau et al. (2021) |  Reviews initiatives for applying FAIR principles in managing health data. |  Health Data |  Not Applicable
Sadeh et al. (2023) |  Examines FAIR data practices’ role in global mental health research. |  Mental Health Research |  Not Applicable
Stanciu (2023) |  Analyzes FAIR principles in healthcare data, emphasizing cybersecurity. |  Health Data |  Partially (InstructGPT)
Raycheva et al. (2023) |  Addresses challenges in FAIR application and legal aspects in European rare disease databases for ML technologies. |  Health Data |  Not Applicable
Vesteghem et al. (2019) |  Elaborates on FAIR principles in precision oncology research. |  Health Data |  Not Applicable
Jacobsen et al. (2020) |  Offers a broad perspective on FAIR principles’ implementation across various fields. |  Multi-Domain |  Not Applicable
Our Study |  Evaluates the application of FAIR data principles in the creation and training of datasets for LLMs. This study encompasses a range of models including the BERT and GPT families, as well as the LLaMa-2-7b model, highlighting diverse approaches in LLM development. |  News Data |  Yes (BERT , GPT families, LLaMa-2-7b)

Table 1 highlights a focus on health data in existing research on FAIR
principles. Notably, one study incorporates a LLM application (GPT-based
model). Our work, however, stands out for its application of FAIR data
principles to a range of LLMs, including BERT-large, GPT, and LLaMa-2-7b. This
approach broadens the scope and generalizability of our research.

##  2 FAIR Data Principles: Theoretical Background and Significance

The FAIR data principles significantly influence data management practices
across various fields. These include physics Dungkek (2022), health Inau et
al. (2023), environmental science Partescano et al. (2021), pharmaceuticals
Wise et al. (2019), chemistry Jeliazkova et al. (2023), computer science Axton
et al. (2016), research and development Wilcox (2018), and clinical studies
Vesteghem et al. (2019). These principles ensure data is well-organized,
accessible, and reusable in different research domains and applications. In AI
and Natural Language Processing (NLP), the implementation of interdisciplinary
strategies Huerta et al. (2023) and structured approaches Bernabé et al.
(2023) is crucial to uphold ethical data practices Shmueli et al. (2021) and
ensure fairness in AI models Bateni et al. (2022). Given the constraints of
space, we present a concise overview of each FAIR principle, as depicted in
Figure 1. Additionally, a compliance checklist is provided in Table 2333For
hyperlinks to the technologies and APIs referenced in the table, refer to
Appendix .1.

Figure 1: FAIR Data Principles: Key Aspects of Findability, Accessibility, Interoperability, and Reusability in Data Management.  Table 2: Comprehensive Checklist for Ensuring FAIR Principles Compliance, Along with Associated Tools and Practices. Principle |  Features |  Compliance list |  Tools and Practices
---|---|---|---
Findability |  F1: Rich and descriptive metadata
F2: Standardized data indexing
F3: Documented data sources
F4: Advanced search functionalities
F5: Persistent identifiers like DOIs |  F1: Metadata includes titles, authors, abstracts, keywords, and affiliations.
F2: Use of standardized taxonomy and ontology for indexing.
F3: Clear documentation of data sources and collection methodologies.
F4: Implementation of advanced search tools and interfaces.
F5: Assignment of persistent identifiers such as DOIs. |  Metadata Management: Apache Atlas, Collibra, ORCID
Persistent Identifiers: CrossRef for DOIs
Data Indexing: Elasticsearch, Apache Solr, DSpace
Search Interfaces: Algolia, Apache Lucene
Data Repositories: NCBI, RE3data, CKAN, Dataverse, Zenodo, Figshare, EPrints,
ResearchGate, Academia.edu
Accessibility |  A1: Clear data access protocols
A2: Long-term data preservation
A3: Open access platforms
A4: Standardized APIs for data access
A5: Data availability in accessible formats |  A1: Detailed access instructions and authentication processes.
A2: Use of reliable digital preservation services like CLOCKSS or Portico.
A3: Data deposited in open repositories such as Figshare or Zenodo.
A4: APIs conform to standards such as OpenAPI for ease of use.
A5: Data provided in multiple formats (e.g., CSV, JSON, XML) to ensure usability. |  API Tools: OpenAPI, GraphQL, RESTful interfaces
Data Preservation: Archivematica, LOCKSS, Zenodo, Figshare
Cloud Storage: Amazon S3, Google Cloud Storage, Microsoft Azure
Ethical Access: OneTrust, TrustArc
Interoperability |  I1: Standard data formats
I2: Common communication protocols
I3: Data exchange standards
I4: Tools for data transformation and mapping |  I1: Data conforms to community-recognized standards (e.g., MIAME, Ecological Metadata Language).
I2: Support for protocols such as OAI-PMH for metadata harvesting.
I3: Use of frameworks like Schema.org for structured data.
I4: Availability of services like XSLT or OpenRefine for data conversion and mapping. |  Data Formats: JSON, XML, CSV, DICOM, GenBank
Protocols: HTTP, SOAP, REST, gRPC
Data Standards: RDF, HL7 FHIR, ISO/IEC standards
Transformation Tools: XSLT, Talend, Informatica, Apache NiFi
Ontology Systems: OWL, SPARQL, XOD
Specialized Databases: IEDB
Reusability |  R1: Detailed metadata for context
R2: Data curated for future use
R3: Adherence to ethical standards
R4: Licensing frameworks
R5: Consideration of societal impacts |  R1: Comprehensive metadata including experimental conditions, methodologies, and provenance.
R2: Curating data with clear versioning and update records.
R3: Compliance with GDPR and other privacy regulations.
R4: Use of licenses like Creative Commons to clarify user rights.
R5: Assessment and documentation of data impact on society and potential biases. |  Metadata Standards: Dublin Core, DataCite, schema.org
Data Curation: CKAN, DSpace, Omeka
Ethical Frameworks: Responsible AI, OpenAI Ethics Guidelines, AI4ALL
Licensing Tools: Creative Commons
Data Provenance tools: Provenance Tracking PROV-DM

###  2.1 Findability

Findability, a crucial aspect of the FAIR data principles, ensures data and
resources are easily locatable and accessible Findlay and Seah (2020). It
involves creating detailed metadata, using persistent identifiers, and
promoting efficient data indexing and searchability. This enhances
discoverability by both humans and machines. Some contributions to data
findability within the FAIR framework include Bernabé et al. (2023), which
explores strategies for FAIRification of data to enhance findability, and
Santos et al. (2023), investigating a model for the FAIR digital object
framework focused on digital objects’ findability. The application of FAIR
principles in AI, particularly the importance of findability in AI datasets,
is discussed in Huerta et al. (2023). Furthermore, Götz (2023) considers the
trustworthiness of data repositories, emphasizing the critical role of
findability in these environments.

