# Data Augmentation and Classification of Sea-Land Clutter for Over-the-
Horizon Radar Using AC-VAEGAN

Xiaoxuan Zhang, Zengfu Wang, Kun Lu, and Quan Pan This work was in part
supported by the National Natural Science Foundation of China (grant no.
61790552) and Natural Science Basic Research Plan in Shaanxi Province of China
(2021JM-06). Xiaoxuan Zhang, Zengfu Wang, Quan Pan are with the School of
Automation, Northwestern Polytechnical University, and the Key Laboratory of
Information Fusion Technology, Ministry of Education, Xi’an, Shaanxi, 710072,
China. Kun Lu is with Nanjing Research Institute of Electronics Technology and
the Sky-Rainbow United Laboratory, Nanjing, Jiangsu, 210039, China. E-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>quanpan@nwpu.edu.cn). (Corresponding author: Zengfu Wang.)

###### Abstract

In the sea-land clutter classification of sky-wave over-the-horizon-radar
(OTHR), the imbalanced and scarce data leads to a poor performance of the deep
learning-based classification model. To solve this problem, this paper
proposes an improved auxiliary classifier generative adversarial network (AC-
GAN) architecture, namely auxiliary classifier variational autoencoder
generative adversarial network (AC-VAEGAN). AC-VAEGAN can synthesize higher
quality sea-land clutter samples than AC-GAN and serve as an effective tool
for data augmentation. Specifically, a 1-dimensional convolutional AC-VAEGAN
architecture is designed to synthesize sea-land clutter samples. Additionally,
an evaluation method combining both traditional evaluation of GAN domain and
statistical evaluation of signal domain is proposed to evaluate the quality of
synthetic samples. Using a dataset of OTHR sea-land clutter, both the quality
of the synthetic samples and the performance of data augmentation of AC-VAEGAN
are verified. Further, the effect of AC-VAEGAN as a data augmentation method
on the classification performance of imbalanced and scarce sea-land clutter
samples is validated. The experiment results show that the quality of samples
synthesized by AC-VAEGAN is better than that of AC-GAN, and the data
augmentation method with AC-VAEGAN is able to improve the classification
performance in the case of imbalanced and scarce sea-land clutter samples.

###### Index Terms:

Generative adversarial network, Data augmentation, Imbalanced and scarce
samples, Clutter classification, Over-the-horizon-radar, Deep learning.

##  I Introduction

As a crucial system for remote sensing, sky-wave over-the-horizon-radar (OTHR)
is widely used in military and civilian fields [1, 2, 3, 4]. The sea-land
clutter classification of OTHR is the process of identifying whether the
background clutter of each range-azimuth cell is originated from land or sea.
Matching the derived classification results with a prior geographic
information then provides coordinate registration parameters for target
localization, which has the potential to improve target localization accuracy
at a low cost [5, 6].

So far, there have been considerable work on sea-land clutter classification.
Turley _et al._ [7] defined the energy ratio between the dominant and sub-
dominant Bragg resonance peak as a Bragg ratio test statistic, which plays as
a key feature for sea-land clutter classification. Jin _et al._ [8] proposed a
support vector machine-based sea-land clutter classification method by
analyzing three kinds of features of sea-land clutter. In [7, 8], careful
human engineering and considerable domain expertise are needed to design a
feature extractor that transform the sea-land clutter spectrum data into a
suitable feature vector so that the classifier can classify it into a specific
category. The recently emerging sea-land clutter classification methods based
on deep convolutional neural network (DCNN) have achieved remarkable results,
of which feature extractor is not designed by human engineers, but learned
from data using a general-purpose learning procedure, thus avoiding the
drawbacks of the methods in [7, 8]. Li _et al._ [9] proposed a DCNN with
multiple hidden layers for sea-land clutter classification, of which
performance is superior to those of [7, 8]. Besides, sea-land clutter
classification based on spectrum data with multi-resolution and multi-scale
characteristics was studied. Zhang _et al._ [10] leveraged feature maps of
different levels to classify sea-land clutter at different scales, and
proposed a DCNN method with cross-scale transfer learning. Li _et al._ [11]
proposed a cross-scale DCNN sea-land clutter classification method based on
the idea of algebraic multi-grid and interpolation-related image downsampling.
However, the DCNN-based methods in [9, 10, 11] did not consider the problem of
imbalanced or scarce sea-land clutter spectrum data. In the context of sea-
land clutter classification, imbalanced sea-land clutter spectrum data means
that the number of training samples of different classes is imbalanced; scarce
sea-land clutter spectrum data means that only a small number of labeled
training samples are available.

Existing deep learning-based methods require a large number of labeled
training data for the classifier to learn a considerable amount of free
parameters for accurate prediction. Although it is easy to collect a large
quantity of sea-land clutter data of OTHR, the manual labeling work is quite
cumbersome. The lack of labeled training data for some or all classes often
make it difficult for the classifier to perform optimally. Therefore, it is
necessary to improve the performance of the classifier considering imbalanced
and scarce sea-land clutter data.

