#
Machine Learning for Quantum Computing Specialists

Daniel Goldsmith and M M Hassan Mahmud
Digital Catapult

(April 2024)

###### Abstract

Quantum machine learning (QML) is a promising early use case for quantum
computing. There has been progress in the last five years from theoretical
studies and numerical simulations to proof of concepts. Use cases demonstrated
on contemporary quantum devices include classifying medical images [1] and
items from the Iris dataset [2], classifying [3] and generating [4]
handwritten images, toxicity screening [5], and learning a probability
distribution [6]. Potential benefits of QML include faster training [2] and
identification of feature maps not found classically [7]. Although, these
examples lack the scale for commercial exploitation, and it may be several
years before QML algorithms replace the classical solutions, QML is an
exciting area.

This article is written for those who already have a sound knowledge of
quantum computing and now wish to gain a basic overview of the terminology and
some applications of classical machine learning ready to study quantum machine
learning. The reader will already understand the relevant relevant linear
algebra, including Hilbert spaces, a vector space with an inner product.

##  1 Introduction

This review is written for those who already some quantum computing knowledge
and need to gain a basic overview of machine learning before learning about
quantum machine learning (QML). There has been progress in the last five years
from theoretical studies and numerical simulations to proof of concepts. Use
cases demonstrated on contemporary quantum devices include classifying medical
images [1] and items from the Iris dataset [2], classifying [3] and generating
[4] handwritten images, toxicity screening [5], and learning a probability
distribution [6]. Potential benefits of QML include faster training [2] and
identification of feature maps not found classically [7].

You may be encouraged to read that “a quantum feature map may enable a quantum
kernel classifier on NISQ machines”, but left wondering what a “feature map”,
a “kernel” and a “classifier” are. You may find it promising that “a possible
early use case for quantum machine learning is to use quantum computers to
generate the latent spaces for GANs” and be puzzled by the terms “GAN model”,
and “latent space”. This review explores these and other important machine
learning concepts at a high level. Key concepts are highlighted in bold when
introduced. This is not an exhaustive text book on Machine Learning and you
are signposted to additional sources of information to supplement this review.
In the interests of brevity we pass over some popular machine learning
algorithms like Random forests 111In this article we adopt a hybrid approach
of citing references because of the intended mode of dissemination as an
online article. For well known concepts from computing, mathematics or quantum
physics, we just use an in text hyperlink to an online resource. For books or
research papers, we include proper citations..

This review was written by a quantum computing technologist who six months ago
knew that Machine Learning could separate pictures of cats from pictures of
dogs, but not much more. Since then I have been guided by my expert mentor, Dr
M M Hassan Mahmud who has given me great advice and tuition, and has carefully
reviewed this document. You are in safe hands!

###  1.1 Quantum Computing has given you a good start to machine learning

You may already understand most of the mathematics needed for machine learning
from quantum computing. You probably understand the linear algebra on which
much of machine learning is built, because the same linear algebra is used in
quantum computing, albeit with complex numbers. Most likely you already
understand a Hilbert space as a vector space equipped with an inner product
because a quantum state can be represented as a vector in Hilbert space. The
inner product is a key concept in machine learning, and you have probably
already has seen that the inner product of two quantum states $\psi$ and
$\phi$, $\langle\psi|\phi\rangle$, gives a measure of the overlap, or
simularity between the two states. You will understand that a quantum state
can be represented as a vector, and an operation on a quantum state can be
represented as a matrix that produces a new vector representing a different
quantum state. Matrices and vectors are ubiquitous in machine learing. The
symbols for matrices and vectors are shown in bold in this review.

You will understand that a matrix A is Hermitian if
$\boldsymbol{A}=\boldsymbol{A}^{\dagger}$ , where $\boldsymbol{A}^{\dagger}$
represents taking the transpose of the matrix, and then the complex conjugate
of each matrix element. In machine learning the equivalent to the Hermitian of
a matrix is a symmetric matrix, as complex numbers are not normally relevant.
A matrix is symmetric if $\boldsymbol{A}=\boldsymbol{A}^{T}$ where
$\boldsymbol{A}^{T}$ represents the transpose of a matrix. We will see that
covariance matrices are symmetric.

You will understand that a square matrix
$\boldsymbol{A}\in\mathscr{R}^{n\times n}$ has eigenvalues
$\lambda_{i}\in\mathscr{R}$ and corresponding eigenvectors
$\boldsymbol{x}_{i}\in\mathscr{R}^{n}$ such that:

$\boldsymbol{A}\boldsymbol{x}_{i}=\lambda_{i}\boldsymbol{x}_{i}$ (1)

You will also have some knowledge of probability. If you would like to brush
up on the underlying maths, then we recommend the on-line book Mathematics for
Machine Learning [8]. In fact, add a few sections on complex numbers, and this
book could be called _“Mathematics for Machine Learning and Quantum
Computing”_.

You may also be familiar with Python, and may have used Jupyter notebooks.
Many Machine learning libraries such as Pytorch and Tensorflow use Python. If
you have not worked with Python a good introductory book is “ _Python Crash
Course”_ [9]. If you install Anaconda then Python and Jupyter will be
downloaded as part of the installation. Jupyter notebooks are straightforward
to pick up. Python code can be written directly into code cells, and text,
including mathematical formulae, can be written in Markdown cells. The code
can be changed directly on your computer to gain insight into the example.

This review provides links to Jupyter notebooks hosted on the GitHub
repository for our quantum technology access programme (QTAP) [10] showcasing
the comprehensive and easy to learn Scikit Learn machine learning package.
Machine Learning is a practical subject and it is really useful to work on
real examples. Kaggle has over 50,000 public datasets available for free that
you can practice on.

Most likely you are reading this review because you are interested in Quantum
Machine Learning (QML). We will point out use cases for QML. Be aware though,
that is not a complete review of the current status of QML. The examples
illustrate the text and point to additional references. Hopefully this will
whet your appetite for more. When reading the literature on QML you need to be
aware of the following caveats: 1\. We are in the age of Noisy Intermediate
Scale Quantum devices (NISQ) with high error rates and few qubits. 2\. In
theory these errors can be corrected using logical qubits where a quantum
state is held across many physical qubits and a corrupted physical qubit can
be detected and corrected by referring to the other uncorrupted physical
qubits, but there are significant challenges to using quantum error correction
for useful computations. 3\. At some stage in the future the expectation is
that we will have the fault tolerant quantum computers with large numbers of
error corrected logical qubits predicted by Preskill [11]. Increasing the
number of physical qubits and getting error correction working properly will
take at least a decade, and probably much longer. 4\. There is no commercially
available quantum memory to store quantum states, meaning that it is difficult
to carry out quantum calculations that take longer than the short qubit
decoherence times. Without quantum memory, converting machine learning data
into quantum states will be very difficult on today’s noisy devices. 5\. Many
quantum computing algorithms, such as Grover’s search algorithm [12], the
Harrow Hassidim Lloyd (HHL) [13] algorithm for matrix inversion and Quantum
Phase Estimation [14], that might be useful for QML, are not feasible until
fault tolerant quantum computers become widely available.

##  2 What is Artificial Intelligence?

Artificial Intelligence (AI) dates back to the earliest days of computing. AI
can be defined as “ _the set of technologies that solve problems that require
human like intelligence_ ”. What is regarded as “ _human like intelligence_ ”
ratchets up over time, and as we will see, much deployed AI is now just
regarded as software. So the unofficial definition of AI is “ _anything that
humans can do but computers cannot do … yet!_ ”.

The book _Artificial Intelligence, A Modern Approach_ [15] gives a good
overview of AI. Our colleague at Digital Catapult, Dr Robert Elliot Smith, has
written an important book “ _Rage Inside The Machine_ ” [16]. As well as
highlighting the involvement of AI algorithms in driving intolerance, Rob
Smith puts AI and ML into their historical context, which I found valuable.
Another very useful book is “ _Machine Learning with Quantum Computers_ ” by
Maria Schuld and Francesco Petruccione [17]. This books covers both machine
learning and quantum computing, and looks at quantum machine learning
algorithms likely to be useful in the future. Maria Schuld has posted
insightful videos on YouTube, for example, Is quantum advantage the right goal
for QML?

I also found the book “ _Machine Learning for Absolute Beginners_ ” [18] by
Oliver Theobald useful as it gave a good basic overview of ML.

AI has its roots in computer science and draws from diverse fields including
logic, probability theory, statistics, neuroscience, information theory,
cognitive science, linguistics and even biology.

Machine Learning is a subset of AI, and machine learning algorithms have the
ability to learn from patterns in data without being explicitly programmed.
There is a useful MIT OpenCourseWare Introduction to Machine Learning
available on YouTube.

A further subset of machine learning is Deep Learning, in which artificial
neural networks adapt to learn from vast amounts of data. Deep learning
methods have had tremendous success in the real world in recent years, and a
lot of the current excitement around AI/ML can be attributed to their success.
There is another great MIT OpenCourseWare Introduction to Deep Learning also
available on YouTube.

The relationship between the different components of Artificial Intelligence
is shown in Figure 1:

Figure 1: How Artifical Intelligence, Machine Learning and Deep Learning fit
together

###  2.1 My first experience of AI - the VAT “expert system’

When I was working as a auditor in the early nineties we used an AI VAT
“expert system” to give our clients VAT advice. We interviewed the client, and
asked pre-defined questions like “do you use cars for business purposes?”,
“what is your turnover?”, input the answers into our COMPAQ microcomputer
shown in Figure 2 and printed out the results on a dot matrix printer to give
to the client.

