# A Survey on Data Selection for LLM Instruction Tuning

Jiahao Wang1,2 Equal contribution and shared co-first authorship. Bolin Zhang1
∗ Qianlong Du2 Jiajun Zhang2 Corresponding author. Dianhui Chu1
1Harbin Institute of Technology
2Institute of Automation, Chinese Academy of Sciences
<EMAIL_ADDRESS>{brolin<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Instruction tuning is a vital step of training large language models (LLM), so
how to enhance the effect of instruction tuning has received increased
attention. Existing works indicate that the quality of the dataset is more
crucial than the quantity during instruction tuning of LLM. Therefore,
recently a lot of studies focus on exploring the methods of selecting high-
quality subset from instruction datasets, aiming to reduce training costs and
enhance the instruction-following capabilities of LLMs. This paper presents a
comprehensive survey on data selection for LLM instruction tuning. Firstly, we
introduce the wildly used instruction datasets. Then, we propose a new
taxonomy of the data selection methods and provide a detailed introduction of
recent advances,and the evaluation strategies and results of data selection
methods are also elaborated in detail. Finally, we emphasize the open
challenges and present new frontiers of this task.

##  1 Introduction

Large language models (e.g. PaLMChowdhery et al. (2023), GPT-4 OpenAI (2023),
and LLaMa Touvron et al. (2023)) exhibit remarkable proficiency across a broad
spectrum of language understanding and generation tasks, adhering to human
instructions with effectiveness and safety. During training process, LLMs
typically involves two fundamental steps Ouyang et al. (2022a): pretraining on
the large-scale corpus, fine-tuning on instruction datasets. In these steps,
fine-tuning on instruction datasets, also referred to instruction tuning,
plays a crucial role on aligning LLMs with human instruction. Through training
LLMs on (instruction, output) datasets, instruction tuning effectively bridges
the gap between LLMs and various of human intents Zhang et al. (2023).
Specifically, instruction tuning could align the output of LLMs to human
preferences, which enhances the controllability and safety of LLMs. Another
benefit is instruction tuning can enable large models to adapt more quickly to
specific domains or learn specialized knowledge without requiring significant
computational resources and architectural changes.

ln the early researches, the work of instruction tuning mainly focus on
construct large-scale instrcution dataset, and creating instruction datasets
can be done in two primary ways. One is transforming text-label pairs from
existing annotated natural language datasets to instruction-output pairs
through templates, such as P3Sanh et al. (2022). Another method involves using
LLMs like GPT-3.5-Turbo to generate outputs for given instructions, such as
self-instructWang et al. (2023). Although there are various large-scale
instruction dataset created by kinds of methods,they often have limitations in
terms of quantity, diversity, and creativity. Additionally, enhancing the
ability to follow instructions and handling unexpected responses on a large
scale of instruction datasets is a current issue that remains to be resolved.

Therefore, selecting the appropriate dataset is crucial for the instruction
fine-tuning phase. While instruction fine-tuning primarily relies on a large
volume of data, research such as LIMAZhou et al. (2023a) indicates that data
quality is more critical than quantity. They demonstrated a significant
performance improvement in LLMs using only 1k high-quality instruction data.
This finding suggests that LLMs have already acquired world knowledge during
the pre-training phase, and the instruction tuning stage requires only a small
amount of high-quality instruction data to generate high-quality responses.

Manual instruction data selection often involves high costs and introduces
human bias. As a result, creating automated methods for efficiently selecting
instruction data has become crucial. However, this task is challenging due to
the complex factors and multidimensional considerations involved. For example,
it is difficult to assess the quality of individual instructions and ensure
the overall diversity of the selected data. Another challenge is reducing
costs and improving the efficiency of the selection process. In light of these
factors, various data selection methods have been developed.Some methods use a
system of indicators to evaluate individual data points, while others rely on
trainable LLMs or powerful external LLMs. These exploit the LLMs’ own
capabilities to select instructions. Additionally, methods that use smaller
models and design a comprehensive process to achieve a balanced effect across
all aspects are noteworthy. These methods have shown promising results. For
instance, the IFDLi et al. (2023a) method significantly outperforms the Alpaca
model by using only about 5% of the Alpaca dataset, and it also surpasses the
WizardLM model by about 10%. Using high-quality subsets for fine-tuning not
only enhances the instruction-following skills of LLMs but also significantly
reduces computational costs and time.

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where level=3text width=8em,font=,, where level=4text width=5em,font=,, [ Data
Selection for Instruction Tuning, ver [Instruction
Sets (§2) [ Alpaca Taori et al. (2023), WizardLM Xu et al. (2023a),
Dolly-v2 Conover et al. (2023), P3 Sanh et al. (2022),
LIMA Zhou et al. (2023a), self-instruct Wang et al. (2023) , leaf, text
width=37em ] ] [Data Selection
Methods (§3) [System of
indicators
(§3.1) [ INSTRUCTMINING Cao et al. (2023), InstructionGPT-4 Wei et al. (2023),
DQ Zhou et al. (2023b), , leaf, text width=37em ] ] [Trainable
LLMs (§3.2) [ IFD Li et al. (2023a), Instruction Backtranslation Li et al.
(2023b),
Nuggets Li et al. (2023c), DIVERSEEVOL Wu et al. (2023),
TEGIT Chen et al. (2023c), ActiveIT Kung et al. (2023) , leaf, text width=37em
] ] [Powerful
LLMs (§3.3) [ AlpaGasus Chen et al. (2023b), INSTAG Lu et al. (2023), LIFT Xu
et al. (2023b),
DEITA Liu et al. (2023), tree-instruct Zhao et al. (2023), WaveCoder Yu et al.
(2023) , leaf, text width=37em ] ] [Small
Models (§3.4) [ MoDS Du et al. (2023), Coreset-based Selection Chen et al.
(2023a), , leaf, text width=37em ] ] ] [Evaluation
Methods (§4) [ Wining rate (§4.1), Inner Comparison (§4.2), External
Comparison (§4.3), leaf, text width=37em ] ] ] ]

Figure 1: Overview of Data Selection for LLM Instruction Tuning.

This paper provides a comprehensive review of existing works on data selection
methods for the instruction tuning of LLM. To facilities the community, we
maintain a paper list111https://github.com/Bolin97/awesome-instruction-
selector, collect commonly instruction sets for data selection. Section 2
describes the mainstream datasets with different source and construction
methods used in instruction tuning and Section 3 describes four types data
selection methods in details: indicators sets, trainable LLMs, powerful LLMs
and small models. Section 4 presents the evaluation methods and shows the
results of different instruction selection methods. Section 5 summarizes the
paper and emphasize the open challenges and future direction of the
instruction selection.

##  2 Instruction Sets

Various instruction tuning datasets (e.g. Self-Instruct and Alpaca), generated
by LLMs, offer a wealth of samples without human labor, but their data quality
depends on the performance of LLMs and is uncertain. Conversely, the manually
curated datasets (e.g. LIMA and Dolly) obtain higher quality through
meticulous human selection but are potentially influenced by human biases.
Alternative dataset construction methods, like prompt mapping and evol-
instruct, aim to enhance dataset quality and diversity but introduce new
challenges in quality assurance. This variability in dataset construction and
sourcing significantly affects data quality, highlighting the importance of
careful data selection for the instructional tuning of LLM. This section
describes the scale and construction procedures of several commonly
instruction tuning datasets.

Self-instruct, created by Wang et al. (2023), which consists of 52,000
training and 252 testing instructions. Initial instructions from seed tasks
were selected, categorized, and diversified using InstructGPT Ouyang et al.
(2022b) to generate inputs and outputs through output-first or input-first
strategies. Post-processing refined the dataset for uniqueness and relevance,
providing a versatile resource for natural language processing applications.

Alpaca, created by Taori et al. (2023) which consists of 52,002 samples and is
used to fine-tune LLaMA for the ability of instruction-following. Based on the
techniques of Wang et al. (2023), the samples are generated by employing text-
davinci-003.

WizardLM, created by Xu et al. (2023a), which consists of 250,000 samples
generated by the evolutionary algorithms. Two types of algorithms (depth
evolution and breadth evolution) are used to enhance the complexity and scope
of basic instructions, generating more complex and diverse high-quality
instruction data by ChatGPT

LIMA, created by Zhou et al. (2023a), which consists of 1,000 training
samples, 300 test samples and 50 development samples. In addition to the
manually authored samples, samples collected from Q&A websites are strictly
selected by human. Despite its modest scale, LIMA stands out for its
meticulous compilation and design. LLMs fine-tuned on LIMA show remarkable
ability in following instructions and adapting to unseen tasks.

Dolly-v2, created by Conover et al. (2023) with 15,000 instructions, involving
various tasks like brainstorming, classification, QA, and summarization.
Employees author (prompt, response) pairs manually. They are restricted to
using Wikipedia and advised against using web sources or generative AI for
crafting response.

P3, created by Sanh et al. (2022), which integrates 170 NLP datasets with
2,052 prompts. These prompts, also known as task templates, convert
conventional NLP tasks like question answering or text classification into
natural language input-output pairs. The P3 dataset itself is formed by
randomly selecting prompts from PromptSource and organizing data into triplets
of inputs, answer choices, and targets.

##  3 Data Selection methods

Formally, define an instruction dataset $X$ with a size of $n$, where
$X=\\{x_{1},x_{2},\dots,x_{n}\\}$, and each $x_{i}$ represents an instance of
instruction fine-tuning data. To employ a specific instruction data selection
method $\pi$ from $X$ and choose a subset $S_{\pi}^{(m)}$ with a size of
$m$Liu et al. (2023), we then use a predefined evaluation metric $Q$ to assess
the quality of $S_{\pi}^{(m)}$. The obtained subset quality, as measured by
the evaluation metric, can gauge the effectiveness of the chosen instruction
data selection method. The process of designing the selection method can be
seen as:

$S^{(m)}_{\pi}=\pi\left(X\right)$ (1)
$\pi^{*}=\arg\max_{\pi}Q\left(S^{(m)}_{\pi}\right)$ (2)

The classification of instruction data selection methods is based on what
scoring rules the method uses and what model base it employs. The methods can
be divided into the following four categories: methods based on system of
indicators, trainable LLMs, powerful LLMs like ChatGPT and small models.

