#  $\mu$-cf2vec: Representation Learning for Personalized Algorithm Selection
in Recommender Systems

Tomas Sousa-Pereira  Tiago Cunha  Carlos Soares

###### Abstract

Collaborative Filtering (CF) has become the standard approach to solve
recommendation systems (RS) problems. Collaborative Filtering algorithms try
to make predictions about interests of a user by collecting the personal
interests from multiple users. There are multiple CF algorithms, each one of
them with its own biases. It is the Machine Learning practitioner that has to
choose the best algorithm for each task beforehand. In Recommender Systems,
different algorithms have different performance for different users within the
same dataset. Meta Learning (MtL) has been used to choose the best algorithm
for a given problem. Meta Learning is usually applied to select algorithms for
a whole dataset. Adapting it to select the to the algorithm for a single user
in a RS involves several challenges. The most important is the design of the
metafeatures which, in typical meta learning, characterize datasets while
here, they must characterize a single user. This work presents a new meta-
learning based framework named $\mu$-cf2vec to select the best algorithm for
each user. We propose using Representation Learning techniques to extract the
metafeatures. Representation Learning tries to extract representations that
can be reused in other learning tasks. In this work we also implement the
framework using different RL techniques to evaluate which one can be more
useful to solve this task. In the meta level, the meta learning model will use
the metafeatures to extract knowledge that will be used to predict the best
algorithm for each user. We evaluated an implementation of this framework
using MovieLens 20M dataset. Our implementation achieved consistent gains in
the meta level, however, in the base level we only achieved marginal gains.

Machine Learning, Recommender Systems, Meta Learning, Collaborative Filtering,
Representation Learning, Information Retrieval

##  1 Introduction

Collaborative Filtering (CF) has become the standard approach to solve
recommendations problems [1]. CF algorithms are relatively simple to implement
while having a satisfactory performance in many applications, when compared to
other algorithms. Different algorithms have different characteristics, and
biases. Their performance varies so choosing the right algorithm is a step
towards improving the performance.
We have many algorithms that not only perform differently across different
datasets, but also even for different users of the same dataset.
Discrimination in recommendations is originated from different sources:
underlying biases in the input data, or the result of recommendation
algorithms [2]. Different algorithms have different impacts on the
recommendations for different groups of users. The algorithm selection problem
has been addressed using Meta Learning (MtL)techniques. The problem is modeled
using a set of metafeatures to describe the task and the performance of
algorithms using an evaluation measure to describe its behavior. The meta
learning model will extract knowledge about the task, and will be used to
select the best algorithm.
However, the definition of suitable metafeatures is a hard problem. The
metafeatures need to be as meaningful as possible to the task. The handcrafted
metafeatures efficiency continues to be questioned, since it is difficult to
evaluate if they really represent the data. Representation Learning shares the
same goal of the metafeatures, i.e, learn good representation about the data.
In Representation Learning we try to extract compact representations from
heterogeneous types of data in a numerical representation that can be reused
in other learning tasks.
We aim to solve the problem of different algorithms that have a different bias
in users. To do so, we propose to use Meta Learning. When using Meta Learning
we have to extract features about the task we want to solve. We suggest using
Representation Learning techniques to retrieve these features. To the best of
our knowledge, this solution is the first of its kind: using Representation
Learning for Personalized Algorithm Selection in Recommender Systems.
This work has two main contributions - $\mu$-cf2vec: Representation Learning
for Personalized Algorithm Selection in Recommender Systems - which is a
framework that uses Representation Learning techniques to retrieve user
embeddings, and will be used by a Meta Learning model to solve the algorithm
selection problem on the user level. The second contribution is an empirical
study of the proposed framework. We implemented a version of the $\mu$-cf2vec
framework.

##  2 Related Work

This work uses metalearning techniques to solve the algorithm selection
problem on the user level. The metalearning techniques use metafeatures that
were retrieved using Representation Learning techniques. To the best of our
knowledge, there is no existent research on the problem we aim to solve.

This work proposes to use Representation Learning to extract metafeatures.
This idea has been used in different implementations [3] [4]. However, both of
these implementations use representation learning on the dataset level.

Our work also proposes to solve the metalearning task of algorithm selection
problem on the user level.

There is research in algorithm selection on the user level [5]. This approach
uses hand crafted metafeatures to characterize each user. It also uses a set
of base learners, each one of them with their own bias. The meta-model will
then use the performance label and the handcrafted metafeatures to predict the
performance of each algorithm for a new user.

Research has also been done on meta learning at user level using CTR model
[6]. Given a collection of models, the goal is to select the best model for
each user. This work uses: model and task agnostic techniques to extract meta
features. The meta model will output the best base model for each user by
learning the relationships from the metafeatures.

##  3 Approach

###  3.1 Problem definition

Collaborative Filtering is the standard approach to tackle recommendation
systems problems. We expect the performance of CF algorithms to vary not only
for datasets, but even within the same dataset, for instance, it is different
for different users. For each user there is an algorithm that obtains the best
performance. The goal is to predict the best algorithm for each user; by doing
this we could reduce the biases of the algorithms against groups of users.

We can use Meta Learning (MtL) to address the algorithm selection problem.
From the performances of each algorithm we can use MtL to address different
issues: label ranking, selecting algorithm, predicting algorithms
performances, selecting hyperparameters and others. For MtL be able to solve
the algorithm selection problem on the user-level, we need to characterize the
user. This characterization intends to create representation about the users.
The representations (metafeatures) will allow the meta models to extract
knowledge and recommend the best algorithm for each user.

###  3.2 Framework Overview

To try to tackle the problems previously addressed we propose a framework
named $\mu$-cf2vec: Representation Learning for Personalized Algorithm
Selection in Recommender Systems. This framework proposes the usage of
Representation Learning Techniques to automatically retrieve expressive meta
features that the meta learning models will use to solve the algorithm
selection problem on the user level. Figure 1 presents the procedure this
framework follows.

Figure 1: Overview of the $\mu$-cf2vec procedure

The procedure presented in figure 1 have some resembles to the traditional
Meta Learning procedure. However, if we analyse carefully we can observe major
diferences. While the traditional MtL tasks try to solve the algorithm
selection problem on the dataset level, here we propose to solve on the user
level. In traditional MtL procedure we would characterize the dataset, and we
would recommend the best algorithm for each dataset. In this framework we aim
to select the best algorithm for each user. To do so, we have to evaluate the
algorithms for each user and characterize the user. The meta learner will then
use these information to recommend the best algorithm for each user.

###  3.3 Metadata Collection

Given a CF dataset $d_{i}$ – that has a set of users $U$, that have
interactions with a set of items $I$ – we can create a metadataset. Each row
of the metadataset relates to an user and it is composed by the metafeatures,
and the label. The metafeatures are a set of $k$ values that characterize the
user. The metatarget is a label $l_{t}$ that can be chosen according to the
task.

Since this framework proposes to solve a meta learning problem, we need to
train and evaluate our models on all users. The splitting strategy is
different when comparing with the CF tasks. In this MtL we need to
characterize each user, therefore we need interactions from each user. Instead
of dividing all the users in three groups: train, validation and test; we
divide each user’s ratings in train, validation, and test. This splitting is
done by the framework.

####  3.3.1 Base Level Evaluation

From a set of base learners $B$, different users have different performances
for different base learners. The base learners are trained using the training
interactions from all users. The validation interactions are used to tune the
parameters and the test interactions are used to make an unbiased evaluation
of the model. There is an evaluation of every base learner for each user.

The evaluation is linked to the metatarget of each user. This evaluation is
chosen accordingly to the task we want to solve. The metatarget represents the
meta-variable that the meta learner wishes to understand or predict i.e. the
algorithm with the best performance for a given task. We can select the
metatarget to rank the base learners, select the best base learner, predict
the performance of each base learner and others. This makes the framework
flexible. We can select the meta target to optimize the task we want to solve.

####  3.3.2 Retrieving metafeatures

One of the contributions of this framework is using Representation Learning to
retrieve metafeatures at the user level. The RL techniques allow us to
automatically generate metafeatures for each user. Using this approach we
remove the need of manually creating metafeatures. Therefore, $\mu$-cf2vec
proposes to use Representation Learning techniques to retrieve the
metafeatures for each user.

Using these techniques to automatically retrieve metafeatures for each user,
is firstly introduced in this framework. $\mu$-cf2vec creates representations
for all users using RL techniques. These representations are created using the
train interactions from all users. The metafeatures created by applying these
RL techniques are going to be used by the meta model to select the best
algorithm for each user.

####  3.3.3 Inference

In real world systems, this framework has to be trained before its deployment.
The training will be used to select the best meta learners for the task.
During the training process the representation learning technique is chosen.
The evaluations of the multiple base learners are also obtained in this stage.
The training stage will create the metadataset and select the best meta model.

After training the framework, it can be deployed and will automatically update
itself. The framework will retrieve the user embeddings periodically, using
the newly collected user-item interactions. These embeddings will be retrieved
using the best RL technique found on the training step. The meta learner will
then extract knowledge which will provide the ability to make personalized
algorithm recommendations. For the new users or existing users, for which the
system is not able to compute embeddings, the framework will use a naive
strategy by recommending the Most Popular algorithm.

##  4 Experimental setup

This sections presents the implementation of the previously proposed
framework.

###  4.1 Base Level

####  4.1.1 Base learners

We used the Fast Python Collaborative Filtering Framework [7] from which we
are going to use the following algorithms:

  * •

Alternating Least Squares (ALS)

  * •

Bayesian Personalized Ranking (BPR)

  * •

Logistic Matrix Factorization (LMF)

  * •

Most popular (to be used as baseline)

Using these algorithms we can capture different biases. We have model-based
and memory-based Collaborative Filtering techniques. The first three base
learners are matrix factorization techniques. They have a parameter that is
the number of latent factors to be computed by each model. Once we decided on
the splitting strategy and the base learners we would use, we then needed to
understand the performance of the base learners to decide upon the split data
percentages.

####  4.1.2 Dataset

In this implementation we decided to use the MovieLens 20m to tackle this
problem. We transformed it into an implicit dataset by setting a threshold for
ratings greater than 3.5. For this implementation we are not going to consider
the Cold Start Problem. For this reason we will remove all users and items
with less than 10 interactions. To evaluate the base learners we decided to
user NDCG@K.

There are multiple different strategies to split the data. This is a
metalearning task, hence we want the meta model to be trained on all users.
The meta learning tasksrequire all the users to train the model. Each users’
ratings are divided in the three groups: train, validation and test. In meta
learning tasks it is relevant to define the different percentages of each
group. These are selected according to the problem we want to solve. In the
meta level we only use the training ratings. The other two groups are only
used in the base level.

We have 4 different algorithms and we still may face the problem that in some
users none of the algorithms will have a good performance. There may exist a
group of users, for which all of the base learners have a NDCG of 0. We are
going to call this class of users _zeroes_. To select the best data split we
retrieved the NDCG of all models for each user. We selected $K$ = 30, 70%
training, 10% validation and 20% for the testing. These values were selected
by finding a balance between the number of users belonging to class _zeroes_
and the value of $K$. Since choosing a high value of $K$ reduces the number of
_zeroes_ users, and the _zeroes_ users can introduce errors in our models.

####  4.1.3 Obtaining User-level metafeatures based on embeddings

In this work we are going to use state of the art techniques in Representation
Learning: Denoising Auto-Encoders for Top-N Recommender Systems [8] and
Variational Autoencoders for Collaborative Filtering [9].