Figure 2: The COMPAQ microcomputer - Thanks to the Norsk Tekisk Museum CC BY-
SA 3.0

With hindsight our “expert system” suffered from all the issues of early
expert systems identified by Rob Smith in his book [16]. It needed a lot of
time from very expensive VAT and computing experts to design and build. It was
fragile, and quite often the carefully studied output simply advised the
client to consult with human experts.

This “expert system” needed to be explicitly programmed for each and every
situation it would encounter. In the next example we will consider a simple
machine learning model that learns from a pattern in the data to predict
values for unseen data.

A machine learning model consists of an algorithm which has been optimised on
training data to make useful predictions of unseen data.

###  2.2 An introductory example - linear regression

I have placed a Jupyter Notebook on GitHub demonstrating an example of linear
regression using the Scikit Learn package. This is an example of supervised
learning, where the algorithm predicts values based on examples provided.
Although relatively straightforward, this example illustrates many of the
points relevant for more complex machine learning models. This algorithm pre-
dates computers, since for small data sets the calculations can be done by
hand.

####  2.2.1 Aim of the linear regression example

The aim of this basic machine learning example is to predict the company
turnover from the number of employees.

####  2.2.2 Loading the data using pandas

In my Notebook Pandas, a Python data analysis tool, is used to load and
process the dataset: a list of UK quantum technology companies with the
company turnover, numbers of employees and latitudes and longitudes. The
dataset is loosely based on real UK data. A plot of employees against turnover
from the dataset is shown in Figure 3:

Figure 3: Plot of employee data against company turnover: the raw data

####  2.2.3 Machine Learning terminology used to describe the data

The company turnover, numbers of employees, latitude and longitude are called
features. Each line of the file is called a sample. Each sample can be
regarded as a 4 dimensional vector, with one dimension for each feature,
allowing concepts from linear algebra to be used. The set of all feature
vectors is know as the design matrix. In this example the latitudes and
longitudes are not used. The company turnover is a label in this example and
the number of employees is an input. In any data analysis “Garbage In, Garbage
Out” applies, so the data is first listed, plotted on a graph to check for
outliers and any records with unavailable data are removed before proceeding.
Whilst data analysis, and data science more generally, are very important to
machine learning in practice, it is too large a subject to be considerd
further in this review.

####  2.2.4 Training and test data

Next, the data is split into two components, training data which makes up 70%
of the data, and test data, which makes up the remaining 30%. The model is
trained with the training data, and then makes predictions which are validated
against the test data, showing how good the model is at generalising from the
training data to the test data, a key feature of all models.

More complicated models need validation data, as well as training and test
data.

####  2.2.5 The hypothesis space

Every machine learning problem has a family of candidate solutions chosen by
the ML engineer. Each family of solutions, for example, polynomial functions,
trigonometric functions and linear functions, is called a hypothesis space.
The set of hypothesis spaces are limited only by our imagination. The aim of a
particular ML algorithm is to use the training data to find the best solution
in a given hypothesis space.

There is no way of being sure which is the best hypothesis space and algorithm
to use. In fact, the “no free lunch” theorem implies that there is no single
best machine learning algorithm. All algorithms work well in some cases and
badly in other.

In this example, if we write $x$ as the number of employees and $\hat{y}$ as
the predicted turnover, then the hypothesis spaces are different families of
predictor functions that returns a prediction $\hat{y}=f(x)$ from an input
$x$.

####  2.2.6 The inductive bias

To get a tractable ML algorithm it is necessary to make specific assumptions
about the form of the mapping from inputs to labels as shown in Figure 4.
These assumptions are called the inductive bias. The hypothesis space encodes
these assumptions mathematically. Just as we all have sub-conscious biases
which means that we may make incorrect decisions without realising it, we may
also incorrectly tie our models to a mistaken assumption.

In this example it is assumed that the predictor function is linear, with
$f(x)=w_{1}x+w_{0}$. The model needs to estimate $w_{1}$, the slope of the
line, and $w_{0}$ the offset.

Figure 4: The inductive bias identifies the hypothesis space to be used from
all possible hypothesis spaces

It is normal in ML to regard the parameters of a model as a vector
$\boldsymbol{\theta}$. Here $\boldsymbol{\theta}=[w_{1},w_{0}]^{T}$. I have
followed the common convention in ML of writing a column vector as the
transpose of the row entries, which saves space. The predictor function
returns the prediction from an input $\boldsymbol{x}$ and parameters
$\boldsymbol{\theta}$ and is writen as
$f(\boldsymbol{x},\boldsymbol{\theta})$.

####  2.2.7 Choosing and optimising a loss function

Once we have chosen our hypothesis space we need to find the best parameters,
or hypothesis, to use. To find the best parameters we need a way to decide
whether one hypothesis is better than another. This is done through a loss
function which measures the quality of prediction made by our ML model. The
loss function $\mathscr{L}(\boldsymbol{\theta}$) determines the usefulness of
the predictions, for example, in estimating turnover. So, choosing a loss
function is a crucial decision when setting up a ML problem. In this example,
the loss function for each sample is chosen as the square of the distance
between the actual turnover $y$ and the predicted turnover $\hat{y}$:

$\mathscr{L}(y,\boldsymbol{\theta})=||y-\hat{y}||^{2}=||y-f(x,\boldsymbol{\theta})||^{2}$

The loss function is shown in Figure 5

Figure 5: The loss function shown diagramatically

The parameters are chosen to minimise the sum of the loss function over all
the samples in the training set,

$\mathscr{L}(\boldsymbol{\theta})=\sum_{i}\mathscr{L}(y_{i},\boldsymbol{\theta})$

which is a “least squares” optimisation problem. There is an exact solution
under certain conditions.

If we consider the nth sample we can write $x_{n}$ as a vector
$\boldsymbol{x}_{n}$. Here $\boldsymbol{x}_{n}=[x_{n},1]^{T}$. This means that
we can neatly write $f(x_{n})=\boldsymbol{x}_{n}^{T}\boldsymbol{\theta}$, the
vector dot product, or inner product, of the input vector and the parameter
vector. If there are N samples we can write the design matrix as
$\boldsymbol{X}=[\boldsymbol{x}_{1},....,\boldsymbol{x}_{n},...\boldsymbol{x}_{N}]^{T}\in\mathscr{R}^{N\times
2}$, and the observed results vector
$\boldsymbol{y}=[y_{1},....,y_{i},....y_{N}]^{T}\in\mathscr{R}^{N}$. All these
different matrices and vectors are illustrated in Figure 6

Figure 6: This diagram show the dataset with features and labels, the results
vector, design matrix, and the parameter vectors

Using this notation it is shown in Chapter Nine of Mathematics for Machine
Learning [8] that by differentating $\boldsymbol{\theta}$ a minimum of the
loss function is found when

${\boldsymbol{\theta}}=(\textbf{X}^{T}\textbf{X})^{-1}\textbf{X}^{T}\textbf{y}$

.

In other words this is a matrix inversion problem where an equation of form
$\boldsymbol{A}\boldsymbol{\theta}=\boldsymbol{b}$ with
$\boldsymbol{A}=\boldsymbol{X}^{T}\boldsymbol{X}$ and
$\boldsymbol{b}=\boldsymbol{X}^{T}\boldsymbol{y}$ needs to be solved. Figure 7
below shows the dataset, and the best fit line for the training data against
the test data calculated by SciKit Learn:

Figure 7: Plot of employee data against company turnover: a best fit straight
line from the training data against the test data

####  2.2.8 Useful linear regression documentation

Details of the LinearRegression class used are found on the SciKit Learn
(abbreviated to sklearn) documentation. There are notes on linear models in
the SkiKit Learn user notes.  The matrix inversion problem above could, in
principle, be solved by the quantum HHL algorithm. However, this is likely to
require fault tolerant quantum computers, so the caveats above are relevant.

###  2.3 Overfitting

In the example above we predicted unseen data points using a best fit straight
line $f(x)=w_{1}x+w_{0}$. We might think that we could increase the accuracy
of our prediction by changing our inductive bias to use a different function
from a different hypothesis space, a polynomial:

$f(x)=w_{6}x^{6}+w_{5}x^{5}+w_{4}x^{4}+w_{3}x^{3}+w_{2}x^{2}+w_{1}x+w_{0}$

This predictor function has more parameters, and so might fit the dataset
better. In the second example I have done exactly that, using some
manufactured data. However, there is a snag. Now, the best-fit curve shown in
Figure 8 has too many parameters: it fits the training data well, but does not
generalise to the unseen test data as well as the simple straight line. This
is called over-fitting For some data-sets the fit to the polynomial might give
better generalisation than the straight line, but here the performance is
worse.

Figure 8: Plot of employee data against company turnover : the polynomial has
over-fitted the data

One solution to overfitting is regularisation. Early stopping, feature
selection and ensembling are other solutions widely used. With regularisation
the predictor function could be simplified, using a straight line, not a
polynomial. Alternatively a penalty term $\lambda||\boldsymbol{\theta}||^{2}$
can be applied to the loss function. This works because the magnitude of the
parameter values can become larger with overfitting.

###  2.4 Clustering

The examples so far have been examples of supervised learning: we have fed our
model training examples, with input values and labels (employee numbers), and
then our model has been able to predict the label for unseen data.

If an AI system is to learn about its environment, it must derive patterns in
the data without any examples. This is called unsupervised learning, and is
often very useful for exploratory data analysis. An example of this is
clustering.

In the third example we use clustering to investigate the latitudes and
longitudes of the UK quantum technology companies that we met first in the
linear regression example.

####  2.4.1 Aim of the clustering example

The aim of the clustering example is to group the companies into clusters so
that companies in the same cluster are close together.