###  3.1 Methods based on System of indicators

Methods employing a metric set system directly identify multiple metrics
$I_{1},I_{2},\dots,I_{n}$, thereby establishing a comprehensive set of
metrics. Each metric within this set is defined by a specific calculation
formula. Notably, certain metrics might leverage deep learning techniques for
feature extraction from datasets, which are, in essence, metric forms. These
metrics facilitate the computation of individual scores for data instances,
denoted as $score_{ij}=I_{i}(x_{j})$. These scores are subsequently
instrumental in developing a more robust metric set system.

$score_{j}=G\left(I_{1}(x_{j}),I_{2}(x_{j}),\dots,I_{n}(x_{j})\right)$ (3)

Once established, the metric set system can be directly utilized to compute
scores for each data instance within the dataset $X$. By establishing suitable
thresholds, this system facilitates the selective inclusion of data based on
their respective scores:

$S_{\pi}=\\{x|G(x)>\tau\\}$ (4)

Cao et al. (2023) introduce INSTRUCTMINING, a linear rule-based method for
assessing the quality of instructional data. This approach initially
identifies key natural language metrics, such as instruction length,
perplexity, reward scores, KNN-iReimers and Gurevych (2019); Dong et al.
(2011), among others. These metrics are subsequently utilized to develop a
linear equation. To explore the correlation between data quality and these
metrics, and to define the parameters of equation, comprehensive fine-tuning
experiments are conducted. Diverse quality datasets are partitioned into
subsets, combined, and then employed in the fine-tuning of large-scale models.
Quality labels for each subset are derived by evaluating performance of the
model on a test set. The least squares method is applied to these experimental
outcomes to estimate the parameters in INSTRUCTMINING. This involves fitting a
linear equation to the loss observed when evaluating the model against the
test set.Once the parameters have been determined, this formula can be
utilized to calculate the quality of instructions, thereby facilitating data
selection.

Wei et al. (2023) present InstructionGPT-4, a data selection method for
multimodal large model fine-tuning. It surpasses MiniGPT-4Zhu et al. (2023) in
various assessments using less data. The first step, metrics like the CLIP
ScoreRadford et al. (2021), instruction length, and others are used. Visual
and textual data are encoded into vectors and then subjected to dimensional
reduction, which is regarded as special metrics. These metrics are combined
into a vector $e$. The vector is then fed to a trainable data selector, such
as a multi-layer perceptron or a self-attention network. This approach,
similar to Cao et al. (2023) in calculating quality labels, employs clustering
algorithms to segment datasets. Quality labels are assigned post fine-tuning
and evaluation of each subset.

Zhou et al. (2023b) introduce the DQ method, which is an innovative data
compression technique for large-scale computer vision datasets, but it has
been adapted for use in the LLM domain as well. The approach involves several
key steps: Initially, a gain function $P(x)$ is defined,

$P(x)=\sum_{p\in S}||f(p)-f(x)||^{2}_{2}-\sum_{p\in D\setminus
S}||f(p)-f(x)||^{2}_{2}$ (5)

incorporating a feature function $f(.)$ — analogous to a metric — and using
the current subset $S$ and the entire dataset $D$. This gain function
essentially forms a metric set function $G(.)$. The dataset is then
iteratively partitioned into non-overlapping subsets, guided by the gain
function to maximize the defined gain. Subsequently, a representative sample
from each subset is uniformly selected, ensuring coverage of the entire
dataset while optimizing for maximum data diversity. This approach prioritizes
maintaining the overall diversity of the dataset.

###  3.2 Methods based on Trainable LLMs

This section outlines the use of trainable LLMs, such as LLaMa, for developing
computation formulas in data selection processes. The LLM, serving as a
trainable data selector, processes and assigns scores to each piece of
instruction fine-tuning data.

$score_{i}=LLM_{trainable}(x_{i})$ (6)

This method not only concentrates on analyzing individual instructions but
also underscores the necessity of synchronizing data selection with the
functionality of the large model employed for fine-tuning. Subsequent sections
will detail specific methodologies.

Li et al. (2023a) propose the IFD method. It hypothesizes that LLMs can
initially learn to recognize instructions from carefully selected data,
enhancing their ability to assess broader dataset quality and estimate
instruction-following difficulty. Initially, the method involves fine-tuning
an LLM on a small, clustered subset of the instruction dataset to develop
basic instruction-following skills. A novel metric, ”Instruction Following
Difficulty (IFD),” is then introduced to assess the challenge of responding to
specific instructions. IFD compares response quality without specific
instructions. The Conditioned Answer Score is defined as:

$s_{\theta}(A|Q)=\frac{1}{N}\sum_{i=1}^{n}logP(w_{i}|Q,w_{1},\dots,w_{i-1})$
(7)

This score evaluates alignment of the model with the correct answer post-
instruction (Q), considering the influence of generating the answer (A). The
final IFD is:

$r_{\theta}(Q,A)=\frac{s_{\theta}(A|Q)}{s_{\theta}(A)}$ (8)

The IFD score quantifies challenges of the model for each sample. By setting
an IFD score threshold, specific instructions are selected for initial LLM
pre-training, resulting in a refined model.

Li et al. (2023b) present Instruction Backtranslation, a method for generating
and filtering instructions. Starting with a baseline instruction-following
model and a web corpus, the model generates instructions for each web
document, forming a dataset. Then the model is fine-tuned with seed
instructions to gain basic capabilities. And it autonomously scores each
instruction, with those exceeding a set threshold forming a high-score subset
for further fine-tuning. This iterative process enhances instruction
generation and filtering efficiency.

Li et al. (2023c) propose the Nuggets framework , which implements a dual-
phase approach. Initially, it employs a variety of predefined tasks to
evaluate the proficiency of a LLM across multiple scenarios, a process known
as zero-shot scoring. Subsequently, each entry in the instruction dataset is
utilized as a unique prompt in a one-shot manner. These prompts are presented
before the predefined tasks, and the performance of LLM is reassessed, a step
termed one-shot scoring. This approach harnesses the disparity between one-
shot and zero-shot scores to calculate a definitive ’gold score’ for each
instruction. Upon acquiring gold scores for all instructions, those
constituting the highest-scoring subset are chosen as the ’gold subset’. This
subset is then directly employed in the fine-tuning of the model. This
methodology exploits the inherent contextual learning capabilities of
extensive models.

Wu et al. (2023) introduce the DIVERSEEVOL mechanism ,which represents an
innovative iterative data selection strategy. It leverages large-scale models
like LLaMa to generate embedding vectors for instructional data. The mechanism
employs the k-center-greedy algorithmSener and Savarese (2018a) to facilitate
diversity in selecting a subset of data for fine-tuning the LLaMa model. This
procedure is repetitively applied, progressively enlarging the chosen subset,
culminating in the creation of a high-quality instructional dataset.

Chen et al. (2023c) propose the TEGIT method, offering a novel approach to
generate premium instruction fine-tuning data. Particularly notable is their
method of filtering instruction data. Utilizing ChatGPT, a small corpus of
documentation is transformed into a format suitable for instruction data,
forming a meta-dataset. This dataset then serves to train two Llama2 models -
one functioning as a task generator and the other as a task discriminator. The
role of generator is to devise tasks from provided texts, while the
discriminator evaluates these tasks, ensuring their quality.

Kung et al. (2023) present Active Instruction Tuning, which is a unique method
focusing on task sensitivity selection, with the goal of enhancing large model
fine-tuning using fewer tasks while improving out-of-task generalization. This
technique introduces the concept of Prompt Uncertainty, which is determined by
generating k perturbed instructions through the random deletion of words from
an original instruction. The output of LLM probability deviation is then
averaged for these k perturbed instructions. Tasks that exhibit higher Prompt
Uncertainty are prioritized for instruction fine-tuning, using the extent of
Prompt Uncertainty as a measure of task uncertainty.

###  3.3 Methods based on Powerful LLMs like ChatGPT

This section introduces the method of using powerful LLMs such as GPT-4 and
ChatGPT as data selectors. The approach primarily involves designing prompt
templates and leveraging the capabilities of LLMs to assess the quality of
instruction data.

$S_{\pi}=\\{x|ChatGPT(score|prompt,x),score>\tau\\}$ (9)

Chen et al. (2023b) presented ALPAGASUS as an innovative data filtering
methodology aimed at improving efficiency and accuracy in Instruction-
Following Task (IFT) data curation. This approach utilizes a well-designed
prompt applied to ChatGPT, to assess the quality of each data tuple,
comprising instruction, input, and response. The methodology focuses on
excluding tuples that fall below a predefined quality threshold. When this
filtering process was applied to a substantial dataset, it was observed that a
considerable portion of the data was compromised by quality issues. Notably,
the application of the LLM-based filtering process led to the development of a
model that surpassed the performance of the original model, which was trained
using the unfiltered dataset with instruction-based fine-tuning.

Lu et al. (2023) introduce an automated instruction tagging method (INSTAG),
which employs ChatGPT to generate detailed, open-ended labels for instructions
and convince the diversity and complexity of subset. The process involves
annotating data with labels that reflect the semantic and intent of each
instruction and normalizing them for a label-based selection approach. This
approach follows a complexity-first diversification sampling strategy. It
starts by sorting queries in descending order of label count and then
iteratively adds queries to a subset based on the uniqueness of their labels,
until the subset reaches a desired size N. The result is a curated set of
queries with a higher number of labels, indicating greater complexity and
diversity.

To enhance the distribution and quality of datasets, Xu et al. (2023b) propose
the method LIFT to reduce the redundancy of samples. It involves two stages:
expanding dataset distribution and curating the diversity and quality of the
dataset. Initially, ChatGPT enhances the data by generating diverse
instructions and vectorizing them. A subset is then selected based on row
variances. Secondly, ChatGPT scores instructions for accuracy,
interpretability, clarity, difficulty, and length. The initial subset is re-
selected based on these scores.

