# Latent Fisher Discriminant Analysis

Gang Chen
Department of Computer Science and Engineering
SUNY at Buffalo
<EMAIL_ADDRESS>

###### Abstract

Linear Discriminant Analysis (LDA) is a well-known method for dimensionality
reduction and classification. Previous studies have also extended the binary-
class case into multi-classes. However, many applications, such as object
detection and keyframe extraction cannot provide consistent instance-label
pairs, while LDA requires labels on instance level for training. Thus it
cannot be directly applied for semi-supervised classification problem. In this
paper, we overcome this limitation and propose a latent variable Fisher
discriminant analysis model. We relax the instance-level labeling into bag-
level, is a kind of semi-supervised (video-level labels of event type are
required for semantic frame extraction) and incorporates a data-driven prior
over the latent variables. Hence, our method combines the latent variable
inference and dimension reduction in an unified bayesian framework. We test
our method on MUSK and Corel data sets and yield competitive results compared
to the baseline approach. We also demonstrate its capacity on the challenging
TRECVID MED11 dataset for semantic keyframe extraction and conduct a human-
factors ranking-based experimental evaluation, which clearly demonstrates our
proposed method consistently extracts more semantically meaningful keyframes
than challenging baselines.

##  1 Introduction

Linear Discriminant Analysis (LDA) is a powerful tool for dimensionality
reduction and classification that projects high-dimensional data into a low-
dimensional space where the data achieves maximum class separability [10, 12].
The basic idea in classical LDA, known as the Fisher Linear Discriminant
Analysis (FDA) is to obtain the projection matrix by minimizing the within-
class distance and maximizing the between-class distance simultaneously to
yield the maximum class discrimination. It has been proved analytically that
the optimal transformation is readily computed by solving a generalized
eigenvalue problem [12]. In order to deal with multi-class scenarios [10], LDA
can be easily extended from binary-case and generally used to find a subspace
with $d-1$ dimensions for multi-class problems, where $d$ is the number of
classes in the training dataset. Because of its effectiveness and
computational efficiency, it has been applied successfully in many
applications, such as face recognition [3] and microarray gene expression data
analysis. Moreover, LDA was shown to compare favorably with other supervised
dimensionality reduction methods through experiments [20].

However, LDA expects instance/label pairs which are surprisingly prohibitive
especially for large training data. In the last decades, semi-supervised
methods have been proposed to utilize unlabeled data to aid classification or
regression tasks under situations with limited labeled data, such as
Transductive SVM (TSVM) [21, 14] and Co-Training [4]. Correspondently, it is
reasonable to extend the supervised LDA into a semi-supervised method, and
many approaches [5, 26, 20] have been proposed. Most of these methods are
based on transductive learning. In other words, they still need instance/label
pairs. However, many real applications require bag-level labeling [1], such as
object detection [11] and event detection [19].

In this paper, we propose a Latent Fisher Discriminant Analysis model (or LFDA
in short), which generalizes Fisher LDA model [10]. Our model is inspired by
MI-SVM [1] or latent SVM [11] and multiple instance learning problems [8, 17].
On the one hand, recently applications in image and video analysis require a
kind of bag-level label. Moreover, using latent variable model for this kind
of problem shows great improvement on object detection [11]. On the other
hand, the requirement of instance/label pairs in the training data is
surprisingly prohibitive especially for large training data. The bag-level
labeling methods are a good solution to this problem.

MI-SVM or latent SVM is a kind of discriminative model by maximizing a
posterior probability. Our model unify the discriminative nature of the Fisher
linear discriminant with a data driven Gaussian mixture prior over the
training data in the Bayesian framework. By combining these two terms in one
model, we infer latent variables and projection matrix in alternative way
until convergence. We demonstrate this capability on MUSK and Corel data sets
for classification, and on TRECVID MED11 dataset for keyframe extraction on
five video events [19].

##  2 Related Work

Linear Discriminant Analysis (LDA) has been a popular method for dimension
reduction and classification. It searches a projection matrix that
simultaneously maximizes the between-class dissimilarity and minimizes the
within-class dissimilarity to increase class separability, typically for
classification applications. LDA has attracted an increasing amount of
attention in many applications because of its effectiveness and computational
efficiency. Belhumeur et al proposed PCA+LDA [3] for face recognition. Chen et
al projects the data to the null space of the within-class scatter matrix and
then maximizes the between-class scatter in this space [6] to deal with the
situation when the size of training data is smaller than the dimensionality of
feature space. [22] combines the ideas above, maximizes the between-class
scatter matrix in the range space and the null space of the within-class
scatter matrix separately and then integrates the two parts together to get
the final transformation. [25] is also a two-stage method which can be divided
into two steps: first project the data to the range space of the between-class
scatter matrix and then apply LDA in this space. To deal with non-linear
scenarios, the kernel approach [21] can be applied easily via the so-called
kernel trick to extend LDA to its kernel version, called kernel discriminant
analysis [2], that can project the data points nonlinearly. Recently, sparsity
induced LDA is also proposed [18].

However, many real-world applications only provide labels on bag-level, such
as object detection and event detection. LDA, as a classical supervised
learning method, requires a training dataset consisting of instance and label
pairs, to construct a classifier that can predict outputs/labels for novel
inputs. However, directly casting LDA as a semi-supervised method is
challenging for multi-class problems. Thus, in the last decades, semi-
supervised methods become a a hot topic. One of the main trend is to extend
the supervised LDA into a semi-supervised method [5, 26, 20], which attempts
to utilize unlabeled data to aid classification or regression tasks under
situations with limited labeled data. [5] propose a novel method, called Semi-
supervised Discriminant Analysis, which makes use of both labeled and
unlabeled samples. The labeled data points are used to maximize the
separability between different classes and the unlabeled data points are used
to estimate the intrinsic geometric structure of the data. [20] propose a
semi-supervised dimensionality reduction method which preserves the global
structure of unlabeled samples in addition to separating labeled samples in
different classes from each other. Most of these semi-supervised methods model
the geometric relationships between all data points in the form of a graph and
then propagate the label information from the labeled data points through the
graph to the unlabeled data points. Another trend prefers to extent LDA into
an unsupervised senarios. For example, Ding et al propose to combine LDA and
K-means clustering into the LDA-Km algorithm [9] for adaptive dimension
reduction. In this algorithm, K-means clustering is used to generate class
labels and LDA is utilized to perform subspace selection.

Our solution is a new latent variable model called Latent Fisher Discriminant
Analysis (LFDA), which complements existing latent variable models that have
been popular in the recent vision literature [11] by making it possible to
include the latent variables into Fisher discriminant analysis model. Unlike
previous latent SVM [11] or MI-SVM [1] model, we extend it with prior data
distribution to maximize a joint probability when inferring latent variables.
Hence, our method combines the latent variable inference and dimension
reduction in an unified Bayesian framework.

##  3 Latent Fisher discriminant analysis

We propose a LFDA model by including latent variables into the Fisher
discriminant analysis model. Let ${\mathcal{X}}=\\{{\bf x_{1}},{\bf
x_{2}},...,{\bf x_{n}}\\}$ represent $n$ bags, and the corresponding labels
${\mathcal{L}}=\\{l_{1},l_{2},...,l_{n}\\}$. For each ${\bf
x_{i}}\in\mathcal{X}$, ${\bf x_{i}}$ can be treated as a bag (or video) in
[1], and its label $l_{i}$ is categorical and assumes values in a finite set,
e.g. $\\{1,2,...,C\\}$. Let ${\bf x_{i}}\in\mathbb{R}^{d\times n_{i}}$, which
means it contains $n_{i}$ instances (or frames), ${\bf
x_{i}}=\\{x_{i}^{1},x_{i}^{2},...,x_{i}^{n_{i}}\\}$, and its $j^{th}$ instance
is a vector in $\mathbb{R}^{d}$, namely $x_{i}^{j}\in\mathbb{R}^{d}$. Fisher’s
linear discriminant analysis pursue a subspace $\mathcal{Y}$ to separate two
or more classes. In other words, for any instance $x\in\mathcal{X}$, it
searches for a projection $f:x\rightarrow y$, where
$y\in\mathbb{R}^{d^{\prime}}$ and $d^{\prime}\leq d$. In general, $d^{\prime}$
is decided by $C$, namely $d^{\prime}=C-1$. Suppose the projection matrix is
$\mathcal{P}$, and $y=f(x)=\mathcal{P}x$, then latent Fisher LDA proposes to
minimize the following ratio:

$\displaystyle\centering({\mathcal{P}^{*}})=\operatornamewithlimits{arg\,min}_{\mathcal{P},z}J({\mathcal{P},z})=\operatornamewithlimits{arg\,min}_{\mathcal{P},z}\textrm{trace}\left(\frac{\mathcal{P}^{T}\Sigma_{w}(x,z)\mathcal{P}}{\mathcal{P}^{T}\Sigma_{b}(x,z)\mathcal{P}}+\beta\mathcal{P}^{T}\mathcal{P}\right)\@add@centering$
(1)

where $z$ is the latent variable, $\beta$ is an weighting parameter for
regularization term. The set $z\in Z(x)$ defines the possible latent values
for a sample $x$. In our case, $z\in\\{1,2,...,C\\}$. $\Sigma_{b}(x,z)$ is
between class scatter matrix and $\Sigma_{w}(x,z)$ is within class scatter
matrix. However, LDA is dependent on a categorical variable (i.e. the class
label) for each instance $x$ to compute $\Sigma_{b}$ and $\Sigma_{w}$. In our
case, we only know bag-level labels, not on instance-level labels. To minimize
$J({\mathcal{P}},z)$, we need to solve $z(x)$ for any given $x$. This problem
is a chicken and egg problem, and can be solved by alternating algorithms,
such as EM [7]. In other words, solve $\mathcal{P}$ in Eq. (1) with fixed $z$,
and vice versa in an alternating strategy.