####  2.4.2 K-Means clustering

A tool called K-Means clustering is used to group the data into 10 clusters,
as shown in Figure 9. In this algorithm $K=10$ initial centroids are chosen
randomly. Data points are assigned to clusters so that the distance from data
points to the initial centroids is minimised. Then $K=10$ new centroids are
chosen at the centre of the new clusters. The data points are reassigned, and
the process continues iteratively, until there are no further re-assignments
of data points to new clusters.

A key point is that the “distance” is the Euclidean distance in two dimension,
where the distance between vectors $\boldsymbol{a}=(a_{x},a_{y})$ and
$\boldsymbol{b}=(b_{x},b_{y})$ is
$\sqrt{(a_{x}-b_{x})^{2}+((a_{y}-b_{y})^{2}}$.

The “centroids” of the clusters found are shown as crosses.

Figure 9: Plot of UK quantum technology company locations, and their K-means
centroids, with K = 10

If we look at the plot of these centroids on a map of the UK shown in Figure
10 we see that the quantum technology companies are clustered around Quantum
Hubs at Birmingham, Glasgow, Oxford and important Universities at Cambridge,
London, Bristol, Belfast, Liverpool and Manchester. There are also companies
in the South West and North East. The machine learning algorithm has found
these clusters without any external help.

Figure 10: Plot of the K=10 Means centroids for UK quantum technology
companies on a map of the UK

There is more information on SciKit learn. If you would like to practice your
clustering there are some artificial and real world datasets provided by
Thomas Barton available on GitHub.

###  2.5 Increasing dimensionality - the kernel trick

The next example is a binary classification supervised learning problem. I
have placed a Jupyter Notebook example on GitHub.

####  2.5.1 Aim of the support vector classifier example

The aim of a support vector classifier is to predict labels for unseen data.
The input is a two dimensional co-ordinate vector, the labels are binary
$y_{i}\in\\{-1,1\\}$, and as explained below, the data points can be separated
by a linear threshold.

####  2.5.2 The support vector classifier

In Figure 11 different colours have been given to training data points with
different labels.

Figure 11: Plot of the training data for the support vector classifier example

The algorithm used, the support vector classifier, is based on an inner
product, which gives a measure of similarity between two vectors. In this
example the inner product is defined as the dot product and the inner product
of two vectors $\boldsymbol{x}$ and $\boldsymbol{x}^{\prime}$ is:

$\langle\boldsymbol{x},\boldsymbol{x}^{\prime}\rangle=\boldsymbol{x}^{T}\boldsymbol{x}^{\prime}$

An inner product is a map from two vectors in the data domain $\mathscr{X}$ to
a real number which meets certain axioms.

$\kappa:\mathscr{X}\times\mathscr{X}\rightarrow\mathscr{R}$

Just like the inner product in quantum mechanics, the inner product in Machine
Learning gives a measure of similarity between two vectors. Using the inner
product allows access to geometrical concepts such as the angle between two
vectors, and their length.

In this example a support vector classifier uses the dot product to find a
threshold, a line to classify the new input data. Any data above the threshold
line is classified as red, and anything below the line is classified blue.
Figure 12 shows that the support vector classifier gives accurate predictions
for the test data. The graph also shows the threshold and the margin, the
distance between the threshold and the closest observation to the threshold.
The margin is shown as two dashed lines.

Figure 12: Plot of predictions showing the threshold, and margins

In support vector classification, the threshold is found by maximising the
margin. The inner product plays a fundamental role in solving this
optimisation problem.

###  2.6 The support vector machine

In all the examples so far there is an explicit assumption that similar points
are “close” in two dimensional space, either because they are separated by a
small Euclidean distance, or have a normalised inner product for the data
position vectors close to one.

In the next example similar points are not necessarily close in two
dimensions. I have placed a Jupyter Notebook on GitHub illustrating the Kernel
trick, which is a general method to solve classification problems where the
data points can’t be separated by a linear classifier, as shown in Figure 13.

Figure 13: Plot of training data for the support vector machine example

####  2.6.1 Aim of the support vector machine example

The aim of the support vector machine example is to separate data points that
can’t be separated by a linear classifier.

####  2.6.2 The kernel trick

As explained in “ _Machine Learning with Quantum Computers_ ” by Maria Schuld
and Francesco Petruccione [17] the support vector machine algorithm requires a
kernel, an inner product which gives a measure of similarity between two
vectors, but in this case the inner product is between two vectors in a higher
dimension feature space $\mathscr{F}$ defined by a feature map
$\phi:\mathscr{X}\rightarrow\mathscr{F}$ The kernel $\kappa$ for two vectors
$\boldsymbol{x}$ and $\boldsymbol{x}^{\prime}$ is defined as:

$\kappa(\boldsymbol{x},\boldsymbol{x}^{\prime})=\langle\phi(\boldsymbol{x}),\phi(\boldsymbol{x}^{\prime})\rangle_{\mathscr{f}}$

In this example a feature map $\phi$ is defined as

$\phi((x_{1},x_{2})^{T}=(x_{1},x_{2},0.5*(x_{1}^{2}+x_{2}^{2}))^{T}$

The feature map is used to transform the dataset from a two dimensional plane
into three dimensional space, as shown in Figure 14. The points are now
separable. The threshold is a two dimensional plane perpendicular to the
z-axis.

Figure 14: Plot of data in three dimensions

It is not necessary to actually do a mapping and hold the information in the
extra dimensions introduced. All that is needed is to compute the kernel
function between two points $x$ and $x^{\prime}$ and return the result, which
is a real number. This is called the kernel trick. This can be seen in the
Notebook where a new kernel function has been written to be called by the
SciKit Learn support vector classifier, which does not “know” about the new
diminension introduced. The accurate predictions by the support vector machine
classifier are shown in Figure 15.

Figure 15: Plot of predictions

If we added more dimensions to our example the threshold would become a
hyperplane in several dimensions. There is a great YouTube video on Support
Vector machines.

There is a link between the kernel function and the quantum inner product. The
kernel function is defined as:
$\kappa(\boldsymbol{x},\boldsymbol{x}^{\prime})=\langle\phi(\boldsymbol{x}),\phi(\boldsymbol{x}^{\prime})\rangle_{\mathscr{f}}$
This is very similar to the quantum inner product:
$\langle\phi(\boldsymbol{x})|\phi(\boldsymbol{x}^{\prime})\rangle$

####  2.6.4 The quantum kernel classifier in the NISQ era

The quantum inner product can be estimated by a simple circuit, and as
mentioned in the introduction, “a quantum feature map may enable a quantum
kernel classifier on NISQ machines”. However, there are still considerable
challenges in translating the classical data to a quantum form. Maria Schuld
has written an insightful paper about quantum kernel methods [19].

Useful progress has been made on NISQ machines. A 2022 paper by Boris Albrect
and others [5] describes a quantum kernel running on a 32 qubit neutral atom
quantum device. In this toxicity screening experiment, molecules were labelled
as either toxic or not toxic, and the QML algorithm classified unseen
molecules, with input data being a graph representing the arrangement of atoms
in the molecule. They achieved results comparable to the best classical
algorithms.

The quantum kernel used is explained in a 2021 paper by Louis-Paul Henry and
others [20]. Each atom in the molecules screened was represented by a qubit,
and the graph of the molecular structure determined which qubits were allowed
to interact. The quantum state of the system was updated by applying a series
of Hamiltonians. The precise form of the Hamiltonian pulse sequence was found
through training. An observable was measured at the end of the updates,
yielding a bit string. From this bit string a probability distribution
$\mathscr{P}(\mathscr{G})$ was built up for each graph by repeating the
measurements several times.

Once probability distributions had been obtained for each graph, the kernel of
two graphs $\mathscr{G}$ and $\mathscr{G^{\prime}}$ was calculated classically
as:

$\kappa(\mathscr{G},\mathscr{G^{\prime}})=exp[JS(\mathscr{P},\mathscr{P^{\prime}})]\in{[0.5,1]}$

$JS$ is the Jensen-Shannon divergence, a measure of the distance between the
two probability distributions $\mathscr{P}$ and $\mathscr{P^{\prime}}$. This
kernel was then used in a support vector machine, as described above.

In a 2018 paper Vojtech Havlicek and others [21] describe using 5 qubit super-
conducting quantum devices to experimentally realise two other quantum
kernels.

###  2.7 Reducing problem dimensionality - Principal Component Analysis

In the example above we increased the dimensionality of the clustering problem
to make points linearly seperable and then used a support vector machine to
find the threshold hyperplane. In the next example we will use Principal
Component Analysis (PCA), an unsupervised algorithm, that decrease the
dimensionality of the data to find the key features as shown in Figure 16. The
mathematics used is interesting, but not essential, and an Appendix explains
how the co-variance matrix is diagonalised, and the principal components are
then identified as the eigenvectors with the largest eigenvalues.

Figure 16: In Principal Component Analysis the original data is compressed,
retaining the key information

If we plot a graph of the data against the principle components then we can
see patterns that we would otherwise miss. An example from SciKit Learn used
the famous IRIS dataset where four features of the flowers are considered:
sepal length, sepal width, petal length and petal width. When the
dimensionality of the problem is reduced to two dimensions by PCA, it can be
seen in Figure 17 that there are three clusters of data, representing three
different species of Iris - Setosa, Versicolour and Virginica.

Figure 17: Results of the PCA analysis of the IRIS dataset from SciKit Learn

There is more information about PCA on SciKit Learn.