###  2.2 Accessibility

Accessibility, as defined by the FAIR data principles, is the ease of
obtaining and using data once located Boeckhout et al. (2018). It involves
long-term data preservation, ethical access protocols, and ensuring data
remains retrievable and usable over time. Discussions on trust in data
repositories as part of accessibility are covered in Wang and Savard (2023),
which also provides a comprehensive understanding of FAIR principles,
including accessibility. The application of FAIR principles to research
software, with a focus on accessibility, is explored in Lamprecht et al.
(2020). How the Fedora platform supports FAIR data principles, particularly
accessibility, is discussed in Wilcox (2018). Challenges and opportunities in
implementing FAIR principles, including accessibility, in Brazilian data
science are addressed in Sales et al. (2020). These works emphasize the
significance of incorporating accessibility considerations into research
software, platform support, and implementation challenges.

###  2.3 Interoperability

Interoperability, as outlined in the FAIR data principles, refers to the
ability of different data systems to work together efficiently da Silva Santos
et al. (2016). It requires standardized data formats and protocols, enabling
easy data exchange and integration across diverse systems. The Immune Epitope
Database (IEDB) and its focus on interoperability in implementing FAIR
principles are discussed in Vita et al. (2018). The eXtensible ontology
development (XOD) principles and tools, which support interoperability in
ontology development, are explored in He et al. (2018). Challenges and
strategies for implementing FAIR data principles, including interoperability,
in data science are addressed in Sales et al. (2020). A framework and metrics
for evaluating the FAIRness of data, with a focus on interoperability, are
provided in Wilkinson et al. (2018). The GO FAIR initiative, which aims to
accelerate the adoption of FAIR principles including interoperability, is
discussed in Schultes et al. (2018). All these works underscore the role of
interoperability in advancing responsible and efficient data management in
various scientific and technological fields.

###  2.4 Reusability

Reusability, a key component of the FAIR data principles, ensures data is
stored and documented for future retrieval and reuse Anguswamy and Frakes
(2012). It encompasses rich metadata, legal and ethical considerations, and
potential societal impact. A structured approach to planning data
FAIRification, with an emphasis on reusability, is discussed in Bernabé et al.
(2023). A model for digital objects under FAIR principles, focusing on
reusability, is offered in Santos et al. (2023). The application of FAIR
principles in AI, particularly regarding the reusability of AI datasets, is
explored in Huerta et al. (2023). The importance of designing workflows with
reusability as a primary goal is discussed in Wolf et al. (2021). All these
works emphasize that data should not only be archived for future use but also
be ethically and efficiently usable, contributing long-term value to the
scientific and technological data ecosystem.

An exploration of FAIR principles, including detailed insights on reusability,
is presented in Jacobsen et al. (2020).

##  3 Data Management Challenges in Large Language Models

The development of LLMs poses a complex array of data management challenges.
These models engage with extensive and intricate datasets, that necessitates
advanced strategies for efficient data organization and accessibility Zhao et
al. (2023b). Ensuring high-quality data is crucial; this requires
representative, well-curated datasets to construct models that are ethically
robust and broadly applicable Bender et al. (2021). Addressing privacy and
ethical concerns mandates stringent data governance to comply with ethical
norms and safeguard individual privacy Wang et al. (2023). The accuracy of
data annotation and labeling is also critical for model reliability Raza and
Schwartz (2023), for standardized and transparent practices Monarch (2021).
Additionally, the balance between data accessibility and the protection of
proprietary information is key, which aligns with legal and ethical guidelines
Jobin et al. (2019). Finally, adhering to data protection laws is vital for
legal compliance and maintaining the ethical integrity of AI technologies,
essential for preserving public confidence Raji et al. (2021). We offer a
structured overview of data management challenges and requirements in LLM
development in Figure 2 and summarize them below.

Figure 2: Data Management Challenges in Large Language Models

Understanding the Context: Enhanced NLP algorithms are vital for improving
contextual awareness in LLMs Xi et al. (2023). The ability of these models to
discern and interpret subtle and complex contexts is key in elevating their
intelligence and practical applicability. There is a growing need for
algorithms capable of understanding linguistic subtleties, cultural nuances,
and the varied implications of context in human language.

Accuracy and Reliability: The issue of LLMs occasionally generating factually
incorrect responses or experiencing ’hallucinations’ Ji et al. (2023)
underscores the importance of robust fact-checking mechanisms. Enhancing the
trustworthiness and truthfulness of these models requires advanced
verification algorithms and data validation methods to ensure accuracy and
reliability.

Ethical and Fair Use: Addressing inherent biases in LLM responses is critical
Deshpande and Sharp (2022); Wang et al. (2023); Raza et al. (2022). The
development of advanced bias detection algorithms is crucial, involving
systems that identify and mitigate biases in training data and model
responses, promoting fairness and equity Raza et al. (2023). This requires a
concerted effort to understand and reflect diverse perspectives, ensuring that
LLMs are used ethically, promoting fairness and equity in their outputs.

Interactivity and Personalization: The tendency of current LLMs to provide
generic responses points to the need for more sophisticated adaptive learning
algorithms He et al. (2023). These algorithms should be capable of learning
from individual user interactions, preferences, and feedback, thus offering
increasingly personalized and relevant responses. The goal is to develop
models that can understand and adapt to individual user needs, enhancing the
overall user experience through personalization.

Language and Cultural Sensitivity: The limitations in supporting low-resource
languages call for the expansion of language models and cultural datasets
Chang et al. (2023). Enhancing the representation of diverse languages and
cultures in LLMs is essential for promoting global inclusivity and cultural
sensitivity . This involves not only expanding the datasets with more
languages but also ensuring that these models understand and respect cultural
nuances and contexts.

Efficiency and Scalability: A major challenge in LLM development is the high
computational resource requirement Zhao et al. (2023b). To address this, it is
necessary to develop and implement optimization strategies that maintain or
improve the performance of LLMs while significantly reducing their
computational demands. This could involve innovations in model architecture,
data processing techniques, and hardware optimizations, enabling more
efficient and scalable deployment of LLMs.

Research and Development Directions Collaboration with academic institutions,
ongoing model training and updates, and user-centric design approaches are
essential for the continuous improvement of LLMs. This ensure that the models
serve the diverse and evolving needs of users.

##### Mapping FAIR principles to Data Management Challenges in LLMs

: These above-summarized challenges of managing data for LLMs are
multifaceted, but the FAIR data principles can offer some targeted solutions.
For example, Findability not only simplifies the task of locating relevant
data in extensive datasets but also aids in streamlining the research process.
Enhanced metadata and persistent identifiers improve the efficiency of data
retrieval and analysis, crucial for developing sophisticated LLMs.
Interoperability ensures that different data formats and systems can work
together seamlessly, which is critical for integrating varied data sources
into LLM training.It also supports the development of more contextually aware
NLP algorithms by enabling the synthesis of diverse data types and sources,
thus contributing to more nuanced and accurate model outputs.