For the Variational Auto-encoders (VAE) we are going to use the code from
[10]. Since this model is designed for Collaborative Filtering it splits all
the users in Train (80%), Validation (10%) and Test (10%). In our approach, as
it is a Meta Learning task, we use all the users in every step and in each
user we split the user ratings in Train (70%), Validation (10%) and Test
(20%). The splitting strategy of the [10] is integrated with the model,
however, we need to obtain the metafeatures for all users. To retrieve the
embeddings we fed the model [10] with all of our training data. We let the
model train and split the data in its own partitions. After the model was
trained we obtained the embeddings for all the users. The model creates a
representation of size 200 for each user. All the users training interaction
enter the VAE. These users are automatically splitted in 80% for training, 10%
for validation and 10% for test. The training users are used to train the
models and the parameters are tuned with the validation users. After the model
is trained we retrieve the embeddings for all the users.

For the Collaborative Denoising Auto-Encoders (CDAE) we are going to use the
DRECPY [11] library. In the original paper of this model the author
facilitated the retrieval of the factors for each user. The attribute $V$ has
the user embeddings. This attribute has a dimension of $UxK$, U being the
number of users and K the hidden factors. We are going to use two different
hidden factors: 50 because it is the number of hidden factors used by the
Author, and 200 hidden factors. We retrieve the embeddings of size 200 to make
a fair comparison with the Variational Auto-encoders which have size 200.

We are going to have three metadatasets: VAE, CDAE50, and CDAE200. Each one of
them is built using the metafeatures obtained by RL techniques with the same
name, e.g., the VAE dataset is built using metafeatures retrieved with the
Variational Auto-Encoder and with the metatarget that is the name of the base
learner with the best performance.

###  4.2 Meta-level Experimental Setup

These metadatasets are going to be used to solve the meta-level problem. We
are going to build meta models which will leverage these metadatasets to
select the best algorithm for each user. To achieve this, we are going to
implement and evaluate a wide variety of models. We will then select the most
suitable for our problem. The meta learners we are going to use are the
following:

  * •

Logistic Regression [12]

  * •

Multi Layer Perceptron[13]

  * •

Support Vector Machine [14]

  * •

Random Forest [15]

  * •

Light GBM [16]

These algorithms have different characteristics and different biases. We also
tried to use Knn however due to the characteristics of the data we couldn’t
achieve good results. We performed a grid search to tune the hyper parameters
of the meta models. We also retrieved the results for normalized data using
z-score normalization. We used Cross Validation with 5 Folds, since this
technique allows a deeper look into the model performance. Comparing with the
traditional train-test split, this approach reduces the variance. Together
with the cross validation we used a different variety of metrics: accuracy,
precision, recall and all their averages. These metrics are acquired in the
meta level and provide analysis for the performance of the meta models in each
class. To evaluate the performance of the meta models in the base level, we
are going to understand the impact of the meta models’ predictions in the base
level performance. The impact in base level performance will be in terms of
NDCG and the intuition behind it is the following: assume for user $u_{i}$ ALS
has a NDCG of 0.4, BPR a NDCG of 0.35, and Most popular a NDCG 0.2. The best
algorithm might be the ALS but if our meta model selects BPR, the overall
performance for that user would be similar to the correct label ALS.
Nevertheless, if the meta model selects Most Popular, the performance on that
user will be affected.

##  5 Research Questions

###  5.1 RQ1: Can we use Representation Learning to obtain useful metafeatures
on the User Level?

This project addresses several new challenges. It is the first, to the best of
our knowledge, that uses meta learning together with Representation Learning
in order to solve the algorithm selection problem on user level. Using
Representation Learning techniques to automatically retrieve metafeatures for
each user, was firstly introduced in this work. This RQ will answer the
question whether it is possible to retrieve expressive metafeatures about the
users.

We know that we have five classes, so we can assume that a completely random
model would have an accuracy of 20%. Table 1 presents the distribution of the
classes, where ALS is the biggest with a value of 26,37%. For our model to
have an impact on solving this problem, we have to be better than a random
model (that has an accuracy of 20%) and have to improve the result of only
recommending one class (better than 26,37%). $\mu$-cf2vec proposes to remove
the biases presented in CF algorithms, hence the meta models have to predict
more than one class. Figure 2, which presents the accuracy for the models with
normalization, can help us answer this question.

Figure 2: Meta Level Accuracy score on normalized datasets Algorithms | # users | Percentage
---|---|---
ALS | 34215 | 26,37
Most Popular | 29088 | 22,42
BPR | 25925 | 19,98
Zeroes | 21277 | 16,40
LMF | 19252 | 14,84
Total | 129757 | 100
Table 1: Proportions of each class

In the CDAE datasets the models achieved an accuracy lower or equal to 26,37%.
Table 2 presents the confusion matrix for the Logistic Regression model for
the normalized CDAE50 dataset. The confusion matrix indicates that the model
only recommends one class, _ALS_. The recall of this class is 1, while in the
other classes it is around 0. The accuracy of the models using both CDAE
metadatasets is the same as the percentage that the ALS class presents in
table 1. The results of the other meta models are similar.

| ALS | BPR | LMF | MostPopular | Zeroes
---|---|---|---|---|---
ALS | 6837 | 0 | 0 | 4 | 2
BPR | 5182 | 0 | 0 | 1 | 1
LMF | 3847 | 0 | 0 | 2 | 1
MostPopular | 5812 | 0 | 0 | 3 | 3
Zeroes | 4249 | 0 | 0 | 3 | 4
Table 2: Logistic Regression meta model confusion matrix for the normalized
CDAE50 dataset.

When using the models on the VAE metadataset, every meta model improves the
accuracy score of 26,37%. The best model is the Light GBM with normalized
data, achieving an accuracy of 32,48%. When analyzing the results in table 3
we can identify some aspects. The LMF class is the one with the worst overall
score, which can be due to the low proportion of the class as stated in the
table 1. This model has a best overall score in the class _zeroes_ , even
though it is the fourth in amount of appearances. The precision score and the
F1 score for this class are the best in comparison with the other classes. In
the metric recall it is in second place, just behind the most common class
_ALS_. These results indicate that the _zeroes_ class affect the performance
of the model. Since ideally we would want to recommend different algorithms,
i.e. it would be better if the the meta models have better performance in
other classes instead of the _zeroes_ class. There is another model in this
same dataset that has almost the same accuracy but its recommendations
spectrum is different.

| Precision | Recall | F1
---|---|---|---
ALS | 0.31 | 0.55 | 0.40
BPR | 0.31 | 0.21 | 0.25
LMF | 0.23 | 0.02 | 0.04
MostPopular | 0.32 | 0.31 | 0.31
Zeroes | 0.40 | 0.40 | 0.40
– | – | – | –
Macro avg | 0.31 | 0.30 | 0.28
Weighted avg | 0.32 | 0.32 | 0.30

Table 3: LightGBM classification report on normalized VAE

The Random Forest meta model has an accuracy of 31,85%. While analyzing the
results in the table 4 we have a new scenario in comparison with the previous
meta model. Random Forest model has a better score when predicting the class
_ALS_. The metrics F1 and Recall on the class _ALS_ are the overall best. The
overall performance of this model has improved in comparison with Light GBM.

| Precision | Recall | F1
---|---|---|---
ALS | 0.30 | 0.75 | 0.42
BPR | 0.36 | 0.13 | 0.19
LMF | 0.40 | 0.0014 | 0.0028
MostPopular | 0.34 | 0.21 | 0.26
Zeroes | 0.40 | 0.30 | 0.34
– | – | – | –
Macro avg | 0.36 | 0.28 | 0.24
Weighted avg | 0.35 | 0.32 | 0.26

Table 4: Random Forest classification report on normalized VAE

In table 1 we can observe that the classes are not evenly distributed. To
overcome this problem, we used the SMOTE technique to create synthetic data.
Although we expected to achieve better results with the use of this technique,
the results presented in figure 3 do not indicate that. In figure figure 3 the
performance is consistently worst when compared to the figure 2.

Figure 3: Meta Level Accuracy score on normalized datasets

Based on these results we can deduce that on the meta level we can achieve an
improvement in the performance. It is possible to use Representation Learning
techniques to make personalized algorithm selection in recommender systems.
From these results we can infer some points: the best overall meta model is
the Light GBM (it has the best accuracy and the best performance for the
_zeroes_ class), although the Random Forest has its own advantages (it has a
good performance for the _ALS_ class). Through Light GBM, we can infer that,
although class _zeroes_ is the fourth most common class, it has the best
performance for this model. That shows how this class can mislead the results.

###  5.2 RQ2: Which Representation Learning Technique is the best?

As stated previously this project introduces a completely new approach to the
algorithm selection problem on the user level. We proposed to use meta
learning combined with Representation Learning, in order to retrieve meta
features. Later, these meta features will be used by the meta model to try to
predict the best algorithm for each user. For this project we have chosen two
techniques: Variational Auto-encoders and Denoising Auto-encoders.

From these two techniques we have three experimental setups as stated
previously. The VAE metadaset used the user embeddings retrieved with the
Variational Auto-encoder and has a size of 200. The CDAE dataset is created
with users representations acquired by the Denoising Auto-encoders. We have
two different datasets from these techniques: CDAE50 and CDAE200. The original
paper of this implementation found out that the optimal number of factors is
50, therefore we started by obtaining the results with 50 factors. However,
the VAE dataset is composed by representations with size 200. To have an head-
to-head comparison between the two techniques we retrieved the CDAE
representations with size 200.

To identify which of the techniques is more suitable to solve this problem, we
have to understand what their performance is. We have different techniques as
stated previously: VAE and CDAE. From the figures presented before (2 and 4)
the VAE metadataset achieves better results in every single meta model. These
results are consistent in all metadatasets: normalized or non normalized, with
or without zeroes and with or without the SMOTE technique. The results
presented indicate that the Variational Auto-encoder is the better
implementation in this situation.

Figure 4: Base Level Performance on normalized datasets

###  5.3 RQ3: What is the impact on the base level performance achieved by
$\mu$-cf2vec?

$\mu$-cf2vec is a framework which aims to improve the recommendations’
performance of every user. As stated previously, the meta is able to predict
the best algorithm for each user with an accuracy higher than the baselines.
However, we need to look deeper into the performance affected by the users, in
order to understand whether the model can improve the performance for the
users or not.

In table 5 we have the upper and lower bounds for our model. The _Perfect
NDCG_ is the upper bound of our model. This value is the average of NDCG of
the best algorithm (with higher NDCG) for each user. The other rows represent
the value of the NDCG of a meta model that has only selected one single
algorithm (ALS, BPR, etc). One thing we have to revisit is that the class
_zeroes_ has zero NDCG in every algorithm. As stated previously $\mu$-cf2vec
will recommend the Most Popular algorithm to users identified as _zeroes_.
This is the policy of the framework to address users that we don’t have enough
information to make good recommendations. Thus, the goal of our meta model is
to improve the value of 0,27 presented in table 5. If we improve this score,
we can infer that the performance is better than if we only used the ALS.

Algorithms | NDCG Evaluation
---|---
Perfect NDCG | 0,46
choosing only ALS | 0,27
choosing only BPR | 0,21
choosing only LMF | 0,17
choosing only Most Popular | 0,21

Table 5: Base level performance in terms of NDCG.

From figure 4 we can observe some interesting aspects. At the meta level the
VAE dataset was always the best. However, at the base level, only Light GBM
and Random Forest achieved better results with VAE. For the other algorithms
the CDAE datasets achieved a better perfromance when comparing to the VAE
performance.

When using the VAE dataset, which can be more expressive as stated previously,
the meta models have a wide spectrum of recommendations. While in the meta
level we could have improvements by using the VAE and recommending different
algorithms, in the base level it can be worse. For instance, the MLP meta
model has an accuracy of around 30% using the VAE dataset compared to an
accuracy of around 26% for both CDAE datasets. From this result we could
expect that in the base level the performance would also be better when using
the VAE dataset. However, the figure 4 indicates the opposite: the base level
performance in MLP using the VAE is worse than both CDAE. While using the VAE
dataset the MLP is recommending different algorithms, in the CDAE datasets it
is only recommending _ALS_. In this case it achieves a better performance.