###  2.8 Deep learning

Deep learning is inspired by biological models, such as the brain, in which
signals are processed by individual neurons connected to thousands of other
neurons, which “fire” dependent on the signals received. In deep learning
these neurons are modelled by networks. A very simple network is shown in
Figure 18:

Figure 18: A simple network with four inputs

In this example there are four inputs and each of these is adjusted by a
weight. The weight adjusted inputs are summed, and a bias $b$ is applied. If
the input is a vector $\boldsymbol{x}$ and the weights are given by
$\boldsymbol{W}$ so that the sum $z$ is given by

$z=\boldsymbol{W}^{T}\boldsymbol{x}+b$

The output will then depend on some activation function, $g(z)$. Rectified
linear units use the activation function $g(z)=max(0,z)$ so there is a linear
output if the sum is greater than zero. Other functions, such as the sigmoid
$S(z)$, which produces an output between zero and one are also used in some
problems and have an output

$S(z)=\frac{1}{1+e^{-z}}$

The power of deep learning comes from using large numbers of nodes connected
together. In Figure 19 two hidden layers are added and each arrow is
associated with a weight. The idea behind this is that the hidden layers might
enable aspects of the image, like lines, to be trigger activiations in the
first layer, and these are combined so that geometrical shapes, like
triangles, triggger activations in the second layer, and so on with more
abstract levels of detail, until objects like hand written numbers can be
classified.

Figure 19: A network with two hidden layers

In a supervised learning classification problem a loss function is set to zero
if the classification is successful, and to one if it fails. This loss
function is a complex function of the various weights and activation functions
in the model. Backpropagation can be used to adjust the model weights
according to the gradient of the loss function with respect to the given
parameter. A feedback loop can be used to modify parameters based on the
results on previous calculations, to minimise the loss function.

Larger versions of these types of networks have been very successful at a wide
range of problems including complex image and video analysis and generation,
solving complex decision making problems at or beyond human level capability,
natural language processing and generation.

If you would like to work through an example there is a great PyTorch Tutorial
that uses the FashionMNIST dataset to train a neural network. The model
predicts if an input image belongs to one of 10 classes such as T-shirt,
Trouser and Pullover. The tutorial starts with an introduction to Tensors,
which are used to hold a representation of the network. Then the tutorial
walks through loading and transforming the data, building a neural network,
and automatic differentiation to optimise the model parameters. The dataset
comprises $28\times 28$ grayscale image and an associated label from one of
the 10 classes. The model has an input has 784 features, one for each
grayscale value, and hidden layers with 512 nodes.

###  2.9 Generative Adversarial Networks

You may have read that deepfakes can be generated by Generative Adversarial
Networks, or GANS. A deepfake is a digital object such as an image, or video
clip, produced by Machine Learing techniques.

A GAN is made up of two deep learning models that compete against each other.
A number of real examples, such as images, are available. As shown in In
Figure 20, the Generator Model generates new examples and the Discriminator
model classifies the examples as real or fake. Both models are trained. The
Generator is trained to produce fakes that fool the Discriminator. The
Discriminator is trained to identify real examples from fakes. Ultimately the
Generator can produce examples that seem very realistic.

Figure 20: A GAN. The Generator is optimised to produce fakes that fool the
Discriminator. The Discriminator it optimised to identify real examples from
fakes

In Figure 20 above the latent space is a random seed that ensures that each
Generator example is different.

As stated in the introduction “a possible early use case for quantum machine
learning is to use quantum computers to generate the latent spaces for GAN
models”. A Boson Sampling quantum computer can be used to generate the latent
space, with promising results. An alternative approach, set out in a 2019
paper [22] adds a Quantum Boltzman machine to the Discriminator to improve
performance.

##  3 To conclude

This review has described how the “human like intelligence” of Artificial
Intelligence (AI) has evolved from Expert Systems to Machine Learning (ML)
algorithms that can learn from data. In a simple linear regression example we
looked at a hypothesis space of linear predictor functions selected from a
large range of possible prediction functions by the inductive bias.

We defined a loss function to measure the quality of the solution, and
observed that many ML problems are optimisation problems in which parameters
are varied to minimise the loss function. We showed how having too many
parameters can lead to overfitting, where the predictions tracked noise in the
training data too closely, and did not generalise properly.

We then moved from the linear regression supervised learning example, where
labels are provided for each example, to an unsupervised learning clustering
example, where the algorithm learnt to group the points into clusters without
predefined labels for the data.

Next, we looked at a support vector classifier which found a linear threshold
between two groups of points. This example was generalised to a support vector
machine where the data set, when plotted in two dimensions, could not be
separated by a linear threshold. Instead the Kernel trick was used to
introduce a higher dimension in which the data points could be separated by a
hyper-plane.

Instead of increasing the number of dimensions, in Principal Component
Analysis (PCA) the number of dimension was reduced. Linear algebra techniques
were used to diagonalise the covariance matrix, and we recognised that the
eigenvectors with the largest eigenvalues (the diagonal entries) contained
most of the information about the data, allowing a simplified analysis in two
or three dimensions.

Finally, we look at deep-learning, which is inspired by biology-like networks,
and generalised this to a Generative Adversarial Networks, where a Generator
is locked in a games-theory inspired competition with a Discriminator. Even a
human can struggle to identify the Generator’s fakes from the real images used
in training.

We have also shown where quantum computing might be relevant, and raised some
important caveats about the readiness of the technology.

We hope that you have found this review valuable. If you have any questions or
feedback please contact the quantum team at Digital Catapult
<EMAIL_ADDRESS>
### Acknowledgements

Many thanks to Robert Elliott Smith for his helpful comments.

## Appendix - The mathematics of Principal Component Analysis

This Appendix explains how the covariance matrix is diagonalised, and the
principal components are identified as the eigenvectors with the largest
eigenvalues.

Following Mathematics for Machine Learning [8] we consider a set of $N$ data
vectors $x_{n}$, each of dimension $D$, with mean zero. The dataset can be
written
$\mathscr{X}=\left\\{{\boldsymbol{x}_{1},,\boldsymbol{x}_{n},,\boldsymbol{x}_{N}}\right\\}$
where $\boldsymbol{x}_{n}\in\ \mathscr{R}^{D}$. Because the mean is zero the
data covariance matrix is a $D\times D$ matrix defined as:

$\boldsymbol{S}=\frac{1}{N}\sum_{n=1}^{N}\boldsymbol{x}_{n}\boldsymbol{x}_{n}^{T}$

The data covariance matrix is a symmetric matrix. The diagonal elements
represent the variance, a measure of how much variability there is in each
dimension of the data. The other matrix elements represent the covariances, a
measure of how the variability in one data dimension is linked to variablity
in the other. This matrix is very similar to the density matrix $\rho$ in
quantum mechanics where the sum of the diagonals is normalised to one, and the
off-diagonal terms hold details of correlations:

$\rho=|\psi\rangle\langle\psi|$

There is a well known theorem from linear algebra that a symmetric matrix like
$\boldsymbol{S}$ can be diagonalised and we can always write:

$\boldsymbol{S}=\boldsymbol{P}\boldsymbol{D}\boldsymbol{P}^{-1}$

$\boldsymbol{D}$ is a diagonal matrix and it is often the case that some of
the diagonal entries are much larger than others. Conventionally
$\boldsymbol{D}$ is written with the largest eigenvector $\lambda_{1}$ at the
top, and the rest in descending order:

$\boldsymbol{D}=\begin{bmatrix}&\lambda_{1}&0&.&.&.\\\ &0&\lambda_{2}&0&.&.\\\
&0&0&\lambda_{3}&.&.\\\ &.&.&.&.&.\\\ &0&.&.&.&\lambda_{D}\end{bmatrix}$

If we multiply both sides of the equation
$\boldsymbol{S}=\boldsymbol{P}\boldsymbol{D}\boldsymbol{P}^{-1}$ above by
$\boldsymbol{P}$ from the right then we get.

$\boldsymbol{S}\boldsymbol{P}=\boldsymbol{D}\boldsymbol{P}$

and considering the column vectors of $\boldsymbol{P}$ :
$\boldsymbol{p}_{1},\boldsymbol{p}_{2},\boldsymbol{p}_{M}$ then we see these
satisfy an eigenvector equation:

$\boldsymbol{S}\boldsymbol{p}_{i}=\lambda_{i}\boldsymbol{p}_{i}$

The column vectors $\boldsymbol{p}_{i}$ are all eigenvectors of
$\boldsymbol{S}$. Those associated with the largest diagonal entries are the
principal components. Moreover these eigenvectors are row vectors of
$\boldsymbol{P}^{-1}$ since it is known that is known that
$\boldsymbol{P}^{T}=\boldsymbol{P}^{-1}$.

If we take the first M rows of $\boldsymbol{P}^{-1}$ this $MXN$ matrix will
project the data onto a lower dimension $MxM$ subspace whilst retaining much
of the information. The matrix multiplication is shown in Figure 21

Figure 21: A vector $\boldsymbol{z}$ in reduced M dimensional space can be
produced by multiplying a D dimensional data vector $\boldsymbol{x}$ by the
transpose of the first M eigenvectors of the covariance matrix.