In image classification, one of the most effective ways to deal with scarce
samples is data augmentation [12, 13, 14]. Traditional methods for image data
augmentation perform a certain of geometric affine transformations on training
data, such as translation, rotation, mirroring, clipping and scaling, etc.,
which can improve the performance of classifiers without actually collecting
more data. However, since the sea-land clutter data is radar echo signal, the
above-mentioned methods for image data augmentation may destroy the
classification-related features of sea-land clutter. One straightforward way
to deal with imbalanced samples is random sampling, which includes random
oversampling and random undersampling [15, 16, 17]. Random oversampling refers
to randomly copying samples from the minority class to balance original
dataset. Since some samples appear repeatedly in the randomly oversampled
dataset, the classification model is prone to overfitting. Conversely, random
undersampling refers to randomly dropping samples from the majority class to
balance original dataset. Since the randomly undersampled dataset may lose
some useful information, the classification model only learns part of the
overall mode.

As an unsupervised and semi-supervised implicit probability density generative
model, generative adversarial network (GAN) has been successfully applied to
data augmentation because of its powerful data synthesis capability [18, 19].
GAN learns data intrinsic distribution through an adversarial game of
generator and discriminator to synthesize fake samples that are consistent
with original data distribution [20]. The first fully connected generator and
discriminator adversarial architecture provides a theoretical guarantee for
the subsequent development of GAN [20]. Up to now, GAN has evolved into a
variety of new architectures. Deep convolutional generative adversarial
network [21] combines DCNN architecture with GAN architecture, which improves
the training stability and the quality of samples synthesized by GAN.
Variational autoencoder generative adversarial network (VAEGAN) [22] is an
extension of GAN of which decoder and generator share the same network, while
improving the quality of samples synthesized by variational autoencoder (VAE)
and GAN. As unsupervised generative models, however, the GANs in [20, 21, 22]
cannot guide networks to synthesize specific class. Conditional generative
adversarial network (CGAN) [23] performs supervised training by introducing
class attribute, which can control the network to synthesize sample with
specific class. Semi-supervised generative adversarial network (SGAN) [24]
extends supervised and unsupervised GAN architectures to a semi-supervised
architecture, and realizes the synthesis and classification of samples by
training a small amount of labeled data and a large amount of unlabeled data.
Auxiliary classifier generative adversarial network (AC-GAN) [25] combines the
network characteristics of CGAN and SGAN, and can output the class which the
synthetic sample belongs to while conditionally synthesizing the sample. On
the basis of AC-GAN, improved AC-GAN architectures are developed for data
augmentation to solve the problem of imbalanced and scarce samples. Data
augmentation generative adversarial network (DAGAN) [18], based on image CGAN,
takes data from a source domain and learns to take any data item and
generalize it to generate other within-class data items, which can perform
data augmentation on scarce samples. Balancing generative adversarial network
(BAGAN) [26] applies class conditioning in latent space to drive the
generation process towards minority class, which can perform data augmentation
on imbalanced samples.

The samples synthesized by GAN have potential to balance the minority class of
unbalanced and scarce sea-land clutter samples, which can then act as a
powerful supplement for the data augmentation of sea-land clutter
classification. To this end, we propose an improved AC-GAN architecture via
combining VAEGAN with AC-GAN, namely, auxiliary classifier variational
autoencoder generative adversarial network (AC-VAEGAN). The overall network
structure of AC-VAEGAN is the same as that of VAEGAN, which consists of three
parts: encoder (En), decoder (De)/generator (G), and discriminator
(D)/classifier (C). The difference is that the input of G of AC-GAN is the
combination of random noise sequence and class attribute, while the proposed
AC-VAEGAN attaches an additional input combination of random noise sequence
encoded by En of VAEGAN and class attribute. Since the random noise sequence
of the latter is obtained by coding of original data, the combination of the
coding sequence and the class attribute is better than that of the former,
which makes the training of AC-VAEGAN more efficient than that of AC-GAN. On
the other hand, AC-VAEGAN can be regarded as an improved VAEGAN, which can
specify the class while synthesizing samples. The proposed AC-VAEGAN is then
applied to data augmentation of sea-land clutter classification of OTHR.

The quality of synthetic samples is particularly important. The sea-land
clutter samples are radar echo signals, of which performance cannot be
evaluated only by the methods from GAN domain. To this end, we propose a
combination of traditional evaluation of GAN domain and statistical evaluation
of signal domain to evaluate the quality of synthetic samples. From the
perspective of evaluation of GAN domain, GAN-train and GAN-test [27] are
adopted to evaluate the diversity and fidelity of synthetic samples. From the
perspective of statistical evaluation, absolute distance (AD), cosine
similarity (CS) and Pearson correlation coefficient (PCC), are proposed to
evaluate the quality of synthetic samples from three aspects including signal
energy difference, signal direction difference, and signal correlation.

In experiment validation, we prepare three kinds of sea-land clutter datasets:
original dataset, imbalanced dataset, and scarce dataset. Based on original
dataset, the quality of samples synthesized by AC-VAEGAN and AC-GAN is
evaluated comparatively. Experiment results show that the quality of samples
synthesized by AC-VAEGAN is better than that of AC-GAN in terms of the above
evaluation metrics. Furthermore, the effectiveness of AC-VAEGAN data
augmentation method is verified based on imbalanced and scarce datasets.  In
summary, our main contributions are four-fold:

  * •

For the first time, a novel GAN architecture, namely AC-VAEGAN, is proposed.
AC-VAEGAN has the advantages over either of AC-GAN/VAEGAN architecture,
enabling VAEGAN to synthesize sample with specified class, and AC-GAN to
synthesize samples with higher quality; this method is of independent
interest.

  * •

An evaluation method combining traditional evaluation of GAN domain and
statistical evaluation of signal domain is proposed to evaluate the quality of
synthetic samples. The evaluation metrics include GAN-train, GAN-test, AD, CS,
and PCC.