Accessibility involves not just the availability of data but also the
mechanisms of access control, ensuring that data is shared responsibly. It
balances the need for open data to promote innovation and collaboration
against the need to protect sensitive information and intellectual property, a
key concern in AI development. Reusability extends the value of data beyond
its initial purpose. By ensuring that data practices comply with legal and
regulatory standards, it guarantees that data remains relevant, reliable, and
ethical for future use. This is particularly important in LLM development,
where data may be reused for multiple models or purposes over time. We provide
a mapping of these data management challenges in LLMs to FAIR data principles
in Appendix .4.

##  4 Framework for FAIR Data Principles Integration in LLM Development

###  4.1 Rationale for Integrating FAIR Data Principles in LLM Development

Our research is dedicated to enhancing the training of LLMs by embedding FAIR
data principles directly into their training methodologies. The central
objective is to develop training datasets for LLMs that are designed to comply
with FAIR principles. To realize this, we have devised a framework that
interweaves FAIR principles throughout the entire model development lifecycle
of LLMs, as depicted in Figure 3. This approach is not limited to just data
preparation but extends to every stage of the LLM development process. In the
following sections, we will elaborate on each phase of this proposed LLM
lifecycle, connecting each step specifically to our case study which
concentrates on the creation of a FAIR-compliant dataset.

Figure 3: FAIR principles integrated into the LLM lifecycle

###  4.2 Case Study: Developing a FAIR-Compliant Dataset

To address the risk of LLMs perpetuating societal biases, our case study
focuses on developing a dataset with a conscientious design that proactively
identifies biases within the data prior to the training of these models. The
dataset and model card are available here 444Anonymous due to double blind
review.

In this case study, “bias" is identified as a tendency within the dataset
leading to the unfair representation of specific groups or outcomes Ntoutsi et
al. (2020). This bias may take the form of linguistic bias Raza et al. (2024),
where the dataset could unfairly favor or disfavor certain protected groups,
identified by race, gender, or ethnicity, in terms of languages or dialects.
It might also manifest as stereotypes Nadeem et al. (2021), toxicity Barikeri
et al. (2021), or misinformation Raza and Ding (2022). Such biases typically
originate from the methods used in data collection, processing, or usage. As a
result, they can cause these LLMs to replicate or magnify such biases May et
al. (2019). We consider a diversity of bias dimensions and aspects in our case
study, as outlined in Figure 4, which shows a broad spectrum of biases
including ageism, occupational jargon barriers, political rhetoric, and more.
This dataset can serve as a template or example for creating datasets with
various objectives for training LLMs while ensuring compliance with FAIR
principles.

Figure 4: Biases across Multiple Dimensions Explored in this Study

####  4.2.1 Data Collection and Curation for LLM Development

##### FAIR Principles Achieved

: Findability, Accessibility, Interoperability, and Reusability.

In line with the FAIR data principles, our dataset is meticulously structured
to enhance its Findability, featuring comprehensive metadata to facilitate
easy discovery. This dataset, sourced from a variety of channels such as news
feeds and websites between January and May 2023, encompasses over 50,000
filtered entries. For data curation, we utilized various feeds and hashtags,
including #MediaBias, #SocialJustice, #GenderEquality, #RacialInjustice,
#CulturalDiversity, #AgeismAwareness, #ReligiousTolerance, and
#EconomicDisparity, to ensure a wide representation of social issues. This
diversity of news articles not only broadens accessibility but also enriches
the dataset relevance to current social discourses.

The detailed metadata includes crucial information such as dataset title,
description, authors, date of creation, version, and keywords that reflect the
dataset scope and purpose. These keywords (e.g., LLMs, Training, Biases, News
Media, NLP) are strategically chosen to enhance the dataset retrieval
efficiency across various research portals. Additionally, the metadata
specifies the data type, whether textual, numerical, or otherwise, further
aligning with the Interoperability principle by facilitating data integration
across different systems.

This readability test is crucial for determining the level of education
required to comprehend our dataset texts. As illustrated in Figure 5, the
distribution of Gunning Fog Index scores in our dataset shows a normal
distribution with a mean score of 7.79. This indicates that the majority of
our texts are suitable for readers with at least an eighth-grade education
level. Such readability analysis is instrumental in aligning our dataset with
the FAIR principle of Accessibility, ensuring that the content is
comprehensible to the intended audience, thus enhancing the usability of the
dataset in training LLMs.

Figure 5: Histogram of the Gunning Fog Index on FAIR-Complaint Dataset. The
x-axis denotes the Gunning Fog Index scores, reflecting text complexity, and
the y-axis represents the number of samples with each score.

Table 3: Data Quality Analysis Summary

Aspect |  Automated Analysis |  Expert Review
---|---|---
Accuracy |  96.5% |  Contextually accurate
Completeness |  93.0% |  High with occasional updates
Consistency |  98.0% |  Uniform across samples

The annotation of this dataset is a sophisticated combination of human
expertise and algorithmic efficiency, involving a team of five domain experts
and two machine learning-trained students. This approach aligns with recent
studies Liu et al. (2023); Shu et al. (2023) that advocate for it as a
superior alternative to traditional crowd-sourcing methods. For bulk
annotation, we utilized an LLM-based model (specifically, the paid API of
GPT-3 turbo), followed by human review to effectively resolve ambiguities.
This process is enhanced by periodic expert reviews and inter-annotator
agreement (IAA) checks. The quality of our annotations is evidenced by a
Cohen’s Kappa score consistently above 0.75. Examples of IAA agreements across
main bias dimensions are provided in Appendix .3, illustrating the robustness
of our annotation methodology.

The robustness of our dataset for LLM training is established through a dual-
stage quality assessment: automated analysis for initial screening of 10,000
statistically significant entries, and expert review of 500 stratified random
samples for deeper insights. This method ensures the dataset accuracy,
completeness, and consistency, crucial for LLM training, as detailed in Table
3.

For Interoperability, the data is formatted for seamless integration for
various ML tasks, such as binary and multi-label classifiers, question
answering (QA) system and debiased language generation task (debiasing), as
detailed in Table 4. This enables researchers to use this data in diverse
analytical contexts, facilitating cross-domain research and development. The
dataset schema and examples on dataset formats are given in Appendix .2.

Table 4: Specialized Data Formats for Interoperability Data Format | Model Type
---|---
Classification Format | Binary Classifier
CoNLL Format | Multi-label Token Classification
SQuAD Format | Question Answering System
Counterfactual Formatting | Debiasing Model
Sentiment Analysis Format | Sentiment Classifier
Toxicity Classification Format | Toxicity Classifier

Furthermore, we have stored our dataset in repositories such as Huggingface,
Zenodo, and Figshare, which not only adhere to metadata standards but also
ensure its long-term Accessibility. Overall, our data collection and curation
strategy aligns with the FAIR principles and enhance the practical utility of
our dataset for LLM development.

####  4.2.2 Model Training and Algorithm Development

##### FAIR Principles Achieved

: Interoperability and Reusability.