Liu et al. (2023) introduce DEITA method, which integrates a multifaceted
approach for selecting instruction data, focusing on complexity, quality and
diversity. Complexity describes factors such as the length, difficulty, and
complexity of the instruction. And quality captures the accuracy of the
output. Utilizing the WizardLM technique, ChatGPT is employed to augment
instructions, which are then evaluated for complexity and quality. These
evaluations involve scoring instructions based on complexity, using a
specially trained complexity scorer, and assessing the output quality. Each
instruction in the dataset is assigned complexity scores ($c$) and quality
scores ($q$), and a comprehensive score is calculated by multiplying these two
metrics. The dataset is then organized according to these comprehensive scores
and vectorized for further analysis.

To ensure diversity, a subset is created by adding samples that exceed a set
distance threshold ($\tau$) from their nearest neighbor in the subset. This
process continues until the subset reaches a predetermined size.

Zhao et al. (2023) propose the tree-instruct method , which improves the
quality of instruction by enhance the complexity of it. Liu et al. (2023) uses
this method to measure instruction complexity and sets thresholds for
instruction filtering. Tree-instruct employs GPT-4 to generate semantic parse
trees for instruction data, using the number of nodes in the tree as a measure
of complexity. The complexity is enhanced by adding nodes to the tree, and
then GPT-4 is used to convert the new tree back into natural language,
resulting in new high-quality instructions.

Yu et al. (2023) present WaveCoder, a code-focused LLM enhanced through an
instruction improvement technique. Its training incorporates generated data,
with the data filtering phase being particularly significant. Following data
generation, a LLM-based discriminator, leveraging GPT-4, evaluates instruction
data against established criteria divided into subtopics. This approach
enables finer control over the filtering process, effectively eliminating low-
quality instruction instances.

###  3.4 Methods based on Small Models

This section introduces methods involving the use of external small models as
scorers, or the transformation of instructions into embedding vectors using
small models followed by further processing. Generally, these approaches are
quite comprehensive. The scoring filtering or embedding generation by Small
Models often serves only as a part of the entire methodological process.

Du et al. (2023) introduce the MoDS method, focusing on instruction selection
through three criteria: quality (instructional data fidelity), coverage
(variety of instruction types), and necessity ( impact of instruction on LLM
fine-tuning). The process unfolds in four key steps:

Firstly, employing a reward model to evaluate the instruction dataset quality,
selecting a subset denoted as $D_{h}$, comprising instructions that surpass a
predefined quality threshold.Secondly,Utilizing the k-center-greedy
algorithmSener and Savarese (2018a) to identify seed instructions, which
ensures a diverse and representative sample of the instruction
dataset.Thirdly,fine-tuning a pre-trained LLM with the seed instructions.
Subsequently, this refined model is applied to $D_{h}$ to generate a new
dataset, $D_{inference}$. This dataset is then evaluated using the reward
model, which identifies instructions critical for learning of the LLM by
focusing on those with lower scores. A threshold is established to select
Augmented Instruction Data, which is specifically tailored to enhance
performance of the model.Finally, the seed instructions with the Augmented
Instruction Data to form a high-quality instruction subset that is poised to
facilitate effective fine-tuning of the LLM.

Chen et al. (2023a) proposes a coreset-based and task-related data selection
method: firstly, obtain sentence embeddings of samples by pre-trained language
models (like BERT), then select the center point by applied unsupervised
clustering on these embeddings, and finally use the KCenterGreedy Sener and
Savarese (2018b) algorithm to retrieve core samples from the given dataset.
This method effectively reduces the required training data volume, maintaining
or potentially enhancing model performance.

Table 1: Performances of different selection methods on wining rate.

Selection methods Model pairs for comparison Training set Benchmark for
testing (WS) Vicuna/Koala/WizardLM/Self-inst/LIMA total WS IFD Li et al.
(2023a) llama-7b(5%), llama-7b(full) alpaca 1.125/0.97/1.077/1/1.1 1.04
llama-7b(10%), llama-7b(full) 1.037/1.055/1.114/1.123/1.103 1.097
llama-7b(15%), llama-7b(full) 1/1.038/1.114/1.027/1.09 1.064 llama-7b(10%),
llama-7b(full) WizardLM 1.1625/1.1278/1.1147/1.0278/1.1067 1.0971
llama2-7b(5%), llama2-7b(full) alpaca 1.5875/1.4889/1.4266/1.2937/1.4733
1.4311 Random Sampling llama-7b(5%), llama-7b(full) alpaca - 0.9
InstructionGPT4 Wei et al. (2023) miniGPT4(6%), miniGPT4(full) cc_sbu_align -
1.167 Alpagsaus Chen et al. (2023b) llama-7b(9k), llama-7b(full) alpaca
1.2125/1.0222/1.0596/1.0556/- 1.0658 llama-7b(9k), llama-7b(3k)
1.1/1.183/1.082/1.17/- 1.074 llama-7b(9k), llama-7b(6k)
1.05/1.072/1.05/1.087/- 1.082 llama-13b(9k), llama-13b(full)
1.2125/1.0167/1.133/1.0198/- 1.074 MoDS Du et al. (2023) llama2-7b(2k),
llama2-7b(full) alpaca 1.7125/1.5111/1.4725/1.369/1.4933 1.4786
InstructionMining Cao et al. (2023) llama-7b(2k), llama-7b(full) dolly - 1.088

  * •

’-’ denotes values not reported in the original paper, and model(x) denotes
tuning the model on the given training set using either x samples or x
percentage of samples.

Table 2: Inner comparisons of LLM tuned on subsets with itself tuned on full
sets.

Selection methods | Training set(samples) | Base model | Benchmarks for testing
---|---|---|---
total WS
activeIT Kung et al. (2023) | selfinstruct(2k) | llama-7b | 1.107
selfinstruct(full) | 1.293
|  |  | BBH/DROP/MMLU/Human-Eval/Avg
DQ Zhou et al. (2023b) | alpaca(20%) | llama-7b | 32.7/26.7/39.8/9.2/27.1
alpaca(full) | 32.9/26.3/41.6/10/27.7
|  |  | Vicuna RS/Vicuna WTR/Koala RS/ Koala WTR
DIVERSEEVOL Wu et al. (2023) | Dolly(1k) | llama-7b | 79.69/20/62.29/6.67
Dolly(full) | 73.84/5/57.9/3.33
SelfIns(1k) | 79.16/7.5/66.95/6.11
SelfIns(full) | 73.03/2.5/69.5/3.89
|  |  | HellaSwag/ARC/TruthfulQA/MMLU
LIFT Xu et al. (2023b) | ✗ | Mistral-7B | 0.823/0.602/0.426/0.627
Platypus(15k random) | 0.82/0.607/0.438/0.625
Platypus(15k) | 0.844/0.643/0.49/0.645
|  |  | RTE/CB/ANLI R1/ANLI R2/ANLI R3
coreset Chen et al. (2023a) | P3(0.5) | Galactica-1.3b | 74.73/73.21/49.6/41.9/43.75
P3(full) | 76.17/75/44/35.7/39.42

  * •

dataset(x) denotes tuning the base model on the x samples of the given set.
✗denotes either the absence of a training set or a base model.

Table 3: External comparisons of LLM tuned on subsets with the other LLM.

Models | Training set(samples) | Base model | MT-bench | AlpacaEval
---|---|---|---|---
gpt-4 | ✗ | ✗ | 8.99 | 95.28
gpt-3.5-turbo | ✗ | ✗ | 7.94 | 91.36
alpaca-13b | alpaca | llama-13b | 4.53 | -
NUGGETSLi et al. (2023c) | alpaca(7.5k) | llama-7b | 5.34 | -
TAGLM-13b-v1.0 Lu et al. (2023) | mixture(6k) | llama-13b | 6.44±0.04 | 72.8
TAGLM-13b-v2.0 Lu et al. (2023) | llama2-13b | 6.55±0.02 | -
Instruction Length | llama-13b | 5.89 | -
Random Sampling | 5.84 | -
IFDLi et al. (2023a) | 5.91 | -
Instruction NodeZhao et al. (2023) | 5.65 | -
DEITA Liu et al. (2023) | mixture(10k) | llama2-13b | 6.79 | 81.09

  * •

’mixture’ denotes a combined dataset of WizardLM(Alpaca), WizardLM(ShareGPT),
UltraChat, and ShareGPT.

##  4 Evaluation methods and Result Analysis

The effectiveness of data selection methods depends on the quality of the
subset filtered from the given dataset. To measure the quality of the subset,
the LLMs fine-tuned on the subset are evaluated on different benchmarks in
different approaches, which can be divided into three categories: wining rate,
inner comparison and external comparison.

###  4.1 Wining Rate

To evaluate the effectiveness of the dataset selection methods, the winning
rate of the LLM-sub compared to the base LLM is calculated as:

$(Num(win)-Num(lose))/Num(all)+1$ (10)

LLM-sub represents the LLM fine-tuned on the subset that filtered from the
training set by the selection methods while the base LLM often involves 2
types: i)fine-tuned on the full training set and ii) fine-tuned on the same-
scale subset filtered by the regular selections (e.g. random sampling and
instruction length). In the formular 10, Num(win) represents the number of
wining cases, Num(lose) represents the number of losing cases and Num(all)
represents the number of all cases in the testing benchmarks.

The outputs of LLM-sub and the base LLM are rated by the judge on a scale from
1 to 10, and usually GPT4 is employed as the judge. To address the positional
bias of the judge, Li et al. (2023a) send the outputs of these two LLMs to the
judge twice with different orders. According to Li et al. (2023a), the wining
cases refer to LLM-sub outperforming the base LLM in both times or winning in
one and tying in the other. The losing cases refer to LLM-sub lagging behind
the base LLM in both times or tying in one and losing in the other. The wining
rates of different selection methods on the testing benchmarks are summarized
in Table 1.

###  4.2 Inner Comparison

To evaluate the effectiveness of the dataset selection methods simply and
directly, LLM-sub is compared with the same LLM, but it is fine-tuned on
either the full training set or on a same-scale subset filtered by the regular
selections. We refer to this evaluation method as inner comparison, because it
only compares the LLM fine-tuned on a subset to itself. The inner comparison
performances of different selection methods on the testing benchmarks are
summarized in Table 2.