###  3.1 Updating $z$

Suppose we have found the projection matrix $\mathcal{P}$, and corresponding
subspace $\mathcal{Y}=\mathcal{P}\mathcal{X}$, where $\mathcal{Y}=\\{{\bf
y_{1}},{\bf y_{2}},...,{\bf y_{n}}\\}$. Instead of inferring latent variables
at instance-level in latent SVM, we propose latent variable inference at
clustering-level in the projected space $\mathcal{Y}$. That means all elements
in the same cluster have same label. Such assumption is reasonable because
elements in the same cluster are close to each other. On the other hand,
cluster-level inference can speed up the learning process. We extend mixture
discriminative analysis model in [13] by incorporating latent variables over
all instances for an given class. As in [13], we assume each class $i$ is a
$K$ components of Gaussians,

$\centering
p(x|\lambda_{i})=\sum_{j=1}^{K}\pi_{i}^{j}g(x|\mu_{i}^{j},\Sigma_{i}^{j})\@add@centering$
(2)

where $x$ is a $d$-dimensional continuous-valued data vector (i.e. measurement
or features); $\pi_{i}=\\{\pi_{i}^{j}\\}_{j=1}^{K}$ are the mixture weights,
and $g(x|\mu_{i}^{j},\Sigma_{i}^{j})$, $j\in[1,K]$, are the component Gaussian
densities with $\mu_{i}=\\{\mu_{i}^{j}\\}_{j=1}^{K}$ as mean and
$\Sigma_{i}=\\{\Sigma_{i}^{j}\\}_{j=1}^{K}$ as covariance. $\lambda_{i}$ is
the parameters for class $i$ which we need to estimate,
$\lambda_{i}=\\{\pi_{i},\mu_{i},\Sigma_{i}\\}$.

Hence, for each class $i\in\\{1,2,\dots,C\\}$, we can estimate $\lambda_{i}$
and get its $K$ subsets $S_{i}=\\{S_{i}^{1},S_{i}^{2},...,S_{i}^{K}\\}$ with
EM algorithm. Suppose we have the discriminative weights (or posterior
probability) for the $K$ centers in each class,
$w_{i}=\\{w_{i}^{1},w_{i}^{2},...,w_{i}^{K}\\}$, which are the posterior
probability determined by the latent FDA and will be discussed later. We
maximize one of the following two equations:

Maximizing a posterior probability:
$\displaystyle\centering\mu_{i}^{j}=\operatornamewithlimits{arg\,max}_{\mu_{i}^{j}\in\mu_{i},j\in[1,K]}w_{i}=\operatornamewithlimits{arg\,max}_{\mu_{i}^{j}\in\mu_{i},j\in[1,K]}p(z_{i}|\mu_{i},\mathcal{P})\@add@centering$
(3a)

Maximizing the joint probability with prior:

$\displaystyle\centering\mu_{i}^{j}=\operatornamewithlimits{arg\,max}_{\mu_{i}^{j}\in\mu_{i},j\in[1,K]}(\pi_{i}\circ
w_{i})\@add@centering$ (3b)

where $z_{i}$ is the latent label assignment, $\pi_{i}$ is the prior
clustering distributions for $\lambda_{i}$ in class $i$, $w_{i}$ is the
posterior (or weight) determined by kNN voting (see further) in the subspace
and $\circ$ is the pointwise production or Hadamard product. We treat Eq. (3a)
as the latent Fisher discriminant analysis model (LFDA), because it takes the
same strategy as the latent SVM model [1, 11]. As for Eq. (3b), we extend LFDA
by combining the both factors (representative and discriminative) together,
and find the cluster $S_{i}^{j}$ in class $i$ by maximizing Eq. (3b). In a
sense, Eq. (3b) considers the prior distribution from the training dataset,
thus, we treat it as the joint latent Fisher discriminant analysis model
(JLFDA) or LFDA with prior. In the nutshell, we propose a way to formulate
discriminative and generative methods under the unified Bayesian framework. We
comparatively analyze both of these models (Section 4).

Consequently, if we select the cluster $S_{i}^{j}$ with the mean $\mu_{i}^{j}$
which maximizes the above equation for class $i$, we can relabel all samples
$x$ positive for class $i$ and the rest negative, subject to
$y={\mathcal{P}}x$ and $y\in S_{i}^{j}$. Then, we construct a new training
data ${\mathcal{X^{+}}}=\\{{\bf x_{1}^{+}},{\bf x_{2}^{+}},...,{\bf
x_{n}^{+}}\\}$, with labels ${\mathcal{L^{+}}}=\\{{\bf z_{1}^{+}},{\bf
z_{2}^{+}},...,{\bf z_{n}^{+}}\\}$, where ${\bf x_{i}^{+}}=S_{i}^{j}$ for
class $i$ with $n_{i}^{j^{\prime}}$ elements, and its labels ${\bf
z_{i}^{+}}=\\{z_{i}^{1},z_{i}^{2},...,z_{i}^{n_{i}^{j^{\prime}}}\\}$ on
instance level. Obviously, ${\bf x_{i}^{+}}\subseteq{\bf x_{i}}$ and
${\mathcal{X^{+}}}$ is a subset of ${\mathcal{X}}$. The difference between
${\mathcal{X^{+}}}$ and ${\mathcal{X}}$ lies that every element
$x_{i}^{+}\in{\bf x_{i}^{+}}$ has label $z(x_{i}^{+})$ decided by Eq.
(LABEL:eq:Eq3sub), while ${\bf x_{i}}\subset\mathcal{X}$ only has bag level
label.

###  3.2 Updating projection $\mathcal{P}$

When we have labels for the new training data ${\mathcal{X^{+}}}$, we use the
Fisher LDA to minimize $J({\mathcal{P}},z)$. Note that Eq. (1) is invariant to
the scale of the vector $\mathcal{P}$. Hence, we can always choose
$\mathcal{P}$ such that the denominator is simply
$\mathcal{P}^{T}\Sigma_{b}\mathcal{P}={\bf 1}$. For this reason we can
transform the problem of minimizing Eq. (1) into the following constrained
optimization problem [10, 12, 24]:

$\displaystyle\centering{\mathcal{P}^{*}}\@add@centering$
$\displaystyle=\operatornamewithlimits{arg\,min}_{{\mathcal{P}}}\textrm{trace}\left(\mathcal{P}^{T}\Sigma_{w}(x,z)\mathcal{P}+\beta\mathcal{P}^{T}\mathcal{P}\right)$
$\displaystyle\quad\textrm{s. t.
}{\mathcal{P}^{T}\Sigma_{b}(x,z)\mathcal{P}}={\bf 1}$ (4)

where 1 is the identity matrix in $\mathbb{R}^{d^{\prime}\times d^{\prime}}$.
The optimal Multi-class LDA consists of the top eigenvectors of
$(\Sigma_{w}(x,z)+\beta)^{\dagger}\Sigma_{b}(x,z)$ corresponding to the
nonzero eigenvalues [12], here $(\Sigma_{w}(x,z)+\beta)^{\dagger}$ denotes the
pseudo-inverse of $\Sigma_{w}(x,z)+\beta$. After we calculated $\mathcal{P}$,
we can project ${\mathcal{X^{+}}}$ into subspace ${\mathcal{Y^{+}}}$. Note
that in the subspace ${\mathcal{Y^{+}}}$, any $y^{+}\in{\mathcal{Y^{+}}}$
preserves the same labels as in the original space. In other words,
${\mathcal{Y^{+}}}$ has corresponding labels ${\mathcal{L^{+}}}$ at element
level, namely $z(y^{+})=z(x^{+})$.

In general, multi-class LDA [24] uses kNN to classify new input data. We
compute $w_{i}$ using the following kNN strategy: for each sample
$x\in{\mathcal{X}}$, we get $y=\mathcal{P}x$ by projecting it into subspace
${\mathcal{Y}}$. Then, for $y\in{\mathcal{Y}}$, we choose its $N$ nearest
neighbors from ${\mathcal{Y^{+}}}$, and use their labels to voting each
cluster $S_{i}^{j}$ in each class $i$. Then, we compute the following
posterior probability:

$\displaystyle\centering w_{i}^{j}\@add@centering$
$\displaystyle=p(z_{i}=1|\mu_{i}^{j})=p(\mu_{i}^{j}|z_{i}=1)p(z_{i}=1)$
$\displaystyle=p(z_{i}=1)\frac{p(\mu_{i}^{j},z_{i}=1)}{\sum_{i=1}^{C}p(\mu_{i}^{j},z_{i}=1)}$
(5)

It counts all $y\in{\mathcal{Y}}$ fall into $N$ nearest neighbor of
$\mu_{i}^{j}$ with label $z_{i}$. Note that kNN is widely used as the
classifier in the subspace after LDA transformation. Thus, Eq. (3.2) consider
all training data to vote the weight for each discriminative cluster
$S_{i}^{j}$ in every class $i$. Hence, we can find the most discriminative
cluster $S_{i}^{j}$, s.t. $w_{i}^{j}>w_{i}^{k}$, $k\in[1,K],k\neq j$.