When inspecting figure 4, we can observe two models where the base level
performance is better than if we only used _ALS_. The Random Forest and the
Light GBM models both achieve marginal gains over recommending the best single
class.

From these results we only achieved marginal gains in the base level
performance. Even though the VAE dataset achieved the best meta level
performance, this was not true on the base level. In fact, in the base level
(in some cases) it is better to use the CDAE datasets. Sometimes it is better
to select one single algorithm to achieve the best performance, than selecting
multiple algorithms.

##  6 RQ4: What is the impact of the zeroes class on $\mu$-cf2vec?

As expected, there are some users for which none of the algorithms considered
provide any useful recommendation. These users belong to the class _zeroes_ ,
since the maximum NDCG that all base learners algorithms’ could achieve was
zero.The users with this class are a group of users that we do not have enough
information on to solve their problem. This can be related to the Cold Start
Problem.

In research question 5.1 we presented that even though the _zeroes’_ class is
in fourth place, in terms of occurrences, most of the models could detect this
class more easily than the BPR and Most Popular (although both are more
frequent). These results might indicate that these users are misleading the
models performance. $\mu$-cf2vec aims to recommend the best algorithm for each
user. Since the _zeroes_ users do not have good performance in any algorithm,
we decided to remove them. This will allow us to understand their impact on
the meta models performance. Table 6 presents the proportions of data after we
removed all the users that belong to the class of _zeroes_. This analysis was
made to understand the impact of the class _zeroes_.

Algorithms | # users | Percentage
---|---|---
ALS | 34215 | 31,54
Most Popular | 29088 | 26,81
BPR | 25925 | 23,90
LMF | 19252 | 17,75
Total | 108480 | 100
Table 6: Proportions of each class

Figures 5 and 6 present the accuracy and base level performance for each
algorithm in all metadatasets without users belonging to class _zeroes_. Since
we removed the users labeled as _zeroes_ , the distribution of the classes has
changed, as we can see in table 6. When comparing to the distribution with all
users (table 1) we can highlight some points. The perfect meta model would
increase the NDCG evaluation by 20%. The base level lower bound is the most
common class and it still is _ALS_ with a percentage of 31,54%. The overall
improvement in the base level performance was around 20%. These values by
themselves show the impact of the users in the overall metrics. These are the
reference results we have to use when analyzing the ones without the users
labeled as _zeroes_.

Figure 5: Accuracy score on normalized datasets, without zeroes Figure 6: Base
Level Performance onnormalized datasets, without zeroes

The meta level accuracy scores are similar to the results with all users. The
VAE dataset still has the best overall performance when comparing to the CDAE.
The best models are the Light GBM and the Random Forest achieving an accuracy
of around 35%. This accuracy showed some improvements, however, we have to
recall that the new percentage of the most common algorithm _ALS_ is 31,54%.
From these results the improvement is not significant. For the CDAE datasets
the results are similar, and the meta models only recommend the most common
class, achieving an accuracy of around 31,54%.

The base level performance results in figure 6 are similar to the results with
all users (shown by figure 4). The only two models that improved the base
level performance are the Random Forest and the Light GBM, when comparing to
the ones that only used the most common class. These marginal gains where
achieved by using VAE dataset, which reinforces the hypothesis that the VAE
technique is able to achieve better results. One aspect where the results in
figure 6 differ from the ones in figure 4 is the fact that the Logistic
Regression and the MLP are closer to the lower boundary. This is explained by
the fact that False _zeroes_ users identified before are now being identified
as _ALS_. The CDAE dataset has similar results when using all users. All the
models are recommending the _ALS_ class, hence the base level performance will
be around the same as the _ALS_ class.

We can conclude, from these results and the ones in the research question 5.1,
that this organically created class can bring problems to this framework.
There are different ways to try to minimize the effects of this class. They
will be discussed in the future work.

##  7 Conclusion

$\mu$-cf2vec proposes to solve the algorithm selection on the user level. To
do so, it uses a Meta Learning approach by learning from prior experience. The
meta learners leveraged from a dataset which is built with the users
metafeatures and the user label. The metafeatures are automatically retrieved
by using Representation Learning techniques. Using these techniques we removed
the need of manually selecting the metafeatures, a process that can be slow
and inefficient. The meta target is related to the algorithms we want to
recommend. The meta models use the metafeatures to recommend the best
algorithm for each user.

This implementation of the framework generated metafeatures from two
Representation Learning techniques: Variational Auto-encoders and Denoising
Auto-encoders. In this implementation the metatarget is obtained by selecting
the name of the algorithm that achieved the best performance for each user. We
used a set of 5 meta models to try to solve the algorithm selection problem on
the user level. In every meta model the dataset with metafeatures obtained
using Variational Auto-encoders achieved the best meta level results. From
this study we can conclude that we are able to achieve meta level improvements
by using Representation Learning (using VAE). The meta models Light GBM and
Random Forest are the ones with the best overall performance. In the base
level we only achieved marginal gains. The organically created _zeroes_ class
has a negative impact on the meta models performance. We can relate this class
to the cold start problem. This implementation is the first of its kind when
addressing Personalized Algorithm Selection in Recommender Systems.

###  7.1 Limitations and Future Work

$\mu$-cf2vec proposes an innovative approach to solve Personalized Algorithm
Selection in Recommender Systems. This implementation of the framework has
some limitations that can lead to future work:

  * •

In the implementation of the $\mu$-cf2vec framework we only used a single
dataset. To validate this framework, we have to test it in more datasets.

  * •

We only used two Representation Learning techniques. However, there are many
different techniques. In future work we should use different RL techniques.

  * •

We did a grid search to find the best parameters, however there is a chance
that the hyperparameters we have chosen aren’t the best for the task we aim to
solve. Not having the optimal parameters for a meta model can have a negative
impact on the performance.

  * •

The base learners selected are a limitation of this work. The framework
$\mu$-cf2vec wants to remove the bias of the CF algorithms. We have to choose
a variety of base learners to leverage different biases. To address this
limitation we should have a more diverse group of base learners.

## References

  * Su and Khoshgoftaar [2009] Xiaoyuan Su and Taghi M. Khoshgoftaar.  A survey of collaborative filtering techniques, 2009.
  * Mansoury et al. [2019] Masoud Mansoury, Bamshad Mobasher, Robin Burke, and Mykola Pechenizkiy.  Bias disparity in collaborative recommendation: Algorithmic evaluation and comparison, 2019.
  * Cunha et al. [2018] Tiago Cunha, Carlos Soares, and André C. P. L. F. de Carvalho.  cf2vec: Collaborative filtering algorithm selection using graph distributed representations.  _CoRR_ , abs/1809.06120, 2018.  URL http://arxiv.org/abs/1809.06120.
  * Jomaa et al. [2019] Hadi S. Jomaa, Lars Schmidt-Thieme, and Josif Grabocka.  Dataset2vec: Learning dataset meta-features, 2019.
  * Collins et al. [2018] Andrew Collins, Jöran Beel, and Dominika Tkaczyk.  One-at-a-time: A meta-learning recommender-system for recommendation-algorithm selection on micro level.  _CoRR_ , abs/1805.12118, 2018.  URL http://arxiv.org/abs/1805.12118.
  * Luo et al. [2020] Mi Luo, Fei Chen, Pengxiang Cheng, Zhenhua Dong, Xiuqiang He, Jiashi Feng, and Zhenguo Li.  Metaselector: Meta-learning for recommendation with user-level adaptive model selection, 2020.
  * Frederickson [2017] Ben Frederickson.  Fast python collaborative filtering for implicit datasets, 2017.  URL http://dx.doi.org/10.17632/n6sjkpy87f.5#file-8b2203cf-ad2a-4b48-8de4-909387df278b.
  * Wu et al. [2016] Yao Wu, Christopher DuBois, Alice Zheng, and Martin Ester.  Collaborative denoising auto-encoders for top-n recommender systems, 02 2016.
  * Liang et al. [2018] Dawen Liang, Rahul G. Krishnan, Matthew D. Hoffman, and Tony Jebara.  Variational autoencoders for collaborative filtering, 2018.
  * NVIDIA [2019] NVIDIA, 2019.  URL https://github.com/NVIDIA/DeepLearningExamples/tree/master/TensorFlow/Recommendation/VAE-CF.
  * Colaço et al. [2020] Fábio Colaço, Márcia Barros, and Francisco M. Couto.  Drecpy: A python framework for developing deep learning-based recommenders.  In _Fourteenth ACM Conference on Recommender Systems_ , RecSys ’20, page 675–680, New York, NY, USA, 2020. Association for Computing Machinery.  ISBN 9781450375832.  doi: 10.1145/3383313.3418483.  URL https://doi.org/10.1145/3383313.3418483.
  * Defazio et al. [2014] Aaron Defazio, Francis Bach, and Simon Lacoste-Julien.  Saga: A fast incremental gradient method with support for non-strongly convex composite objectives, 2014.
  * Hinton [1989] Geoffrey E. Hinton.  Connectionist learning procedures.  _Artificial Intelligence_ , 40(1):185 – 234, 1989\.  ISSN 0004-3702.  doi: https://doi.org/10.1016/0004-3702(89)90049-0.  URL http://www.sciencedirect.com/science/article/pii/0004370289900490.
  * Platt [1999] John C. Platt.  Probabilistic outputs for support vector machines and comparisons to regularized likelihood methods.  In _ADVANCES IN LARGE MARGIN CLASSIFIERS_ , pages 61–74. MIT Press, 1999.
  * Breiman [2001] Leo Breiman.  Random forests.  _Mach. Learn._ , 45(1):5–32, October 2001.  ISSN 0885-6125.  doi: 10.1023/A:1010933404324.  URL https://doi.org/10.1023/A:1010933404324.
  * Ke et al. [2017] Guolin Ke, Qi Meng, Thomas Finley, Taifeng Wang, Wei Chen, Weidong Ma, Qiwei Ye, and Tie-Yan Liu.  Lightgbm: A highly efficient gradient boosting decision tree.  In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, _Advances in Neural Information Processing Systems 30_ , pages 3146–3154. Curran Associates, Inc., 2017.  URL http://papers.nips.cc/paper/6907-lightgbm-a-highly-efficient-gradient-boosting-decision-tree.pdf.


# Embedded Class-I Solution of Compact Stars in $f(R)$ Gravity with
Karmarkar Condition

Tayyaba Naz<EMAIL_ADDRESS>National University of Computer and
Emerging Sciences,
Lahore Campus, Pakistan.  Ammara Usman<EMAIL_ADDRESS>National
University of Computer and Emerging Sciences,
Lahore Campus, Pakistan.  M. Farasat Shamir<EMAIL_ADDRESS>National
University of Computer and Emerging Sciences,
Lahore Campus, Pakistan.

###### Abstract

This paper’s main aim is to investigate the existence of a new classification
of embedded class-I solutions of compact stars, by using Karmarkar condition
in $f(R)$ gravity background. To achieve that goal, we consider two different
models of the $f(R)$ theory of gravity for the static spherically symmetric
spacetime by considering anisotropic matter distribution. Further, we employ
Karmarkar condition to relate the two components of metric potentials $g_{rr}$
and $g_{tt}$. We assume a particular model for one metric potential and obtain
the second one by Karmarkar condition. Moreover, we also calculate the values
of constant parameters by using the observational data of these compact stars,
namely, $Vela~{}X-1,$ $PSR~{}J1614-2230,$ $4U~{}1608-52,$ $Cen~{}X-3$ and
$4U~{}1820-30.$ We perform different physical tests like variational behavior
of energy density and pressure components, stability and equilibrium
conditions, energy constraints, mass function and adiabatic index to check the
viability of $f(R)$ gravity models. All these physical attributes indicate the
consistent behavior of our models. Our investigation also suggests that $f(R)$
theory of gravity appears as a suitable theory in describing the viability of
a new classification of embedded class-I solutions of compact objects.