The model training and algorithm development phase is crucial for adhering to
FAIR principles, particularly interoperability and reusability. We
strategically deploy various LLMs, fine-tuned for tasks ranging from sentiment
analysis and QA to debiasing (language generation), and develop them with a
modular design for enhanced Reusability across projects.

Interoperability is achieved by using common frameworks like TensorFlow and
PyTorch and standardizing data formats for inputs and outputs. Every task
performed by LLMs, such as classification and debiasing, is executed following
the preparation of specific training formats tailored to each task. This
ensures seamless integration of our models with various systems and datasets.
We further enhance this with the inclusion of model cards for each LLM,
providing essential information such as model purpose, performance, and usage
guidelines, thus aiding in understanding and adoption.

We prepare API documentation that provides details for every aspect of
training and development, including parameter settings and algorithm
modifications. This transparency facilitates scientific validation and
progress Jacobsen et al. (2020). We also foster community collaboration by
sharing our findings, models, and tools. The goal is to contribute to public
repositories, and participating in AI research platforms.

####  4.2.3 Model Evaluation and Validation

##### FAIR Principles Achieved

: Reusability and Accessibility.

In our evaluation and validation phase, we prioritize the FAIR principles of
Reusability and Accessibility to ensure that our models and findings can be
widely utilized and understood. Transparency in reporting and the provision of
accessible documentation are key aspects of our methodology. All results,
including bias analysis and model benchmarking, are made publicly available,
allowing for comprehensive observation and application by the wider research
community. An example of this approach is presented in our detailed bias
analysis heatmap (Figure 6) and the benchmarking table (Table 5), which
illustrate our commitment to these principles.

Figure 6: Heatmap Visualization: the prevalence and intensity of different
types of biases, such as ageism, gender, and political, across various
classifications like bias, non-biased, toxic, and sentiment within a dataset.

In Table 5, we present performance metrics across various tasks, demonstrating
the effectiveness of our classifiers in toxicity detection, bias
classification, sentiment analysis, multi-label token classification, and QA
capabilities. Additionally, we integrated a debiasing process using LLaMa 2B
Chat Touvron et al. (2023), re-evaluating our models on the same test set
post-debiasing. This allowed us to assess the impact of debiasing on reducing
toxicity and bias, and its influence on sentiment analysis. The debiasing
process notably improved scores and indicated the efficacy of the debiasing
intervention.

Table 5: Comprehensive Analysis of Language Model Performance and Impact of
Debiasing: the first section details performance metrics of various models on
different tasks, fine-tuned for optimal settings. The second section
highlights the effect of debiasing on toxicity, bias classification, and
sentiment scores.

Performance Metrics of Language Models. Higher the score, better the
performance.
---
Model (Type) | Accuracy | Precision | F1-Score
Binary Bias Classifier (BERT-large-uncased) | 92% | 89% | 90%
NER Model (RoBERTa-large-uncased) | 95% | 93% | 94%
QA System (BERT-base-uncased) | 88% | 87% | 86%
Sentiment Classifier (BERT-large-uncased) | 90% | 91% | 90%
Toxicity Classifier (BERT-large-uncased) | 91% | 90% | 91%
LLaMa2 (7b-chat-hf) | 93% | 94% | 93%
Impact of Debiasing. Lower scores post-debiasing is better.
Metric | Pre-Debiasing Score | Post-Debiasing Score |
Toxicity Score | 75% | 24% |
Bias Classification Score | 80% | 55% |
Sentiment Score | -0.3 | 0.2 |

The results in the table demonstrate that all models initially exhibit strong
classification performance. However, after the debiasing process, a noticeable
decline in classification performances is observed across these models when
applied to the debiased dataset. This suggests that the original training was
heavily reliant on biased data, and the introduction of debiased data disrupts
the model’s classification ability, leading to a reduction in accuracy. This
result aligns very much with the recent safety benchmarking in recent works
Touvron et al. (2023), which shows there is a trade-off highlighting the
impact of debiasing on model performance.

Overall, these models’ evaluation and validation phase, emphasize transparent
reporting and public availability of models and the detailed results, aligns
with the FAIR principles of Reusability and Accessibility.

####  4.2.4 Deployment and Ongoing Monitoring

##### FAIR Principles Achieved

: Accessibility, Reusability, and Findability.

In the post-deployment phase, we emphasize the FAIR principles of
Accessibility, Reusability, and Findability. Our approach includes providing
comprehensive documentation and clear usage guidelines to enhance the model
findability and Accessibility. We employ version control systems to ensure
easy tracking and retrieval of the latest model versions, supporting both
findability and Reusability.

To facilitate user engagement and model effectiveness, we offer extensive
training materials and support, making the models more accessible to a diverse
user base. Regular reporting of performance metrics offers insights into the
models’ ongoing effectiveness and areas for improvement, aiding in their
reusability. We also provide integration support with various tools and
platforms, ensuring that our models can be easily adopted in different
technological contexts. These practices collectively ensure that our deployed
models adhere to the FAIR principles, maintaining their utility and efficacy
in a dynamic, user-centric environment.

####  4.2.5 Community Engagement and Collaborative Development

##### FAIR Principles Achieved

: Findability, Accessibility, Interoperability, and Reusability.

In our efforts to foster community engagement and collaborative development,
we align with the FAIR principles effectively. We provide open-source
development to facilitate collaborative innovation. Our datasets are provided
in accessible formats such as JSON and CSV and are indexed with rich metadata.
This approach ensures that our LLMs are developed in environments conducive to
collaborative progress. The datasets are disseminated under the Creative
Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) license
Creative Commons (2023), aligning our work with the FAIR principles of data
stewardship.

##### Upholding FAIR Principles

We provide a comprehensive analysis in Table 6, showcasing our commitment to
upholding the FAIR data principles to enhance the dataset’s usefulness for
both present and future research endeavors.

Table 6: FAIR Compliance in Dataset and LLM Development Principle |  FAIR Compliance Criteria |  Met |  Remarks (Implementation Details)
---|---|---|---
Findability |  F1: Rich, descriptive metadata
F2: Standardized data indexing
F3: Documented data sources
F4: Persistent identifiers for data and models |  ✓ |  F1: Metadata includes descriptions for data and models.
F2: Data and models indexed with relevant keywords.
F3: Clear documentation of data sources and model development.
F4: Unique identifiers for data and models.
Accessibility |  A1: Clearly defined access protocols
A2: Long-term data and model preservation
A3: Data and models available in accessible formats |  ✓ |  A1: Open-access repository for data and models.
A2: Long-term preservation of data and models.
A3: Data in CSV, JSON, XML; Models accessible via APIs.
Interoperability |  I1: Use of standard data and model formats
I2: Adherence to data and model exchange standards |  ✓ |  I1: Data in standard formats; Models compatible with common ML frameworks.
I2: Adheres to standard data and model exchange protocols.
Reusability |  R1: Detailed metadata for data and models
R2: Adherence to ethical and transparency standards
R3: Licensing information for data and model usage |  ✓ |  R1: Metadata includes usage guidelines for data and models.
R2: Focus on ethical data use and transparent model training.
R3: Data licensed under CC BY-NC 4.0; Models with open-source licenses.