Algorithm. We summarize the above discussion in pseudo code. To put simply, we
update ${\mathcal{P}}$ and $z$ in an alternative manner, and accept the new
projection matrix ${\mathcal{P}}$ with LDA on the relabeled instances. Such
algorithm can always convenge in around 10 iterations. After we learned matrix
$\mathcal{P}$ and $\\{\lambda_{i}\\}_{i=1}^{C}$ by maximizing Eq. (3), we can
use them to select representative and discriminative frames from video
datasets by nearest neighbor searching.

Algorithm 1

Input: training data ${\mathcal{X}}$ and its labels $\mathcal{L}$ at video
level, $\beta$, $K$, $N$, $T$ and $\epsilon$.
Output: $\mathcal{P}$, $\\{\lambda_{i}\\}_{i=1}^{C}$

1: Initialize $\mathcal{P}$ and $w_{i}$;

2: for $Iter=1$ to $T$ do

3: for $i=1;i<=C;i++$ do

4: Project all the training data ${\mathcal{X}}$ into subspace ${\mathcal{Y}}$
using ${\mathcal{Y}}$=$\mathcal{P}{\mathcal{X}}$;

5: For each class $i\in[1,C]$, using Gaussian mixture model to partition its
elements in the subspace, and compute $\lambda_{i}$ =
$\\{\pi_{i},\mu_{i},\Sigma_{i}\\}$;

6: Maximize Eq. (3) to find $S_{i}^{j}$ with center $\mu_{i}^{j}$;

7: Relabel all elements positive in the cluster $S_{i}^{j}$ for class $i$
according to Eq. (LABEL:eq:Eq3sub);

8: end for

9: Update $z$ and construct the new subset ${\mathcal{X^{+}}}$ and its labels
$\mathcal{L}^{+}$ for all $C$ classes;

10: Do Fisher linear discriminant analysis and update $\mathcal{P}$

11: if $\mathcal{P}$ converge (change less than $\epsilon$), then break

12: Compute $N$ nearest neighbors for each training data, and calculate
discriminative weight $w_{i}$ for each class $i$ according to Eq. (3.2).

13: end for

14: Return ${\mathcal{P}}$ and cluster centers $\\{\lambda_{i}\\}_{i=1}^{C}$
learned respectively for all $C$ classes;

###  3.3 Convergence analysis

Our method updates latent variable $z$ and then $\mathcal{P}$ in an
alternative manner. Such strategy can be attributed to the hard assignment of
EM algorithm. Recall the EM approach:

$\displaystyle\centering\mathcal{P^{*}}=\operatornamewithlimits{arg\,max}_{\mathcal{P}}p(\mathcal{X},\mathcal{L}|\mathcal{P})=\operatornamewithlimits{arg\,max}_{\mathcal{P}}\sum_{i=1}^{C}p(\mathcal{X},\mathcal{L}|z_{i})p(z_{i}|\mathcal{P})\@add@centering$
(6)

then the likelihood can be optimized using iterative use of the EM algorithm.

###### Theorem 3.1

Assume the latent variable $z$ is inferred for each instance in $\mathcal{X}$,
then to maximize the above function is equal to maximize the following
auxiliary function

$\displaystyle\centering\mathcal{P}=\operatornamewithlimits{arg\,max}_{\mathcal{P}}\sum_{i=1}^{C}p(z_{i}|\mathcal{X},\mathcal{L},\mathcal{P}^{\prime})ln\bigg{(}p(\mathcal{X},\mathcal{L}|z_{i})p(z_{i}|\mathcal{P})\bigg{)}\@add@centering$
(7)

This proof can be shown using Jensen’s inequality.

###### Lemma 3.2

The hard assignment of latent variable $z$ by maximizing Eq. (3) is a special
case of EM algorithm.

  * Proof

$\displaystyle\sum_{i=1}^{C}p(z_{i}|\mathcal{X},\mathcal{L},\mathcal{P}^{\prime})ln\bigg{(}p(\mathcal{X},\mathcal{L}|z_{i})p(z_{i}|\mathcal{P})\bigg{)}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{C}p(z_{i}|\mathcal{X},\mathcal{L},\mathcal{P}^{\prime})ln(p(z_{i}|\mathcal{X},\mathcal{L},P))+\sum_{i=1}^{C}p(z_{i}|\mathcal{X},\mathcal{L},\mathcal{P}^{\prime})ln(p(\mathcal{X},\mathcal{L}|\mathcal{P}))$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{C}p(z_{i}|\mathcal{X},\mathcal{L},\mathcal{P}^{\prime})ln(p(z_{i}|\mathcal{X},\mathcal{L},\mathcal{P}))+ln(p(\mathcal{X},\mathcal{L}|\mathcal{P}))$
(8)

Given $\mathcal{P}^{\prime}$, we can infer the latent variable $z$. Because
the hard assignment of $z$, the first term in the right hand side of Eq.
(Proof) assigns $z_{i}$ into one class. Note that
$p(z|\mathcal{X},\mathcal{L},\mathcal{P})ln(p(z|\mathcal{X},\mathcal{L},\mathcal{P}))$
is a monotonically increasing function, which means that by maximizing the
posterior likelihood $p(z|\mathcal{X},\mathcal{L},\mathcal{P})$ for each
instance, we can maximize Eq. (Proof) for the hard assignment case in Eq. (3).
Thus, the updating strategy in our algorithm is a special case of EM
algorithm, and it can converge into a local maximum as EM algorithm. Note that
in our implement, we infer the latent variable in cluster level. In other
words, to maximize $p(z_{i}|\mathcal{X},\mathcal{L},\mathcal{P}^{\prime})$, we
can include another latent variable $\pi_{j},j\in[1,K]$. In other words, we
need to maximize
$\sum_{j=1}^{K}p(z_{i},\pi_{j}|\mathcal{X},\mathcal{L},\mathcal{P}^{\prime})$,
which we can recursively determine the latent variable $\pi_{i}$ using an
embedded EM algorithm. Hence, our algorithm use two steps of EM algorithm, and
it can converge to a local maximum. Refer [23] for more details about the
convergence of EM algorithm.

###  3.4 Probabilistic understanding for the model

The latent SVM model [11, 1] propose to label instance $x_{i}$ in positive
bag, by maximize $p(z(x_{i})=1|x_{i})$, which is the optimal Bayes decision
rule.

Figure 1: Example of graphical representation only for one class (event).
$h_{i}^{j}$ is the hidden variable, $x_{i}^{j}$ is the observable input,
$y_{i}^{j}$ is the projection of $x_{i}^{j}$ in the subspace, $j\in[1,n_{i}]$,
and $n_{i}$ is the number of total training data for class $i$. The $K$
cluster centers $\mu_{i}=\\{\mu_{i}^{1},\mu_{i}^{2},...,\mu_{i}^{K}\\}$ is
determined by both $\pi_{i}$ and $w_{i}$. The graphical model of our method is
similar to GMM model in vertical. By adding $z_{i}$ into LDA, the graphical
model can handle latent variables.

Similarly, Eq. (3a) takes the same strategy as latent SVM to maximize a
posterior probability. Moreover, instead of only maximizing the $p(z=1|x)$, we
also maximize the joint probability $p(z=1,x)$, using the Bayes rule,
$p(z=1,x)=p(x)p(z=1|x)$. In this paper, we use Gaussian mixture model to
approximate the prior $p(x)$ in this generative model. We argue that to
maximize a joint probability is reasonable, because it considers both
discriminative (posterior probability) and representative (prior) property in
the video dataset. We give the graphical representation of our model in Fig.
(1).

##  4 Experiments and results

In this section, we perform experiments on various data sets to evaluate the
proposed techniques and compare it to other baseline methods. For all the
experiments, we set $T$ = 20 and $\beta=40$; and initialize uniformly weighted
$w_{i}$ and projection matrix $\mathcal{P}$ with LDA.

###  4.1 Classification on toy data sets

The MUSK data sets111www.cs.columbia.edu/ andrews/mil/datasets.html are the
benchmark data sets used in virtually all previous approaches and have been
described in detail in the landmark paper [8]. Both data sets, MUSK1 and
MUSK2, consist of descriptions of molecules using multiple low-energy
conformations. Each conformation is represented by a 166-dimensional feature
vector derived from surface properties. MUSK1 contains on average
approximately 6 conformation per molecule, while MUSK2 has on average more
than 60 conformations in each bag. The Corel data set consists three different
categories ( elephant , fox , tiger ) , and each instance is represented with
230 dimension features, characterized by color, texture and shape descriptors.
The data sets have 100 positive and 100 negative example images. The latter
have been randomly drawn from a pool of photos of other animals. We first use
PCA reduce its dimension into 40 for our method. For parameter setting, we set
$K$=3, $T$ = 20 and $N=4$ (namely the 4-Nearest-Neighbor (4NN) algorithm is
applied for classification). The averaged results of ten 10-fold cross-
validation runs are summarized in Table (1). We set LDA222we use the bag label
as the instance label to test its performance and MI-SVM as our baseline. We
can observe that both LFDA and JLFDA outperform MI-SVM on MUSK1 and Fox data
sets, while has comparative performance as MI-SVM on the others.