Keywords: Compact star, Karmarkar, $f(R)$ theory.
PACS: 04.50.Kd, 04.20.Jb, 40.40.Dg.

##  I Introduction

Compact objects are usually designated as neutron stars, white dwarfs and
black holes in astrophysics. Neutron stars and white dwarfs are born in the
result of gravitational collapse which generally happen due to degeneracy
pressure of the relativistic objects. These object emerged huge densities
values because these stars were considered to be massive objects but
volumetrically smaller. Although, we do not know the precise characteristics
of these kinds of compact stars, but these objects are presumed to be heavy
stars having tiny radius. Every kinds of stellar objects are mostly
acknowledged as degenerate stars, except black holes. In general relativity
($GR$) the study of configured equilibrium of compact stars is valuable
because of the massive nature and high densities of these objects. The study
of compact objects in the framework of $GR$ and modified gravitational
theories has always been considered as a topic of great interest in
astrophysics. To investigate celestial compact structure modeling, we need an
exact solution to Einstein field equations (EFE). In $1916,$ Schwarzschild Sch
obtained the EFE solution for the interior structure of compact objects. In
this regard, various modified gravitational theories have been presented in
order to obtain the complicated exact solutions of EFE. In the framework of
observational data, Tolman Tolman  and Oppenheimer Oppen  investigated some
realistic models of non-transversable stellar objects and claimed that the
physical characteristics of these objects represent the relationship between
the internal pressure and the gravitational force which eventually leads in a
state of equilibrium structures. In the study of stars internal configuration,
this phenomenon has considerable importance and provides realistic results in
various occasions. Further, compact celestial structure was examined by Baade
and Zwicky Baa . According to their study the supernova might transform into
smaller compact objects after the observation of strongly magnetized spinning
neutrons. Moreover, Ruderman Rud  identifies for the first time that at the
center of the stellar object the nuclear density exhibits anisotropic
behavior.
A perfect fluid was assumed to be a source of celestial structures for the
creation of stellar objects. These objects are commonly defined as ultra-
dense, isotropic, celestial bodies that seem to be spherically symmetric.
Whereas, isotropy is assumed as a desired attribute, still it does not have a
general feature of compact stars. The concept of non-zero anisotropy in a
stellar configuration was first presented by Bowers and Liang Bow . Moreover,
Ruderman Rud  investigated the possibility that stars might come with huge
densities $10^{15}~{}g/cm^{3},$ when the nuclear matter is anisotropic in
nature. In fact, anisotropic matter distribution plays a crucial role in
narrating the inner representation and development phase of the relativistic
stellar objects. In anisotropic fluid distribution background, a lot of
literature is available Bhar1 ; Bhar2 ; Mau ; MH ; Her1 . In anisotropic
distribution, the pressure component of the fluid sphere divides into two
components, radial and transverse.
One of the most promising topics of modern research in cosmology is the
accelerated expansion of the universe. Cosmologists argue that the
accelerating expansion of the universe depends on dark energy and dark matter
which retains negative pressure. As a substitute of $GR$, different
gravitational theories have been presented to unfold the mystery behind dark
energy issues. These gravitational theories are recognized as modified
theories of gravity. Some of these modified theories are
$f(R),~{}f(R,T),~{}f(G),~{}f(R,G).$ Among these valuable theories, $f(R)$ is
one of the most simplest and popular theory, obtained as an arbitrary function
of Ricci scalar. This theory was proposed by Buchdahl Buch . Later on, Nojiri
and Odintsov Odintsov1  demonstrated some models of $f(R)$ theory of gravity
by placing curvature as a function of Ricci scalar and the outcomes of their
considered models are quite viable and stable. Further, Starobinsky Star
presented an interesting class of $f(R)$ theory of gravity models that showed
the physically acceptable results in laboratory testing of the solar
structure. Some $f(R)$ gravity model was proposed by Hu and Sawicki Hu  by
ignoring the cosmological constant and their study evident some interesting
results in regard to accelerating expansion phenomena. The viability of
physical attributes of compact stars by taking exponential type models of
$f(R)$ theory of gravity was discussed by Cognola et al. Cog . In modified
theories of gravity beyond $GR$ and its Hilbert-Einstein action,
diffeomorphism invariance and the Bianchi identities violation has attracted a
lot of interest. Hamity and Barraco Ha  derived the generalized Bianchi
identities for the non-linear $f(R)$ gravity to throw some light on the issue
that $f(R)$ theory of gravity generates higher than second order equations of
motion and violates Bianchi identities. Further, Wang et al. Wa  confirmed the
local energy-momentum conservation of Bianchi identities by establishing the
equivalence relation between Palatini $f(R)$ and the Brans-Dicke gravity.
Moreover, Koivisto Ko  explored a composition of $f(R)$ gravity and the
generalized Brans-Dicke gravity and claimed covariant conservation from both
the metric tensors and the Palatini variational techniques.
For physically stable models, one could use an analytical approach of EFE and
assume the family of a four dimensional manifold and transform it into
Euclidean space. The embedding family of curved geometry into geometries of
higher dimensions is considered to create various new exact solutions in
astrophysical stellar systems. Schlai S1  designated the embedding issue on
geometrically important spacetimes for the very first time. In this regard,
Nash Nash  presented the isometric embedding theorem. Several features of
anisotropic compact objects by employing embedding class one approach have
been discussed in literature 101 ; 103 ; 104 ; 105 ; 106 ; 107 ; 108 ; 109 . A
class of non-static fluid distribution along with non-vanishing acceleration
was studied by Gupta and Gupta GG . Further, Gupta and Sharma GS  assumed
plane symmetric metric to explore the embedding class-I solutions of non-
static perfect fluid. The embedding class constraint yields a differential
equation in static spherically symmetric geometry relating the two components
of metric potentials, known as Karmarkar condition Kar . A lot of work Mau2 ;
Abb ; Wah ; Mus  has been done related to Karmarkar condition, defined as
$R_{1414}R_{2323}=R_{1212}R_{3434}+R_{1224}R_{1334}$. The Karmarkar condition
develops a connection between two parts of metric tensors $g_{rr}$ and
$g_{tt}$ for a spherical static symmetric fluid distribution. Maurya et al.
Mau3  and Bhar et al. Bhar  studied the EFE by adopting the Karmarkar
condition and composing numerous classes of embedded class-I solutions. They
also observed that these outcomes exhibit stable nature and might be helpful
in exploring the internal structure of the stellar objects. Later, in the
background of $f(G)$ gravity, Sharif and Saba Saba  investigated the charged
anisotropic solutions and their derived solutions are physically consistent
and stable. In this regard, a class of embedded solutions by adopting
Karmarkar condition is recently presented by Upreti et al. Up .
In the background of $GR$, Maurya et al. 102  probe an anisotropic compact
star using embedding class-I approach. They studied different features of
compact stars in the presence of anisotropic fluid distribution and claimed
that obtained outcomes describe the internal core of stellar objects. Further,
Bhar et al. Bhar4  presented a new relativistic anisotropic compact star model
which is physically acceptable for embedding spacetime and can be used to
describe the interior solution of stellar objects. Moreover, the relativistic
model for anisotropic compact stars in $GR$ background was studied by Prasad
et al. Pra . Their results indicate that by adopting embedding class-I
condition the obtained relativistic stellar structure is physically
reasonable. Recently, Mustafa et al. Sha2  introduced new exact solutions of
EFE in the framework of Bardeen black hole spacetime by using the well known
Karmarkar condition. Their chosen model demonstrated the well-behaved nature
under the particular values of the parameters. By getting motivated from
literature, in this particular paper, we extended the concept of Bhar et al.
Bhar4  in the background of $f(R)$ theory of gravity and analyzed the physical
attributes of the compact stars, namely, $Vela~{}X-1,$ $PSR~{}J1614-2230,$
$4U~{}1608-52,$ $Cen~{}X-3$ and $4U~{}1820-30.$ To meet our goal, we assume
two different $f(R)$ theory of gravity models by employing the Karmarkar
condition in the framework of anisotropic pressure. We consider a particular
model for one of the metric potential $g_{rr}$ and by adopting the Karmarkar
condition, we construct the second metric potential $g_{tt}$ of spherically
symmetric spacetime.
The outline of the current article is organized as follows: In the next
segment, the $f(R)$ field equations have been developed by using Karmarkar
condition with an anisotropic matter source. Two of viable $f(R)$ gravity
models along with boundary constraints have been introduced in segment III. In
section IV, we compute the constant values by using the matching conditions.
In section V, we discuss the physical attributes of the compact stars in
detail. In the last portion, we provide the final verdict and conclusion of
our study.

##  II Modified Field Equations

First, we are proceeding to develop the field equations in $f(R)$ gravity
context. For this purpose, we consider the action of $f(R)$ gravity Nod
defined as

$S=\int\Big{[}\frac{f(R)}{2\kappa}+\mathcal{L}_{m}\Big{]}\sqrt{-g}d^{4}x.$ (1)

Here, $f$ is a function of Ricci scalar and $\mathcal{L}_{m}$ is the
Lagrangian matter. Varying the action identified in (1) regarding the metric
potential $g_{\eta\zeta},$ we obtain the succeeding $f(R)$ gravity field
equation

$FR_{\eta\zeta}-\frac{1}{2}f(R)g_{\eta\zeta}-\nabla_{\eta}\nabla_{\zeta}F+g_{\eta\zeta}\Box
F=\kappa T_{\eta\zeta}.$ (2)

Thus, $F=\frac{df(R)}{dR}.$ Although, $\nabla_{\eta}$ and $\Box$ indicate
covariant derivative and D’Alembertian notation, i.e.
$\Box\equiv\nabla_{\eta}\nabla^{\zeta}$. The stress-energy momentum tensor is
presented as

$T_{\eta\zeta}=(\rho+p_{r})u_{\eta}u_{\zeta}+p_{t}g_{\eta\zeta}+(p_{r}-p_{t})\mathcal{X}_{\eta}\mathcal{X}_{\zeta},$

where $\rho$, $p_{r}$ and $p_{t}$ represent the energy density, radial and
transverse pressure, respectively. Here, $u_{\eta}$ and $\mathcal{X}_{\eta}$
are four velocity vectors, which satisfy the relations
$u^{\eta}u_{\eta}=-\mathcal{X}^{\eta}\mathcal{X}_{\eta}=1$. Further, we assume
a static spherically symmetric line element defined by

$ds^{2}=e^{\nu(r)}{dt}^{2}-e^{\lambda(r)}{dr}^{2}-r^{2}({d\theta}^{2}+sin^{2}\theta{d\phi}^{2}).$
(3)

Here, ${\nu(r)}$ and ${\lambda(r)}$ are the functions of radial coordinate
only. We assume that our stress-energy tensor is anisotropic in nature and
relativistic geometrized units $G=c=1$, which implies that $\kappa=\frac{8\pi
G}{c^{4}}$ reduces to $8\pi.$ The $f(R)$ gravity field equations for spacetime
(3) yield