###  4.3 External Comparison

Another simple and straightforward type of evaluation method is external
comparison, which compares LLM-sub with the external LLMs (i.e. differ from
the model of LLM-sub) on different testing benchmarks. The external comparison
performances of different selection methods on the testing benchmarks are
summarized in Table 3.

###  4.4 Result Analysis

The proposed selection methods outperform the regular selections, which proves
the significance of the data selection in instruction tuning. As shown in
Tabel 1 and 2, TAGLM-13b-v1.0 and IFD outperform the regular selections on MT-
benc based on instruction length and random sampling when tuning llama-13b on
the mixture dataset, and Alpagsaus and IFD outperform the random sampling on
total WS when tuning llama-7b on alpaca.

The more advanced LLM (llama2-7b) achieves higher performance than the
standard LLM (llama-7b), both of which are fine-tuned on the same subset.
Table 3 shows that TAGLM-13b-v2.0 outperforms TAGLM-13b-v1.0, and Table 1
shows that llama2-7b(5%) outperforms llama-7b(5%) when employing IFD selection
method. These improvements are attributed to the inherent sophistication of
the advanced LLMs, leading to higher learning efficiency on the subset.

Instruction tuning the specific LLM on a larger subset does not necessarily
guarantee performance improvement. This may be related to the inherent
features of the selection methods. As shown in Table 1, the performance of
llama-7b does not improve with the increase of the subset size when applying
the method IFD in the training set alpaca. However, the performance of
llama-7b improves with the increase of the subset size when applying the
method Alpagsaus in the same training set.

##  5 Conclusions and Open Challenges

This paper provides a comprehensive overview of methods and challenges on data
selection of instruction tuning, emphasizing the crucial role of high-quality
data on the fine-tuning stage. We showcase some existing datasets along with
their corresponding construction methods. These datasets suffer from issues
such as imbalanced data distribution and inconsistent data quality. Based on
this foundation, we present the existing four types of data selection methods.
Methods relying solely on metric sets could effectively assess the quality of
individual data, but lack consideration of complex dataset characteristics.
Trainable large model methods that design scoring formulas could choose data
suited to the LLM itself, and Methods using external powerful LLMs like GPT-4
for scoring have exceptional data selection capabilities. Furthermore, methods
utilizing small models usually designed with multiple modules, considering
various aspects. Finally we introduce how to evaluate the data selection
methods, including wining rate, inner comparison and external comparison.
Despite existing methods achieving commendable performance, several challenges
persist.

There is a notable lack of uniform evaluation standards. In Section 4, we have
presented multiple evaluation methods and various benchmarks concerning LLMs.
Various data selection methods usually choose various evaluation standard,
which has caused considerable difficulty in determining which method has more
advantages. Future research could be devoted to establishing a reasonable,
comprehensive, and automatic evaluation method for selection methods, thereby
unifying the evaluation process for methods.

Handling large volumes of data often proves to be inefficient and exhibit a
heavy reliance on powerful LLMs. When the scale of the instruction dataset to
be filtered is too large, existing methods generally take too long to process,
especially when using LLMs to extract instruction features or perform related
metric calculations. Processing hundreds of thousands instruction data with
LLMs can be very time-consuming.Furthermore, when using powerful LLMs like
GPT4 API to process hundreds of thousands of instructions, the cost is very
high. Future work should explore the use of smaller models and strive to match
the selection capabilities of LLMs.

Existing models and methods for evaluating data quality primarily focus on
English and general domains, lacking models for other languages and selection
methods in specific domains.Future research should explore instruction quality
assessment models for other languages, and investigate the design of task-
specific selection methods. This aims to enhance the performance of methods
across various scales, domains, and languages of instruction sets.

The challenges provide promising directions for further exploration,
potentially leading to breakthroughs.In conclusion, it is our hope that these
analytical efforts and the provided overview can further contribute to
enhancing the abilities of LLMs in following human instructions.

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The minimal entanglement wedge cross section in the GMMG/GCFT flat holography

Mohammad Reza Setare 111E-mail<EMAIL_ADDRESS>, Meisam Koohgard 222E-mail:
<EMAIL_ADDRESS>
Department of Science,
Campus of Bijar, University of Kurdistan, Bijar, Iran

We focus on a proper candidate for the entanglement wedge in asymptotically
flat bulk geometries that are described by the generalized minimal massive
gravity (GMMG) in the context of the flat holography. To this end, we describe
the boundary by two dimensional Galilean conformal field theory (GCFT) at the
bipartite mixed state of the two disjoint intervals. We make a conjecture on
the minimal entanglement wedge cross section (EWCS) and we find that the
results are consistent with the previous computations on the holographic
entanglement negativity.

Keywords: Entanglement wedge cross section, GMMG, Holographic entanglement
negativity, bipartite mixed state,

##  1 Introduction

The entanglement entropy is a reliable measure to determine the quantum
entanglement between the bipartite pure states. This measure is not a proper
one for the mixed quantum states; the quantum correlations have more involved
structures than the pure states [1, 2]. A computable measure of the
entanglement has been introduced in [3] (see also [4, 5, 6, 7]) to
characterize the correlation between the mixed states. This is a logarithmic
negativity that can be computed by taking the trace norm of the mixed density
matrix which is a partially transposed matrix in related to the subsystems. A
classical approach based on a replica method to compute the negativity in QFT
has been introduced in [2] and this method has been used for the various
bipartite states in two dimensional conformal field theories ($CFT_{2}$’s) in
[8, 9, 10, 11, 12].

A holographic computation of the entanglement negativity [13, 14, 15] has been
motivated by the holographic conjecture of Ryu and Takayanagi [16, 17] in
computation of the universal part of the entanglement entropy of a $CFT_{d}$
which is dual to an $AdS_{d+1}$ in the bulk side of the duality. In the Ryu
and Takayanagi’s conjecture, a codimension-2 minimal surface (as RT surface)
in the bulk is homologous to the subsystem in the boundary and its area is
considered proportional to the universal part of the entanglement entropy.
Along with the progress in the characterization of the holographic
entanglement entropy in $CFT_{d}$’s, a proposal for the holographic
entanglement negativity in the $AdS_{2+1}/CFT_{2}$ was developed in [14, 18,
19]. This proposal was based on finding out some geodesics on the bulk
homologous to some combinations of the intervals in the boundary.

Another approach to compute the holographic entanglement negativity have been
introduced in [20] using the minimal cross sectional area of entanglement
wedges. The entanglement wedge [21, 22, 23] is a natural region in the bulk
that is distinct from the causal wedge [21, 22, 23, 24, 25, 26] which are both
corresponding to the boundary configuration. The former is a bulk region that
its dual in the boundary is the reduced density matrix of the CFT and the
minimal entanglement wedge cross section has been proposed as a measure of the
entanglement of purification (EoP) [27, 28, 29, 30]. In the theory of quantum
information, the entanglement of purification that receives both classical and
the quantum contributions is a proper measure for both of the classical and
quantum correlations in the bipartite mixed states [31]. This feature of the
EoP is in contrast to the entanglement entropy. The minimal entanglement wedge
cross section as a measure in relation to other entanglement measures for
example the odd entanglement entropy [32] and the reflected entropy [33, 34,
35, 36] is an interesting quantity that could be investigated in the quantum
entanglement.

The two dimensional Galilean conformal field theories ($GCFT_{2}$) were
proposed as the non-relativistic version of the corresponding conformal field
theories in two dimensional spacetime by using the Inonu-Wigner contraction of
the $CFT_{2}$ algebra [37, 38, 39, 40]. Computation of the entanglement
entropy of CFTs in [40, 41] was extended to find out the entanglement
negativity of the $GCFT_{2}$ in [42] through the replica technique. The non-
relativistic flat holography between the bulk gravity and boundary quantum
field theory has been extended in [43]. The three dimensional generalized
minimal massive gravity (GMMG) [44] could be an option for the bulk
description of the two dimensional Galilean conformal algebra (2d GCA) with
asymmetric central charges. GMMG model is a theory that avoids the bulk-
boundary clash and as a Minimal Massive Gravity (MMG) [45] has some positive
energy excitations around the vacuum that are the maximally $AdS_{3}$. The
GMMG’s central charges are positive in the dual CFT. In contrast to GMMG, the
Topologically Massive Gravity (TMG) [46, 47] and the cosmological extension of
New Massive Gravity [48] that are constructed previously with local degrees of
freedom in three dimensional spacetime could not avoid the aforementioned
class. GMMG model is a general extension of MMG that can be constructed from
pure Einstein gravity by adding the Chern-Simons (CS-) term and another term
with a negative cosmological constant.

In the study of the proper measures for the entanglement between the mixed
states, the holographic entanglement negativity of the bipartite states in
$GCFT_{2}$ has been considered in [49]. The authors in [49] investigated the
asymptotically flat (three dimensional) geometries in the bulk side of the
holography and they considered the three dimensional Einstein gravity and the
topologically massive gravity (TMG) dual to a class of $GCFT_{2}$’s. Their
study on the entanglement negativity was based on the evaluation of some
extremal curves homologous to the appropriate configurations of the bipartite
state in the boundary of the holography. These are generalizations of the RT
surfaces and are known as the swing surfaces in [50, 51]. In [52], we extended
the measurement of the entanglement negativity to another flat and no-
relativistic holography that we considered the GMMG in the asymptotically flat
background of the bulk and the $GCFT_{2}$ in the bipartite configuration of
the boundary. We utilized the replica technique in the computation of the
entanglement negativity and the results match with the results in [53] in the
large central charge limit.

The computation of the entanglement negativity in the $AdS_{3}/CFT_{2}$
duality has been developed through the computation of the bulk entanglement
wedge cross section (EWCS) in [20, 54]. The development of this approach to
flat holography has been done by a novel construction in [55]. In this
construction, a new approach to computation of the EWCS has been extended in
some asymptotically flat three dimensional gravities in the bulk side of the
duality. Based on the relation between the minimal entanglement wedge cross
section found in [55] in the $TMG/GCFT_{2}$ framework, we make a conjecture on
the structure of the minimal EWCS and we find the holographic entanglement
negativity in $GMMG/GCFT_{2}$ framework. As a consistency check we find the
result in this work the same as the result on the $GMMG/GCFT_{2}$ on our
previous work in [52].