Data set | inst/Dim | MI-SVM | LDA | LFDA | JLFDA
---|---|---|---|---|---
MUSK1 | 476/166 | 77.9 | 70.4 | 81.4 | 87.1
MUSK2 | 6598/166 | 84.3 | 51.8 | 76.4 | 81.3
Elephant | 1391/230 | 81.4 | 70.5 | 74.5 | 79.0
Fox | 1320/230 | 57.8 | 53.5 | 61.5 | 59.5
Tiger | 1220/230 | 84.0 | 71.5 | 74.0 | 80.5
Table 1: Accuracy results for various methods on MUSK and Corel data sets. Our
approach outperform LDA on both datasets, and we get better result than MI-SVM
on MUSK1 and Fox data set.

###  4.2 Semantic keyframe extraction

We conduct experiments on the challenging TRECVID MED11
dataset333http://www.nist.gov/itl/iad/mig/med11.cfm. It contains five events:
attempting a board trick feeding an animal, landing a fish, wedding ceremony
and working on a woodworking project. All of five events consist of a number
of human actions, processes, and activities interacting with other people
and/or objects under different place and time. At this moment, we take 105
videos from 5 events for testing and the remaining 710 videos for training.
For parameters, we set $K=10$ and $N=10$. We learned the representative
clusters for each class, and then use them to find semantic frames in videos
with the same labels. Then we evaluation the semantic frames for each video
through human-factors analysis—the semantic keyframe extraction problem
demands a human-in-the-loop for evaluation. We explain our human factors
experiment in full detail in experiment setup. Our ultimate findings
demonstrate that our proposed latent FDA with prior model is most capable of
extraction semantically meaningful keyframes among latent FDA and competitive
baselines.

Video representation. For all videos, we extract HOG3D descriptors [16] every
25 frames (about sampling a frame per second). To represent videos using local
features we apply a bag-of-words model, using all detected points and a
codebook with 1000 elements.

Benchmark methods. We make use of SVM as the benchmark method in the
experiment. We take the one-vs-all strategy to train a linear MI-SVM
classifier using $SVM^{light}$ [15], which is very fast in linear time, for
each kind of event. Then we choose 10 frames for each video which are far from
the margin and close to the margin on positive side. For the frames chosen
farthest away from the margin, we refer it SVM(1), while for frames closest to
the margin we refer it SVM(2). We also randomly select 10 frames from each
video, and we refer it RAND in our experiments.

Experiment setup Ten highly motivated graduate students (range from 22 to 30
years’ old) served as subjects in all of the following human-in-the-loop
experiments. Each novel subject to the annotation-task paradigm underwent a
training process. Two of the authors gave a detailed description about the
dataset and problem, including its background, definition and its purpose. In
order to indicate what representative and discriminative means for each event,
the two authors showed videos for each kind of event to the subjects, and make
sure all subjects understand what semantic keyframes are. The training
procedure was terminated after the subject’s performance had stabilized. We
take a pairwise ranking strategy for our evaluation. We extract 10 frames per
video for 5 different methods (SVM(1), SVM(2), LFDA, JLFDA and RAND)
respectively. For each video, we had about 1000 image pairs for comparison. We
had developed an interface using Matlab to display two image pair and three
options (Yes, No and Equal) to compare an image pair each time. The students
are taught how to use the software; a trial requires them to give a ranking:
If the left is better than the right, then choose ’Yes’; if the right is
better than the left, choose ’No’. If the two image pair are same, then choose
’Equal’. The subjects are again informed that better means a better semantic
keyframe. The ten subjects each installed the software to their computers, and
conducted the image pair comparison independently. In order to speed up the
annotation process, the interface can randomly sample 200 pairs from the total
1000 image pairs for each video, and we also ask subjects to random choose 10
videos from the test dataset.

Experimental Results We have scores for each image pair. By sampling 10 videos
from each event, we at last had annotations of 104 videos. It means our
sampling videos got from 10 subjects almost cover all test data (105 videos).
Table (2) shows the win-loss matrix between five methods by counting the
pairwise comparison results on all 5 events. It shows that JLFDA and LFDA
always beat the three baseline methods. Furthermore, JLFDA is better than LFDA
because it considers a prior distribution from training data, which will help
JLFDA to find more representative frames. See Fig. (3) for keyframes extracted
with JLFDA.

Figure 2: Comparison of 5 methods for five events. Higher value, better
performance.

Method | Win-Loss matrix
---|---
JLFDA |  FLDA |  RAND |  SVM(1) |  SVM(2)
JLFDA |  - |  3413 |  2274 |  2257 |  3758
LFDA |  2957 |  - |  2309 |  2230 |  3554
RAND |  2111 |  2175 |  - |  1861 |  2274
SVM(1) |  2088 |  2270 |  2010 |  - |  2314
SVM(2) |  3232 |  3316 |  2113 |  2125 |  -

Table 2: Win-Loss matrix for five methods. It represents how many times
methods in each row win methods in column.

We compared the five methods on the basis of Condorcet voting method. We treat
’Yes’, ’No’ and ’Equal’ are voters for each method in the image pairwise
comparison process. If ’Yes’, we cast one ballot to the left method; else if
’No’, we add a ballot to the right method; else do nothing to the two methods.
Fig. (2) shows ballots for each method on each event. It demonstrates our
method JLFDA always beat other methods, except for E004 dataset. We also
compared the five methods based on Elo rating system. For each video, we
ranked the five methods according to Elo ranking system. Then, we counted the
number of No.1 methods in each event. The results in Table (3) show that our
method is better than others, except E004. Such results based on Elo ranking
is consistent with Condorcet ranking method in Fig. (2). E004 is the wedding
ceremony event and our method is consistently outperformed by the SVM baseline
method. We believe this is due to the distinct nature of the E004 videos in
which the video scene context itself distinguishes it from the other four
events (the wedding ceremonies typically have very many people and are
inside). Hence the discriminative component of the methods are taking over,
and the SVM is able to outperform the Fisher discriminant. This effect seems
more likely due to the nature of the five events in the data set than the
proposed method intrinsically.

Method | the number of No.1 method in each event
---|---
E001 | E002 | E003 | E004 | E005
JLFDA | 6 | 7 | 7 | 3 | 7
LFDA | 6 | 4 | 4 | 5 | 1
SVM(1) | 4 | 4 | 4 | 2 | 4
SVM(2) | 6 | 3 | 1 | 7 | 6
RAND | 2 | 2 | 4 | 3 | 2
Table 3: For each video, we ranked the five methods according to Elo ranking
system. Then, we counted the number of No.1 method one video level in each
event. For example, E002 has total 20 videos, and JLFDA has rank first on 7
videos, while RAND has rank first on only two videos. Higher value, better
results. It demonstrates that our method is more capable at extracting
semantically meaningful keyframes.
---
Figure 3: Sample keyframes from the first five events (each row from top to
down): (a) snowboard trick, (b) feeding animal, (c) fishing, (d) marriage
ceremony, and (e) wood making. Each row indicates sample results from the same
videos for each event. It shows that our method can extract the representative
and discriminative images for each kind of events. In other words, we can
decide what’s happen when we scan the images.

##  5 Conclusion

In this paper, we have presented a latent Fisher discriminant analysis model,
which combines the latent variable inference and dimension reduction in an
unified framework. Ongoing work will extend the kernel trick into the model.
We test our method on classification and semantic keyframe extraction problem,
and yield quite competitive results. To the best of our knowledge, this is the
first paper to study the extraction of semantically representative and
discriminative keyframes—most keyframe extraction and video summarization
focus on representation summaries rather than jointly representative and
discriminative ones. We have conducted a thorough ranking-based human factors
experiment for the semantic keyframe extraction on the challenging TRECVID
MED11 data set and found that our proposed methods are able to consistently
outperform competitive baselines.

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# Energy extraction from Janis-Newman-Winicour naked singularity

Vishva Patel<EMAIL_ADDRESS>International Center for Cosmology,
Charusat University, Anand-388421 (Gujarat), India  Kauntey Acharya
<EMAIL_ADDRESS>International Center for Cosmology, Charusat
University, Anand-388421 (Gujarat), India  Parth Bambhaniya
<EMAIL_ADDRESS>International Center for Cosmology, Charusat University,
Anand-388421 (Gujarat), India  PDPIAS, Charusat University, Anand-388421
(Gujarat), India  Pankaj S. Joshi<EMAIL_ADDRESS>International Center
for Cosmology, Charusat University, Anand-388421 (Gujarat), India
International Centre for Space and Cosmology, School of Arts and Sciences,
Ahmedabad University, Ahmedabad-380009 (Gujarat), India

###### Abstract

In general, energy extraction methods such as the Penrose process and the
magnetic Penrose process are thought to be reliant on the existence of an
ergoregion. Inside an ergoregion, there are negative energy states that allow
a particle to extract energy and escape to an observer at infinity. In this
paper, we considered the electromagnetic field in the rotating Janis-Newman-
Winicour (JNW) spacetime. This concept is feasible because an accretion disc
forms an electromagnetic field around compact objects. After that, we briefly
examine negative energy orbits and their significance in energy extraction.
The ergoregion is absent in a rotating JNW geometry, but we show that the
effective ergoregion is there. The change in a negative energy orbit
concerning the magnetic field (B), spin parameter (a), and electric charge (Q)
is analyzed. We find that the total energy extraction efficiency within this
process can be around $60\%$ for the rotating JNW naked singularity.