## References

  * [1] Natansh Mathur, Jonas Landman, Yun Yvonna Li, Martin Strahm, Skander Kazdaghli, Anupam Prakash, and Iordanis Kerenidis.  Medical image classification via quantum neural networks.  http://arxiv.org/abs/2109.01831, 9 2021\.
  * [2] Amira Abbas, David Sutter, Christa Zoufal, Aurélien Lucchi, Alessio Figalli, and Stefan Woerner.  The power of quantum neural networks.  Nature Computational Science, 1:403 – 409, 2020.
  * [3] Edward Farhi and Hartmut Neven.  Classification with quantum neural networks on near term processors.  https://arxiv.org/abs/1802.06002, 2 2018\.
  * [4] He Liang Huang, Yuxuan Du, Ming Gong, Youwei Zhao, Yulin Wu, Chaoyue Wang, Shaowei Li, Futian Liang, Jin Lin, Yu Xu, Rui Yang, Tongliang Liu, Min Hsiu Hsieh, Hui Deng, Hao Rong, Cheng Zhi Peng, Chao Yang Lu, Yu Ao Chen, Dacheng Tao, Xiaobo Zhu, and Jian Wei Pan.  Experimental quantum generative adversarial networks for image generation.  Physical Review Applied, 16:024051, 8 2021.
  * [5] Boris Albrecht, Constantin Dalyac, Lucas Leclerc, Luis Ortiz-Gutiérrez, Slimane Thabet, Mauro D’Arcangelo, Vincent E. Elfving, Lucas Lassablière, Henrique Silvério, Bruno Ximenez, Louis-Paul Henry, Adrien Signoles, and Loïc Henriet.  Quantum feature maps for graph machine learning on a neutral atom quantum processor.  http://arxiv.org/abs/2211.16337, 11 2022\.
  * [6] Marcello Benedetti, Brian Coyle, Mattia Fiorentini, Michael Lubasch, and Matthias Rosenkranz.  Variational inference with a quantum computer.  Physical Review Applied, 16, 3 2021.
  * [7] Michele Grossi, Noelle Ibrahim, Voica Radescu, Robert Loredo, Kirsten Voigt, Constantin von Altrock, and Andreas Rudnik.  Mixed quantum–classical method for fraud detection with quantum feature selection.  IEEE Transactions on Quantum Engineering, 3:1–12, 2022.
  * [8] Marc Peter Deisenroth, A. Aldo Faisal, and Cheng Soon On.  Mathematics for machine learning.  https://mml-book.com.  Accessed: 2023-01-21.
  * [9] Eric Matthes.  Python Crash Course - A Hands-on, Project-Based Introduction to Programming.  No Starch Press, San Francisco, second edition, 2019.
  * [10] Daniel Goldsmith.  Quantum technology access programme github repository.  https://github.com/digicatapult/QTAP, 2023\.
  * [11] John Preskill.  Fault-tolerant quantum computation.  https://arxiv.org/abs/quant-ph/9712048, 12 1997.
  * [12] Lov K. Grover.  Quantum mechanics helps in searching for a needle in a haystack.  Physical Review Letters, 79:325–328, 6 1997.
  * [13] Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd.  Quantum algorithm for solving linear systems of equations.  Physical Review Letters, 103, 11 2008.
  * [14] A. Yu. Kitaev.  Quantum measurements and the abelian stabilizer problem.  http://arxiv.org/abs/quant-ph/9511026, 11 1995.
  * [15] Stuart Russell and Peter Norvig.  Artificial intelligence a modern approach, third edition.  https://zoo.cs.yale.edu/classes/cs470/materials/aima2010.pdf.  Accessed: 2023-01-21.
  * [16] Robert Elliott Smith.  Rage Inside the Machine: The Prejudice of Algorithms, and How to Stop the Internet Making Bigots of Us All.  Bloomsbury Business, London, 2019.
  * [17] Maria Schuld and Francesco Petruccione.  Machine learning with quantum computers.  https://link.springer.com/book/10.1007/978-3-030-83098-4.  Accessed: 2023-01-21.
  * [18] Oliver Theobald.  Machine Learning for Absolute Beginners. Scatter Plot Press, third edition, 2022.
  * [19] Maria Schuld.  Supervised quantum machine learning models are kernel methods.  https://arxiv.org/abs/2101.11020, 2021\.
  * [20] Louis-Paul Henry, Slimane Thabet, Constantin Dalyac, and Loïc Henriet.  Quantum evolution kernel : Machine learning on graphs with programmable arrays of qubits.  Physical Review A, 104, 7 2021.
  * [21] Vojtech Havlicek, Antonio D. Córcoles, Kristan Temme, Aram W. Harrow, Abhinav Kandala, Jerry M. Chow, and Jay M. Gambetta.  Supervised learning with quantum enhanced feature spaces.  Nature, 567:209–212, 4 2018.
  * [22] Eric R. Anschuetz and Cristian Zanoci.  Near-term quantum-classical associative adversarial networks.  Physical Review A, 100, 5 2019.


# Non-existence of certain type of convex functions on a Riemannian manifold
with a pole

Absos Ali Shaikh1∗, Chandan Kumar Mondal2 and Izhar Ahmad3
1,2Department of Mathematics,
University of Burdwan, Golapbag,
Burdwan-713104,
West Bengal, India<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
3Department of Mathematics and Statistics,
King Fahd University of Petroleum and Minerals,
Dhahran-31261,
Saudi Arabia<EMAIL_ADDRESS>
###### Abstract.

This paper is devoted to the study of non-existence of certain type of convex
functions on a Riemannian manifold with a pole. To this end, we have developed
the notion of odd and even function on a Riemannian manifold with a pole and
proved the non-existence of non-trivial and non-negative differentiable odd
convex function whose gradient is complete. Finally, we have deduced some
isoperimetric type inequality related with convex function.

††footnotetext:  ∗ Corresponding author.
$\mathbf{2010}$ Mathematics Subject Classification: 52A20, 52A30, 52A42,
53B20, 53C20, 53C22.
Key words and phrases: Convex function, odd convex function, even convex
function, manifold with a pole, negative point of a manifold, isoperimetric
type inequality.

##  1\. Introduction

The notion of convexity plays a crucial role in the field of economics,
management science and optimization problems etc. For instance, a (strictly)
convex function on an open set of the Euclidean space has not more than one
minimum. Consequently, the study of convex function and other related concepts
of convexity are essential for both the theoretical and practical point of
view. To model different real world problems, various researchers have
generalized the concept of convexity assumption and many articles have
appeared in the literature involving generalization of convex functions.

Throughout this paper by $M$ we mean a complete Riemannian manifold of
dimension $n$ endowed with some positive definite metric $g$ unless otherwise
stated. To generalize various concepts of Euclidean space to M, straight line
segment is substituted by a geodesic and vector space by $M$. There are many
convex functions with several structural implication on $M$ and such functions
form an important bridge between analysis and geometry. Since convex functions
on a compact manifold reduce to constant, the investigation of such functions
is more interesting on a non-compact manifold. A full discussion of convex
functions on $M$ can be found in Rapcsak [16] and Udrişte [19]. After that
many authors generalized the notion of convexity, see [8, 7, 11, 18].

In the literature of differential geometry the existence of proper convex
functions on a $M$ is a long standing problem. The first solution of this
problem appears from the work of Bishop and O’Neill [2]. They proved that if
$M$ has finite volume, then it does not possess any non-trivial smooth convex
function. Later, Yau [20] modified their result and proved the non-existence
of continuous convex function on such $M$. In 2017, Neto et. al. [13] showed
that if the geodesic flow on $M$ with finite volume is conservative then
convex functions on such $M$ are constants. In the case of infinite volume,
Greene and Wu [4] presented that there always exists a $C^{\infty}$ Lipschitz
continuous strictly convex exhaustion function on a complete non-compact $M$
whose sectional curvature is positive everywhere. Borsuk-Ulam Theorem [12]
implies that real valued continuous function on an unit $n$-dimensional sphere
attains the same value at two antipodal points. By using Borsuk-Ulam Thorem
and negative gradient flow of convex function on $M$, we prove the non-
existence of odd convex functions on $M$ with a pole.

The paper is structured as follows. Section 2 deals with some well known facts
about $M$. Section 3 is concerned with the convex functions on $M$ with pole.
In this section we define the concept of negative of a point and also study
some properties of even and odd convex functions. We apply Borsuk-Ulam theorem
on $M$ and study the behavior of continuous function. The object of this paper
is to prove the non-existence of odd convex functions on $M$ with a pole. We
also obtain a characterization of even convex function on geodesics. In the
last section we obtain some isoperimetric type inequality on $M$ with a pole
whose sectional curvature is bounded by some constant.

##  2\. Notations and Symbols

In this section we have discussed some rudimentary facts of $M$, which will be
used in the sequel (for reference see [9]). The tangent space at the point
$p\in M$ is denoted by $T_{p}M$ and $TM=\cup_{p\in M}T_{p}M$ is called the
tangent bundle of $M$. Let $p,q\in M$ and $\gamma:[a,b]\rightarrow M$ be a
smooth curve such that $\gamma(a)=p$ and $\gamma(b)=q$. The length $l(\gamma)$
of the curve $\gamma$ is given by

$\displaystyle l(\gamma)$ $\displaystyle=$
$\displaystyle\int_{a}^{b}\sqrt{g_{\gamma(t)}(\dot{\gamma}(t),\dot{\gamma}(t))}\
dt$ $\displaystyle=$
$\displaystyle\int_{a}^{b}\left\lVert\dot{\gamma}(t)\right\rVert dt.$

The curve $\gamma$ is said to be a geodesic if
$\nabla_{\dot{\gamma}(t)}\dot{\gamma}(t)=0\ \forall t\in[a,b]$, where $\nabla$
is the Riemannian connection of $g$. For any point $p\in M$, the exponential
map $exp_{p}:V_{p}\rightarrow M$ is defined by

$exp_{p}(u)=\sigma_{u}(1),$

where $\sigma_{u}$ is a geodesic with $\sigma(0)=p$ and
$\dot{\sigma}_{u}(0)=u$ and $V_{p}$ is a collection of vectors of $T_{p}M$
such that for each element $u\in V_{p}$, the geodesic with initial tangent
vector $u$ is defined on $[0,1]$. It is easy to see that for a geodesic
$\sigma$, the norm of tangent vector $\left\lVert\dot{\gamma}(t)\right\rVert$
is constant. A geodesic is said to be normal if its tangent vector is of unit
norm. If for each point $p\in M$, the exponential map is defined at every
point of $T_{p}M$, then $M$ is said to be a complete. Hopf-Rinow theorem
states that there are some equivalent cases of completeness of $M$. Let $p$,
$q\in M$. The distance between $p$ and $q$ is defined by