  * •

The proposed AC-VAEGAN is successfully applied to data augmentation of
imbalanced and scarce sea-land clutter samples of OTHR. To our best knowledge,
the problem of data augmentation of imbalanced and scarce sea-land clutter
samples of OTHR has not been considered in the literature. The classification
model trained with the balanced dataset after data augmentation performs
significantly better than that trained with the imbalanced and scarce
datasets, leading to potential improvement on the coordinate registration of
OTHR.

  * •

Also, we verify the data augmentation and classification performance of AC-
VAEGAN on the publicly available Moving and Stationary Target Acquisition and
Recognition (MSTAR) dataset. This further demonstrates that the general AC-
VAEGAN classification method is applicable not only to one-dimensional remote
sensing signal dataset (e.g., sea-land clutter dataset), but also to two-
dimensional remote sensing image dataset (e.g., MSTAR dataset).

The remainder of this paper is organized as follows. In section II, the
architecture, sub-module details and training procedure of the proposed AC-
VAEGAN are described. In Section III, the evaluation method is proposed to
evaluate the quality of synthetic samples. In Section IV, sea-land clutter
dataset and MSTAR dataset are introduced, and then the quality of samples
synthesized by AC-VAEGAN and AC-GAN is compared. Also, data augmentation
experiment is performed to verify the effectiveness of AC-VAEGAN. Section V
draws the conclusions and highlight some future work.

##  II Sea-Land Clutter Sample Synthesis Using AC-VAEGAN

In this section, we first describe the proposed AC-VAEGAN architecture and the
loss function, and then elaborate on the AC-VAEGAN sub-modules (En, De/G and
D/C) for sea-land clutter sample synthesis. At last, we introduce the training
and algorithmic procedures of AC-VAEGAN.

###  II-A AC-VAEGAN Architecture

The architecture of AC-VAEGAN is shown in Fig. 1. AC-VAEGAN is a semi-
supervised generative model consisting of three parts: En, De/G and D/C. AC-
VAEGAN has the following advantages. Comparing with VAEGAN, AC-VAEGAN can not
only synthesize samples, but also specify the class of synthetic samples.
Comparing with AC-GAN, the inputs of G of AC-VAEGAN contain not only the
combination of random noise sequence $z_{\text{gen}}$ and class attribute $c$,
but also the combination of $z_{\text{deco}}$ encoded by En of VAEGAN and
class attribute $c$, which implies the distribution of $x_{\text{real}}$ with
class $c$ in latent space. Thus, the training of AC-VAEGAN is more efficient
than that of AC-GAN. Also, AC-VAEGAN is easier to synthesize high quality
samples than AC-GAN. The loss function of AC-VAEGAN is defined as follows:

$L=L_{\text{VAE}}+L_{\text{De/G}}+L_{\text{D/C}},$ (1)

where,

$\displaystyle
L_{\text{VAE}}\\!=\\!\min\text{MSE}(x_{\text{deco}},x_{\text{real}})\\!+\\!\text{KLD}[N(\mu_{x_{\text{real}}},\sigma_{x_{\text{real}}}^{2}),N(0,1)],$
$\displaystyle L_{\text{De/G}}=\max L_{C}-L_{S},$ $\displaystyle
L_{\text{D/C}}=\max L_{C}+L_{S},$ $\displaystyle L_{C}=\mathbb{E}[\log
p(C=c|x_{\text{real}})]+\mathbb{E}[\log
p(C=c|x_{\text{deco}},x_{\text{gen}})],$ $\displaystyle
L_{S}\\!=\\!\mathbb{E}[\log
p(S=\text{real}|x_{\text{real}})]\\!+\\!\mathbb{E}[\log
p(S=\text{fake}|x_{\text{deco}},x_{\text{gen}})],$

where $x_{\text{real}}$ are real samples, $x_{\text{deco}}$ are fake samples
synthesized by De, $x_{\text{gen}}$ are fake samples synthesized by G,
$N(0,1)$ represents the standard normal distribution,
$N(\mu_{x_{\text{real}}},\sigma_{x_{\text{real}}}^{2})$ represents the normal
distribution with mean $\mu_{x_{\text{real}}}$ and variance
$\sigma_{x_{\text{real}}}^{2}$, MSE denotes the mean squared error operator,
KLD denotes the Kullback–Leibler divergence operator, $L_{C}$ denotes the log-
likelihood of the correct class, and $L_{S}$ denotes the log-likelihood of the
correct source.

The input of AC-VAEGAN is 1-dimensional sea-land clutter samples of size $N$,
for example, $N=512$. See Section IV-A1 for the details of the sea-land
clutter samples. Since the configuration of sub-modules is related to the size
of the input signal, we next depict the details of the sub-modules by assuming
$N=512$. If $N\neq 512$, the number of layers in the configuration of sub-
modules will be adapted to fit the size of the input signal. The output of AC-
VAEGAN include 1-dimensional real or fake probability. and 3-dimensional
classification results.

Figure 1: The architecture of AC-VAEGAN.

###  II-B Details of AC-VAEGAN Sub-Modules

Next, we elaborate on AC-VAEGAN sub-modules (En, De/G and D/C) for sea-land
clutter sample synthesis.