Key words: Penrose process, Negative energy orbits, Ergoregion, Magnetic
Penrose process

##  I Introduction

The major concerns in high-energy astrophysics these days are the powering of
active galactic nuclei, X-ray binaries, and quasars. To understand the reason
behind these high-energy phenomena such as Gamma Ray Bursts, Fast Radio
Bursts, and the formation of jets, many efforts have been made by researchers
by studying the energy extraction mechanisms. These mechanisms have been
studied for various compact objects, but mainly for black holes. On the other
hand, it is well known now that by the end state of collapsing star both a
visible and hidden singularity can form joshi ; goswami ; mosani1 ; mosani2 ;
mosani3 ; mosani4 ; Deshingkar:1998ge ; Jhingan:2014gpa ; Joshi:2011zm . On
that count, the presence or absence of the event horizon should be responsible
for the distinct physical signature of a black hole and naked singularity. A
number of articles in the literature are devoted to investigating accretion
disks, tidal disruption, gravitational lensing, and timelike and null
geodesics around such compact objects Page:1974he ; Liu:2020vkh ; Liu:2021yev
; Joshi:2013dva ; Bambhaniya:2021ugr ; Tahelyani:2022uxw ; Rahaman:2021kge ;
Harko:2008vy ; Harko:2009xf ; Kovacs:2010xm ; Guo:2020tgv ; Chowdhury:2011aa ;
Lattimer:1976kbf ; Chicone:2004pv ; Madan:2022spd ; Shaikh:2019jfr ;
Shaikh:2018oul ; Paul:2020ufc ; Virbhadra:2007kw ; Gyulchev:2008ff ;
Kala:2020prt ; Sahu:2012er ; Martinez ; Madan:2022spd ; Eva1 ; Eva2 ; tsirulev
; Joshi:2019rdo ; Bambhaniya:2019pbr ; Dey:2019fpv ; Bam2020 ;
Bambhaniya:2022xbz ; Gralla:2019xty ; Vagnozzi:2019apd ; Chen:2022nbb ;
Dey:2020haf ; atamurotov_2015 ; abdujabbarov_2015b ; Li:2021 ; Hu:2020usx ;
Patel:2022vlu ; Kaur:2021wgy ; Saurabh:2020zqg ; Pugliese:2022oes ;
Vagnozzi:2022moj ; Saurabh:2022jjv . As expected, studying these properties
indeed gives a distinct observable signature. In addition to that, for the
energy extraction due to the absence of an event horizon, naked singularity
should give high energy extraction efficiency rate than the black hole.

The exciting idea of energy extraction leads back to 1969 when R. Penrose and
R. M. Floyd gave the rotational energy extraction mechanism Penrose:1971uk ,
in which they considered the ergoregion of Kerr black hole where negative
energy orbits exist. Later on, numerous studies were carried out where the
effects of the magnetic field around the compact objects on the energy
extraction processes were considered which is known as the magnetic Penrose
process. Apart from this, the concept of collision of particles in the Kerr
black hole was also developed, now known as the collisional Penrose process
Wagh:1985vuj ; Tursunov:2019oiq ; Wagh:1989zqa ; Patil:2011yb ; Patil:2014lea
; Patil:2015fua ; Hejda:2021cbk ; Khan:2019gco ; Okabayashi:2019wjs ;
Mukherjee:2018cbu ; Schnittman:2018ccg ; Tanatarov:2016mcs . The radiative
Penrose effect and its potential significance to the formation of relativistic
jets are discussed in Kolos:2020gdc . The general relativistic rotational
energy extraction from the accretion disk has been investigated in
Pugliese:2021ivl . A few other mechanisms of energy extraction such as the
Blandford-Znajek (BZ) Process, and the Blandford Payne process, involving
rotating black holes as a source, also extensively include the effects of
magnetohydrodynamics Bz . There are several other processes where the
collision of particles in the vicinity of static and rotating compact objects
have also been studied which are the Banados-Silk-West (BSW) process and
particle acceleration process Banados:2009pr ; Zaslavskii:2012fh ;
Zaslavskii:2012yp ; Zaslavskii:2012ax ; Pradhan:2016pgn ; Zaslavskii:2020agl ;
Zaslavskii:2022fwv ; Komissarov:2008yh ; Konoplya:2021qll ; Kimura:2021dsa .

In addition to this, many articles show these energy extraction mechanisms for
other classes of compact objects as well along with Kerr black hole. The
Magnetic Penrose process is studied for the Kerr naked singularity in which
energy extraction efficiency is about 150% Kerrnakedsingularity . The energy
extraction using the Super Penrose process is also studied in which
Reissner–Nordström (RN) naked singularity is considered as a central compact
object Zaslavskii:2022nbm . Authors have explored particle acceleration and
energy extraction for singular as well as regular black hole spacetimes
Patil:2012fu ; Nakao:2013uj ; Patel:2022jbk . However, mechanisms related to
rotational energy extraction have not been explored for naked singularity
spacetimes as compared to black hole spacetimes. Therefore in this paper, we
will analyze the presence of an ergoregion in the naked singularity spacetime.

There are multiple studies investigating the behaviour of an ergoregion and
negative energy orbits Chakraborty:2016ipk ; Pani:2010jz ; Zaslavsky:2013dra .
For the JNW naked singularity spacetime, many properties including an
ergoregion have been studied in Karmakar:2017lho , where the authors show that
an ergoregion does not exist in rotating JNW naked singularity spacetime. In
this paper, we consider the rotating JNW naked singularity spacetime in the
electromagnetic field. This configuration can act as a good source that emits
highly energetic radiation. However, we show that energy extraction mechanisms
such as the Penrose process and other rotational energy extraction mechanisms
defined by modifying the Penrose process, do not require the existence of an
ergoregion, rather it mainly depends on the behaviour of negative energy
orbits. Another article also suggests this fact Denardo:1973pyo . When a
rotating compact object in the magnetic field is considered, particles can
extract energy from the rotation of the geometry as well as the magnetic field
around the compact object.

In this paper, the effect of the magnetic field around the rotating JNW naked
singularity spacetime is studied. This concept of considering the magnetic
field around the compact object is certainly interesting, but not surprising.
A magnetic field appears in very small-scale stars for example the Sun. Thus
studies also include magnetic fields around black hole spacetimes as it can
give answers to many questions, mostly related to the high energy phenomena as
one can consider energy released through excitation of charged particles along
the magnetic field lines, as well as phenomena such as magnetic re-connection
(MR) Stuchlik:2019dlx .

This paper is organised as follows: we discuss the JNW naked singularity
metric and electromagnetic four components of the external electromagnetic
field in Section (II). In section (III), we first explain the importance of
negative energy orbits and then analyze the behavior of negative energy orbits
for rotating JNW spacetime with the uniform electromagnetic field. In section
(IV), we discuss the magnetic Penrose process and the behaviour of the energy
extraction efficiency with magnetic field parameter $B$, the spin parameter
$a$, and JNW metric parameter $\nu$. In the last section (V), we discuss the
results and conclusions of this study. Throughout the paper, we have
considered geometrized units. Hence, the gravitational constant (G) and the
speed of light (c) are set equal to one. The signature of the metric is
considered as (-,+,+,+).

##  II Janis-Newman-Winicour naked singularity spacetime

The extended geometry of Schwarzschild spacetime with a massless scalar field
is known as Janis-Newman-Winicour spacetime. The rotating version of the JNW
metric is derived in Solanki:2021mkt , using Newman-Janis Algorithm (NJA)
without complexification. The rotating JNW naked singularity spacetime can be
written as,

$\displaystyle
ds^{2}=-\left(1-\frac{2f}{\rho^{2}}\right)dt^{2}-\frac{4af\sin^{2}\theta}{\rho^{2}}dtd\phi+\frac{\Sigma\sin^{2}\theta}{\rho^{2}}d\phi^{2}$
$\displaystyle+\frac{\rho^{2}}{\Delta}dr^{2}+\rho^{2}d\theta^{2},\,\,\,\,\,\,\,\,\,\,$
(1)

where,

$\displaystyle\nu\,=\,\frac{2\,m}{b},$ (2) $\displaystyle
b=2\,\sqrt{m^{2}\,+\,q_{s}^{2}},$ (3) $\displaystyle
f\,=\,\frac{1}{2}r^{2}\left(\left(1-\frac{2m}{\nu
r}\right)^{-\nu}-1\right)\left(1-\frac{2m}{\nu r}\right),$ (4)
$\displaystyle\rho^{2}\,=\,a^{2}\cos^{2}(\theta)+r^{2}\left(1-\frac{2m}{\nu
r}\right)^{1-\nu},$ (5)
$\displaystyle\Sigma\,=\,(a^{2}\sin^{2}(\theta)+\rho^{2})^{2}-a^{2}\sin^{2}(\theta)\Delta.$
(6) $\displaystyle\Delta=a^{2}-\frac{2mr}{\nu}+r^{2},$ (7)

where $m$, $a$, and $q_{s}$ represent the ADM mass of spacetime, spin
parameter, and scalar field charge respectively. In the spacetime, the limit
for the $\nu$ is $0<\nu<1$ and the scalar field charge is non-zero. The Eq.
(1) is reduced to a Kerr black hole and static JNW naked singularity when
$\nu\,=\,1$ and $a\,=\,0$ respectively.