$d(p,q)=\inf\\{l(\gamma):\gamma\text{ be a curve joining }p\text{ and }q\\}.$

A geodesic $\sigma$ joining $p$ and $q$ is called minimal if
$l(\sigma)=d(p,q)$. Hopf-Rinow theorem provides the existence of minimal
geodesic between two points in a $M$. A smooth vector field is a smooth
function $X:M\rightarrow TM$ such that $\pi\circ X=id_{M}$, where
$\pi:TM\rightarrow M$ is the projection map. The integral curve of the vector
field $X$ at the point $p$ is the solution $($always exists by Picard
existence theorem$)$ of the differential equation

$(1)\qquad\qquad\qquad\qquad\left\\{\begin{array}[]{ll}\dot{x}(t)=X(x(t))\\\
x(0)=p.\\\ \end{array}\right.$

######  Definition 2.1.

[19] A real valued function $f$ on $M$ is called convex if for every geodesic
$\gamma:[a,b]\rightarrow M$ the following inequality holds

$f\circ\gamma((1-t)a+tb)\leq(1-t)f\circ\gamma(a)+tf\circ\gamma(b)\quad\forall
t\in[0,1].$

##  3\. Convex functions on the manifold with a pole

A pole in $M$ is such a point where the tangent space is diffeomorphic to the
whole manifold. Gromoll and Meyer [5] introduced the notion of pole in $M$. A
point $o\in M$ is called a pole [5] of $M$ if the exponential map at $o$ is a
global diffeomorphism and a manifold $M$ with a pole $o$ is denoted by
$(M,o)$. If a manifold possesses a pole then the manifold is complete. Simply
connected complete Riemannian manifold with non-positive sectional curvature
and a paraboloid of revolution are the examples of Riemannian manifolds with a
pole. A manifold with a pole is diffeomorphic to the Euclidean space but the
converse is not true always, see [6].
For any point $x\in\mathbb{R}^{n}$, the negative of $x$ lies on the straight
line passing through the origin and the point, and the distance between $-x$
and origin is same as that of $x$ and the origin. Similarly one can define the
negative of a point in a manifold with a pole by considering the pole as the
origin. Let $(M,o)$ be a manifold with a pole $o$. Then for each $r>0$ and
$v\in\mathbb{S}^{n-1}$ the map $(r,v)\mapsto exp_{o}(rv)$ is a diffeomorphism
from $(0,\infty)\times\mathbb{S}^{n-1}\rightarrow M-\\{o\\}$. Now we define
the negative of a point as follows:

######  Definition 3.1.

For a point $x\in(M,o)$ the negative of $x$ with respect to $o$ is defined by

$-_{o}x=exp_{o}(r_{1}v^{a}_{1}),$

where $x=exp_{o}(r_{1}v_{1})$ for some $r_{1}\in(0,\infty)$ and
$v_{1}\in\mathbb{S}^{n-1}$ and $v^{a}_{1}$ is the antipodal point of $v_{1}$
in $\mathbb{S}^{n-1}$.

######  Definition 3.2.

A function $f:(M,o)\rightarrow\mathbb{R}$ is said to be even with respect to
$o$ if

$f(x)=f(-_{o}x),$

and is called odd with respect to $o$ if

$f(x)=-f(-_{o}x),$

for all $x\in M$.

######  Example 3.1.

Let $o$ and $\bar{o}$ be two antipodal points in $\mathbb{S}^{2}$. Then
$M=\mathbb{S}^{2}-\\{\bar{o}\\}$ is a Riemannian manifold with the canonical
metric induced from $\mathbb{R}^{3}$ and $o$ is the pole. Now each $x\in M$
can be represented by $(x^{i})_{i=1}^{n}$, where
$exp_{o}^{-1}(x)=x^{i}\partial_{i}$ and
$\\{\partial_{1},\cdots,\partial_{n}\\}$ is the standard basis for $T_{o}M$.
If $f:(M,o)\rightarrow\mathbb{R}$ is defined by $f(x)=\sum_{i=1}^{n}x^{i}$ for
$x\in(M,o)$, then it can be easily proved that $f$ is an odd function with
respect to $o$.

######  Example 3.2.

Consider the manifold $(M,o)$, defined in the Example 3.1. Then the distance
function $d_{o}:(M,o)\rightarrow\mathbb{R}$ defined by

$d_{o}(x)=L(\gamma_{x}),$

where $L(\gamma_{x})$ is the length of the minimal geodesic joining $o$ and
$x$. Then $d_{o}$ is an example of even function on $(M,o)$.

In this section an even or odd function means even or odd function with
respect to a pole.
If $M$ is simply connected with non-positive sectional curvature, then for any
point $o\in M$, the square of distance function $d^{2}_{o}$, which is even
function by our definition, is a convex function [9]. Kasue [10] obtained some
conditions of the convexity of distance function on the manifold with a pole.
Naturally the question arises that is there any odd convex function in a
manifold with a pole? In this section we shall give a partial answer to this
question. The main theorem of the paper is as follows:

######  Theorem 3.1.

On a $(M,o)$ with dim$(M)>1$,there does not exist any non-negative
differentiable odd convex function whose gradient is a complete vector field
on $M$.

The following results are needed to established this Theorem.

######  Theorem 3.2 (The Borsuk-Ulam Theorem).

[12] Let $f:\mathbb{S}^{n}\rightarrow\mathbb{R}^{m}$ be any continuous map,
where $m\leq n$. Then there exist two antipodal points
$x,\bar{x}\in\mathbb{S}^{n}$ such that $f(x)=f(\bar{x})$.

Two points $x$ and $\bar{x}$ in $\mathbb{S}^{n-1}$ are said to be antipodal if
$x$ and $\bar{x}$ lie on the same diameter.

######  Lemma 3.3.

Let $f$ be a real valued continuous function on $M$ with dimension $>1$. Then
each neighborhood $U_{p}$ of any point $p\in M$ contains uncountably many
pairs of points where $f$ gives the same value.

###### Proof.

Let $V$ be a neighborhood $($ briefly, nbd $)$ of $p$ and $N_{p}$ be the
normal nbd of $p$. Then there exists a diffeomorphism
$\varphi:(0,R)\times\mathbb{S}^{n-1}\rightarrow V_{p}-\\{p\\}$ defined by
$\varphi(r,v)=exp_{p}(rv)$, where $V_{p}$ is an open nbd of $p$ containted in
$V\cap N_{p}$ and $R\leq inj(p)$, $inj(p)$ is the injective radius of $p$.
Hence the map $h:(0,R)\times\mathbb{S}^{n-1}\rightarrow\mathbb{R}$ defined by
$h(r,v)=f(\varphi(r,v))$ is also continuous. Now for a fixed $r_{0}\in(0,R)$,
the map $\varphi(r_{0},.)$ is continuous and injective. Hence the function
$h(r_{0},.):\mathbb{S}^{n-1}\rightarrow\mathbb{R}$ is continuous. Now by
Borsuk-Ulam theorem there exist two antipodal points $v$ and $\bar{v}$ in
$\mathbb{S}^{n-1}$ such that $h(r_{0},v)=h(r_{0},\bar{v})$. Since
$\varphi(r_{0},.)$ is injective, hence $\xi=\varphi(r_{0},v)$ and
$\eta=\varphi(r_{0},\bar{v})$ are two distinct points in $V_{p}-\\{p\\}$ such
that $f(\eta)=f(\xi)$. Now for $q\in V_{p}-\\{p\\}$ there exists
$r_{1}\in(0,R)$ and $v_{q}\in\mathbb{S}^{n-1}$ such that
$\varphi(r_{1},v_{q})=q$ and $r_{1}\neq r_{0}$. Hence by similar argument
there exists $\bar{q}\in V_{p}-\\{p\\}$ such that $f(q)=f(\bar{q})$. Hence for
each $r\in(0,R)$ there exists a pair in which $f$ gives the same value. Hence
there are uncountably many such pairs in $U$. ∎

A simple observation from the characterization of odd function and Lemma 3.3
is the following:

######  Corollary 3.3.1.

If $f:(M,o)\rightarrow\mathbb{R}$ is an odd function, then $f$ vanishes at
uncountably many antipodal points in $M$.