TABLE I: The configuration of AC-VAEGAN sub-modules Module | Layer | Configuration | Output Size
---|---|---|---
En | 1 | Conv1D, LeakyReLU | 8$\times$256
2-7 | Conv1D, BN1D, LeakyReLU | 512$\times$4
8 | FC, LeakyReLU | 1000
9 | FC | 100
De/G | 1 | DeConv1D, BN1D, ReLU | 512$\times$4
2-7 | DeConv1D, BN1D, ReLU | 8$\times$256
8 | DeConv1D, Tanh | 1$\times$512
D/C | 1 | Conv1D, LeakyReLU | 8$\times$256
2-7 | Conv1D, BN1D, LeakyReLU | 512$\times$4
8 | FC, LeakyReLU | 1000
9_1 | FC, Sigmoid | 1
| 9_2 | FC, Softmax | 3

####  II-B1 Encoder

En consists of a 9-layer neural network: seven 1-dimensional convolutional
(Conv1D) layers are followed by two fully connected (FC) layers. The
architecture and configuration of En are shown in Fig. 2 and Table I,
respectively. All convolutional kernels have the same configuration with size
4, stride 2 and padding 1. As we mentioned above, the input is a real signal
of size 1$\times$512\. After layer 1, the number of channels is increased by
eight times, and the signal size is decreased by one-half. Then after layers
2-7, the number of channels is increased by two times per layer, and the
signal size is decreased by one-half per layer. Next, flatten the resulting
feature vector. Finally, after layers 8-9, output a 100-dimensional feature
vector, which is used to fit the mean $\mu$ or log variance $\log\sigma^{2}$
of normal distribution. One can sample a random noise sequence $z$ from
$N(0,1)$. Leveraging the reparameterization technique [28], a 100-dimensional
random noise sequence $z_{\text{deco}}$ following $N(\mu,\sigma^{2})$ is
obtained.

Figure 2: Details of En sub-module.

####  II-B2 Decoder/Generator

De/G consists of a 8-layer neural network: eight 1-dimensional deconvolutional
(DeConv1D) layers. The architecture and configuration of De/G are shown in
Fig. 3 and Table I, respectively. The convolutional kernel of layer 1 has
configuration with size 4, stride 1 and padding 0. The convolutional kernels
of layers 2-8 have the same configuration with size 4, stride 2 and padding 1.
The input is a 103-dimensional vector [$z_{\text{deco}}$, $c$] or
[$z_{\text{gen}}$, $c$], where $z_{\text{deco}}$ and $z_{\text{gen}}$ are
100-dimensional random noise sequences, and $c$ is a 3-dimensional one-hot
encoding of class attribute. After layer 1, the number of channels becomes
512, and the dimension of the feature vector is four. Then after layers 2-7,
the number of channels is decreased by one-half per layer, and the signal size
is increased by two times per layer. Finally, after layer 8, output a $1\times
512$ reconstructed signal $x_{\text{deco}}$ or generated signal
$x_{\text{gen}}$.

Figure 3: Details of De/G sub-module.

####  II-B3 Discriminator/Classifier

D/C consists of a 9-layer neural network: seven 1-dimensional convolutional
layers are followed by two fully connected layers. The architecture and
configuration of D/C are shown in Fig. 4 and Table I, respectively. Layers 1-8
are consistent with En. Layer 9 is divided into two channels 9_1 and 9_2. The
input is a $1\times 512$ real/reconstructed/generated signal. After layer 9_1,
output a 1-dimensional real or fake probability. After layer 9_2, output a
3-dimensional classification result.

Figure 4: Details of D/C sub-module.

###  II-C Training Procedure of AC-VAEGAN

It is well known that the core idea of VAE is to maximize the variational
lower bound of log-likelihood function through En and De [28], while the core
idea of GAN is to make the network reach Nash equilibrium through “zero-sum
game” of G and D [20]. AC-VAEGAN combines these two ideas. During the training
of AC-VAEGAN, the role of En is to learn data representation from the real
signal $x_{\text{real}}$ and encode it into low-dimensional latent variable
$z_{\text{deco}}=\text{En}(x_{\text{real}})$, the role of De/G is to
decode/generate $z_{\text{deco}}$ from $N(\mu,\sigma^{2})$ or $z_{\text{gen}}$
from $N(0,1)$ together with the fake label $c$ into a high-dimensional fake
signal, i.e., the reconstructed signal
$x_{\text{deco}}=\text{De}(z_{\text{deco}},c)$ or the generated signal
$x_{\text{gen}}=\text{G}(z_{\text{gen}},c)$, and the role of D/C is to
discriminate $x_{\text{real/deco/gen}}$ from the real/reconstructed/generated
signal as a real or fake probability
$p_{\text{real/deco/gen}}=\text{D}(x_{\text{real/deco/gen}})$ or classify it
as a class label
$l_{\text{real/deco/gen}}=\text{C}(x_{\text{real/deco/gen}})$. The detailed
training procedure of AC-VAEGAN is described in Algorithm 1.

Algorithm 1 Training of AC-VAEGAN.