In section 2 we study the minimal EWCS in the AdS/CFT scenario. In the
following, we introduce the GMMG model as the model describing the three
dimensional asymptotically flat space geometry of the bulk side of the
GMMG/GCFT duality and we present an argument on the structure of the minimal
EWCS of the GMMG and its relation with the result in the TMG case. In the
section 3, we focus on the minimal EWCS in the cases where bipartite mixed
with two disjoint intervals are considered at the boundary. First, we find the
minimal EWCS at the Minkowski space of the bulk that is dual to the mixed
bipartite state of $GCFT_{2}$ in the vacuum. Then the minimal EWCS at the flat
space geometry of the bulk is introduced in the following of this section. In
the latter case, the boundary will be introduced by two disjoint mixed
intervals in the $GCFT_{2}$ at a finite temperature state. In the last
section, we present our results of this study.

##  2 EWCS in asymptotically flat space GMMG

The relation between the minimal EWCS and the entanglement negativity in the
$AdS_{d+1}/CFT_{d}$ duality plays an important role in our study in extending
the relation to the GMMG/GCFT duality. For this reason, in this section, we
will briefly review some results in the EWCS in $AdS_{d+1}/CFT_{d}$ framework.
Since our focus in this paper is on the bipartite mixed state of the boundary,
we consider two disjoint intervals $A$ and $B$ in a $CFT_{d}$ at a slice with
constant time. The static geometry of the bulk is considered with $AdS_{d+1}$.
If $\Gamma_{AB}$ is the RT surface for the interval $A\cup B$, the
entanglement wedge $M_{AB}$ is a region bounded blue solid line by the
following boundary [27] (Fig.2.1)

$\partial M_{AB}=A\cup B\cup\Gamma_{AB}.$ (2.1)

The minimal EWCS can be defined as follows [27]

$E_{W}(A:B)=\frac{Area(\Sigma_{AB}^{min})}{4G_{N}}$ (2.2)

where $\Sigma_{AB}^{min}$ is a minimal surface that can split two entanglement
wedges belonging to $A$ and $B$. The $G_{N}$ is Newton’s constant. The minimal
EWCS can be seen in Fig.2.1 by the red colored dashed line area.

BA$\Sigma_{W}^{min}$ Figure 2.1: The region bounded blue solid line is the
entanglement wedge $M_{AB}$ for the bipartite mixed stated $A\cup B$ in the
static $AdS_{3}$. The minimal entanglement wedge cross section is depicted as
the red colored dashed line.

When the subsystems in the $CFT_{d}$’s have a spherical entangling surface,
the holographic entanglement negativity $\mathcal{E}$ has the following direct
relation with the minimal EWCS [20, 54]

$\mathcal{E}=\zeta_{d}E_{W},$ (2.3)

where $\zeta_{d}$ is a constant that its relation for a vacuum state in a
$CFT_{d}$ can be determined as follows [11, 56]

$\displaystyle\zeta_{d}$ $\displaystyle=$
$\displaystyle\bigg{(}\frac{1}{2}x_{d}^{d-2}(1+x_{d}^{2})-1\bigg{)},$
$\displaystyle x_{d}$ $\displaystyle=$
$\displaystyle\frac{2}{d}\bigg{(}1+\sqrt{1-\frac{d}{2}+\frac{d^{2}}{4}}\bigg{)}.$
(2.4)

For the case of the $AdS_{3}/CFT_{2}$, the relation (2.3) is as follows

$\mathcal{E}=\frac{3}{2}E_{W},$ (2.5)

where can be obtained by substituting $d=2$ into (2). In this case all the
entangling surfaces of subsystems have the spherical geometries.

The story in flat holography is somewhat different. The authors in [55], given
the relation between the minimal EWCS and the four-point functions of the
twist fields inserted in the two disjoint intervals, claim that if the
computational channels in the monodromy analysis [20] changes, a non-zero
result for the EWCS can be obtained. The four-point functions are expanded
based on the conformal blocks, and as the channel changes, the dominant term
of the expansion as the channel changes. By changing the dominant term in the
four-point function, a transition from zero to non-zero is obtained in the
minimal EWCS. Such a transition has already been observed in the AdS/CFT case
[34, 57, 58]. In [55] , we do not only see the non-zero minimal EWCS for the
flat holography, but also the relation that is obtained for the EWCS in the
holography has a relation with the entanglement negativity that was obtained
before in [53]. The relationship between the minimal EWCS $E_{W}$ and the
holographic entanglement negativity $\mathcal{E}$ is obtained in the cases of
flat Einstein gravity and the TMG which both are dual to the $GCFT_{2}$. In
both cases, the results are the same as the result in the $AdS_{3}/CFT_{2}$
framework.

In our previous work [52] in the GMMG/GCFT framework, we obtained the
holographic entanglement negativity with the help of the replica technique. In
this paper, we present another approach for computation of the entanglement
negativity with the help of the minimal EWCS. The GMMG model and the TMG model
are similar in some respects in that they include the Chern-Simons and the
Einstein gravity parts, but the former includes the higher derivative gravity
terms. The Lagrangian 3-form of the GMMG model can be written as follows [44]

$L_{GMMG}=L_{TMG}-\frac{1}{m^{2}}\big{(}f.R+\frac{1}{2}e.f\times
f\big{)}+\frac{\alpha}{2}e.h\times h$ (2.6)

where $L_{TMG}$ is the Lagrangian of the TMG and $R$ is the Ricci scalar. $f$
is an auxiliary one-form field, and $m$ is a mass parameter term. $e$ is a
dreibein and $h$ is the auxiliary field. The Chern-Simons (CS-) term in the
$L_{TMG}$ has the following form in the first order formalism

$CS-term:\frac{1}{2\mu}(\omega.d\omega+\frac{1}{3}\omega.\omega\times\omega)$
(2.7)

where $\omega$ is dualised spin-connection. The algebra of the asymptotic
conserved charges of asymptotically $AdS_{3}$ spacetimes in the context of
GMMG is isomorphic to two copies of the Virasoro algebra with the following
right and left central charges [59]

$c_{GMMG}^{+}=\frac{3l}{2G}(-\sigma-\frac{\alpha
H}{\mu}-\frac{F}{m^{2}}+\frac{1}{\mu
l}),~{}~{}~{}c_{GMMG}^{-}=\frac{3l}{2G}(-\sigma-\frac{\alpha
H}{\mu}-\frac{F}{m^{2}}-\frac{1}{\mu l}),$ (2.8)

where $H$ and $F$ are some constants. By taking Inonue-Wigner contraction as
[37, 38], we can find the asymptotic symmetry group as a GCA with the
following central charges

$\displaystyle c_{L}$ $\displaystyle=$ $\displaystyle
c_{GMMG}^{+}-c_{GMMG}^{-}=\frac{3}{\mu G},$ $\displaystyle c_{M}$
$\displaystyle=$
$\displaystyle\frac{(c_{GMMG}^{+}+c_{GMMG}^{-})}{l}=\frac{3}{G}(-\sigma-\frac{\alpha
H}{\mu}-\frac{F}{m^{2}}).$ (2.9)

Based on the argument presented in the appendix A, we are led to make a claim
that the minimal EWCS of the GMMG in the context of the asymptotically flat
geometry can have a similar structure with the minimal EWCS in the flat TMG.
To provide an argument in support of this claim, we present an argument
similar to the one we have already given in relation to negativity in [52]. To
this end, we follow the approach taken by the authors in [61, 62]. The authors
consider the bulk action with a conical singularity of $2\pi\epsilon$ along a
worldline $C$ in the bulk where $\epsilon$ is equal $n-1$. $n$ defines the
number of the copies of the plane that are sewn together to form the boundary
interval in the holography. The entanglement wedge cross section can be
related to the worldline action. The worldline metric could have the following
form

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
e^{\epsilon\phi(\gamma)}\delta_{ab}d\gamma^{a}d\gamma^{b}+(g_{zz}+K_{a}\gamma^{a}+...)dz^{2}$
(2.10) $\displaystyle+e^{\epsilon\phi(\gamma)}U_{a}(\gamma,z)d\gamma^{a}dz,$

where $z$ is a spacelike direction along the cone worldline and $\gamma^{a}$
are two perpendicular flat coordinates. $K_{a}$ as extrinsic curvatures are
the expansion coefficients of $g_{zz}$ along $\gamma^{a}$ directions. $U_{a}$
are some arbitrary functions and $\phi(\gamma)$ is a function that is needed
to regularize the cone. By the technique extended in [61, 62, 63, 64] , we
find the regularized cone action in the GMMG as follows [52]

$\displaystyle S_{cone}|_{GMMG}$ $\displaystyle=$
$\displaystyle-\frac{\epsilon}{4G}(-\sigma-\frac{\alpha
H}{\mu}-\frac{F}{m^{2}})\int_{C}dz\sqrt{g_{zz}}$ (2.11)
$\displaystyle-\frac{i\epsilon}{16\mu
G}\int_{C}dz\epsilon^{ab}\partial_{a}U_{b}+\mathrm{O}(\epsilon^{2})$

where the action have been computed to the first order in $\epsilon$ in the
expansion. The second term in (2.11) is related to the CS-term in the GMMG
Lagrangian. By changing the first term of (2.11) in a covariant manner, the
cone action can written as follows

$\displaystyle S_{cone}|_{GMMG}$ $\displaystyle=$
$\displaystyle-\frac{\epsilon}{4G}(-\sigma-\frac{\alpha
H}{\mu}-\frac{F}{m^{2}})\int_{C}d\tau\sqrt{g_{\mu\nu}\big{(}X(\tau)\big{)}\dot{X}^{\mu}\dot{X}^{\nu}}$
(2.12) $\displaystyle-\frac{i\epsilon}{16\mu
G}\int_{C}dz\epsilon^{ab}\partial_{a}U_{b}+\mathrm{O}(\epsilon^{2})$

where $X^{\mu}(\tau)$ parameterizes the cone tip and
$\dot{X}^{\mu}\equiv\frac{\partial X^{\mu}}{\partial\tau}$.