##  5 Discussion

###  5.1 Limitations of the FAIR Data Principles in Addressing Data Challenges

The FAIR data principles mark a significant step towards enhancing the
openness and efficiency of data usage in research and related fields. Yet,
their effectiveness is constrained by several factors. Primarily, while the
FAIR principles improve data accessibility and usability, they do not
automatically guarantee data quality or validity. The absence of mechanisms
for ensuring accuracy, completeness, or reliability could lead to the spread
of low-quality data, negatively impacting research and decision-making.
Additionally, FAIR principles, though encouraging data sharing, may not
adequately address the ethical and privacy concerns associated with sensitive
data like health records or personal information, which could result in
privacy violations.

Furthermore, the implementation of FAIR principles requires considerable
resources, including necessary infrastructure and expertise. This poses a
significant challenge, especially for smaller institutions or those in
developing countries, potentially exacerbating the digital divide. The broad
application of FAIR principles may not be suitable for all scientific
disciplines, as different fields might require tailored data management
approaches not fully encompassed by these general guidelines. The
effectiveness of these principles also depends heavily on the awareness and
training of data handlers, where a lack of such training can be a major
barrier to adoption. Moreover, the absence of universally accepted standards
for evaluating ’FAIRness’ leads to an inconsistent application of these
principles.

Additionally, the diversity of data formats creates technical challenges in
achieving seamless interoperability, where the emphasis on data sharing might
overshadow other vital aspects of data stewardship, such as privacy
considerations. Finally, the principles can conflict with intellectual
property rights and commercial interests. The concept of free data access and
reuse might challenge proprietary research, leading to resistance from certain
sectors. This necessitates compliance with various legal and regulatory
frameworks, adding another layer of complexity to the application of FAIR
principles.

##### Strategies for Mitigating Limitations and Enhancing Data Utility

In addressing the limitations of FAIR principles and enhancing data utility, a
multifaceted strategy is essential. This involves refining data quality
through rigorous validation processes, ensuring accuracy and reliability.
Simultaneously, balancing data accessibility with ethical and privacy
considerations is crucial, there is need to employ advanced anonymization
techniques and strict privacy protocols. Emphasizing data preservation and
sustainability alongside sharing, broadens the scope of stewardship.
Integrating legal and regulatory compliance is vital for the robust and
ethical utilization of data in research and practice.

###  5.2 Limitation of the Study and Future Perspectives

Our Fair-Compliant dataset, designed to identify and mitigate known biases,
may inadvertently overlook emerging biases, thereby emphasizing the need for
ongoing revision and vigilant monitoring May et al. (2019). Concurrently,
scaling the dataset to match the complexity of advanced LLMs, while adhering
to ethical standards, emerges as a substantial challenge. This endeavor is
compounded by the imperative to mitigate biases in model interpretations,
bolster interpretability, and ensure robust data privacy and security Zhao et
al. (2023a); Chen et al. (2023). Therefore, future research directions should
consider creating dynamic mechanisms for dataset updates Wilson et al. (2023),
advancing bias mitigation techniques, aligning datasets with evolving LLM
technologies, exploring scalable dataset maintenance solutions, and broadening
the spectrum of LLM applications. Integral to this journey is the development
of comprehensive ethical guidelines and the fostering of collaborative
research, both of which are pivotal for the responsible evolution of LLM
technology.

##  6  Conclusion

Our study integrates FAIR data principles into LLM development, enhancing data
management and model training. This has led to a robust and versatile dataset,
useful in various research contexts and offering insights into applying FAIR
principles in LLM training. Despite advancements, there are limitations and
opportunities for future research. These include updating datasets to reflect
emerging trends, improving bias detection, adapting to new LLM architectures,
enhancing model interpretability, and developing ethical guidelines. Our work
contributes to the responsible development of AI, aiming to create more
ethical and effective AI tools that benefit diverse communities.

### Research Ethics and Social Impact

### Ethical Considerations Statement

In conducting this research, we carefully adhered to ethical guidelines to
ensure the integrity and societal value of our work. We ensured the privacy
and confidentiality of data, obtaining consent where required, and employed
measures to safeguard sensitive information. Our methodology was designed to
minimize any potential harm, with a focus on responsible data handling and
analysis. Recognizing the potential implications of our research, we took
steps to ensure that our findings are presented in a manner that is respectful
and considerate of diverse perspectives and experiences.

### Researcher Positionality Statement

Our team’s diverse academic and cultural backgrounds significantly influenced
our approach to this research. Our team comprises members from both academia
and industry, bringing a wealth of different viewpoints to the study. This
diversity enriched our analysis, allowing for a more comprehensive
understanding of the data and its implications. We believe that acknowledging
our positionality is crucial for maintaining transparency and objectivity in
our research.

### Adverse Impact Statement

We recognize the potential adverse impacts of our research, particularly given
the sensitive nature of the dataset prepared for bias identification. While
our aim is to enhance fairness and inclusivity, there is a risk that the data
or findings could be misinterpreted or misapplied in a manner that
inadvertently negatively portrays underrepresented groups. We are committed to
ensuring our case study does not contribute to further stigmatization or
discrimination of any group. We emphasize the importance of using our results
responsibly, particularly in contexts that were not directly intended in our
study, to avoid potential ethical dilemmas or societal harm.

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## Appendices

###  .1 Acronyms and Full Forms

Academia.edu, Academia.edu; AI4ALL, AI4ALL; Algolia, Algolia; Amazon S3,
Amazon Simple Storage Service; Apache Atlas, Apache Atlas; Apache Lucene,
Apache Lucene; Apache NiFi, Apache NiFi; Apache Solr, Apache Solr;
Archivematica, Archivematica; CKAN, Comprehensive Knowledge Archive Network;
Clockss, Controlled LOCKSS; Collibra, Collibra; Creative Commons, Creative
Commons; Crossref, Crossref; DSpace, DSpace; Dublin Core, Dublin Core Metadata
Initiative; Data Provenance Tools, Data Provenance Tools; DataCite, DataCite;
Dataverse, Microsoft Dataverse; EPrints, EPrints; Ecoinformatics,
Ecoinformatics; Elasticsearch, Elasticsearch; FGED, Functional Genomics Data
Society; Figshare, Figshare; GDPR, General Data Protection Regulation; Google
Cloud Storage, Google Cloud Storage; GraphQL, GraphQL; HL7 FHIR, Health Level
Seven Fast Healthcare Interoperability Resources; IEDB, Immune Epitope
Database; ISO/IEC, International Organization for
Standardization/International Electrotechnical Commission; LOCKSS, Lots of
Copies Keep Stuff Safe; NCBI, National Center for Biotechnology Information;
OAI-PMH, Open Archives Initiative Protocol for Metadata Harvesting; Omeka,
Omeka; OneTrust, OneTrust; OpenAI Ethics Guidelines, OpenAI Ethics Guidelines;
OpenAPI, OpenAPI Initiative; OpenRefine, OpenRefine; ORCID, Open Researcher
and Contributor ID; OWL, Web Ontology Language; Portico, Portico; PROV-DM,
Provenance Data Model; RDF, Resource Description Framework; RE3data, Registry
of Research Data Repositories; ResearchGate, ResearchGate; Responsible AI,
Responsible AI; REST, Representational State Transfer; schema.org, Schema.org;
SOAP, Simple Object Access Protocol; SPARQL, SPARQL Protocol and RDF Query
Language; Talend, Talend; TrustArc, TrustArc; XOD, eXtensible ontology
development; XSLT, Extensible Stylesheet Language Transformations; Zenodo,
Zenodo; gRPC, gRPC Remote Procedure Calls.