$8\pi\rho=e^{-\lambda}F\big{(}\frac{\nu^{{}^{\prime\prime}}}{2}+\frac{\nu^{{}^{\prime}}}{r}+\frac{\nu^{{}^{\prime}}(\nu^{{}^{\prime}}-\lambda^{{}^{\prime}})}{4}\big{)}-\frac{1}{2}f(R)-e^{-\lambda}F^{{}^{\prime\prime}}-e^{-\lambda}F^{{}^{\prime}}\big{(}\frac{\lambda^{{}^{\prime}}}{2}+\frac{2}{r}\big{)},$
(4) $8\pi
p_{r}=e^{-\lambda}F\big{(}-\frac{\nu^{{}^{\prime\prime}}}{2}+\frac{\lambda^{{}^{\prime}}}{r}-\frac{\nu^{{}^{\prime}}(\nu^{{}^{\prime}}-\lambda^{{}^{\prime}})}{4}\big{)}+\frac{1}{2}f(R)+e^{-\lambda}F^{{}^{\prime}}\big{(}\frac{\nu^{{}^{\prime}}+2\lambda^{{}^{\prime}}}{2}+\frac{2}{r}\big{)},$
(5) $8\pi
p_{t}=F\big{(}\frac{1-e^{-\lambda}}{r^{2}}+e^{-\lambda}\frac{(\lambda^{{}^{\prime}}-\nu^{{}^{\prime}})}{2r}\big{)}+\frac{1}{2}f(R)+e^{-\lambda}F^{{}^{\prime\prime}}+e^{-\lambda}F^{{}^{\prime}}\big{(}\frac{\nu^{{}^{\prime}}+\lambda^{{}^{\prime}}}{2}+\frac{1}{r}\big{)},$
(6)

here ‘prime’ symbolized as a $r$ derivative. Whereas, the anisotropic
parameter is presented as

$\Delta=8\pi p_{t}-8\pi p_{r}.\\\ $ (7)

By utilizing Eqs. (5) and (6), we receive

$\displaystyle\Delta$ $\displaystyle=$ $\displaystyle
F\big{[}\frac{1-e^{-\lambda}}{r^{2}}+e^{-\lambda}\big{(}\frac{\nu^{{}^{\prime\prime}}}{2}-\frac{(\nu^{{}^{\prime}}+\lambda^{{}^{\prime}})}{2r}+\frac{\nu^{{}^{\prime}}(\nu^{{}^{\prime}}+\lambda^{{}^{\prime}})}{4}\big{)}\big{]}+e^{-\lambda}F^{{}^{\prime\prime}}-e^{-\lambda}F^{{}^{\prime}}\big{(}\frac{\lambda^{{}^{\prime}}}{2}+\frac{1}{r}\big{)}.$
(8)

Moreover, the nature of $\Delta$ is attractive if $p_{r}<p_{t}$ and repulsive
if $p_{r}>p_{t}$ A2 . Further, the metric tensor (3) represents the embedded
class-I family, if it satisfies the Karmarkar condition, i.e.

$R_{1414}R_{2323}=R_{1212}R_{3434}+R_{1224}R_{1334},$ (9)

with $R_{2323}\neq 0$. By employing the well-known Karmarkar condition, the
succeeding differential equation derived for metric (3)

$\lambda^{{}^{\prime}}\nu^{{}^{\prime}}-2\nu^{{}^{\prime\prime}}-{\nu^{{}^{\prime}}}^{2}=\frac{\lambda^{{}^{\prime}}\nu^{{}^{\prime}}}{1-e^{\lambda}},$
(10)

where $e^{\lambda}\neq 1$. Integrating Eq. (10), we get

$e^{\nu}=\Big{[}{{\big{(}A+B\int{\sqrt{e^{\lambda}-1}dr}\big{)}}^{2}}\Big{]}.$
(11)

Here $A$ and $B$ are constant of integration. In order to obtain the embedded
class-I solution, we consider a specific model for the component
$g_{rr}=e^{\lambda(r)}$ Bhar4  which is given as

$e^{\lambda}=1+\frac{a^{2}r^{2}}{(1+br^{2})^{4}}.$ (12)

To demonstrate embedded class-I spacetime solution $a$ and $b$ should be non-
zero. Singh and Pant SP  explored a similar compact star model, in the context
of $GR$, by considering the metric potential
$e^{\lambda}=1+a^{2}r^{2}(1+br^{2})^{x}$ of embedding class-I solution. In
their research, they only assume positive values of $x$ and presented a family
of new exact solutions for relativistic anisotropic stellar objects. Later,
Bhar et al. Bhar4  studied this metric potential by taking $x=-4$ and the
obtained results are well behaved in all respects in the background of $GR$.
Motivated from the work of Bhar et al. Bhar4 , here we extended the analysis
in $f(R)$ gravity context. Manipulating Eqs. (11) and (12), the metric
potential expression becomes

$e^{\nu}=\Big{[}{\Big{(}A-\frac{aB}{2b(1+br^{2})}\Big{)}}^{2}\Big{]}.$ (13)

##  III Viable $f(R)$ Gravity Models

Here, we consider the two realistic and simple viable $f(R)$ gravity models to
analysis the modified field equations presented in Eqs. $(\ref{3})-(\ref{5}).$

###  III.1 Model 1

To study the existence of the compact stellar objects, we first assume an
exponential type $f(R)$ gravity model Cog

$f(R)=R+\alpha(e^{-\beta R}-1),$ (14)

where, $\alpha$ and $\beta$ are any arbitrary constants. The exponential based
$f(R)$ gravity models plays a very crucial role in the context of describing
exponential expansion. A class of exponential, realistic modified gravities
was presented by Cognola et al. Cog  in which they argued that this model (14)
passes all the local tests including non-violation of Newton’s law and
stability of spherical body solution. Nojiri and Odintsov Odintsov1
investigated this model (14) to describe the early-time inflation and late-
time cosmic acceleration in a natural, unified way. It was shown that
exponential type models present a realistic dark energy epoch that is
compatible with local and observational tests Lin . In order to explore the
compact stars existence for our proposed model (14), the values of the
constant parameters are chosen in kind that $\rho$, $p_{r}$ and $p_{t}$
exhibit positive behavior. The physical features of the compact stars for the
model (14) by using Eqs. $(\ref{3})-(\ref{5})$ can be defined by the following
relation as

$\displaystyle\rho=\frac{1}{16\pi rh_{1}^{2}h_{2}}\Big{(}-\alpha(\beta
h_{1})^{2}\left(r\lambda^{\prime}+4\right)h_{2}h_{3}h_{5}+2\alpha r(\beta
h_{1})^{2}h_{2}h_{3}\left(\beta
h_{5}^{2}-h_{6}\right)-rh_{1}^{2}h_{2}\left(\alpha\left(e^{-\beta
h_{4}}-1\right)+h_{4}\right)~{}~{}~{}$
$\displaystyle+2abBrh_{3}\left(rh_{1}\lambda^{\prime}+2br^{2}-6\right)\left(e^{\beta
h_{4}}-\alpha\beta\right)\Big{)},$ (15) $\displaystyle
p_{r}=\frac{1}{8\pi}\Big{(}\frac{\left(e^{\beta
h_{4}}-\alpha\beta\right)h_{3}\left(h_{1}\lambda^{\prime}\left(2Ab\left(br^{2}+1\right)^{2}-aB\right)+2abBr\left(3br^{2}-1\right)\right)}{-rh_{1}^{2}h_{2}}+\alpha\beta^{2}h_{3}h_{5}(\frac{2abBr}{-h_{1}h_{2}}+\lambda^{\prime}~{}~{}~{}$
$\displaystyle+\frac{2}{r})+\frac{1}{2}\left(\alpha\left(e^{-\beta
h_{4}}-1\right)+h_{4}\right)\Big{)},$ (16) $\displaystyle
p_{t}=\frac{1}{16\pi}\Big{(}2\alpha\beta^{2}h_{3}h_{5}\left(\frac{2abBr}{-h_{1}h_{2}}+\frac{\lambda^{\prime}}{2}+\frac{1}{r}\right)+\frac{\left(1-\alpha\beta
e^{-\beta
h_{4}}\right)\left(e^{-\lambda}\left(\frac{4abBr^{2}}{h_{1}h_{2}}-2\right)+re^{-\lambda}\lambda^{\prime}+2\right)}{r^{2}}-2\alpha\beta^{2}h_{3}~{}~{}~{}$
$\displaystyle\left(\beta h_{5}^{2}-h_{6}\right)+\alpha\left(e^{-\beta
h_{4}}-1\right)+h_{4}\Big{)}.$ (17)

###  III.2 Model 2

We also discuss the graphical response of the considered stars for the model
Faul

$f(R)=R-(1-n)\mu^{2}\Big{(}\frac{R}{\mu^{2}}\Big{)}^{n}.$ (18)

Here, $\mu$ and $n$ are arbitrary constant parameters. In literature, the
model (18) used to investigate the effects of cosmic acceleration and the
Chameleon mechanism. Faulkner et al. Faul  established the relation between
the scalar tensor gravity and modified $f(R)$ gravity by using different
models. According to them, the fundamental feature of this model is conformal
coupling. Furthermore, Amendola et al. Ame  discussed the standard matter
dominated epoch for the $f(R)$ dark energy models. They argued that the model
(18) showed interesting results in supernova tests and revealed well-behaved
consistent and stable reactions in the framework of the solar system. For
$f(R)$ gravity model under investigation, we assume the appropriate value of
constants to obtained the physically acceptable results. By making use of Eq.
(18), the explicit relation for the energy density, radial and transverse
pressures have been found to be:

$\displaystyle\rho=\frac{1}{32\pi
rh_{4}^{3}}\Big{(}-2h_{4}h_{5}h_{7}h_{8}\left(r\lambda^{\prime}+4\right)-4rh_{7}h_{8}\left(h_{4}h_{6}+(n-2)h_{5}^{2}\right)+h_{4}^{2}\left(h_{7}h_{8}+e^{-\lambda}h_{4}\right)(\left(4-r\lambda^{\prime}\right)~{}~{}~{}$
$\displaystyle\nu^{\prime}+2r\nu^{\prime\prime}+r\nu^{\prime
2})-2rh_{4}^{3}\left(\mu^{2}(n-1)h_{8}+h_{4}\right)\Big{)},$ (19)
$\displaystyle p_{r}=\frac{1}{32\pi
rh_{4}^{2}}\Big{(}2h_{5}h_{7}h_{8}\left(2r\lambda^{\prime}+r\nu^{\prime}+4\right)+h_{4}\left(h_{7}h_{8}+e^{-\lambda}h_{4}\right)\left(\lambda^{\prime}\left(r\nu^{\prime}+4\right)-r\left(2\nu^{\prime\prime}+\nu^{\prime
2}\right)\right)+2rh_{4}^{2}~{}~{}~{}$
$\displaystyle\left(\mu^{2}(n-1)h_{8}+h_{4}\right)\Big{)},$ (20)
$\displaystyle
p_{t}=\frac{1}{16\pi}\Big{(}\left(r\lambda^{\prime}+2e^{\lambda}-r\nu^{\prime}-2\right)\left(\frac{h_{7}h_{8}}{r^{2}h_{4}(n-1)}+\frac{e^{\lambda}}{r^{2}}\right)+\frac{h_{5}h_{7}h_{8}\left(r\lambda^{\prime}+r\nu^{\prime}+2\right)}{rh_{4}^{2}}+\mu^{2}(n-1)h_{8}+h_{4}~{}~{}~{}$
$\displaystyle+\frac{2h_{7}h_{8}\left(h_{4}h_{6}+(n-2)h_{5}^{2}\right)}{h_{4}^{3}}\Big{)},$
(21)

where,

$h_{1}=(br^{2}+1),~{}~{}~{}h_{2}=aB-2Ab(br^{2}+1),~{}~{}~{}h_{3}=e^{-\lambda-\beta
h_{4}},$ $\displaystyle
h_{4}=\frac{1}{-\left(a^{2}r^{2}+h_{1}^{4}\right)^{2}h_{2}}\Big{(}2a(a^{4}Br^{2}-2a^{3}Abr^{2}+h_{1}-a^{2}B\left(br^{2}-3\right)~{}~{}~{}$
$\displaystyle+h_{1}^{3}+2aAb\left(5br^{2}-3\right)+h_{1}^{4}-2bB\left(br^{2}-3\right)+h_{1}^{6})\Big{)},$
$h_{5}=\frac{\partial h_{4}}{\partial r},~{}~{}~{}h_{6}=\frac{{\partial
h_{4}}^{2}}{{\partial}^{2}r},~{}~{}~{}h_{7}={{\mu}^{2}}n(n-1)^{2}e^{-\lambda},~{}~{}~{}h_{8}=\left(\frac{h_{4}}{\mu^{2}}\right)^{n}.$

###  III.3 Boundary Conditions

Next, we analyze the mandatory conditions for the metric potential, i.e.
$e^{\lambda(0)}=1$ and $((e^{\lambda(r)})^{\prime})_{r=0}=0$. For a well-
behaved $f(R)$ theory of gravity model, the graphical response of $g_{rr}$
should not have any type of singularity and the curvature must be regular.
From Fig. $\ref{Fig:1},$ one can notice that expression $e^{\lambda(r)}$,
given in (12), is physically acceptable as both the metric potential shown
monotonically increasing behavior, and attains maximum value in the boundary.

|  |
---|---|---
Figure 1: Evolution of metric potential $g_{rr}$ and $g_{tt}$.