By the following normal vectors

$\displaystyle n_{1}$ $\displaystyle\equiv$
$\displaystyle\frac{\partial}{\partial\gamma^{1}},$ $\displaystyle n_{2}$
$\displaystyle\equiv$ $\displaystyle\frac{\partial}{\partial\gamma^{2}},$
(2.13)

the $CS$-term in the $S_{cone}$ can be written as follows

$\displaystyle S_{cone}|_{GMMG}$ $\displaystyle=$
$\displaystyle-\frac{\epsilon}{4G}(-\sigma-\frac{\alpha
H}{\mu}-\frac{F}{m^{2}})\int_{C}d\tau\sqrt{g_{\mu\nu}\big{(}X(\tau)\big{)}\dot{X}^{\mu}\dot{X}^{\nu}}$
(2.14) $\displaystyle-\frac{i\epsilon}{4\mu G}\int_{C}d\tau n_{2}.\nabla
n_{1}+\mathrm{O}(\epsilon^{2}),$

where the ”$\nabla n$” is the covariant derivative along the worldline. By
changing the normal vectors to the Lorentzian signature as follows

$\displaystyle\tilde{n}_{1}$ $\displaystyle\equiv$ $\displaystyle
in_{1}=\partial_{t},$ (2.15) $\displaystyle\tilde{n}_{2}$
$\displaystyle\equiv$ $\displaystyle n_{2},$ (2.16)

the action could have the following form

$S_{cone}|_{GMMG}=-\epsilon\frac{c_{M}}{12}L^{extr}-\epsilon\frac{c_{L}}{12}\Delta\eta^{extr},$
(2.17)

where we have used the following relations for the extremal curve $L^{extr}$
and the extremized boost $\Delta\eta^{extr}$, respectively as follows

$L^{extr}=\int_{C}d\tau\sqrt{g_{\mu\nu}\big{(}X(\tau)\big{)}\dot{X}^{\mu}\dot{X}^{\nu}}$
(2.18)

and

$\Delta\eta^{extr}=\int_{C}d\tau\tilde{n}_{2}.\nabla\tilde{n}_{1}+\mathrm{O}(\epsilon^{2}),$
(2.19)

$L^{extr}$ is the lengths of the extremal curve that is homologous to the
entangling surfaces at the boundary. $\Delta\eta^{extr}$ is the extremized
boost will be explained in the next section.

Given the effect that worldline action receives from the Chern-Simmons term in
(2.17), we make a conjecture that such an effect is also found in the minimal
entanglement wedge cross section for the flat GMMG as follows

$E_{W}=\frac{c_{M}}{12}L^{extr}+\frac{c_{L}}{12}\Delta\eta^{extr}.$ (2.20)

This result is similar to the TMG/GCFT one in [55] and the effect of the
differences of the GMMG model is reflected only in the central charges in the
minimal EWCS relation and that there is no change in the computations of
$L^{extr}$ and $\Delta\eta^{extr}$. To support this conjecture, we consider
configurations with two disjoint intervals in the following section. We find
that by using the relation (2.20), the results are consistent with our
previous work in [52].

##  3 EWCS for two disjoint intervals

The authors in [55] consider a general case for a bipartite mixed disjoint
intervals in the $CFT_{2}$ which is dual to $AdS_{3}$. Through the following
contraction on the spacetime coordinates at the boundary,

$t\to t,~{}~{}~{}~{}x\to\epsilon x~{}~{}~{}(with~{}~{}~{}\epsilon\to 0),$
(3.1)

they obtain a result of the minimal EWCS for the bulk could have an
asymptotically flat geometry. Using the above limit, the GCA algebra
generators can be built from the CFT Virasoro generators [65]. The results of
[55] on minimal EWCS of a generic form of two disjoint intervals can be
compared with the results of [49] on the entanglement negativity of the
intervals. By this comparison, it can be confirmed that when the geometry of
the bulk is considered to be an asymptotically flat one, and there is a
$GCFT_{2}$ at the boundary, the following relation can be used between the
entanglement negativity and the minimal EWCS [55]

$\mathcal{E}=\frac{3}{2}E_{W}.$ (3.2)

Since the GMMG/GCFT duality we are discussing in this paper has a similar
configuration to the one mentioned above, we also use the above relation
between the entanglement negativity and the minimal EWCS in our computations.

###  3.1 EWCS for two disjoint intervals in the vacuum

To find the EWCS for the bipartite mixed states, we describe the boundary in
the $GCFT_{2}$ vacuum that is dual to the bulk flat 3-dimensional spacetime.
The bulk spacetime can be described by the Minkowski metric in Eddington-
Finkelstein coordinates as follows

$ds^{2}=-du^{2}+dr^{2}+r^{2}d\phi^{2},$ (3.3)

where $\phi$ is the angular coordinate and $u=t-r$ is the retarded time. The
two intervals
$A=\big{[}\big{(}u_{1},\phi_{1}\big{)},\big{(}u_{2},\phi_{2}\big{)}\big{]}$
and
$B=\big{[}\big{(}u_{3},\phi_{3}\big{)},\big{(}u_{4},\phi_{4}\big{)}\big{]}$
are considered in the cylindrical coordinates. The planar coordinates $(x,t)$
in the boundary have the following relation with the cylindrical coordinates

$x=e^{i\phi},~{}~{}~{}t=iue^{i\phi}.$ (3.4)

If the asymptotically flat geometry of the bulk described by the pure Einstein
gravity that only has one central charge $c_{M}$ in the dual field theory, the
minimal EWCS can be obtained as follows

$E_{W1}=\frac{c_{M}}{12}L^{extr}(p_{b},p^{\prime}_{b})$ (3.5)

where $L^{extr}(p_{b},p^{\prime}_{b})$ is the length of the extremal curve
which is homologous to A and B intervals (this extremal curve has been
depicted by the blue dashed line in Fig.3.1). The non-trivial entanglement
wedge has been shown by the $M_{AB}$ region in Fig.3.1. The red lines in
Fig.3.1 lie on the null planes extending from the endpoints of the intervals.
$p_{1}$, $p_{2}$, $p_{3}$ and $p_{4}$ lie on the intersection of the null
planes. The extremal curves connecting $p_{1}(p_{2})$ and $p_{4}(p_{3})$ lie
on the null planes extending from $\partial_{1}A(\partial_{2}A)$ and
$\partial_{2}B(\partial_{1}B)$. The curve that connects $p_{1}$ to $p_{4}$ is
called $\gamma_{14}$ (and so is $\gamma_{23}$ for the points $p_{2}$ and
$p_{3}$). $L^{extr}(p_{b},p^{\prime}_{b})$ is the extremized length of the
curve connects two arbitrary points $p_{b}$ and $p^{\prime}_{b}$ on the
$\gamma_{14}$ and $\gamma_{23}$, respectively.

$M_{AB}$$M_{AB}$$p_{1}$$\partial_{1}A$$\partial_{2}A$$\partial_{1}B$$\partial_{2}B$$p_{3}$$p_{4}$$p_{b}$$p^{\prime}_{b}$$E_{W}$$B$$A$$p_{2}$
Figure 3.1: The connected entanglement wedge construction for two disjoint
intervals $A$ and $B$ in a $GCFT_{2}$ which is dual to an asymptotically flat
spacetime that can be described by the Einstein gravity.

Another effective contribution to the actions of TMG and the GMMG is related
to the Chern-Simons terms. By modification of the bulk description by CS-term,
the minimal EWCS can get a contribution from the CS-term. In the flat-TMG/GCFT
case, we are encountered with two central charges and the $c_{L}$, in addition
to $c_{M}$, becomes non-zero. The non-zero $c_{L}$ corresponds to the fact
that the primary twist operators involved in determining the n-point
correlation functions have non-zero spins [60, 61]. These operators correspond
to spinning particles in the bulk space. The spinning particles move in a
straight line in the Minkowski spacetime (based on the proof made in [60]).
These straight lines are the same extremized curves that are considered in
calculating the entanglement wedge. The effect of the Chern-Simons term is
that a normal frame is mounted any point on the bulk [60, 61]. This frame is
expressed in terms of the coordinates, and timelike vector $v$ and spacelike
vector $\tilde{v}$ that are normal to the trajectory of the particle in the
bulk. Thus, the action of a moving particle is not only related to the curve
length that connects the endpoints of the trajectory $X^{\mu}$, but also to
another factor, which is related to this frame changes. Therefore, the action
also depends on the product of normal vectors $v_{i}.v_{f}$, which is obtained
by the parallel transporting $v_{f}$ to $v_{i}$ along the $X^{\mu}$. This
shift in the action is presented as a boost, which is reflected in the
following relation [61]

$\Delta\eta(v_{i},v_{f})=\int_{C}ds(\tilde{v}.\nabla
v)=\cosh^{-1}(-v_{i}.v_{f}).$ (3.6)

Since the minimal EWCS in the first case with $c_{L}=0$ just depends on the
minimal length of the curve homologous to the intervals, when $c_{L}$ is non-
zero, the Chern-Simons boosts are added to the particle action. The CS-
contribution to the extremal EWCS can be found as follows

$E_{W}^{CS}=\frac{c_{L}}{12}\Delta\eta^{extr}(v_{b},v^{\prime}_{b}),$ (3.7)

where $\Delta\eta^{extr}$ is the extremized boost that is needed to take the
parallel transport $v^{\prime}_{b}$ and $v_{b}$ along the bulk entanglement
wedge.

Given the effect that Chern-Simons term has, in the construction of the
entanglement wedge in the $GCFT_{2}$ dual to the flat TMG, there is a new
ingredient compared to the Einstein gravity case. The ingredients are timelike
normal vectors at each bulk point that have been depicted in Fig.3.2.