Initialize: En, De/G, D/C. Let $K$ be the total number of iterations.

for $k=1,\ldots,K$ do

Train D/C:

(1) Sample $m$ real signals
$\\{x_{\text{real}}^{1},x_{\text{real}}^{2},\dots,x_{\text{real}}^{m}\\}$ from
dataset with real labels $\\{c_{real}^{1},c_{real}^{2},\dots,c_{real}^{m}\\}$;

(2) Encode $m$ codings
$\\{z_{\text{deco}}^{1},z_{\text{deco}}^{2},\dots,z_{\text{deco}}^{m}\\}$ from
En:

$z_{\text{deco}}^{i}=\text{En}(x_{\text{real}}^{i})$

(3) Sample $m$ fake labels
$\\{c_{\text{fake}}^{1},c_{\text{fake}}^{2},\dots,c_{\text{fake}}^{m}\\}$ from
label space;

(4) Decode $m$ fake signals
$\\{x_{\text{deco}}^{1},x_{\text{deco}}^{2},\dots,x_{\text{deco}}^{m}\\}$ from
De:

$x_{\text{deco}}^{i}=\text{De}(z_{\text{deco}}^{i},c_{\text{fake}}^{i})$

(5) Sample $m$ latent vectors
$\\{z_{\text{gen}}^{1},z_{\text{gen}}^{2},\dots,z_{\text{gen}}^{m}\\}$ from
noise prior $p(z)$;

(6) Generate $m$ fake signals
$\\{x_{\text{gen}}^{1},x_{\text{gen}}^{2},\dots,x_{\text{gen}}^{m}\\}$ from G:

$x_{\text{gen}}^{i}=\text{G}(z_{\text{gen}}^{i},c_{\text{fake}}^{i})$

(7) Update D/C by $L_{D/C}$.

Train De/G and En:

(8) Repeat Steps (2)-(6);

(9) Update De/G and En by $L_{\text{VAE}}+L_{\text{De/G}}$.

end for

##  III Evaluation Method of Synthetic Sea-Land Clutter Samples

The sea-land clutter samples synthesized by AC-VAEGAN are used for data
augmentation, so the quality of the synthetic samples is particularly
important. To our best knowledge, there is no a unified standard for
evaluating the quality of samples synthesized by GAN. In GAN domain, a number
of quantitative evaluation metrics have been proposed to evaluate the quality
of synthetic samples, including Inception Scores (IS) [29], Mode Scores (MS)
[30], Fréchet Inception Distance (FID) [31], Maximum Mean Discrepancy (MMD)
[32], Earth Mover’s Distance (EMD) [33, 34], and GAN-train/GAN-test [27], etc.
Different metrics measure different aspects of GAN. Abundant empirical
experiments have shown that different conclusions on the quality of the
synthetic samples may be draw from different evaluation metrics. Therefore, it
is necessary to choose appropriate metrics for real application scenarios.

In our experiment, the sea-land clutter samples used are spectrum data of
OTHR. The above metrics of GAN domain evaluate the quality of synthetic
samples from the visual perspective of an image, which is not entirely
appropriate to sea-land clutter samples. To this end, a combination of
traditional evaluation of GAN domain and statistical evaluation of signal
domain is proposed to evaluate the quality of sea-land clutter samples
synthesized by AC-VAEGAN.

Considering traditional evaluation of GAN domain, two metrics, GAN-train and
GAN-test, are used to evaluate the quality of synthetic sea-land clutter
samples. GAN-train treats synthetic samples as training data to train
classifier and real samples as test data to obtain classification accuracy,
which is used to evaluate the diversity of synthetic samples. Intuitively,
this measures the difference between the learned (i.e., synthetic clutter)
distribution and the target (i.e., real clutter) distribution. A good GAN-
train performance shows that the synthetic samples are diverse enough.
However, GAN-train also requires a sufficient fidelity, as otherwise the
classifier will be impacted by the sample quality. GAN-test treats real
samples as training data to train classifier and synthetic samples as test
data to obtain classification accuracy, which is used to evaluate the fidelity
of synthetic samples. A good GAN-test with a high value denoting that the
synthetic samples are a realistic approximation of the distribution of real
clutter.

Considering statistical evaluation of signal domain, three metrics, AD, CS and
PCC, are proposed to evaluate the quality of synthetic sea-land clutter
samples. As a direct distance metric, AD measures the absolute distance
between samples, which is used to evaluate the signal energy difference
between synthetic samples and real samples. The value range of AD is
$[0,+\infty)$, and the smaller the value, the better the synthetic samples. CS
measures the directional similarity between samples, which is used to evaluate
the difference of signal direction between synthetic samples and real samples.
The value range of CS is $[-1,1]$, and the larger the value, the better the
synthetic samples. The most common metric used to measure the similarity of
radar signals is correlation function. PCC is introduced to evaluate the
degree of linear correlation between synthetic samples and real samples. The
value range of PCC is $[-1,1]$, and the larger the value, the better the
synthetic samples.

##  IV Experiment and Evaluation

In this section, we verify the effectiveness of the proposed AC-VAEGAN on two
datasets, one of which is sea-land clutter dataset and the other is MSTAR
dataset. The purpose of conducting experiment on MSTAR dataset is to
demonstrate that AC-VAEGAN has a wider range of applications.