######  Theorem 3.4 (Mean value theorem).

[1] Let $f$ be a real valued differential function on $M$. Then, for every
pair of points $p,q\in M$ and every minimal geodesic path
$\sigma:[0,1]\rightarrow M$ joining $p$ and $q$, there exists $t_{0}\in[0,1]$
such that

$f(p)-f(q)=d(p,q)df_{\sigma(t_{0})}(\dot{\sigma}(t_{0})).$

In particular, $|f(p)-f(q)|\leq\left\lVert
df(\sigma(t_{0}))\right\rVert_{\sigma(t_{0})}d(p,q).$

######  Lemma 3.5.

If $f$ is a non-trivial real valued differentiable convex function on $M$,
then there exists a geodesic $\sigma:[0,l]\rightarrow M$, for some $l>0$, and
a point $t_{2}\in[0,l]$ such that

(1) $\langle grad(f)(\sigma(t_{2})),\dot{\sigma}(t_{2})\rangle\geq 1.$

###### Proof.

Let us consider the following negative gradient flow of the non-trivial convex
function $f:M\rightarrow\mathbb{R}$:

$\left\\{\begin{array}[]{ll}\dot{x}(t)=-\text{grad }f(x(t))\\\ x(0)=x.\\\
\end{array}\right.$

The value of $f$ decreases along the orbit of $grad(f)$. Now

$\displaystyle f(x)-f(x(t))$ $\displaystyle=$
$\displaystyle-\int_{0}^{t}\frac{d}{dt}f(x(t))dt$ $\displaystyle=$
$\displaystyle-\int_{0}^{t}\left\lVert grad(f(x(t)))\right\rVert^{2}dt$
$\displaystyle=$
$\displaystyle\int_{0}^{t}\left\lVert\dot{x}(t)\right\rVert^{2}dt.$

Let $\sigma:[0,t_{0}]\rightarrow M$ be the minimal geodesic connecting $x$ and
$x(t_{0})$ and parametrized in the interval $[0,t_{0}]$. Then

$\displaystyle d(x,x(t_{0}))$ $\displaystyle=$
$\displaystyle\int_{0}^{t_{0}}\left\lVert\dot{\sigma}(t)\right\rVert dt$
$\displaystyle=$ $\displaystyle\left\lVert\dot{\sigma}(0)\right\rVert t_{0}$
$\displaystyle\leq$
$\displaystyle\int_{0}^{t_{0}}\left\lVert\dot{x}(t)\right\rVert dt$
$\displaystyle\leq$
$\displaystyle\sqrt{t_{0}}\Big{(}\int_{0}^{t_{0}}\left\lVert\dot{x}(t)\right\rVert^{2}\Big{)}^{\frac{1}{2}}dt\quad[\text{
Using H\"{o}lder's inequality}]$ $\displaystyle=$
$\displaystyle\sqrt{t_{0}}\Big{(}f(x)-f(x(t_{0}))\Big{)}^{\frac{1}{2}}$
$\displaystyle=$
$\displaystyle\sqrt{t_{0}}[d(x,x(t_{0}))df(\sigma(t_{1}))(\dot{\sigma}(t_{1}))]^{\frac{1}{2}}\
\text{for some }0<t_{1}<t_{0}.$

The last inequality is obtained by using mean value theorem. From the above
discussion we get

$\displaystyle d(x,x(t_{0}))$ $\displaystyle\leq$ $\displaystyle
t_{0}df(\sigma(t_{1}))\dot{\sigma}(t_{1})$
$\displaystyle\left\lVert\dot{\sigma}(0)\right\rVert t_{0}$
$\displaystyle\leq$ $\displaystyle t_{0}\langle
grad(f(\sigma(t_{1})),\dot{\sigma}(t_{1})\rangle.\quad[\text{ since $\sigma$
is the minimal geodesic,}]$
$\displaystyle\left\lVert\dot{\sigma}(0)\right\rVert$ $\displaystyle\leq$
$\displaystyle\langle grad(f(\sigma(t_{1})),\dot{\sigma}(t_{1})\rangle.$

Since $\sigma$ is a geodesic, hence
$\left\lVert\dot{\sigma}(0)\right\rVert=\left\lVert\dot{\sigma}(t)\right\rVert\
\forall t\in[0,t_{0}].$ So, we get

(2) $\left\lVert\dot{\sigma}(t)\right\rVert\leq\langle
grad(f(\sigma(t_{1})),\dot{\sigma}(t_{1})\rangle\ \forall t\in[0,t_{0}].$

Due to the convexity of $f:M\rightarrow\mathbb{R}$ for $0\leq\lambda\leq 1$,
we get

(3) $f\circ\sigma((1-\lambda)t_{1}+\lambda
t_{0})\leq(1-\lambda)f(\sigma(t_{1}))+\lambda f(\sigma(t_{0})).$

Now taking limit $\lambda\rightarrow 0$, we get

$\lim\limits_{\lambda\rightarrow 0}\frac{f\circ\sigma((1-\lambda)t_{1}+\lambda
t_{0})-f(\sigma(t_{1}))}{\lambda}\leq f(\sigma(t_{0}))-f(\sigma(t_{1})).$

Hence,

(4) $\langle
grad(f)(\sigma(t_{1})),(\dot{\sigma}(t_{1}))\rangle\leq\frac{f(\sigma(t_{0}))-f(\sigma(t_{1}))}{t_{0}-t_{1}}.$

So, from equation (2) we say that

(5)
$\left\lVert\dot{\sigma}(t)\right\rVert\leq\frac{f(\sigma(t_{0}))-f(\sigma(t_{1}))}{t_{0}-t_{1}}.$

Now again applying mean value theorem to the right hand side of the above
equation we get

(6) $\displaystyle\left\lVert\dot{\sigma}(t)\right\rVert$ $\displaystyle\leq$
$\displaystyle\frac{1}{t_{0}-t_{1}}d(\sigma(t_{0}),\sigma(t_{1}))df(\sigma(t_{2}))(\dot{\sigma}(t_{2})),\quad\text{for
some }t_{1}<t_{2}<t_{0},$ (7) $\displaystyle\leq$
$\displaystyle\left\lVert\dot{\sigma}(t)\right\rVert
df(\sigma(t_{2}))(\dot{\sigma}(t_{2})).$

Hence

(8) $\langle grad(f)(\sigma(t_{2})),\dot{\sigma}(t_{2})\rangle\geq 1.$

∎

######  Corollary 3.5.1.

Let $f:M\rightarrow\mathbb{R}$ be a convex function such that $grad(f)$ is a
complete vector field on $M$. Then equation (1) holds for all geodesics on
$M$.

###### Proof.

This can easily be proved by taking the minimal geodesic with initial point
$x$ and final point $x(1)$. ∎

######  Lemma 3.6.

Let $f:(M,o)\rightarrow\mathbb{R}$ $($ dim $M>1)$ be a non-negative odd convex
function with $grad(f)$ as a complete vector field. Then for every geodesic
$\sigma:[0,1]\rightarrow M$ with $\sigma(0)=o$, there exists a point
$\sigma(t_{0})$ for some $t_{0}\in[0,1]$ such that $f(\sigma(t_{0}))=1$.

###### Proof.

Let $f:(M,o)\rightarrow\mathbb{R}$ be an odd convex function such that
$grad(f)$ as a complete vector field on $M$. Then by virtue of Lemma 3.5,
every geodesic $\sigma:[0,1]\rightarrow M$ has a point $q=\sigma(t_{0}),$
where $0<t_{0}<1$, such that

(9) $\langle grad(f)(\sigma(t_{0})),\dot{\sigma}(t_{0})\rangle\geq 1.$

By convexity of $f$ we get

(10) $\langle grad(f)(\sigma(t_{0})),\dot{\sigma}(t_{0})\rangle\leq
f(exp_{\sigma(t_{0})}\dot{\sigma}(t_{0}))-f(\sigma(t_{0})).$

If $f$ is non-negative, then from the above two equations we get

$f(exp_{\sigma(t_{0})}\dot{\sigma}(t_{0}))\geq 1.$

Since $f$ is an odd convex function, so $f(o)=0$. Hence by intermediate value
theorem, on the geodesic $\sigma$ with initial point $o$, there is a point
where $f$ takes the value $1$. The point
$exp_{\sigma(t_{0})}\dot{\sigma}(t_{0})$ depends on the choice of geodesic
$\sigma$. Hence for different geodesics we get different points. ∎

###### Proof of Theorem 3.1:.

Let $f:(M,o)\rightarrow\mathbb{R}$ be a non-negative differentiable odd convex
function such that $grad(f)$ is complete. Then for any two points $p,q\in M$
and for the unique minimal geodesic $\sigma:[0,1]\rightarrow M$ with
$\sigma(0)=p$ and $\sigma(1)=q$, we get

$f(\sigma(t))\leq A\ \forall t\in[0,1],\text{ where }A=(1-t)f(p)+tf(q).$

Now by taking negative in both sides of the above equation, we get

$-f(\sigma(t))\geq-A\ \forall t\in[0,1],$

which implies that

$f(-\sigma(t))\geq(1-t)f(-p)+tf(-q)\ \forall t\in[0,1],$

where $-\sigma:[0,1]\rightarrow M$ is the unique minimal geodesic such that
$-\sigma(0)=-p$ and $-\sigma(1)=-q$. But since $f$ is convex, it follows that

$f(\sigma(t))=A,$

where $p,q\in M$ and $\sigma:[0,1]\rightarrow M$ is the minimal geodesic with
initial point $p$ and final point $q$. Again in view of Corollary 3.3.1, there
exists a point $p\in M$ such that $f(p)=0$. So, by using convexity of $f$, we
get

(11) $f(\sigma(t))=0\ \forall t\in[0,1],$

where $\sigma(t)$ is the minimal geodesic with $\sigma(0)=o$ and
$\sigma(1)=p$, which contradicts to the Lemma 3.6. Hence there does not exist
any non-negative differentiable odd convex function in $(M,o)$ whose gradient
is a complete vector field. ∎

######  Lemma 3.7.

[15] Let $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ be a continuous function
whose lower level sets are bounded. Then $f$ has the global minimum.

######  Theorem 3.8.

Let $f:\mathbb{R}^{n}(\simeq T_{o}M)\rightarrow\mathbb{R}$ be a continuous
function whose lower level sets are bounded. Then the function
$h:(M,o)\rightarrow\mathbb{R}$ defined by

$h(x)=f(exp_{o}^{-1}(x)),\text{ for }x\in M,$

has the global minimum.