There are multiple methods to consider the different profiles of the magnetic
field around the compact object as the exact distribution of the magnetic
field is unknown. Here, we consider an external magnetic field with intensity
B which is asymptotically uniform Stuchlik:2019dlx . The field lines are
oriented along the z-axis, i.e., orthogonal to the equatorial plane of the
spacetime geometry. This can be described using the electromagnetic four
components $A_{\mu}$ having two nonzero components of the form,

$\displaystyle
A_{t}=\frac{B}{2}(g_{t\phi}+2ag_{tt})-\frac{Q}{2}g_{tt}-\frac{Q}{2},$ (8)
$\displaystyle
A_{\phi}=\frac{B}{2}(g_{\phi\phi}+2ag_{t\phi})-\frac{Q}{2}g_{tt},$ (9)

where $Q$ is the externally induced charge in the rotating JNW geometry. In
the next section, we derive the equations of motion for the charged particle
to observe the behavior of negative energy orbits in the rotating JNW
geometry, in an external asymptotically uniform magnetic field $B$.

##  III Negative energy orbits (NEO)

Penrose and Floyd showed that how particles can extract the energy from an
ergoregion of a Kerr black hole, where the negative energy orbits are also
possible. Later on, Wheeler suggested that the Penrose process can act as a
possible mechanism to explain the high energy phenomena in the universe. Thus
many research works are carried out in the domain of rotational energy
extraction and discuss the ergoregion of compact objects Pani:2010jz ;
Chakraborty:2016ipk ; Zaslavsky:2013dra . However, studies suggest that energy
extraction requires the existence of negative energy orbits Denardo:1973pyo .
The ergoregion is not always necessary for energy extraction phenomena.
Therefore in this section, we first discuss the negative energy orbits.

Negative energy orbits are energy states in which the total energy of the
particle is negative with respect to an observer at infinity. It can be easily
understood for classical mechanics, where one can consider a system of a
central object and a particle. If the particle is far enough from the central
object, the potential energy of a particle far from the central object is
taken to be zero, as the point of zero potential energy can be chosen
arbitrarily. Thus if the particle is near the central object with zero kinetic
energy, the total energy of the particle can be considered to be negative. If
the particle gets the energy equivalent to the negative energy, it will escape
to the infinite distance where it possesses the zero potential energy i.e. it
will be free from the gravitational influence of the central object.

(a) $\nu$ = 0.3, a = 0.9, Q = 0.5, $L$ = -13.

(b) $\nu$ = 0.3, a = 0.9, Q = 0.9, $L$ = -13.

(c) $\nu$ = 0.5, a = 0.1, Q = 0.1, $L$ = -3.

(d) $\nu$ = 0.5, a = 0.9, Q = 0, $L$ = -20.

(e) $\nu$ = 0.5, a = 0.9, Q = 0.5, $L$ = -13.

(f) $\nu$ = 0.5, a = 0.9, Q = 0.9, $L$ = -13.

(g) $\nu$ = 0.7, a = 0.9, Q = 0, $L$ = -20.

(h) $\nu$ = 0.7, a = 0.9, Q = 0.5, $L$ = -15.

(i) $\nu$ = 0.7, a = 0.9, Q = 0.9, $L$ = -15.

(j) $\nu$ = 0.9, a = 0.9, Q = 0, $L$ = -20.

(k) $\nu$ = 0.9, a = 0.9, Q = 0.5, $L$ = -20.

(l) $\nu$ = 0.9, a = 0.9, Q = 0.9, $L$ = -20.

Figure 1: The above figures show the change in the total energy with respect
to the radial distance in the rotating JNW geometry. The different values on
the right side of the plots indicate the intensity of the magnetic field (B)
around the compact object. The mass of the object is taken as unity for all
different values of $\nu$.

However, in special relativity, a timelike particle always possesses rest mass
energy. Even if the particle is far enough from any central object, it will
have non-zero rest mass energy. If the particle is near a Schwarzschild black
hole, it needs energy equivalent to the rest mass energy to escape the
gravitational influence of the Schwarzschild black hole. But in the case of
the Kerr black hole, when the particle is inside the ergoregion, the particle
possesses rest mass energy as well as energy because of the frame-dragging
effect. In addition to that, presence of electromagnetic field also increases
the energy of the particle near the central compact object. In this case, the
particle require energy equivalent to the sum of its rest mass energy and
energy because of other properties such as rotation and electromagnetic field
to escape from the gravitational influence of the central compact object. This
energy state of the particle is known as the negative energy state. Since the
particle possesses the negative energy state only in certain orbits, those
orbits are known as negative energy orbits. For the Kerr black hole, the
negative energy orbits exist only inside the ergoregion.

In the Penrose process, a particle splits into two particles inside an
ergoregion of the Kerr black hole. One particle enters into an orbit with a
negative energy state and falls inside the central black hole, while the other
particle gets the energy equivalent to the frame-dragging energy of the
particle, in addition to the rest mass energy and thus escapes to infinity. As
mentioned for Kerr black hole case, the negative energy orbits exist only
inside the ergoregion, thus Penrose process is defined only inside the
ergoregion.

Following this, one can understand that a particle can possess a negative
energy state because of certain properties of the compact object such as
rotation, and electromagnetic. Other particle can extract this energy and
escape to infinity. Thus in this section, we first discuss the negative energy
orbits of a particle around a rotating compact object in the presence of a
general electromagnetic field tensor $A_{\mu}$. We then discuss the negative
energy orbits for a charged particle around a rotating JNW naked singularity
in an asymptotically uniform magnetic field.

Consider a rotating compact object metric $g_{\mu\nu}$ surrounded with an
electromagnetic field denoted by general electromagnetic four components
$A_{\mu}$. For the simplicity, we consider that a particle with charge $q$ and
non-zero mass $m$ is moving in an equatorial plane. The conserved energy and
angular momentum can be written as,

$\displaystyle-E\,=\,g_{tt}\dot{t}+g_{t\phi}\dot{\phi}+q\,A_{t},$ (10)
$\displaystyle L\,=\,g_{\phi\phi}\dot{\phi}+g_{t\phi}\dot{t}+q\,A_{\phi}.$
(11)

From Eqs. (10) and (11), expressions of four velocity of the charged particle
$u^{t}$ and $u^{\phi}$ can be written as,

$\displaystyle
u^{\phi}=-\left(\frac{Eg_{t\phi}+\,L\,g_{tt}}{g_{t\phi}^{2}-g_{tt}g_{\phi\phi}}\right)-q\left(\frac{A_{t}g_{t\phi}-A_{\phi}g_{tt}}{g_{t\phi}^{2}-g_{tt}g_{\phi\phi}}\right),$
(12) $\displaystyle
u^{t}=-\left(\frac{Eg_{\phi\phi}+\,L\,g_{t\phi}}{g_{tt}g_{\phi\phi}-g_{t\phi}^{2}}\right)-q\left(\frac{A_{t}g_{\phi\phi}-A_{\phi}g_{t\phi}}{g_{tt}g_{\phi\phi}-g_{t\phi}^{2}}\right).$
(13)

From $u_{\alpha}\,u^{\alpha}=-1$ we can write the equation of total energy for
the motion of a charged particle in an equatorial plane at any constant radius
($\theta\,=\,\pi/2$),

$g_{tt}\,(u^{t})^{2}\,+\,2\,g_{t\phi}\,(u^{t})\,(u^{\phi})\,+\,g_{\phi\phi}\,(u^{\phi})^{2}\,=\,-1.$
(14)

Using Eq. (12), and (13) and solving it for $E$, we get general equations that
describe the energy states of the particle in orbits around the compact
object.

Using the Eq. (1) and expressions of four components of the electromagnetic
four vectors from Eq. (8) as well as Eq. (9) in Eq. (14) to obtain the
expression for energy states of the charged particle around the rotating JNW
naked singularity, which is given in the Appendix (A).

Here, we consider the negative energy orbits $E\,<\,0$ for $L\,<\,0$, and the
charge of a moving particle and the charge of the electromagnetic four
components are such that the condition $q\,Q\,<\,0$ is maintained. The reason
behind taking $L\,<\,0$ is, in the rotating compact object two scenarios are
considered for the motion of a particle. In the first scenario, a particle is
moving closer to the compact object in the same direction as the object’s spin
($L\,>\,0$). The second one is, a particle coming from the opposite side of
the objects’ spin ($L\,<\,0$). Hence, a particle coming from the same
direction as the object’s spin will eventually move in the object’s direction.
While the particle coming from the opposite direction, have to experience the
frame-dragging effect and after that, it also goes in the direction of the
object’s spin. In a static case, due to the absence of a frame-dragging effect
the results with $L\,<\,0$ and $L\,>\,0$ are similar. The other condition with
charge is because we considered the electric charge in the geometry. As per
the nature of an electric charge, having the same charges will turn into a
repulsive effect. To avoid that the $q\,Q\,<\,0$ condition is considered.