###  IV-A Dataset

####  IV-A1 Sea-Land Clutter Dataset

To verify the effectiveness of AC-VAEGAN, we adopt sea-land clutter dataset as
the benchmark dataset, which is the spectrum of clutter obtained by OTHR.
Fig. 5 shows the spectrum map of full range of a beam from OTHR under a good
ionospheric condition.  We take a collection of spectrum of a range-azimuth
cell as sea-land clutter dataset, as shown in Fig. 6. Fig.
6LABEL:sub@fig:sea_real is an example of sea clutter. The first-order Bragg
peak of sea clutter is formed by the Bragg resonance scattering of high-
frequency electromagnetic waves emitted by OTHR and ocean waves, showing
double peaks symmetrical to zero frequency. Fig. 6LABEL:sub@fig:land_real is
an example of land clutter. Since land is stationary, it is observed that a
single peak appears near zero frequency. Fig. 6LABEL:sub@fig:sea-land_real is
an example of sea-land boundary clutter, which combines the features of sea
clutter and land clutter, showing triple peaks. In addition to the examples
shown in Fig. 6, the sea/land clutter dataset also includes samples under a
relatively poor ionospheric condition or complex environment, such as samples
with Doppler shifts, spread Doppler spectrum and radio-frequency interference
(see Fig. 12). Note that there are more complex sea/land clutter if the
ionospheric condition is very poor. In this paper, we limit ourselves to the
sea/land clutter obtained under not very poor ionospheric condition, that is,
the sea/land clutter that is not hard for human experts to identify. We leave
the classification of more complicated sea/land clutter as our future work.
The detailed description of sea-land clutter dataset used to evaluate AC-
VAEGAN is shown in Table II, in which training data is used for the training
of AC-VAEGAN and the evaluation of synthetic samples, and test data is only
used for the evaluation of synthetic samples and do not participate in any
training process.

Figure 5: The spectrum map of full range of a beam.

(a)

(b)

(c)

Figure 6: Sea-land clutter dataset. (a) An example of sea clutter. (b) An example of land clutter. (c) An example of sea-land boundary clutter. TABLE II: Detailed Description of Sea-Land Clutter Dataset Class | Attribute | Label | Quantity | Training Quantity | Test Quantity
---|---|---|---|---|---
1 | sea | 0 | 1000 | 700(70%) | 300(30%)
2 | land | 1 | 1000 | 700(70%) | 300(30%)
3 | sea-land | 2 | 1000 | 700(70%) | 300(30%)

Further, we randomly divide the original sea-land clutter dataset into
imbalanced and scarce datasets, in which the test data is consistent with the
original test data, and only some samples of one or more classes are excluded
from the original training data. See Table III. The data augmentation
performance of AC-VAEGAN is verified based on the above datasets.

TABLE III: Sea-Land Clutter Training Data for Data Augmentation NO. | Sea | Land | Sea-Land
---|---|---|---
1 | 700(100%) | 700(100%) | 140(20%)
2 | 700(100%) | 140(20%) | 700(100%)
3 | 140(20%) | 700(100%) | 700(100%)
4 | 350(50%) | 350(50%) | 350(50%)
5 | 210(30%) | 210(30%) | 210(30%)
6 | 140(20%) | 140(20%) | 140(20%)

####  IV-A2 MSTAR Dataset

Besides, the effectiveness of AC-VAEGAN is verified on the MSTAR dataset,
which was collected by the Sandia National Laboratory (SNL) in a project
jointly sponsored by the Defense Advanced Research Projects Agency (DARPA) and
the Air Force Research Laboratory (AFRL) [35, 36]. As shown in Fig. 7, the
publicly available MSTAR dataset (128$\times$128 pixels) includes ten
different types of ground targets (BMP2, BTR70, T72, 2S1, BRDM2, D7, BTR60,
T62, ZIL131, ZSU234). It was collected by using an X-band SAR sensor in one
foot resolution spotlight mode, full aspect coverage (in the range from
$0\degree$ to $360\degree$). The detailed description of MSTAR dataset used to
evaluate AC-VAEGAN is shown in Table IV, in which the data collected at
$17\degree$ depression angles are used for training, and the data collected at
$15\degree$ depression angles are used for testing.

Further, we randomly divide the original MSTAR dataset into imbalanced and
scarce datasets, in which the test data is consistent with the original test
data, and only some samples of one or more classes are excluded from the
original training data. See Table V. The data augmentation performance of AC-
VAEGAN is verified based on the above datasets.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

Figure 7: MSTAR dataset: optical images (top) and the corresponding SAR images (bottom). (a) BMP2. (b) BTR70. (c) T72. (d) 2S1. (e) BRDM2. (f) D7. (g) BTR60. (h) T62. (i) ZIL131. (j) ZSU234. TABLE IV: Detailed Description of MSTAR Dataset Class | Attribute | Label | Quantity | Training Quantity | Test Quantity
---|---|---|---|---|---
1 | BMP2 | 0 | 428 | 233 | 195
2 | BTR70 | 1 | 429 | 233 | 196
3 | T72 | 2 | 428 | 232 | 196
4 | 2S1 | 3 | 573 | 299 | 274
5 | BRDM2 | 4 | 572 | 298 | 274
6 | D7 | 5 | 573 | 299 | 274
7 | BTR60 | 6 | 451 | 256 | 195
8 | T62 | 7 | 572 | 299 | 273
9 | ZIL131 | 8 | 573 | 299 | 274
10 | ZSU234 | 9 | 573 | 299 | 274
TABLE V: MSTAR Training Data for Data Augmentation NO. | BMP2 | BTR70 | T72 | 2S1 | BRDM2 | D7 | BTR60 | T62 | ZIL131 | ZSU234
---|---|---|---|---|---|---|---|---|---|---
1 | 50 | 75 | 100 | 125 | 150 | 175 | 200 | 225 | 250 | 275
2 | 50 | 50 | 100 | 100 | 150 | 150 | 200 | 200 | 250 | 250
3 | 50 | 50 | 50 | 50 | 50 | 75 | 100 | 125 | 150 | 175
4 | 150 | 150 | 150 | 150 | 150 | 150 | 150 | 150 | 150 | 150
5 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100
6 | 50 | 50 | 50 | 50 | 50 | 50 | 50 | 50 | 50 | 50

###  IV-B Samples Synthesis

####  IV-B1 Synthesis of Sea-Land Clutter Samples

The experiment environment used in this work for sea-land clutter sample
synthesis and subsequent data augmentation is shown in Table VI.