###### Proof.

Since $o\in M$ is a pole of $M$, hence the map $exp_{o}^{-1}:M\rightarrow
T_{o}M$ is a global diffeomorphism. Now there is a natural isomorphism between
$T_{o}M$ and $\mathbb{R}^{n}$ by using the standard orthonormal basis, so we
can consider the function $f$ from $T_{o}M$ to $\mathbb{R}$. Since the lower
level set of $f$ is bounded hence it has the global minimum $u_{0}\in T_{o}M$.
We claim that $x_{0}=exp_{o}(u_{0})$ is the global minimum point of $M$.
Suppose there exists a point $y_{0}\in M$ such that $h(x_{0})>h(y_{0})$. But
$h(y_{0})=f(exp_{o}^{-1}(y_{0}))$. So by taking $v_{0}=exp_{o}^{-1}(y_{0})$ we
get that $f(u_{0})>f(v_{0})$, which is a contradiction since $u_{0}$ is the
global minimum. Hence $x_{0}$ is the global minimum point of $h$. ∎

Let $f:(M,o)\rightarrow\mathbb{R}$ be an even convex function. Then
$f(x)=f(-_{o}x)\ \forall x\in M$. Let $\sigma:[0,1]\rightarrow M$ be a
geodesic such that $\sigma(0)=x$ and $\sigma(1)=-_{o}x$. Then by convexity we
get

$f(\sigma(t))\leq(1-t)f(x)+tf(-_{o}x)=f(x).$

Hence

$\lim\limits_{t\rightarrow 0}\frac{f(\sigma(t))-f(x)}{t}=\langle
grad(f)(\sigma(0)),\dot{\sigma}(0)\rangle\leq 0.$

Consider the geodesic $\tilde{\sigma}:[0,1]\rightarrow M$ defined by
$\tilde{\sigma}(t)=\sigma(1-t)$. Then
$\dot{\tilde{\sigma}}(t)=-\dot{\sigma}(1-t)\ \forall t\in[0,1].$ Similarly, we
get $\langle grad(f)(\tilde{\sigma}(0)),\dot{\tilde{\sigma}}(0)\rangle\leq 0.$
This implies that $\langle grad(f)(\sigma(1)),\dot{\sigma}(1)\rangle\geq 0.$
Thus by intermediate value theorem there exists $t_{0}\in[0,1]$ such that
$\langle grad(f)(\sigma(t_{0})),\dot{\sigma}(t_{0})\rangle=0.$ Hence by
changing the parameter of geodesic, we can conclude the following:

######  Theorem 3.9.

Let $f:(M,o)\rightarrow\mathbb{R}$ be an even convex function. Then for each
normal geodesic $\sigma:[0,l]\rightarrow M$ connecting $x$ and $-_{o}x$, there
exists a point $\sigma(t_{0})$, for $0<t_{0}<l$, such that

(12) $\langle grad(f)(\sigma(t_{0})),\dot{\sigma}(t_{0})\rangle=0.$

##  4\. Isoperimetric type inequality in manifold with a pole

If $f:M\rightarrow\mathbb{R}$ is a convex function, then Greene and Wu [3]
proved that $f$ is subharmonic, i.e., $\Delta f\geq 0$. There is a mean value
inequality on a Riemannian manifold [17, Section 6]. It compares the value of
a subharmonic function at the center of a ball with its mean value in the ball

######  Theorem 4.1.

[17] Let $B_{R}(p)$ be a geodesic ball in $M$. Suppose that the sectional
curvature $K_{M}\leq k$ for some constant $k$ and $R<inj(M,g)$. Then for any
real valued smooth function $f$ with $\Delta f\geq 0$ and $f\geq 0$ on $M$,

(13) $f(p)\leq\frac{1}{V_{k}(R)}\int_{B_{R}(p)}fdV,$

where $V_{k}(R)=\text{Vol}(B_{R},g_{k})$ is the volume of a ball of radius $R$
in the space form of constant curvature $k$, $dV$ is the volume form and
$inj(M,g)$ is the injective radius of $M$.

We will use the above theorem in the manifold with a pole to prove an
isoperimetric type inequality. Any ratio of volume and surface area of a
compact subset is called isoperimetric type inequality, for more information
see [14]. Since $(M,o)$ is a manifold with a pole, its injective radius is
infinite. Here we will compare the volume of an unit ball in a manifold whose
sectional curvature is a constant $k$ with surface area of unit ball in a
manifold with sectional curvature less than $k$ with respect to a convex
function.

######  Theorem 4.2.

Let $(M,o)$ be a Riemannian manifold with a pole with sectional curvature
$K_{M}\leq k$, for some constant $k$, and $f:(M,o)\rightarrow\mathbb{R}$ be a
smooth convex function such that $f\geq 0$. Then

(14)
$2\leq\frac{\omega_{M}}{V_{k}}+\frac{2}{V_{k}}\int_{\partial(B(o))}\frac{f(x)}{f(o)}dx,$

where $\omega_{M}$ is the surface area of unit ball in $M$ and $V_{k}$ is the
volume of unit ball in the space form of constant curvature $k$ and $B(o)$ is
the unit ball with center $o$.

###### Proof.

Since $f$ is convex, so $f$ is subharmonic function, i.e., $\Delta f\geq 0$.
Now taking the unit ball $B(o)$ with center $o$ and applying the above theorem
we get

(15) $\displaystyle f(o)$ $\displaystyle\leq$
$\displaystyle\frac{1}{V_{k}}\int_{B(o)}fdV,$ (16) $\displaystyle=$
$\displaystyle\frac{1}{V_{k}}\int_{\partial
B(o)}\Big{\\{}\int_{0}^{1}f\circ\gamma_{x}(t)dt\Big{\\}}dx,$

where $\gamma_{x}:[0,1]\rightarrow M$ is the minimal geodesic such that
$\gamma_{x}(0)=o$ and $\gamma_{x}(1)=x$ for each $x\in\partial B(o)$. By using
convexity of $f$ we obtain

$\displaystyle f(o)$ $\displaystyle\leq$
$\displaystyle\frac{1}{V_{k}}\int_{\partial
B(o)}\Big{\\{}\int_{0}^{1}((1-t)f(o)+tf(x))dt\Big{\\}}dx$ $\displaystyle=$
$\displaystyle\frac{1}{V_{k}}\int_{\partial
B(o)}\Big{\\{}f(o)\int_{0}^{1}(1-t)dt+f(x)\int_{0}^{1}tdt\Big{\\}}dx$
$\displaystyle=$ $\displaystyle\frac{1}{2V_{k}}\int_{\partial
B(o)}(f(o)+f(x))dx$ $\displaystyle=$
$\displaystyle\frac{\omega_{M}}{2V_{k}}f(o)+\frac{1}{V_{k}}\int_{\partial(B(o))}f(x)dx.$

Since $f\geq 0$, hence dividing both sides by $f(o)$ we get our result. ∎

## Acknowledgement

The second author greatly acknowledges to The UGC, Government of India for the
award of JRF.

## References

  * [1] Azagra, D., Ferrera, J. and Lopez-Mesas, F., Nonsmooth analysis and Hamilton-Jacobi equations on Riemannian manifolds, J. Funct. Anal., 220(2) (2005), 304–-361.
  * [2] Bishop, R. L. and O’Neill, B., Manifolds of negative curvature, Trans. Amer. Math. Soc., 145 (1969), 1–49.
  * [3] Greene, R. E. and Wu. H., On the subharmonicity and plurisubharmonicity of a geodesic convex function, Indiana Univ. Math. J., 22 (1971), 641–653.
  * [4] Greene, R. E. and Wu. H., $C^{\infty}$ convex functions and manifolds of positive curvature, Acta Mathematica, 137(1) (1976), 209–245.
  * [5] Gromoll, D. and Meyer, W., On complete open manifolds of positive curvature, Ann. Math., 90(1) (1969), 75–90.
  * [6] Itoh, M., Some geometrical aspects of Riemannian manifolds with a pole, Tsukuba J. Math., 4(2) (1980), 291–301.
  * [7] Iqbal, A., Ahmad, I. and Ali, S., Some properties of geodesic semi-E-convex functions, Nonlinear Anal., 74 (2011), 6805–6813.
  * [8] Iqbal, A., Ali, S. and Ahmad, I., On geodesic E-convex sets, geodesic E-convex functions and E-epigraphs, J. Optim. Theory Appl., 155(1) (2012), 239–251.
  * [9] Jost, J., Riemannian geometry and geometric analysis, Springer, Berlin, 2011.
  * [10] Kasue, A., On Riemannian manifolds with a pole, Osaka J. Math., 18 (1981), 109–113.
  * [11] Kilicman, A. and Saleh,W., Some properties of geodesic semi E-b-vex functions, Open Math., 13 (2015), 795–804.
  * [12] Matoušek, J., Using the Borsuk-Ulam theorem, Springer-Verlag, Berlin, 2003.
  * [13] Neto, C. J. X., Melo, Í. D. L. and Sousa, P. A. A., Convexity and some geometric properties, J. Optim. Theory Appl., 173 (2017), 459–470.
  * [14] Osserman, R., The isoperimetric inequality, Bull. Amer. Math. Soc., 86(6) (1978), 1182–1238.
  * [15] Pavić, Z. and Novoselac, V., Investigating an over-determined system of linear equations by using convex function, Hacet. J. Math. Stat., 46(5) (2017), 865–874.
  * [16] Rapcsak, T., Smooth nonlinear optimization in $\mathbb{R}^{n}$, Kluwer Academic Publisher, 1997.
  * [17] Schoen, R. and Yau, S. T., Lectures on differential geometry. Conference Proceedings and Lecture Notes in Geometry and Topology, I. International Press, Cambridge, MA, 1994.
  * [18] Shaikh, A. A., Iqbal, A. and Mondal, C. K., Some results on $\varphi$-convex functions and geodesic $\varphi$-convex functions, arXiv:1706.05841 [math.DG] (2017).
  * [19] Udrişte, C., Convex functions and optimization methods on Riemannian manifolds, Kluwer Academic Publisher, 1994.
  * [20] Yau, S. T., Non-existence of continuous convex functions on certain Riemannian manifolds, Math. Ann., 207 (1974), 269–270.