In Fig. (1), the negative energy states in the static JNW geometry are shown
for different values of the metric parameter $\nu$, magnetic field parameter
$B$, induced charge $Q$, and spin parameter $a$. It is clear that the energy
state of the particle is negative before a certain radial distance for given
values spin parameter $a$, magnetic field intensity parameter $B$, and the
metric parameter $\nu>0.1$. As we increase the values of either of the
parameter among metric parameter $\nu$, the spin parameter $a$, and magnetic
field parameter $B$, keeping the values of the other parameters constant, the
region of negative energy orbits increases. As discussed above about negative
energy orbit, we can expect that energy extraction efficiency would increase
with this increasing behavior of negative energy orbits as the particle will
get more energy to escape to infinity. This expected behavior of energy
extraction efficiency is also observed in the next section. In the range of
$0.5\,<\,\nu\,<\,1$, the negative energy orbits exist for every range of
angular momentum. The detailed discussion regarding Fig. (1) is given in
section (V).

(a) $\nu$ = 0.2, a = 0.9, r = 10.01.

(b) $\nu$ = 0.9, a = 0.9, r = 2.23.

(c) $\nu$ = 0.2, a = 0.9, B = 0.9.

(d) $\nu$ = 0.9, a = 0.9, B = 0.9.

(e) $\nu$ = 0.5, Q = 0.9, B = 0.5.

(f) $\nu$ = 0.9, Q = 0.9, B = 0.9.

Figure 2: The above Figures show the change in the energy extraction
efficiency with respect to the radial distance $r$ (Figs. (2(a)), (2(b))) and
the magnetic field intensity parameter $B$ (Figs. (2(c)), (2(d))) in the
rotating JNW naked singularity. The different values on the right side of the
Figs. (2(a), 2(b), 2(c), 2(d)) indicates the variation in an electric charge
parameter $(Q)$ around the compact object. The density plots (Figs. (2(e)),
(2(f))) indicate the change in energy extraction efficiency with respect to
the radius (r) and spin parameter (a). The bar on the right side of the Figs.
(2(e), 2(f)) represents the energy extraction efficiency ($\eta$) in
percentage. The mass is taken as unity.

##  IV Energy extraction

In this section, we study the energy extraction mechanism and behavior of
energy extraction efficiency for the charged particle around the JNW naked
singularity in an asymptotically uniform magnetic field. The mechanism we
consider here is the modified version of the Penrose process known as the
Magnetic Penrose process, as we are considering the presence of the magnetic
field. The magnetic Penrose process is defined in the Wagh:1985vuj ;
Tursunov:2019oiq ; Wagh:1989zqa . Hence, here we present a quick overview of
the magnetic Penrose process.

Let us assume that a charged particle $q_{1}$ with energy $E_{1}$ and mass
$m_{1}$ is in the vicinity of the rotating JNW naked singularity which splits
into two particles with charge $q_{2}$, energy $E_{2}$, mass $m_{2}$ and
particle with charge $q_{3}$, energy $E_{3}$, mass $m_{3}$. Assume that, a
particle with charge $q_{3}$ and energy $E_{3}$ enters the negative energy
state $E_{3}<0$ and falls into the naked singularity while the other particle
with charge $q_{2}$ and energy $E_{2}$ gains this energy and escapes to
infinity with $E_{2}>0$ (as the energy is conserved). Taking the conservation
laws of different quantities into consideration during this phenomenon, one
can write,

$E_{(1)}=E_{(2)}+E_{(3)},\hskip 22.76228ptq_{(1)}=q_{(2)}+q_{(3)},$
$L_{(1)}=L_{(2)}+L_{(3)},\hskip 22.76228ptM_{(1)}=M_{(2)}+M_{(3)}.$

The canonical momentum is also conserved,

$P_{(1)}^{\mu}=P_{(2)}^{\mu}+P_{(3)}^{\mu}.$

From the conservation law, one can define a relationship for the four velocity
$u^{t}$ and $u^{\phi}$ as the angular velocity of the particles,

$\Omega=\frac{u^{\phi}}{u^{t}}=\frac{\dot{\phi}}{\dot{t}},$

and from the conservation $u^{\phi}=\Omega\beta/\alpha$ where, $\alpha$ and
$\beta$ are,

$\alpha\,=\,\epsilon+\frac{q\,A_{t}}{m},\hskip
22.76228pt\beta\,=\,g_{tt}\,+\,\Omega\,g_{t\phi},$

where $\epsilon=E/m$ and by doing the algebraic manipulation one can get the
expression for the energy of the escaping particle as,

$E_{2}\,=\,\chi\,(E_{1}\,+\,q_{1}\,A_{t})\,-\,q_{2}\,A_{t},$ (15)

where, $\chi$ is,

$\chi\,=\,\frac{\Omega_{1}\,-\,\Omega_{3}}{\Omega_{2}\,-\,\Omega_{3}}\frac{\beta_{2}}{\beta_{1}},\hskip
22.76228pt\beta_{i}\,=\,g_{tt}\,+\,\Omega_{i}\,g_{t\phi},$

where, $\Omega_{i}$ is the angular velocity of the $i^{th}$ particle which can
be derived from the geodesic Eq. (14). The angular velocity of a different
particle is,

$\displaystyle\Omega_{1}\,=\,\frac{-g_{t\phi}(\pi^{2}_{v}\,+\,g_{tt})+\sqrt{(\pi^{2}_{v}\,+\,g_{tt})(g_{t\phi}^{2}\,-\,g_{tt}g_{\phi\phi})\pi^{2}_{v}}}{\pi^{2}_{v}g_{\phi\phi}+g_{t\phi}^{2}},$
(16)
$\displaystyle\Omega_{2}\,=\,\frac{-g_{t\phi}\,-\,\sqrt{g_{t\phi}^{2}-g_{tt}g_{\phi\phi}}}{g_{\phi\phi}},$
(17)
$\displaystyle\Omega_{3}\,=\,\frac{-g_{t\phi}\,+\,\sqrt{g_{t\phi}^{2}-g_{tt}g_{\phi\phi}}}{g_{\phi\phi}},$
(18)

here, $\pi_{v}\,=\,-(\epsilon\,+\,q\,A_{t}/m)$. Using the usual definition of
efficiency,

$\eta\,=\,\frac{E_{3}\,-\,E_{1}}{E_{1}}=\frac{-E_{2}}{E_{1}},$

using the above expression and by doing algebraic manipulation with Eq. (15),
we get the efficiency of the magnetic Penrose process in the form,

$\eta\,=\,\frac{\chi\,q_{1}\,A_{t}\,-\,q_{3}\,A_{t}}{E_{1}}+\chi\,-\,1.$ (19)

The energy extraction efficiency in the magnetic Penrose process is reduced to
the original Penrose process in the absence of an electromagnetic field. Figs.
(2) represent the change in efficiency with respect to the radial distance $r$
(Figs. (2(a)), (2(b))) and the magnetic field intensity $B$ (Figs. (2(c)),
(2(d))) with different electric charge (Q). The detailed discussion is given
in section (V).

##  V Discussion and Conclusions

In this paper, we investigate the energy extraction phenomena from a rotating
JNW naked singularity case. For this, we considered rotating JNW naked
singularity in a uniform magnetic field. We briefly discussed the importance
of negative energy orbits in energy extraction and considered the negative
energy states for a charged particle moving in this geometry. In order to
understand the magnetic Penrose process we considered the splitting of a
charged particle in JNW naked singularity surrounded by an electromagnetic
field. The following are the outcomes of this study:

  * •

In Misner:1973prb , it is given that for in-falling neutral particle for Kerr
black hole, the negative energy orbits exist inside the ergoregion. Whereas,
for in-falling charged particle, the region of the negative energy orbits is
larger than the ergoregion. This region is known as effective ergoregion. For
energy extraction previously, it was shown by Ruffini that the existence of
negative energy orbits is necessary. Apart from that, it is known and in
Karmakar:2017lho  authors show that there does not exist ergoregion in the
rotating JNW naked singularity. However, in our study, using the uniform
magnetic field around the rotating JNW naked singularity, we show the presence
of negative energy orbits for the charged particles. Hence, one can say that
there exists an effective ergoregion in the rotating JNW naked singularity.

  * •

We proved that the negative energy orbits exist for $0.1\,<\,\nu\,<\,1$ with
the conditions $E\,<\,0$, $L\,<\,0$ and the charge of moving particle (q) and
induced electromagnetic charge (Q) in the geometry are such that $q\,Q\,<\,0$
in the rotating JNW naked singularity surrounded by the uniform
electromagnetic field. Figs. (1) shows the change in the energy (E) with
respect to the radial distance (r) for different parameters. The region of the
negative energy orbits increases as the value of the parameter $\nu$ is
increased and also with the increasing magnetic field intensity parameter
($B$) and spin parameter ($a$). The reason behind this behavior is, with the
increasing value of the magnetic field intensity parameter and spin parameter,
the required energy for the particle to escape to infinity also increases as
discussed in the section (III). The Figs. (1) of negative energy orbits for
$\nu=1$ resemble the plots of negative energy orbits for the Kerr geometry
cases.

  * •

One may see that in the absence of the induced electric charge, the negative
energy orbits are exists from Figs. (1(d), 1(g), 1(j)). Apart from that, when
the magnetic field reduces to zero then there is an absence of the negative
energy orbits. This concludes that negative energy orbits exist in the
rotating JNW geometry in the presence of a magnetic field or electromagnetic
field. The induced electric charge might play a crucial role in the higher
negative energy orbit region or energy extraction efficiency but is not
necessary for the existence of the negative energy orbit or the energy
extraction.