$\partial_{1}A$$\partial_{2}A$$\partial_{1}B$$\partial_{2}B$$B$$A$$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{b}$$v^{\prime}_{b}$
Figure 3.2: The entanglement wedge construction for two disjoint intervals in
a $GCFT_{2}$ dual to flat TMG. In this figure, the CS contribution to the
$X^{\mu}$ bulk trajectory of the spinning particle has been depicted by the
timelike vectors that are the ingredients of the normal frames in each points
of the bulk

The CS-contribution in the minimal EWCS can be obtained by extremizing the
boost $\Delta\eta$ that is needed to drag the normal frame through the
timelike vectors $v_{b}$ and $v^{\prime}_{b}$. Since the action structure of
the two models TMG and GMMG is similar in terms of Chern-Simons terms, we make
a conjecture the contribution mentioned above in the minimal EWCS to be
considered in the flat-GMMG/GCFT duality as well.

In this paper, instead of describing the bulk geometry with the TMG, we want
to describe it with the GMMG. The GMMG as an extension of the pure Einstein
gravity has three important differences [44]: The CS deformation term, the
higher derivative term and the term with a negative cosmological constant. The
GMMG model and the TMG model are similar in some respects in that they include
the Chern-Simons and the Einstein gravity parts, but the former includes the
higher derivative gravity terms. Despite the differences, we guess that in the
GMMG case, the extremal EWCS must consist of two parts, one similar to (3.5)
and the other similar to (3.7). Although we note that in the first part, we do
not just encounter the pure Einstein gravity and there is the higher
derivative term, we present an argument in the Appendix A that can be
convincing to support our conjecture. Meanwhile, the CS-part of the action in
the GMMG is not the same in all respects as the CS-term in TMG, and there are
some differences. The conjecture we present in this paper requires to be
proved, but in the rest of this article, we provide evidences that it is true.
The extremal EWCS for the flat GMMG that is dual to GCFT with two bipartite
intervals is consistent with our previous calculations of the holographic
negativity in the flat holography scenario in [52] and this can be a reliable
evidence for the conjecture provided in this paper.

For two disjoint intervals $A$ and $B$ in the $GCFT_{2}$ vacuum that is dual
to three dimensional Minkowski spacetime (3.3), the minimal curve length
between $p_{b}$ and $p^{\prime}_{b}$ can be considered in the pure Einsein
gravity case as follows [55]

$L^{extr}(p_{b},p^{\prime}_{b})=\bigg{|}\frac{\mathcal{X}}{\sqrt{\mathcal{T}}(1-\mathcal{T})}\bigg{|}$
(3.8)

which needs to be proven independently in the flat GMMG, because there is the
higher derivative term in the GMMG than the Einstein gravity case.
$\mathcal{T}$ and $\mathcal{X}/\mathcal{T}$ are the cross-ratios that can be
defined as follows

$\displaystyle\mathcal{T}$ $\displaystyle=$
$\displaystyle\frac{t_{12}t_{34}}{t_{13}t_{24}},$
$\displaystyle\frac{\mathcal{X}}{\mathcal{T}}$ $\displaystyle=$
$\displaystyle\frac{x_{12}}{t_{12}}+\frac{x_{34}}{t_{34}}-\frac{x_{13}}{t_{13}}-\frac{x_{24}}{t_{24}}.$
(3.9)

In the limit $\mathcal{T}\to 1$ which is characteristic of the t-channel
computations in the monodromy analysis, we can find the minimal EWCS for the
bulk dual of the GCFT with mixed bipartite disjoint intervals in the vacuum as
follows

$\displaystyle E_{W1}$ $\displaystyle=$
$\displaystyle\frac{c_{M}}{12}L^{extr}(p_{b},p^{\prime}_{b})$ (3.10)
$\displaystyle=$ $\displaystyle-\frac{1}{4G}(\sigma+\frac{\alpha
H}{\mu}+\frac{F}{m^{2}})L^{extr}(p_{b},p^{\prime}_{b}),$ $\displaystyle=$
$\displaystyle-\frac{1}{4G}(\sigma+\frac{\alpha
H}{\mu}+\frac{F}{m^{2}})\big{(}\frac{\mathcal{X}}{\sqrt{\mathcal{T}}(1-\mathcal{T})}\big{)},$
$\displaystyle=$ $\displaystyle-\frac{1}{4G}(\sigma+\frac{\alpha
H}{\mu}+\frac{F}{m^{2}})\big{(}\frac{x_{13}}{t_{13}}+\frac{x_{24}}{t_{24}}-\frac{x_{14}}{t_{14}}-\frac{x_{23}}{t_{23}}\big{)},$

where we have substituted (2) into (3.5) for the GMMG model. The minimal boost
between $v_{b}$ and $v^{\prime}_{b}$ can preserve its shape as in the flat TMG
case as follows [55]

$\Delta\eta^{extr}=2\log\frac{1+\sqrt{\mathcal{T}}}{\sqrt{1-\mathcal{T}}},$
(3.11)

where in the t-channel limit ($\mathcal{T}\to 1$), the above equation can be
chaned as follows

$\Delta\eta^{extr}=2\log 2-\log(1-\mathcal{T}).$ (3.12)

Substituting (3.12) and the $c_{L}$ central charge (2) into (3.7), we find the
CS-contribution to the minimal EWCS for two disjoint intervals in the vacuum
as follows

$E_{W}^{CS}=\frac{1}{4\mu G}\log\frac{t_{13}t_{24}}{t_{14}t_{23}}.$ (3.13)

Using (3.10) and (3.13), we find the minimal EWCS in the asymptotically flat
space bulk in the GMMG that is dual to the two intervals in the $GCFT_{2}$ as
follows

$\displaystyle E_{W}$ $\displaystyle=$ $\displaystyle E_{W1}+E_{W}^{CS}$
(3.14) $\displaystyle=$ $\displaystyle-\frac{1}{4G}(\sigma+\frac{\alpha
H}{\mu}+\frac{F}{m^{2}})\big{(}\frac{x_{13}}{t_{13}}+\frac{x_{24}}{t_{24}}-\frac{x_{14}}{t_{14}}-\frac{x_{23}}{t_{23}}\big{)}+\frac{1}{4\mu
G}\log\frac{t_{13}t_{24}}{t_{14}t_{23}}.$

Substituting the result (3.14) on the minimal EWCS into the proposal (3.2),
the holographic entanglment negativity $\mathcal{E}$ of two intervals at zero
temperature $GCFT_{2}$ that is dual to the flat GMMG, can be found as follows

$\mathcal{E}=-\frac{3}{8G}(\sigma+\frac{\alpha
H}{\mu}+\frac{F}{m^{2}})\big{(}\frac{x_{13}}{t_{13}}+\frac{x_{24}}{t_{24}}-\frac{x_{14}}{t_{14}}-\frac{x_{23}}{t_{23}}\big{)}+\frac{3}{8\mu
G}\log\frac{t_{13}t_{24}}{t_{14}t_{23}}.$ (3.15)

This result is the same as the holographic entanglement negativity for two
disjoint intervals configurations at zero temperature holographic $GCFT_{2}$
in [52].

###  3.2 EWCS for two disjoint intervals at a finite temperature

In this section, the two intervals $A$ and $B$ are considered in a thermal
$GCFT_{2}$ on a cylinder with the temperature that is equal to its inverse
circumference. The bulk metric can be given by three dimensional flat space
cosmology as follows [40, 41, 66]

$ds^{2}=Mdu^{2}-2dudr+r^{2}d\phi^{2},$ (3.16)

where $M$ has the following relation with the temperature of the $GCFT_{2}$

$M=\big{(}\frac{2\pi}{\beta}\big{)}^{2}.$ (3.17)

If the bulk geometry is considered by the pure Einstein gravity, the minimal
curve length between $p_{b}$ and $p^{\prime}_{b}$ in the t-channel limit
($T\to 1$) can be found as follows [55]

$\displaystyle L^{extr}(p_{b},p^{\prime}_{b})$ $\displaystyle=$
$\displaystyle\frac{\pi
u_{13}}{\beta}\coth\big{(}\frac{\pi\phi_{13}}{\beta}\big{)}+\frac{\pi
u_{24}}{\beta}\coth\big{(}\frac{\pi\phi_{24}}{\beta}\big{)}$ (3.18)
$\displaystyle-$ $\displaystyle\frac{\pi
u_{14}}{\beta}\coth\big{(}\frac{\pi\phi_{14}}{\beta}\big{)}-\frac{\pi
u_{23}}{\beta}\coth\big{(}\frac{\pi\phi_{23}}{\beta}\big{)},$

where it is needed to be proved in the GMMG scenario with the higher
derivative deformation in related to the pure Einstein gravity. Using the
corresponding central charge (2) in the $GCFT_{2}$ which is dual to the GMMG
in 3d asymptotically flat spacetime and the relation (3.5), the extremal EWCS
in the GMMG case, regardless of the CS-contribution, can be found as follows

$\displaystyle E_{W1}$ $\displaystyle=$
$\displaystyle-\frac{1}{4G}(\sigma+\frac{\alpha
H}{\mu}+\frac{F}{m^{2}})\bigg{[}\frac{\pi
u_{13}}{\beta}\coth\big{(}\frac{\pi\phi_{13}}{\beta}\big{)}$ (3.19)
$\displaystyle+$ $\displaystyle\frac{\pi
u_{24}}{\beta}\coth\big{(}\frac{\pi\phi_{24}}{\beta}\big{)}-\frac{\pi
u_{14}}{\beta}\coth\big{(}\frac{\pi\phi_{14}}{\beta}\big{)}-\frac{\pi
u_{23}}{\beta}\coth\big{(}\frac{\pi\phi_{23}}{\beta}\big{)}\bigg{]}.$

To consider the CS-contribution in the minimal EWCS, the timelike vectors as
new ingredients can be changed in the FSC geometry [53, 55]. We make a
conjecture that the minimal boost between $v_{b}$ and $v^{\prime}_{b}$ can
have a similar relation to the TMG case as follows [55]