###  .2 Dataset Schema and Formats

The dataset has multiple attributes like Text, Dimension, Biased Words,
Aspect, Bias Label, Sentiment, Toxic, Identity Mention, Debasied Text.

"Text": "Lawyers are always manipulative and cannot be trusted.",
"Dimension": "Professional Integrity",
"Biased Words": "always, manipulative, cannot be trusted",
"Aspect": "Trustworthiness of lawyers",
"Bias Label": "BIASED",
"Sentiment": "Negative",
"Toxic": "Yes",
"Identity Mention": "Lawyers",
"Debiased Text": "Trustworthiness is an individual trait and varies among
professionals, including lawyers."

We provide examples for each data format, demonstrating the use of dataset
attributes in diverse machine learning models:

Table 1: Examples of Dataset Formats

Format |  Example
---|---
Classification |  { "Text": "Politicians are often corrupt.", "Bias Label": "BIASED", "Sentiment": "Negative" }
CoNLL |  { "Text": "Doctors are always caring.", "Biased Words": ["always", "caring"], "Identity Mention": "Doctors" }
SQuAD |  { "Text": "Why are lawyers untrustworthy?", "Aspect": "Trustworthiness", "Debiased Text": "Trustworthiness varies individually." }
Counterfactual |  { "Text": "Young people are irresponsible with money.", "Biased Words": ["irresponsible"], "Debiased Text": "Financial habits vary by individual." }
Sentiment/Toxicity |  { "Text": "Politicians are dishonest.", "Sentiment": "Negative", "Toxic": "Yes" }

###  .3 Inter Annotators Agreement

Figure1 illustrates the expert consensus aligned with active learning outcomes
across various bias dimensions, showing the highest agreement in
’Socioeconomic Bias’.

Figure 1: Expert Agreement Across Bias Dimensions. The bar graph quantifies
the concordance between domain experts evaluations and the model’s
predictions.

###  .4 Data Management Challenges in LLMs and Corresponding FAIR Principles

Table 2: Data Management Challenges in LLMs and Corresponding FAIR Principles Data Management Challenge |  FAIR Principle Addressed
---|---
Vast and Complex Datasets |  Findability through detailed metadata and persistent identifiers to enhance data discovery .
Data Quality and Bias |  Accessibility with ethical access protocols to provide high-quality, unbiased data.
Privacy and Ethical Concerns |  Reusability with clear legal and ethical documentation to uphold privacy and ethics.
Data Annotation and Labeling |  Interoperability through standardized data formats for reliable annotation and labeling.
Data Accessibility and Sharing |  Accessibility to ensure a balance between open data sharing and protection of proprietary information .
Legal and Regulatory Compliance |  Reusability to align data management with legal standards for future use .
Contextual Awareness |  Interoperability for enhanced NLP algorithms to accurately capture and interpret context.
Accuracy and Reliability |  Accessibility and Reusability to ensure mechanisms for improved fact-checking and consistent data quality.
Ethical and Fair Use |  Accessibility with advanced bias detection algorithms to promote fairness .
Interactivity and Personalization |  Findability and Accessibility for adaptive learning from user interactions for personalized experiences.
Language and Cultural Sensitivity |  Interoperability to support expanded language models and cultural datasets for inclusivity .
Technical and Scalability |  Reusability for efficient processing strategies that facilitate the scalable use of computational resources.


Classical non - linear operators

in Grand Lebesgue Spaces.

M.R.Formica, E.Ostrovsky and L.Sirota.

Università degli Studi di Napoli Parthenope, via Generale Parisi 13, Palazzo
Pacanowsky, 80132, Napoli, Italy.

e-mail<EMAIL_ADDRESS>

Department of Mathematics and Statistics, Bar-Ilan University,
59200, Ramat Gan, Israel.

e-mail<EMAIL_ADDRESS>

Department of Mathematics and Statistics, Bar-Ilan University,
59200, Ramat Gan, Israel.

e-mail<EMAIL_ADDRESS>

Abstract.

We study in this short report the boundedness of classical non - linear
operators: Nemytskii, Urysohn, Hammerstein acting from one Grand Lebesgue
Space to another one, and deduce some its upper norm estimates.

We bring also some examples to illustrate the exactness of our estimates.

Key words and phrases. Measure and measurable space, Nemytskii, Urysohn,
Hammerstein and other operators, boundedness, ordinary and operators norm,
estimating factorization, power operator, degree, ordinary and mixing
(anisotropic) Lebesgue - Riesz and Grand Lebesgue Spaces (GLS), estimate,
Hölder’s inequality, examples, measurable functions, generating function.

##  1 Statement of problem. Notations. Definitions.

Let $\ (X=\\{x\\},\ {\cal M},\ \mu);\ (Y=\\{y\\},{\cal N},\nu);\ $ be two
measurable spaces equipped with two non - trivial measures correspondingly $\
(\mu,\nu).\ $ Let also $\ U=U[g](x),\ $ where

$f(x):=K[g(\cdot)](x)=K[g(\cdot),x,\cdot],$ (1)

$f:X\to R,\ g:Y\to R$

be an operator, not necessary to be linear.

It is presumed that all the considered functions are measurable.

We intent in this short report to estimate the norms of these operators as an
operator acting between two Grand Lebesgue Spaces (GLS).

We will consider the following three important types of nonlinear operators:
Nemytskii, Urysohn and Hammerstein.

Definitions of operators.

Definition 1. Nemytskii operator, see [8], [29], [33], [43]:

$N[g](x)\stackrel{{\scriptstyle def}}{{=}}n(x,g(x)).$ (2)

Definition 2. Urysohn’s operator, see [27], pp. 423 - 429, [28], [43]:

$U[g](x)\stackrel{{\scriptstyle def}}{{=}}\int_{Y}u(x,y,g(y))\ \nu(dy).$ (3)

Definition 3. Hammerstein’s operator, [16], [25], [26], [29]:

$H[g](x)\stackrel{{\scriptstyle def}}{{=}}\int_{Y}h(x,y)\ w(y,g(y))\ \nu(dy).$
(4)

Here all the introduced functions $\ n(x,y),\ u(x,y,z),\ h(x,y),\ w(x,y)\ $
are measurable and numerical valued.