  * •

In general, when the region of the negative energy orbits increases because of
a change in the value of the magnetic field intensity parameter, spin
parameter, or the metric parameter $\nu$, the energy extraction efficiency
also increases. This is because the escaping particle gets more energy as the
region of negative energy orbits increases. This is also observed in Figs. (2)
as a result of an investigation. The result of increasing energy extraction
efficiency was expected as the region of the negative energy orbits was also
increasing.

  * •

The Figs. ((2(a)), (2(b))) indicates the decreasing rate of the energy
extraction efficiency with respect to the radial distance (r) as one moves
away from the singularity. Figs. (2(c)), (2(d))) show that energy extraction
efficiency increases with increasing value of magnetic field intensity, the
spin parameter. This behaviour is because of the increasing negative energy
orbits region in the presence of magnetic field and rotation as escaping
particle gets more energy. The energy extraction efficiency also increases
with the increasing value of the metric parameter $\nu$. Figs. (2(e), 2(f))
shows the change in energy extraction efficiency rate with respect to the spin
parameter and radial distance. As expected it indicates that for the higher
spin parameter, the efficiency rate is increased compared to the lower spin
parameter.

  * •

The energy extraction efficiency for rotating JNW naked singularity case is
60% for metric parameter $\nu=0.9$, magnetic field intensity $B=0.9$, and spin
parameter $a=1$. While for the Kerr black hole and Kerr naked singularity
case, the efficiency is around $20.7\%$ and $150\%$ respectively. There are
many possible reasons why there exists a high-efficiency rate in the Kerr
naked singularity than rotating JNW naked singularity. The first reason is
that a spin parameter is higher in the Kerr naked singularity $(a>M)$ with the
existence of an ergoregion. While in JNW spacetime, there is no ergoregion but
negative energy orbits are present in so-called effective ergoregion,
therefore the efficiency of energy extraction from the Kerr naked singularity
is higher than JNW naked singularity case. Moreover, we may have to look
further for the relation between the presence of an ergoregion and negative
energy orbits. These fundamental questions need to be explored critically in
future.

###### Acknowledgements.

The authors would like to express their gratitude towards Dr. Daniela Pugliese
for her valuable suggestions and comments. V.P. would like to acknowledge the
support of the SHODH fellowship (ScHeme Of Developing High quality
research—MYSY).

## Appendix A

The expression for the energy states can be written as,

$\displaystyle
E\,=\,\frac{f_{1}+f_{2}-\left(f_{3}\pm\sqrt{\left(f_{4}-f_{5}+f_{6}\right){}^{2}-4f_{7}\left(f_{8}+f_{9}+f_{10}-f_{11}+f_{12}-f_{13}-f_{14}-f_{15}\right)}\right)}{f_{0}},$
(20)

where, $f_{i}$ ($i=0,,,15$) is function of $[B,m,q,Q,\nu,a,L,r]$,

$\displaystyle f_{0}\,=\,2\left(8a^{2}\nu^{2}\left(1-\frac{2m}{\nu
r}\right)^{\nu}-4a^{2}\nu^{2}\left(1-\frac{2m}{\nu r}\right)^{2\nu}-8m\nu
r+4\nu^{2}r^{2}\right),$ (21) $\displaystyle f_{1}\,=\,-16aBm\nu
qr\left(1-\frac{2m}{\nu r}\right)^{\nu}+8m\nu qQr\left(1-\frac{2m}{\nu
r}\right)^{\nu}-8m\nu qQr+4\nu^{2}qQr^{2},$ (22) $\displaystyle
f_{2}\,=\,8a^{3}B\nu^{2}q\left(1-\frac{2m}{\nu
r}\right)^{\nu}+4a^{2}\nu^{2}qQ\left(1-\frac{2m}{\nu
r}\right)^{\nu}+8aB\nu^{2}qr^{2}\left(1-\frac{2m}{\nu
r}\right)^{\nu}+8aL\nu^{2}\left(1-\frac{2m}{\nu r}\right)^{\nu},$ (23)
$\displaystyle f_{3}\,=\,-4a^{2}\nu^{2}qQ\left(1-\frac{2m}{\nu
r}\right)^{2\nu}-8aL\nu^{2}\left(1-\frac{2m}{\nu
r}\right)^{2\nu}+4\nu^{2}qQr^{2}\left(1-\frac{2m}{\nu r}\right)^{\nu},$ (24)
$\displaystyle f_{4}\,=\,16aBm\nu qr\left(1-\frac{2m}{\nu
r}\right)^{\nu}-8m\nu qQr\left(1-\frac{2m}{\nu r}\right)^{\nu}+8m\nu
qQr-4\nu^{2}qQr^{2},$ (25) $\displaystyle
f_{5}\,=\,-8a^{3}B\nu^{2}q\left(1-\frac{2m}{\nu
r}\right)^{\nu}-4a^{2}\nu^{2}qQ\left(1-\frac{2m}{\nu
r}\right)^{\nu}-8aB\nu^{2}qr^{2}\left(1-\frac{2m}{\nu
r}\right)^{\nu}+8aL\nu^{2}\left(1-\frac{2m}{\nu r}\right)^{\nu},$ (26)
$\displaystyle f_{6}\,=\,4a^{2}\nu^{2}qQ\left(1-\frac{2m}{\nu
r}\right)^{2\nu}+8aL\nu^{2}\left(1-\frac{2m}{\nu
r}\right)^{2\nu}+4\nu^{2}qQr^{2}\left(1-\frac{2m}{\nu r}\right)^{\nu},$ (27)
$\displaystyle f_{7}\,=\,8a^{2}\nu^{2}\left(1-\frac{2m}{\nu
r}\right)^{\nu}-4a^{2}\nu^{2}\left(1-\frac{2m}{\nu r}\right)^{2\nu}-8m\nu
r+4\nu^{2}r^{2},$ (28) $\displaystyle f_{8}\,=\,2a^{2}B^{2}m\nu
q^{2}r-4B^{2}m^{2}q^{2}r^{2}+4B^{2}m\nu q^{2}r^{3}-2m\nu q^{2}Q^{2}r,$ (29)
$\displaystyle f_{9}\,=\,-4a^{2}B^{2}m\nu q^{2}r\left(1-\frac{2m}{\nu
r}\right)^{\nu}-4aBm\nu q^{2}Qr\left(1-\frac{2m}{\nu r}\right)^{\nu}-8BLm\nu
qr\left(1-\frac{2m}{\nu r}\right)^{\nu}+8m\nu r\left(1-\frac{2m}{\nu
r}\right)^{\nu},$ (30) $\displaystyle f_{10}\,=\,-2a^{2}B^{2}m\nu
q^{2}r\left(1-\frac{2m}{\nu r}\right)^{2\nu}+4aBm\nu
q^{2}Qr\left(1-\frac{2m}{\nu r}\right)^{2\nu}+4m\nu
q^{2}Q^{2}r\left(1-\frac{2m}{\nu r}\right)^{\nu}-2m\nu
q^{2}Q^{2}r\left(1-\frac{2m}{\nu r}\right)^{2\nu},$ (31) $\displaystyle
f_{12}\,=\,2a^{4}B^{2}\nu^{2}q^{2}\left(1-\frac{2m}{\nu
r}\right)^{\nu}+2a^{3}B\nu^{2}q^{2}Q\left(1-\frac{2m}{\nu
r}\right)^{\nu}+4a^{2}BL\nu^{2}q\left(1-\frac{2m}{\nu
r}\right)^{\nu}+4aL\nu^{2}qQ\left(1-\frac{2m}{\nu r}\right)^{\nu},$ (32)
$\displaystyle f_{13}\,=\,2a^{2}B^{2}\nu^{2}q^{2}r^{2}\left(1-\frac{2m}{\nu
r}\right)^{\nu}+2aB\nu^{2}q^{2}Qr^{2}\left(1-\frac{2m}{\nu
r}\right)^{\nu}+4BL\nu^{2}qr^{2}\left(1-\frac{2m}{\nu
r}\right)^{\nu}+4\nu^{2}r^{2}\left(1-\frac{2m}{\nu r}\right)^{\nu},$ (33)
$\displaystyle f_{14}\,=\,a^{4}B^{2}\nu^{2}q^{2}\left(1-\frac{2m}{\nu
r}\right)^{2\nu}-4aL\nu^{2}qQ\left(1-\frac{2m}{\nu
r}\right)^{2\nu}-4L^{2}\nu^{2}\left(1-\frac{2m}{\nu
r}\right)^{2\nu}+2\nu^{2}q^{2}Q^{2}r^{2}\left(1-\frac{2m}{\nu
r}\right)^{\nu},$ (34) $\displaystyle
f_{15}\,=\,2a^{3}B\nu^{2}q^{2}Q\left(1-\frac{2m}{\nu
r}\right)^{2\nu}+a^{2}B^{2}\nu^{2}q^{2}r^{2}\left(1-\frac{2m}{\nu
r}\right)^{2\nu}-2aB\nu^{2}q^{2}Qr^{2}\left(1-\frac{2M}{\nu
r}\right)^{2\nu}+\nu^{2}q^{2}Q^{2}r^{2}\left(1-\frac{2m}{\nu
r}\right)^{2\nu}.$ (35)

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