$\Delta\eta^{extr}=\log\bigg{[}\frac{\sinh\frac{\pi\phi_{13}}{\beta}\sinh\frac{\pi\phi_{24}}{\beta}}{\sinh\frac{\pi\phi_{14}}{\beta}\sinh\frac{\pi\phi_{23}}{\beta}}\bigg{]}$
(3.20)

Using (3.20) and (2) into (3.7), the CS-contribution in the minimal EWCS for
the holographic dual of the $GCFT_{2}$ at finite temperature can be find as
follows

$E_{W}^{CS}=\frac{1}{4\mu
G}\log\bigg{[}\frac{\sinh\frac{\pi\phi_{13}}{\beta}\sinh\frac{\pi\phi_{24}}{\beta}}{\sinh\frac{\pi\phi_{14}}{\beta}\sinh\frac{\pi\phi_{23}}{\beta}}\bigg{]}.$
(3.21)

Using the minimal EWCS (3.19) and its CS-modification (3.21) that are in the
GMMG scenario, the final form of the extremized EWCS at the finite temperature
case can be find as follows

$\displaystyle E_{W}$ $\displaystyle=$ $\displaystyle E_{W1}+E_{W}^{CS}$
(3.22) $\displaystyle=$ $\displaystyle-\frac{1}{4G}(\sigma+\frac{\alpha
H}{\mu}+\frac{F}{m^{2}})\bigg{[}\frac{\pi
u_{13}}{\beta}\coth\big{(}\frac{\pi\phi_{13}}{\beta}\big{)}+\frac{\pi
u_{24}}{\beta}\coth\big{(}\frac{\pi\phi_{24}}{\beta}\big{)}$ $\displaystyle-$
$\displaystyle\frac{\pi
u_{14}}{\beta}\coth\big{(}\frac{\pi\phi_{14}}{\beta}\big{)}-\frac{\pi
u_{23}}{\beta}\coth\big{(}\frac{\pi\phi_{23}}{\beta}\big{)}\bigg{]}+\frac{1}{4\mu
G}\log\bigg{[}\frac{\sinh\frac{\pi\phi_{13}}{\beta}\sinh\frac{\pi\phi_{24}}{\beta}}{\sinh\frac{\pi\phi_{14}}{\beta}\sinh\frac{\pi\phi_{23}}{\beta}}\bigg{]}$

Substituting the result (3.22) on the minimal EWCS into the proposal (3.2),
the holographic entanglment negativity $\mathcal{E}$ of two intervals at
finite temperature $GCFT_{2}$ that is dual to the flat GMMG, can be found as
follows

$\displaystyle\mathcal{E}$ $\displaystyle=$
$\displaystyle-\frac{3}{8G}(\sigma+\frac{\alpha
H}{\mu}+\frac{F}{m^{2}})\bigg{[}\frac{\pi
u_{13}}{\beta}\coth\big{(}\frac{\pi\phi_{13}}{\beta}\big{)}+\frac{\pi
u_{24}}{\beta}\coth\big{(}\frac{\pi\phi_{24}}{\beta}\big{)}$ (3.23)
$\displaystyle-$ $\displaystyle\frac{\pi
u_{14}}{\beta}\coth\big{(}\frac{\pi\phi_{14}}{\beta}\big{)}-\frac{\pi
u_{23}}{\beta}\coth\big{(}\frac{\pi\phi_{23}}{\beta}\big{)}\bigg{]}+\frac{3}{8\mu
G}\log\bigg{[}\frac{\sinh\frac{\pi\phi_{13}}{\beta}\sinh\frac{\pi\phi_{24}}{\beta}}{\sinh\frac{\pi\phi_{14}}{\beta}\sinh\frac{\pi\phi_{23}}{\beta}}\bigg{]}$

This result is the same as the holographic entanglement negativity for two
disjoint intervals configurations at finite temperature holographic $GCFT_{2}$
in [52].

##  4 Conclusion

In this paper, we considered the minimal entanglement wedge cross section
(EWCS) on a flat holography scenario. We have made a conjecture on the minimal
EWCS for the bipartite mixed states in $GCFT_{2}$ in the vacuum and at a
finite temperature where it is at the boundary of the three dimensional
asymptotically flat GMMG model in (2.6). Based on the argument presented in
the appendix A, we make a conjecture that the minimal EWCS of the GMMG in the
context of the asymptotically flat geometry can have a similar structure with
the minimal EWCS in the flat TMG. This argument is based on a comparison of
the holographic entanglement entropy in the two models and can be used to
support the conjecture we have presented. The GMMG model and the TMG model are
similar in some respects in that they include the Chern-Simons and the
Einstein gravity parts, but the former includes the higher derivative gravity
terms. Despite the differences, we guess that in the GMMG case, the extremal
EWCS must consist of two parts, one similar to (3.5) and the other similar to
(3.7). The first part is related to the minimal EWCS in a pure Einstein
gravity case and the second part is related to the minimal EWCS contribution
from the Chern-Simons (CS-) term of the model. We conclude that if our
conjecture is correct, then the minimal EWCS approach could be another way to
calculate the holographic entanglement negativity in the flat space
holography. To support this claim, we argue that the minimal EWCS and the
holographic entanglement negativity in flat holography such as the
$AdS_{3}/CFT_{2}$ could satisfy the relation (3.2). The results obtained for
the holographic entanglement negativity in the two cases we examined, in
relations (3.15) and (3.23), are the same as our previous results, which we
obtained with another approach in [52]. This correspondence between the
results for the holographic entanglement negativity by two different
approaches can be considered as an evidence for the given conjecture.

##  Appendix A  Comparison of the entanglement entropies of GMMG and TMG

Both the holographic entanglement entropy and the minimal EWCS are related to
the geometric calculations inside the bulk. From this point of view, if we can
provide an argument based on the holographic entanglement entropy, we guess
that the results can be used for the minimal EWCS. As mentioned in the section
3, the GMMG model and the TMG model are similar in some respects in that they
include the Chern-Simons and the Einstein gravity parts, but the former
includes the higher derivative gravity terms. Therefore, comparing the
entanglement entropy of these models in the context of the flat holography can
have good results. We perform the computation in this Appendix by considering
the pure state of a single interval in a $GCFT_{2}$ at the boundary is dual to
a gravity on the asymptotically flat background. We examine the bulk geometry
once by the TMG and again by the GMMG.

The holographic entanglement entropy for the TMG in an asymptotically flat
background which is dual to the $GCFT_{2}$ at the boundary with a single
interval can be found in the literature as follows [41]

$S_{EE}=\frac{c^{\prime}_{M}}{12}\bigg{(}u_{12}\cot\big{(}\frac{\phi_{12}}{2}\big{)}\bigg{)}+\frac{c^{\prime}_{L}}{12}\bigg{(}2\log\big{(}\frac{2}{\epsilon}\sin\frac{\phi_{12}}{2}\big{)}\bigg{)},$
(A.1)

where $c^{\prime}_{L}$ and $c^{\prime}_{M}$ are the central charges of the
$GCFT_{2}$ which is dual to TMG in an asymptotically flat background. By
comparing the calculations performed in [55], it can be seen that

$\displaystyle\Delta\eta^{extr}$ $\displaystyle=$ $\displaystyle
2\log\big{(}\frac{2}{\epsilon}\sin\frac{\phi_{12}}{2}\big{)},$ (A.2)
$\displaystyle L^{extr}$ $\displaystyle=$ $\displaystyle
u_{12}\cot\big{(}\frac{\phi_{12}}{2}\big{)},$ (A.3)

that $L^{extr}$ and $\Delta\eta^{extr}$ are the minimal length of the curve
homologous to the entangling interval and the extremized boost respectively
computed for a single interval configuration at the boundary with
$(u_{1},\phi_{1})$ and $(u_{2},\phi_{2})$ endpoints. By substituting the
$L^{extr}$ and $\Delta\eta^{extr}$ into (A.1), we can change the form of the
TMG holographic entanglement entropy as follows

$S_{EE}=\frac{c^{\prime}_{M}}{12}L^{extr}+\frac{c^{\prime}_{L}}{12}\Delta\eta^{extr},$
(A.4)

where this result is obtained if we have a GCFT with zero temperature at the
boundary and the global Minkowski spacetime in the bulk. We have considered
such a configuration for the GMMG model and the results are presented in [67].
In this paper, the holographic entanglement entropy is computed for the
asymptotically flat GMMG which is dual to a field theory whose the symmetry
generators in the vacuum satisfy the $GCFT_{2}$ generators algebra. The GCFT
is considered at a zero temperature configuration with a single interval as an
entangling surface. The entanglement entropy for the mentioned configuration
of the $GMMG/GCFT$ duality is computed as follows [67]

$S_{EE}=-\frac{1}{4G}\big{(}\sigma+\frac{\alpha
H}{\mu}+\frac{F}{m^{2}}\big{)}\bigg{(}u_{12}\cot\big{(}\frac{\phi_{12}}{2}\big{)}\bigg{)}+\frac{1}{4\mu
G}\bigg{(}2\log\big{(}\frac{2}{\epsilon}\sin\frac{\phi_{12}}{2}\big{)}\bigg{)}.$
(A.5)

By substituting the relation (A.2) and (A.3) into the holographic entanglement
entropy (A.5), we have the following relation

$S_{EE}=\frac{c_{M}}{12}L^{extr}+\frac{c_{L}}{12}\Delta\eta^{extr},$ (A.6)

where we have used the central charge relations (2) into (A.5). Comparing the
relations (A.4) and (A.6), we find that the holographic entanglement entropy
for the flat-TMG and the flat-GMMG for a single interval configuration has a
similar structure. If we consider $L^{extr}$ and $\Delta\eta^{extr}$ as the
result of geometric calculations, we assume that these calculations reach the
same result in both dualities. The effect of the differences between GMMG and
TMG is reflected only in the central charges in the result of the holographic
entanglement entropy.

The computation of $L^{extr}$ and $\Delta\eta^{extr}$ in the cases where the
gravity is dual with the GCFT with a single interval configuration at the
boundary is no different in holographic entanglement entropy case and the
minimal EWCS case. Our conjecture is that in the mixed bipartite
configurations the story is the same and we can use the result of calculations
in TMG to achieve the reliable results for the GMMG.

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