These operators appear in particular in the theory of non - linear Integral
and Partial Differential Equations (PDE), see e.g. [10] \- [13], [41], [42],
and reference therein.

A brief excursus in the theory of Grand Lebesgue Spaces (GLS).

Let $\ (V=\\{v\\},{\cal D},\gamma)\ $ be again measurable space equipped with
non - trivial sigma finite measure $\ \gamma.\ $ Let also $\ (a,b)=\rm const,\
1\leq a<b\leq\infty\ $ and let $\ \psi=\psi(p),\ a<p<b\ $ be certain strictly
positive numerical valued function; which is named as ordinary generating
function. By definition, the Grand Lebesgue Space (GLS) $\
G\psi[V,\psi;a,b]=G\psi\ $ consists on all the numerical valued measurable
functions $\ f:V\to R\ $ having the following finite norm

$||f||G\psi=||f||G\psi[V,\psi;a,b]\stackrel{{\scriptstyle
def}}{{=}}\sup_{p\in(a,b)}\ \left\\{\ \frac{||f||_{p,V}}{\psi(p)}\ \right\\}.$
(5)

Hereafter $\ ||f||_{p,V}\ $ denotes the usually Lebesgue - Riesz norm:

$||f||_{p,V}\stackrel{{\scriptstyle def}}{{=}}\left[\ \int_{V}|f(v)|^{p}\
\gamma(dv)\ \right]^{1/p},\ 1\leq p<\infty.$ (6)

Notation: $\ (a,b)\stackrel{{\scriptstyle def}}{{=}}\ \rm supp\ \psi.\ $

Define formally $\ \psi(p)=\infty,\ $ when $\ p\ \notin\ \rm supp\ \psi.\ $

Definition of the natural generating function. Let $\ f=f(v)\ $ be measurable
numerical valued function such that

$\exists(a,b),\ 1\leq a<b\leq\infty\ \Rightarrow\forall p\in(a,b)\
||f||_{p,V}<\infty.$

The natural generating function $\ \psi[f](p)\ $ is defined as follows:

$\psi[f](p)\stackrel{{\scriptstyle def}}{{=}}||f||_{p,V},\ p\in(a,b).$

Evidently, $\ ||f||G\psi[f]=1.\ $

The general theory of these spaces is represented in many works, see e.g. [1],
[3], [4], [9], [15], [18], [19], [20], [21], [22], [23], [24],[30], [35],[36],
[37], [38] etc. In particular, these spaces are complete, Banach functional
and rearrangement invariant.

It is known, see e.g. [39], that in the case when $\ a=1,b=\infty\ $ the space
$\ G\psi_{1,\infty}\ $ coincides with appropriate exponential Orlicz space.
The belonging of some function $\ h=h(v)\ $ is closely related with its tail
behavior

$T_{h}(t):=\gamma\\{\ v:\ |h(v)|\geq t\ \\},\ t\to\infty.$

##  2 Main result: Nemytskii operator.

We consider here the Nemytskii’s operator (2). Let us impose the following
condition of factorization estimating on the leading function $\ n=n(x,y):\ $

$\exists\beta=\rm const\in[1,\infty),\ \exists\phi=\phi(x),\ x\in X,\
\Rightarrow|n(x,y)|\leq\phi(x)\cdot|y|^{\beta},$ (7)

$\ \phi:X\to R.\ $ Assume further that in (2) [and in (7)]

$g(\cdot)\in G\psi,\ \phi(\cdot)\in G\nu,$ (8)

for suitable generating functions $\ \psi,\ \nu;\ $ so that

$||g||_{p,X}\leq C_{1}\ \psi(p),\ ||\phi||_{q,X}\leq C_{2}\nu(q),$

where $\ C_{1}=||g||G\psi,\ C_{2}=||\phi||G\nu.\ $

Of course, one can choose as such a functions $\ \psi,\ \nu\ $ the natural
ones: $\ \psi(p)=||g||_{p,X},\ \nu(q)=||\phi||_{q,X},\ $ if they there exist.
Then $\ C_{1}=C_{2}=1.\ $

Following, if we introduce the so - called power operator of the degree $\
\beta:\ P_{\beta}[g](x)=|g(x)|^{\beta},$ where as before $\ \beta=\rm
const\geq 1,\ $ then

$||P_{\beta}[g]||_{p,X}=||g^{\beta}||_{p,X}\leq
C_{1}^{\beta}\cdot\psi^{\beta}(\beta p).$

We have:

$|f(x)|\leq|\phi(x)|\cdot|g(x)|^{\beta}.$

One can apply Hölder’s inequality:

$||f||_{r,X}\leq||\phi||_{q,X}\cdot||g^{\beta}||_{p,X},$ (9)

where $\ (p,q,r)\ $ are arbitrary numbers such that $\ p,q,r\geq 1\ $ and
moreover

$\frac{1}{r}=\frac{1}{p}+\frac{1}{q},$

or equally

$q=\frac{pr}{p-r},\hskip 8.53581pt1<r<p.$

Introduce the following auxiliary functions

$W_{a}[\psi,\nu;\beta](p,r)=W_{a}(p,r)\stackrel{{\scriptstyle
def}}{{=}}\nu\left(\ \frac{pr}{p-r}\ \right)\times\psi^{\beta}(\beta p),$ (10)

$W[\psi,\nu;\ \beta](r)=W(r)\stackrel{{\scriptstyle
def}}{{=}}\inf_{p>r}W_{a}(p,r).$ (11)

It follows from the inequality (9)

$||f||_{r,X}\leq C_{1}^{\beta}\ C_{2}\ W_{a}(p,r),\ p>r.$ (12)

To summarize: we have from (12) after optimization over $\ p;\ p\in(r,\
\infty):\ $

Theorem 2.1. Suppose that for some non - trivial segment $\ r\in(c,d):\ 1\leq
c<d\leq\infty\ $ the function $\ W[\psi,\nu;\beta](r)\ $ is finite. Then we
propose in (2) under our notations

$||f||GW[\psi,\nu;\beta]\leq||\phi||G\nu\times[\ ||g||G\psi\ ]^{\beta}.$ (13)

Example 2.1. Let us show by means of bringing examples the exactness of
proposition (13), yet for arbitrary positive value $\ \beta.\ $ Suppose $\
\mu(X)=1,\ \phi(x)=1,\ $ so that the natural function for the $\ \phi(\cdot)\
$ is equal to 1:

$\nu(q)=||\phi||_{q,X}=1,\ q\in[1,\infty).$

Put also $\ \psi(p)=||g||_{p},\ 1\leq p<\infty,\ $ i.e. $\ \psi(\cdot)\ $ is
the natural generating function for $\ g(\cdot).\ $ Choose for definiteness as
a function $\ g=g(x)\ $ certain positive measurable bounded function having
support with finite measure.