TABLE VI: Experiment Environment Environment | Version
---|---
System | Windows 10 (64-bit)
GPU | NVIDIA GeForce RTX 3090
CUDA | 11.6
python | 3.9.0 (in Anaconda 4.11.0)
torch | 1.11.0
torchvision | 0.12.0
numpy | 1.22.3
matplotlib | 3.5.1

The schematic diagram of sea-land clutter sample synthesis and data
augmentation based on AC-VAEGAN is shown in Fig. 8. The three types of clutter
samples in the original sea-land clutter training data of sea, land and sea-
land boundary are used as the input of AC-VAEGAN. By Algorithm 1, fake clutter
samples are synthesized, which are used for subsequent samples quality
evaluation and data augmentation. The hyperparameter configuration of AC-
VAEGAN for sea-land clutter sample synthesis is shown in Table VII.

Figure 8: The schematic diagram of sea-land clutter sample synthesis and data augmentation. TABLE VII: The hyperparameter Configuration of AC-VAEGAN for Sea-Land Clutter Sample Synthesis Configuration | Default
---|---
Training Epoch | 1000
Batch Size | 64
Learning Rate | 0.0001
KLD | -
MSE Loss | -
BCE Loss | -
Cross Entropy Loss | -
Adma Optimizer | beta1:0.5, beta2:0.999
Data Normalization | [-1,1]
Weight Initialization | -

Fig. 9LABEL:sub@fig:loss and Fig. 9LABEL:sub@fig:accuracy plot the loss curve
and the classification accuracy curve of AC-VAEGAN in the first 100 training
epochs for sea-land clutter dataset, respectively, from which we can conclude
that the training of AC-VAEGAN is stable and converges rapidly.

(a)

(b)

Figure 9: Loss curve and classification accuracy curve of AC-VAEGAN for sea-
land clutter dataset. (a) Loss curve. (b) Accuracy curve.

####  IV-B2 Synthesis of MSTAR Samples

Firstly, the convolutional kernel of AC-VAEGAN is adjusted to fit the MSTAR
samples of size $N=128\times 128$. See Table VIII for the corresponding
configuration, where the size of all convolutional kernels is 3 in En and D/C,
the size of convolutional kernels of layers 1-4 is 5, and the size of
convolutional kernel of layer 5 is 8 in De/G. In addition, we add dropout with
the probability of 0.5 in D/C to prevent overfitting. The ten types of MSTAR
samples in the original MSTAR training data of BMP2, BTR70, T72, 2S1, BRDM2,
D7, BTR60, T62, ZIL131 and ZSU234 are used as the input of AC-VAEGAN. By
Algorithm 1, fake clutter samples are synthesized, which are used for
subsequent samples quality evaluation and data augmentation. The
hyperparameter configurations of MSTAR samples are the same as those of sea-
land clutter samples (see Table VII).

TABLE VIII: The configuration of AC-VAEGAN for the MSTAR dataset Module | Layer | Configuration | Output Size
---|---|---|---
En | 1-6 | Conv2D, ReLU | 512$\times$13$\times$13
7 | FC | 100
De/G | 1-4 | DeConv2D, BN2D, ReLU | 64$\times$61$\times$61
5 | DeConv2D, Tanh | 1$\times$128$\times$128
D/C | 1 | Conv2D, LeakyReLU, Dropout | 16$\times$64$\times$64
2-6 |  Conv2D, BN2D, LeakyReLU, Dropout | 512$\times$13$\times$13
7_1 | FC, Sigmoid | 1
7_2 | FC, Softmax | 10

Fig. 10LABEL:sub@fig:loss-2 and Fig. 10LABEL:sub@fig:accuracy-2 plot the loss
curve and the classification accuracy curve of AC-VAEGAN in the first 100
training epochs for MSTAR dataset, respectively. Again, from Fig.
10LABEL:sub@fig:loss-2 and Fig. 10LABEL:sub@fig:accuracy-2, we can conclude
that the training of AC-VAEGAN is stable and converges rapidly.

(a)

(b)

Figure 10: Loss curve and classification accuracy curve of AC-VAEGAN for the
MSTAR dataset. (a) Loss curve. (b) Accuracy curve.

###  IV-C Evaluation of the Synthetic Samples by AC-VAEGAN and AC-GAN

####  IV-C1 Evaluation of the Sea-Land Clutter Samples Synthesized by AC-
VAEGAN and AC-GAN

Figure 11: Network structure of ResNet18 for sea-land clutter sample
classification.

When using GAN-train and GAN-test to evaluate synthetic sea-land clutter
samples, ResNet18 [37] is employed. The network structure and hyperparameter
configurations of ResNet18 for sea-land clutter sample classification are
shown in Fig. 11 and Table IX. ResNet18 achieves 100% training accuracy and
98.87% testing accuracy on the original sea-land clutter dataset.