Further, we list all the necessary constraints, which must be fulfil for the
well-behave nature of the stars.

  * •

The $g_{rr}$ and $g_{tt}$ should be singularities free. Also the anisotropic
fluid sphere should satisfy $e^{\lambda(r=0)}=1$ and $e^{\nu(r=0)}=constant$.

  * •

$\rho,$ $p_{r}$ and $p_{t}$ should always be finite and positive. Further,
these illustration must be maximum at center and decreasing toward the
boundary.

  * •

The gradient of $\rho,$ $p_{r}$ and $p_{t}$ must exhibit negative nature.

  * •

The equation of state should obey the necessary condition, i.e.
$0<w_{r},~{}w_{t}<1.$

  * •

The velocity sound must lies within $[0,1]$, i.e.
$0<{v_{r}}^{2},~{}{v_{t}}^{2}<1.$

  * •

Adiabatic index for anisotropic fluid sphere should be greater the $4/3.$

##  IV Matching Conditions

For the stellar compact objects, the intrinsic boundary metric irrespective of
the geometry (interior or exterior) will remain the same. This phenomena
verify that, whatever the coordinates system covering the surface of the
boundary, the components of the metric tensor will remain continuous. For
stellar compact objects, the Schwarzschild solution is observed as the most
suitable possibility to choose from the various choices of the matching
conditions in the context of $GR$. Jebsen-Birkhoff’s theorem also ensure that
the solution of field equations should be asymptotically flat and remain
static for spherical symmetric spacetime. In modified theories of gravity
case, the outside solution of star may differ from Schwarzschild solution,
when we come to modified TOV equations Tolman ; Oppen  with zero energy
density and pressure. However, this concern may be overcome in modified $f(R)$
gravity with a suitable possibility of $f(R)$ gravity models for non-zero
energy density and pressure in Schwarzschild’s solution. Due to this
phenomena, the Birkhoff theorem fails in modified gravity Far . Many research
articles have been published on matching conditions Ast ; Cooney ; Gan ; Mom
and provide some fascinating results for Schwarzschild solution. To compute
the field equations solution under the limited boundary conditions at $r=R$,
the radial pressure $p_{r}(R)=0$. Here, we match the obtained solution of
interior geometry (3) with the Schwarzschild exterior geometry, given by

$ds^{2}=\big{(}1-\frac{2M}{r}\big{)}{dt}^{2}-\big{(}1-\frac{2M}{r}\big{)}^{-1}{dr}^{2}-r^{2}({d\theta}^{2}+sin^{2}\theta{d\phi}^{2}),$
(22)

The continuity condition for Eqs. (3) and (22) at the boundary $r=R$ yield

${g_{rr}}^{+}={g_{rr}}^{-},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{g_{tt}}^{+}={g_{tt}}^{-},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\frac{\partial{g_{tt}}^{+}}{\partial
r}=\frac{\partial{g_{tt}}^{-}}{\partial r}~{},$ (23)

where “+” identify the exterior geometry and “-” identify the interior
geometry. With the help of Eqs. (3), (22) and (23), we computed the values of
the parameters a, b, A and B, stated as

$a=\frac{(1+bR^{2})^{2}}{R}\sqrt{\frac{\frac{2M}{R}}{1-\frac{2M}{R}}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}b=\frac{4M-R}{R^{2}(9R-20M)},$
(24)
$A=\frac{aB}{2b(1+bR^{2})}+\sqrt{1-\frac{2M}{R}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}B=(\frac{1}{2R})\sqrt{\frac{2M}{R}}.$
(25)

The values of the parameters by assuming the radius and mass of considered
compact stars $Vela~{}X-1,$ $PSR~{}J1614-2230,$ $4U~{}1608-52,$ $Cen~{}X-3,$
$4U~{}1820-30,$ are given in Tab. IV.

Table 1: Constants values of $a$, $b$, $A$ and $B$ for considered compact
stars

.  Star Model Vela X-1 $(S_{1})$ PSR J1614-2230 $(S_{2})$ 4U 1608-52 $(S_{3})$
Cen X-3 $(S_{4})$ 4U 1820-30 $(S_{5})$ $M~{}(M_{\Theta})$ 1.77 $\pm$ 0.08 Gang
1.97 $\pm$ 0.04 Gang  1.74 $\pm$ 0.01 Gang  1.49 $\pm$ 0.08 Gang  1.58 $\pm$
0.06 Gang  $R~{}(km)$ 9.56 $\pm$ 0.08 9.69 $\pm$ 0.2 9.3 $\pm$ 0.10 9.178
$\pm$ 0.13 10.56 $\pm$0.10 $M/R$ 0.272756 0.301209 0.277485 0.242338 0.257507
$a~{}(km)$ 0.120435 0.144972 0.127851 0.104090 0.115014 $b~{}(km)$ 0.000280
0.000732 0.000368 -0.000088 0.000094 $A~{}(km)$ 8.754660 4.339340 7.402050
-21.98830 24.58420 $B~{}(km)$ 0.038589 0.040008 0.040052 0.037919 0.039431

##  V Physical Characteristics of the Compact Stars Models

Here, we discuss the graphical behavior of the considered compact stars in the
context of $f(R)$ gravity models. We studied the graphical illustration of
energy density, pressure components, stability and equilibrium condition,
energy conditions, mass function, adiabatic index, etc. For plotting the
graphs, we consider the same values of constants mentioned in Tab. IV.
Moreover, for Model-1, we fix the constant parameter $\alpha=0.1$ and the
variation of $\beta,$ for different compact stars are given as: $\beta=1.5$
for $S_{1}$, $\beta=1$ for $S_{2}$, $\beta=1$ for $S_{3}$, $\beta=1.5$ for
$S_{4}$ and $\beta=1$ for $S_{5}$. While for Model-2, we fix the constant
parameters $\mu=5$ and $n=3.$

###  V.1 Energy Density and Pressure Movement

In this portion, we represent the graphical response of $\rho,$ $p_{r}$ and
$p_{t}$ for considered $f(R)$ theory of gravity models. The graphical behavior
of the preceding aspects are represented in Fig. 2 and Fig. $\ref{Fig:3}.$
Furthermore, we also present the graphical behavior of gradient of $\rho,$
$p_{r}$ and $p_{t}$. There graphs have been shown in Fig. 4 and Fig.
$\ref{Fig:5}.$

|  |  |
---|---|---|---
Figure 2: Behavior of $\rho$, $p_{r}$ and $p_{t}$ for Model-1. |  |  |
---|---|---|---
Figure 3: Behavior of $\rho$, $p_{r}$ and $p_{t}$ for Model-2.

It is clearly viable from Fig. 2 and Fig. 3 that $\rho,$ $p_{r}$ and $p_{t}$
plots are positive and finite. Further, one can easily observed that the
illustration of an energy density and pressure components attain the highest
values at compact star center and decreases toward surface boundary.

|  |  |
---|---|---|---
Figure 4: Variation of $\frac{d\rho}{dr},$ $\frac{dp_{r}}{dr}$ and $\frac{dp_{t}}{dr}$ for Model-1. |  |  |
---|---|---|---
Figure 5: Variation of $\frac{d\rho}{dr},$ $\frac{dp_{r}}{dr}$ and
$\frac{dp_{t}}{dr}$ for Model-2.

The gradient of $\rho,$ $p_{r}$ and $p_{t}$ presented negative, which can be
seen from Fig. 4 and Fig. $\ref{Fig:5}.$ These facts show the high compactness
nature of the compact stars.

###  V.2 Anisotropy

Next, we discuss the graphical behavior of anisotropy parameter Harko ,
symbolized by $\Delta$ and shown in Eq. (7). For the existence of the compact
stars the anisotropic force must be repulsive A2 . From Fig. 6, it is
worthwhile to mention here that in our study $\Delta>0$, which means that our
models exhibit repulsive nature.

|  |
---|---|---
Figure 6: Evolution of anisotropy.

###  V.3 Energy Conditions

In the existence of the compact stars, energy conditions plays a significant
role. These energy bonds You  are categories as null energy, weak energy,
strong energy and dominant energy conditions symbolized by NEC, WEC, SEC and
DEC, respectively and defined as

$NEC:\rho+p_{r}\geq 0,\rho+p_{t},~{}~{}~{}~{}~{}~{}WEC:\rho\geq
0,\rho+p_{r}\geq 0,\rho+p_{t}\geq 0,~{}~{}~{}~{}~{}~{}SEC:\rho-
p_{r}-2p_{t}\geq 0,~{}~{}~{}~{}~{}~{}DEC:\rho\geq 0,\rho\pm p_{r}\geq
0,\rho\pm p_{t}\geq 0$

It has been observed from Fig. 7 and Fig. 8 that all energy bonds have been
well satisfied for our proposed models.

|  |  |
---|---|---|---
|  |  |
Figure 7: Graphs of energy conditions for Model-1. |  |  |
---|---|---|---
|  |  |
Figure 8: Graphs of energy conditions for Model-2.

###  V.4 Equilibrium Condition

In this subsection, we explore the equilibrium stability of the stellar model
under the presence of gravitational force, hydrostatics force and anisotropic
force. According to the TOV equation Tolman ; Oppen , the equilibrium
condition between all these forces can be described as

$\frac{M_{G}(r)(\rho+p_{r})}{r}e^{\frac{\lambda-\nu}{2}}+\frac{dp_{r}}{dr}-\frac{2}{r}(p_{t}-p_{r})=0.$
(26)

The effective gravitational mass $M_{G}(r)$ is defined as

$M_{G}(r)=\frac{1}{2}\nu^{{}^{\prime}}e^{\frac{\nu-\lambda}{2}}.$ (27)

Putting value of $M_{G}(r)$ in Eq. (26), we get

$\frac{\nu^{{}^{\prime}}}{r}(\rho+p_{r})+\frac{dp_{r}}{dr}-\frac{2}{r}(p_{t}-p_{r})=0,$
(28)

where,
$\mathcal{F}_{g}=-\frac{\nu^{{}^{\prime}}}{r}(\rho+p_{r}),~{}\mathcal{F}_{h}=-\frac{dp_{r}}{dr},~{}\mathcal{F}_{a}=\frac{2}{r}\Delta.$
Here, $\mathcal{F}_{g}$ designates gravitational force, $\mathcal{F}_{h}$
designates hydrostatic force and $\mathcal{F}_{a}$ designates anisotropic
force. For viable $f(R)$ gravity models, all three forces necessarily express
the balancing trend, i.e. for a system to be in equilibrium the joint sum of
three forces should be exactly equal to zero. Consequently, the Eq. (28) yield
F_g+F_h+F_a=0. From Fig. 9, it can be easily seen that all forces are
satisfied the necessary condition of equilibrium.

|  |
---|---|---
Figure 9: Behavior of $\mathcal{F}_{h}$, $\mathcal{F}_{g}$ and
$\mathcal{F}_{a}$ for our proposed models.