Notice that this function $\ p\to\psi(p),\ 1\leq p<\infty\ $ is monotonically
increasing.

On the other words,

$f(x)=g^{\beta}(x),\ x\in X.$

Let us investigate both the sides of the assertion of theorem 2.1 (13) in this
case. We calculate the left hand side taking into account the monotonicity and
equality $\ \nu(q)=1\ $ at the value $\ p=r+0\ $

$W_{a}[\psi,\nu;\beta](r+0,r)=W_{a}(r+0,r)=W(r)=\psi^{\beta}(\beta r),$

We have at the same time for the right hand side of (13) choosing also $\
p=r+0,\ $ so that $\ q=+\infty:\ $

$||g^{\beta}||_{r}=\psi^{\beta}(\beta r),\ 1\leq r<\infty,$

which coincides with the right - hand side (in this example).

##  3 Main result. Urysohn’s operator.

We impose the following condition on the ”kernel” $\ u=u(x,y,z)\ $ in the
definition of the Urysohn’s operator (3): $\ \exists u_{0}=u_{0}(x,y)\in R,\
x\in X,\ y\in Y,\ $

$|u(x,y,z)|\leq u_{0}(x,y)\ |z|^{\beta},\ \exists\beta=\rm const\geq 1.$ (14)

Denote for at the same values $\ (p,q,r)\ $ as before

$\kappa(q,r)\stackrel{{\scriptstyle def}}{{=}}||\ ||\ u_{0}\ ||_{q,Y}\
||_{r,X};$ (15)

here $\ q=pr/(p-r),\ p>r>1.\ $

Assume as above that for some non - trivial generating function $\
\psi(\cdot)\ $

$g(\cdot)\in G\psi.$

We have by virtue of Hölder’s inequality for these values of parameters $\
(p,q,r)\ $

$||U[g]||_{r,X}\leq\left[\ ||g||G\psi\right]^{\beta}\times[\
\psi^{\beta}(\beta p)\ \kappa(pr/(p-r),r)\ ].$ (16)

Introduce the following generation function

$\theta(r)\stackrel{{\scriptstyle def}}{{=}}\inf_{p\in(r,\infty)}\left\\{\
\psi^{\beta}(\beta p)\times\kappa\left(\ \frac{pr}{p-r},r\ \right)\
\right\\}.$

We conclude under our notations

Theorem 3.1.

$||U[g]||G\theta\leq\left[\ ||g||G\psi\ \right]^{\beta}.$ (17)

Remark 3.1. Unimprovability. The equality in the proposition of theorem 3.1
(17) for all the positive values $\ \beta\ $ is attained if for instance $\
\nu(Y)=1=\mu(X),\ g(y)=C=\rm const\in(0,\infty),\ u_{0}(x,y)=1,\ \
\kappa(q,r)=1,\ \psi(p)=1;\ $ then both the sides in (17) are equal to $\
C^{\beta}.\ $

##  4 Main result. Hammerstein’s operator.

The Hammerstein’s operator, which we will rewrite as

$H[g](x)\stackrel{{\scriptstyle def}}{{=}}\int_{Y}h(x,y)\ n(y,g(y))\ \nu(dy),$
(18)

may be represented as a superposition of Nemytskii operator and Urysohn’s one.
Therefore, we retain all the notations and restrictions imposed above on the
functions $\ n(\cdot,\cdot).\ $

We have consequently by means of Hölder’s inequality

$||H[g]||_{r,X}\leq||\ ||\ h\ ||_{q,Y}\
||_{r,X}\times||n(\cdot,g(\cdot))||_{p,Y},$

where as above $\ 1/r=1/q+1/p,\ p,q,r>1,\ p>r.\ $

The functional

$h(\cdot,\cdot)\to||\ ||\ h\ ||_{q,Y}\ ||_{r,X}=||\ ||\ h\ ||\ ||_{q,Y;r,X}$
(19)

is named as ordinary mixed, or anisotropic Lebesgue - Riesz $\ L_{q,r}\ $ norm
of the function $\ h(\cdot,\cdot),\ $ see [6], [7], chapters 1,2. It is a
multivariate generalization of the classical Lebesgue - Riesz norms.

The correspondent multidimensional Grand Lebesgue Mixed (Anisotropic) norm $\
||h(\cdot,\cdot)||G\tau\ $ is defined very similar to the one - dimensional
case

$||h(\cdot,\cdot)||G\tau\stackrel{{\scriptstyle
def}}{{=}}\sup_{(q,r)}\left\\{\ \frac{||h||_{q,r}}{\tau(q,r)}\ \right\\}.$
(20)

Evidently, the space $\ G\tau\ $ relative the norm $\ ||\cdot||G\tau\ $ is
complete bi - rearrangement invariant Banach functional space. Herewith, for
all the admissible values $\ (q,r),\ $ i.e. for which $\ \tau(q,r)<\infty,\ $

$||h||_{q,r}\leq||h(\cdot,\cdot)||G\tau\cdot\tau(q,r).$ (21)

Further,

$||n(\cdot,g(\cdot))||_{p,Y}\leq||\phi||_{s,Y}\cdot||g^{\beta}||_{t,Y},\
1/p=1/r+1/t,\ s=tp/(t-p),\ t>p.$

Thus,

$||H[g]||_{r,X}\leq\upsilon(r;p,t),$ (22)

where

$\upsilon(r;p,t):=||\ ||h\ ||_{q,Y}\
||_{r,X}\times||\phi||_{s,Y}\times||g^{\beta}||_{t,Y}.$ (23)

Introduce the following domain on the positive quadrant $\ D(r)=\\{\ p,t\
\\},\ $ dependent on the real number (parameter) $\ r;\ r>1:\ $ where

$\frac{1}{r}=\frac{1}{p}+\frac{1}{q},\ \frac{1}{s}+\frac{1}{t}=\frac{1}{p},\
p,q,r,s,t>1,\ p>r,\ t>p,$ (24)

and

$q=pr/(p-r),\ p>r;s=tp/(t-p),\ t>p.$

Denote also

$\Delta(r):=\inf_{(p,t)\in D(r)}\upsilon(r;p,t).$ (25)

Notice that if

$h(\cdot,\cdot)\in G\tau,\ \phi(\cdot)\in G\nu,\ g(\cdot)\in G\psi,$ (26)

then

$\upsilon(r;p,t)\leq||h||G\tau\ ||\phi||G\nu\ [\ ||g||G\psi\
]^{\beta}\times\tau(q,r)\nu(s)\ \psi^{\beta}(\beta t).$ (27)

We obtained actually the following proposition.

Theorem 4.1. We conclude in our notations $\ H[g]\in G\Delta\ $ and moreover

$||H[g]||G\Delta\leq 1.$ (28)

_Acknowledgement._ The first author has been partially supported by the Gruppo
Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni
(GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and by
Università degli Studi di Napoli Parthenope through the project “sostegno alla
Ricerca individuale”.

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