###  V.5 Equation of State

In this portion, we compute the equations of state (EoS) for both radial and
transverse pressures. The EoS are presented by two ratios, which are mentioned
as

$w_{r}=\frac{p_{r}}{\rho},~{}~{}~{}~{}~{}~{}~{}~{}w_{t}=\frac{p_{t}}{\rho}.$
(29)

|  |
---|---|---
Figure 10: Evolution of $w_{r}$ and $w_{t}$ for Model-1. |  |
---|---|---
Figure 11: Evolution of $w_{r}$ and $w_{t}$ for Model-2.

From the graphical illustration shown in Fig. 10 and Fig. 11, it can be
noticed that $0<w_{r},~{}w_{t}<1.$ Hence, we can conclude that the variational
responses of EoS are sufficient for a relativistic $f(R)$ gravity model.

###  V.6 Causality Condition

The velocity sound for radial pressure component ${v_{r}}^{2}$ and tangential
pressure component ${v_{t}}^{2}$ are defined as

${v_{r}}^{2}=\frac{dp_{r}}{d\rho},~{}~{}~{}~{}~{}~{}~{}~{}{v_{t}}^{2}=\frac{dp_{t}}{d\rho}.$
(30)

To verify the stability of our chosen system, we use the Herrera concept Her
according to that ${v_{r}}^{2}$ and ${v_{t}}^{2}$ must lies within the range
of $0$ and $1.$ Moreover, we also studied the graphical behavior of our chosen
compact stars for the Andreasson condition And , that is,
$|{v_{r}}^{2}-{v_{t}}^{2}|\leq 1.$ The graphical illustration presented in
Fig. 12 and Fig. 13 confirm that the considered $f(R)$ gravity models are
potentially stable and satisfied both causality condition and Herrera cracking
concept.

|  |  |
---|---|---|---
Figure 12: Behavior of ${v_{r}}^{2},$ ${v_{t}}^{2}$ and $|{v_{r}}^{2}-{v_{t}}^{2}|$ for $f(R)$ gravity Model-1. |  |  |
---|---|---|---
Figure 13: Behavior of ${v_{r}}^{2},$ ${v_{t}}^{2}$ and
$|{v_{r}}^{2}-{v_{t}}^{2}|$ for $f(R)$ gravity Model-2.

###  V.7 Mass Function, Compactness Factor and Redshift Analysis

Using the metric potential $g_{rr}^{-}=g_{rr}^{+}$ the mass function Bhar4  is
obtained as

$\mathcal{M}(r)=\frac{a^{2}r^{3}}{2[(1+br^{2})^{2}+a^{2}r^{2}]}.$ (31)

Next, we calculate the compactness factor Maha  and redshift Harko  which are
given by the following mathematical relation

$\mathcal{U}(r)=\frac{2\mathcal{M}(r)}{r}=\frac{a^{2}r^{2}}{[(1+br^{2})^{2}+a^{2}r^{2}]},$
(32) $\mathcal{Z}=e^{-\frac{\nu}{2}}-1.$ (33)

The graphical response of mass function, compactness parameter and redshift
are given in Fig. $\ref{Fig:14}.$ The behavior of mass function satisfies the
Buchdahl Buchdahl  and Bondi Bon  limit, that is, $\frac{2M}{R}<\frac{8}{9}.$
Furthermore, the mass function and compactness parameter shown monotonically
increasing plots whereas the graphical behavior of redshift is monotonically
decreasing. Thus, we conclude that all the three graphs are stable for our
$f(R)$ gravity models.

|  |  |
---|---|---|---
Figure 14: Behavior of Mass function (left panel), Compactness Factor (middle
panel) and Redshift (right panel).

###  V.8 Adiabatic Index

Adiabatic index studied to describe the stiffness of EoS, for a given energy
density. It provides verification about stability of both relativistic and non
relativistic stellar objects. Chandrasekhar Chand  developed the idea of the
dynamical stability against infinitesimal radial adiabatic perturbation of the
stellar system. His concept has been successfully tested for isotropic as well
as anisotropic stellar spheres, by many authors HeinHil ; Hill ; Hor ; Don ;
Silva ; Bomb . For the stability of the compact stars models the value of
adiabatic index must be greater than $4/3$. The expressions of the adiabatic
index corresponding to radial and transverse pressure components are defined
as

$\gamma_{r}=\frac{\rho+p_{r}}{p_{r}}(\frac{dp_{r}}{d\rho})=\frac{\rho+p_{r}}{p_{r}}{v_{r}}^{2},~{}~{}~{}~{}~{}~{}~{}\gamma_{t}=\frac{\rho+p_{t}}{p_{t}}(\frac{dp_{t}}{d\rho})=\frac{\rho+p_{t}}{p_{t}}{v_{t}}^{2}.$
(34)

Fig. 15 and Fig. $\ref{Fig:16},$ shows that this stability condition satisfied
everywhere for our proposed two models.

|  |
---|---|---
Figure 15: Evolution of $\gamma_{r}$ and $\gamma_{t}$ with respect to “r” for Model-1. |  |
---|---|---
Figure 16: Evolution of $\gamma_{r}$ and $\gamma_{t}$ with respect to “r” for
Model-2.

##  VI Conclusion

In order to explore a new family of embedded class-I solutions in the
anisotropy background, we assume two realistic $f(R)$ gravity models by using
the Karmarkar condition. Since the Karmarkar condition reduced the solution-
generating method of EFE to a single metric potential by providing a
connection between $g_{rr}$ and $g_{tt}.$ For this purpose, we assume a metric
potentials $e^{\lambda}=1+\frac{a^{2}r^{2}}{(1+br^{2})^{4}}$ and by employing
Karmarkar condition we obtained second metric potential given as
$e^{\nu}={\Big{[}A-\frac{aB}{2b(1+br^{2})}\Big{]}}^{2}$, where $a$, $b$, $A$
and $B$ are arbitrary constants. Further, we calculate the unknown constants
by using matching conditions between the interior and Schwarzschild exterior
geometries. To verify that our obtained solution are physically acceptable, we
analyze the physical response of five individual compact stars. The main goal
of our investigation is to generate new solutions of $f(R)$ gravity models in
the frame of anisotropic matter source. The significant outcomes are enlisted
below.

  * •

Fig. $\ref{Fig:1},$ presents that the graphical behavior of both the metric
potential $g_{rr}=e^{\lambda}$ and $g_{tt}=e^{\nu}$ are finite, positive,
singularity free and fulfil the conditions, i.e., $e^{\lambda(r=0)}=1$ and
$e^{\nu(r=0)}={\Big{[}A-\frac{aB}{2b}\Big{]}}^{2}$. It has been observed that
both metric potentials are monotonically increased and attain maximum values
at boundary, which provides evidence that considered models indicate
satisfaction outcomes.

  * •

The variation of $\rho,$ $p_{r}$ and $p_{t}$ regarding “r” for both the models
are finite and regular at the center. One can easily notice from Fig. 2 and
Fig. 3 that the corresponding plots attain highest value at the center and
express decreasing response toward boundary, which confirm that our proposed
models are stable.

  * •

The gradient of $\rho,$ $p_{r}$ and $p_{t}$ has been shown in Fig. 4 and Fig.
$\ref{Fig:5},$ which exhibit the consistent nature of our proposed models.

  * •

From Fig. $\ref{Fig:6},$ we can easily verify that anisotropy remains positive
throughout the compact stars. This behavior shows that the nature of
anisotropy is repulsive which confirms the existence for the compact objects.

  * •

From illustration of energy conditions presented in Fig. 7 and Fig.
$\ref{Fig:8},$ we noted that all energy bonds are satisfied.

  * •

Fig. 9, shows the balancing behavior of ($\mathcal{F}_{g}$),
($\mathcal{F}_{h}$) and ($\mathcal{F}_{a}$) of our physical acceptable
structure.

  * •

For consistent nature of the stellar objects, the graphical response of EoS
should honour the constraints $0<w_{r},~{}w_{t}<1.$ The corresponding
graphical illustration provided in Fig. 10 and Fig. $\ref{Fig:11},$ revealed
the stability of our system.

  * •

For compact stars, the variation of velocity of sounds $v^{2}_{r}$ (for radial
component) and $v^{2}_{t}$ (for transverse component) plots should lies within
$[0,1].$ From Fig. 12 and Fig. 13, we observed that our considered model obeys
the causality conditions. In addition, the Herrera cracking concept i.e.
$0\leq|v^{2}_{r}-v^{2}_{t}|\leq 1$, for both the models are consistent, which
verify the stability of our system.

  * •

It has been observed from Fig. $\ref{Fig:14},$ that the graphs of mass
function and compactness parameter obey monotonically increasing responses,
while on the other hand, the graphical illustrating of redshift analysis
revealed monotonically decreasing behavior, this confirms that our chosen
models execute the stability criteria.

  * •

The adiabatic index values of $\gamma_{r}$ and $\gamma_{t}$ for both models
are greater than $4/3$, shown in Fig. 15 and Fig. $\ref{Fig:16},$ which
reconfirms the stability of our system.

As a concluding remark, in this paper we have favorably presented the well
stable and singularity free stellar system, which is appropriate to exhibit
the anisotropic nature of the compact stars, by using the Karmarkar condition.
Moreover, it is worthwhile to mention here that our presented outcomes are
equivalent to the results which were obtained by Bhar Bhar4  in the context of
$GR$.

## References

  * (1) K. Schwarzschild, Sit. Dts. Aka. Wis. Math. Phys. Ber. 24, 424 (1916).
  * (2) R. C. Tolman, Phys. Rev. 55, 364 (1939).
  * (3) J. R. Oppenheimer, G. M. Volkoff, Phys. Rev. 55, 374 (1939).
  * (4) W. Baade, F. Zwicky, Phys. Rev. 46, 76 (1934).
  * (5) R. Ruderman, Ann. Rev. Ast. Astrophys. 10, 427 (1972).
  * (6) R. L. Bowers, E. P. T. Liang, Astrophy. J. 188, 657 (1974).
  * (7) L. Herrera, N. O. Santos, Phys. Rep. 286, 53 (1997).
  * (8) M. K. Mak, T. Harko, Pro. R. Soc. A 459, 393 (2003).
  * (9) S. K. Maurya, Y. K. Gupta, Astrophys. Spa. Sci. 344, 243 (2013).
  * (10) P. Bhar, Astrophy. Spa. Sci. 356, 309 (2015).
  * (11) P. Bhar, S. K. Maurya, Y. K. Gupta, T. Manna, Eur. Phys. J. A 52, 312 (2016).
  * (12) H. A. Buchdahl, Mon. Not. R. Ast. Soc. 150, 1 (1970).
  * (13) S. Nojiri, S. D. Odintsov, Phys. Rep. 505, 59 (2011).
  * (14) A. A. Starobinsky, JETP Lett. 86, 157 (2007).
  * (15) W. Hu, I. Sawicki, Phys. Rev. D 76, 064004 (2007).
  * (16) G. Cognola et al., Phys. Rev. D 77, 046009 (2008).
  * (17) V. H. Hamity, D. E. Barraco, Gen. Rel. Grav. 25, 461-471 (1993).
  * (18) P. Wang, G. M. Kremer, D. S. M. Alves, X. Meng, Gen. Rel. Grav. 38, 517-521 (2006).
  * (19) T. Koivisto, Cla. Qua. Grav. 23, 4289-4296 (2006).
  * (20) L. Schlai, Ann. di Mat. 5, 170 (1871).
  * (21) J. Nash, Ann. Math. 63, 20 (1956).
  * (22) M. Kohler, K. L. Chao, Z. Nat. Ser. A 20, 1537 (1965).
  * (23) S. K. Maurya, Y. K. Gupta, T. T. Smitha, F. Rahaman, Eur. Phys. J. A 52, 191 (2016).
  * (24) P. Bhar, S. K. Maurya, Y. K. Gupta, T. Manna, Eur. Phys. J. A 52, 312 (2016).
  * (25) D. Deb, S. V. Ketov, S. K. Maurya, MNRAS 485, 5652 (2019).
  * (26) P. K. F. Kuhfittig, Ann. Phys. 392, 63-70 (2018).
  * (27) P. K. F. Kuhfittig, Pra. J. Phys. 92, 75 (2019).
  * (28) S. K. Maurya, S. D. Maharaj, Eur. Phys. J. A 54, 68 (2018).
  * (29) S. K. Maurya, Y. K. Gupta, F. Rahaman, M. Rahaman, A. Banerjee, Ann. Phys. 385, 532 (2017).
  * (30) Y. K. Gupta, R. S. Gupta, Gen. Rel. Grav. 6, 641 (1986).
  * (31) Y. K. Gupta, J. R. Sharma, Gen. Rel. Grav. 28, 1447 (1996).
  * (32) K. R. Karmarkar, Proc. Ind. Aca. Sci. A 27, 56 (1948).
  * (33) G. Mustafa, M. F. Shamir, X. T. Cheng, Phys. Rev. D 101, 10, 104013 (2020).
  * (34) S. K. Maurya, Y. K. Gupta, B. Dayanadan, S. Ray, Eur. Phys. J. C 76, 266 (2016).
  * (35) G. Abbas, S. Qaisar, W. Javed, Can. J. Phys. 97, 4, 374-381 (2019).
  * (36) S. Waheed, G. Mustafa, M. Zubair, A. Ashraf, Sym. 12, 6, 962 (2020).
  * (37) S. K. Maurya, Y. K. Gupta, S. Ray, D. Deb, Eur. Phys. J. C 76, 693 (2016).
  * (38) P. Bhar, Eur. Phys. J. Plus 132, 274 (2017).
  * (39) M. Sharif, S. Saba, Chin. J. Phys. 64, 374-389 (2020).
  * (40) J. Upreti et al., New Astro. 80, 101403 (2020).
  * (41) S. K. Maurya, Y. K. Gupta, Saibal Ray, S. R. Chowdhury, Eur. Phys. J. C 75, 389 (2015).
  * (42) P. Bhar, K. N. Singh, T. Manna, Int. J. Mod. Phys. D 26, 1750090 (2017).
  * (43) A. K. Prasad et al., Astrophys. Spa. Sci. 364, 66 (2019).
  * (44) G. Mustafa, M. F. Shamir, M. Ahmad, Phys. Dark Uni. 30, 100652 (2020).
  * (45) S. Nojiri, S. D. Odintsov, Phys. Rev. D 74, 086005 (2006).
  * (46) M. K. Gokhroo, A. L. Mehra, Gen. Rel. Grav. 26, 75 (1994).
  * (47) K. N. Singh, N. Pant, Eur. Phys. C 76, 524 (2016).
  * (48) E. V. Linder, Phys. Rev. D 80, 123528 (2009).
  * (49) T. Faulkner, M. Tegmark, E. F. Bunn, Y. Mao, Phys. Rev. D 76, 063505 (2007).
  * (50) L. Amendola, D. Polarski, S. Tsujikawa, Phys. Rev. Lett. 98, 131302 (2007).
  * (51) V. Faraoni, Phys. Rev. D 81, 044002 (2010).
  * (52) A. V. Astashenok, S. Capozziello, S. D. Odintsov, J. Cos. Ast. Phys. 12, 040 (2013).
  * (53) A. Cooney, S. D. Deo, D. Psaltis, Phys. Rev. D 82, 064033 (2010).
  * (54) A. Ganguly, R. Gannouji, R. Goswami, S. Ray, Phys. Rev. D 89, 064019 (2014).
  * (55) D. Momeni, R. Myrzakulov, Int. J. Geo. Met. Mod. Phys. 12, 1550014 (2015).
  * (56) T. Gangopadhyay, S. Ray, X. D. Li, J. Dey, M. Dey, Mon. Not. R. Ast. Soc. 431, 3216 (2013).
  * (57) C. G. Bohmer, T. Harko, Cla. Qua. Grav. 23, 6479 (2006).
  * (58) Z. Yousaf, Eur. Phys. J. Plus 132, 276 (2017).
  * (59) L. Herrera, Phys. Lett. A 165, 206 (1992).
  * (60) H. Andreasson, Com. Math. Phys. 288, 715 (2009).
  * (61) M. K. Mak, T. Harko, Proc. R. Soc. Lon. A 459, 393 (2003).
  * (62) H. A. Buchdahl, Phys. Rev. 116, 1027 (1959).
  * (63) H. Bondi, Pro. R. Soc. Lon. A 281, 39 (1964).
  * (64) S. Chandrasekhar, Astrophys. J. 140, 417 (1964).
  * (65) H. Heintzmann, W. Hillebrandt, Ast. Astrophys. 38, 51 (1975).
  * (66) W. Hillebrandt, K. O. Steinmetz, Ast. Astrophys. 53, 283 (1976).
  * (67) D. Horvat, S. Ilijic, A. Marunovic, Cla. Qua. Grav. 28, 025009 (2011).
  * (68) D. D. Doneva, S. S. Yazadjiev, Phys. Rev. D 85, 124023 (2012).
  * (69) H. O. Silva, C. F. B. Macedo, E. Berti, L. C. B. Crispino, Cla. Qua. Grav. 32, 145008 (2015).
  * (70) I. Bombaci, Ast. Astrophys. 305, 871 (1996).


# A remark on Wu’s remark on
holomorphic sectional curvature

Gunnar Þór Magnússon  Hafnarfjörður, Iceland<EMAIL_ADDRESS>
Wu [3] proved that if $g$ and $h$ are Hermitian metrics on a complex manifold
and their holomorphic sectional curvatures satisfy $H_{g}\leq-K_{g}<0$ and
$H_{h}\leq-K_{h}<$ then the holomorphic sectional curvature of their sum
satisfies

$H_{g+h}\leq\frac{-K_{g}K_{h}}{K_{g}+K_{h}}<0.$

The proof shows that given any tangent vector $\xi\in T_{X,x}$ there exists a
holomorphic embedding $f:D\to X$ of the unit disk in $X$ such that $f(0)=x$
and $f^{\prime}(0)=\xi$ and such that the holomorphic sectional curvature of a
given metric at $\xi$ is the curvature of the pullback metric to the disk at
its center. Thus we reduce to the case of one complex variable, which had
already been proven by Grauert and Reckziegel [1].

In this note we point out that there is a direct route to Wu’s theorem by
using an expression for the curvature tensor of the sum of Hermitian metrics
one obtains from Griffiths’ theorem on the curvatures of sub- and quotient
bundles [2]. Our starting point is the following result, which is given as an
exercise in Zheng’s lovely textbook [4].

######  Proposition 1.

Let $E\to X$ be a holomorphic vector bundle over a complex manifold. Let $g$
and $h$ be Hermitian metrics on $E$. Then the curvature tensor of $g+h$ is

$R_{g+h}=R_{g}+R_{h}-\sigma^{*}q,$

where $q$ is a Hermitian metric on $E$ and
$\sigma(\xi)s=D_{g,\xi}s-D_{h,\xi}s$.

###### Proof.

Consider short exact sequence

$0\longrightarrow E\longrightarrow E\oplus E\longrightarrow E\longrightarrow
0,$

where the first nonzero arrow is $j(s)=s\oplus s$ and the second is
$\pi(s\oplus t)=s-t$. We equip $E\oplus E$ with the metric $g\oplus h$, the
subbundle with its pullback $g+h$, and the quotient with the induced metric
$q$. The second fundamental form of $E$ in $E\oplus E$ is then

$\sigma(\xi)s=\pi(D_{g\oplus h,\xi}j(s))=D_{g,\xi}s-D_{h,\xi}s.$

The curvature tensor of $g+h$ is then $j^{*}R_{g\oplus h}-\sigma^{*}q$, which
unravels to what we wanted to show. ∎

It is possible to write down exactly what this “quotient” metric $q$ is; in
fact

$q=((g+h)^{-1}h)^{*}g+((g+h)^{-1}g)^{*}h,$

where we view Hermitian metrics on $T_{X}$ as smooth isomorphisms
$T_{X}\to\overline{T}_{X}$. We won’t need to know this, but it is useful for
analyzing what happens when one of the metrics varies and leads to the basic
estimate $q\leq g+h$.

######  Theorem 2 (Wu [3]).

Let $g$ and $h$ be Hermitian metrics on a complex manifold $X$ and assume both
have nonpositive holomorphic sectional curvature. Then the holomorphic
sectional curvature of $g+h$ is nonpositive, and satisfies

$H_{g+h}\leq\frac{H_{g}H_{h}}{H_{g}+H_{h}}<0$

if the holomorphic sectional curvatures of $g$ and $h$ are negative.

###### Proof.

We know that

$R_{g+h}=R_{g}+R_{h}-\sigma^{*}q$

and that $q$ is positive-definite. Then

$H_{g+h}(\xi)\leq\frac{H_{g}(\xi)|\xi|^{4}_{g}+H_{h}(\xi)|\xi|^{4}_{h}}{|\xi|^{4}_{g+h}}=\frac{H_{g}(\xi)|\xi|^{4}_{g}+H_{h}(\xi)|\xi|^{4}_{h}}{(|\xi|^{2}_{g}+|\xi|^{2}_{h})^{2}}.$

Thus $H_{g+h}\leq 0$ if $H_{g}\leq 0$ and $H_{h}\leq 0$.

As Wu notes, for any positive real numbers $a,b,x,y$ we have

$\frac{xy}{x+y}\leq\frac{xa^{2}+yb^{2}}{(a+b)^{2}}$

by elementary algebra. The inequality is reversed if $x$ and $y$ are negative.
If the holomorphic sectional curvatures of $g$ and $h$ are negative we thus
get

$H_{g+h}\leq\frac{H_{g}H_{h}}{H_{g}+H_{h}}<0.\qed$

One can show that if a metric on a compact manifold has negative (or positive)
holomorphic sectional curvature, then so does any sufficiently small
deformation of the metric. On a Kähler manifold this implies that the set of
Kähler classes with a representative with negative holomorphic sectional
curvature is an open cone inside the Kähler cone. Wu’s theorem further shows
that convex combinations of metrics in this cone are again in it, so the cone
is connected. It would be unexpected if these two cones were the same but I
don’t know of an example where they differ.

## References

  * [1] Hans Grauert and Helmut Reckziegel.  Hermitesche metriken und normale familien holomorpher abbildungen.  Mathematische Zeitschrift, 89(2):108–125, 1965.
  * [2] Phillip A Griffiths.  Hermitian differential geometry and the theory of positive and ample holomorphic vector bundles.  J. Math. Mechanics, 14(1):117–140, 1965.
  * [3] H Wu.  A remark on holomorphic sectional curvature.  Indiana University Mathematics Journal, 22(11):1103–1108, 1973\.
  * [4] Fangyang Zheng.  Complex differential geometry.  Number 18 in AMS/IP Studies in Advanced Mathematics. Amer. Math. Soc., 2000.