TABLE IX: The hyperparameter Configuration of ResNet18 for Sea-Land Clutter Sample Classification Configuration | Default
---|---
Training Epoch | 100
Batch Size | 64
Learning Rate | 0.0001
Cross Entropy Loss | -
Adma Optimizer | beta1:0.9, beta2:0.999
Data Normalization | [-1,1]
Weight Initialization | -

The trained AC-VAEGAN is used to synthesize 700 sea clutter samples, 700 land
clutter samples, and 700 sea-land boundary clutter samples. See Fig. 12 for
the examples of fake samples. Visually, most of the three types of synthetic
samples have their own features, and are consistent with the real samples. We
mix the synthetic samples with real samples, and then ask two human experts to
identify them. The error rates of Expert 1 and Expert 2 misjudging synthetic
samples as real samples are 90% and 93%, real samples as synthetic samples are
10% and 9%, respectively.

(a)

(b)

(c)

Figure 12: The sea-land clutter samples synthesized by AC-VAEGAN: real samples
(top) and synthetic samples (bottom). (a) sea clutter. (b) land clutter. (c)
sea-land boundary clutter.

Further, the quality of the synthetic sea-land clutter samples is evaluated in
terms of the metrics including GAN-train, GAN-test, AD, CS and PCC. The
evaluation results are shown in Table X. As far as GAN-train is concerned, we
mix three types of synthetic sea-land clutter samples together as training
data for ResNet18. Then, the original sea-land clutter test data is used as
the test data for evaluation. The average classification accuracy after
training ResNet18 to a steady state is 76.44%. As far as GAN-test is
concerned, we evaluate synthetic samples with ResNet18 trained on the original
dataset and obtain the classification accuracy of 92.57%. This shows that the
sea-land clutter samples synthesized by AC-VAEGAN have good diversity and
fidelity.

From the perspective of statistical evaluation, the three types of synthetic
sea-land clutter samples and the original training data are statistically
analyzed, and the average evaluation results of AD, CS and PCC are obtained.
Different from GAN-train and GAN-test, the three metrics evaluate the global
properties of synthetic samples from a statistical signal perspective. High
scores are given to the samples synthesized by AC-VAEGAN for all the three
statistical metrics. The value of AD is 0.0160 dB. The value of CS is 0.8988,
which is very close to 1. The value of PCC is 0.7897, showing that the signal
has a strong correlation with real samples.  To further verify that the
proposed statistical metrics are reasonable, we collect another sea-land
clutter dataset according to the same way in Section IV-A1 and evaluate the
degree of similarity between the two real sea-land clutter datasets. The
results are as follows. The value of AD is 0.0069 dB, the value of CS is
0.9428, and the value of PCC is 0.9183. These further verify that AC-VAEGAN is
able to synthesize high quality sea-land clutter samples.

Besides, we compare the quality of sea-land clutter samples synthesized by AC-
VAEGAN and AC-GAN. AC-GAN removes En from AC-VAEGAN, and the rest of network
structure and hyperparameter configurations are the same as AC-VAEGAN. The
evaluation results of AC-GAN are shown in Table X. Comparing with the
evaluation results of AC-VAEGAN and AC-GAN, one can observe that the quality
of sea-land clutter samples synthesized by AC-VAEGAN is better than that of
AC-GAN.

TABLE X: The Evaluation Results of Synthetic Sea-Land Clutter Samples Evaluation Method | Metric | AC-GAN | AC-VAEGAN
---|---|---|---
Traditional Evaluation | GAN-train | 68.22% | 76.44%
GAN-test | 81.29% | 92.57%
Statistical Evaluation | AD | 0.0161 | 0.0160
CS | 0.8036 | 0.8988
PCC | 0.7649 | 0.7897

####  IV-C2 Evaluation of the MSTAR Samples Synthesized by AC-VAEGAN and AC-
GAN

Since MSTAR dataset is an image dataset, we only use GAN-train and GAN-test to
evaluate the quality of the synthetic MSTAR samples. Firstly, the
convolutional kernel of ResNet18 is adjusted to evaluate synthetic MSTAR
samples (see Fig. 13). The hyperparameter configurations of ResNet18 for MSTAR
dataset are the same as those of sea-land clutter dataset (see Table IX).
ResNet18 achieves 100% training accuracy and 96.79% testing accuracy on the
original MSTAR dataset.

Figure 13: Network structure of ResNet18 for MSTAR sample classification.

The trained AC-VAEGAN is used to synthesize 233 BMP2 samples, 233 BTR70
samples, 232 T72 samples, 299 2S1 samples, 298 BRDM2 samples, 299 D7 samples,
256 BTR60 samples, 299 T62 samples, 299 ZIL131 samples, and 299 ZSU234
samples. See Fig. 14 for the examples of fake samples. Again, we remove En and
use AC-GAN to synthesize the same number of MSTAR samples as above. Then, the
quality of MSTAR samples synthesized by AC-VAEGAN and AC-GAN are compared. The
evaluation results are shown in Table XI. Comparing with the evaluation
results of AC-VAEGAN and AC-GAN, one can observe that the quality of MSTAR
samples synthesized by AC-VAEGAN is better than that of AC-GAN.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

Figure 14:  The MSTAR samples synthesized by AC-VAEGAN: real samples (top) and synthetic samples (bottom). (a) BMP2. (b) BTR70. (c) T72. (d) 2S1. (e) BRDM2. (f) D7. (g) BTR60. (h) T62. (i) ZIL131. (j) ZSU234.  TABLE XI: The Evaluation Results of Synthetic MSTAR Samples Evaluation Method | Metric | AC-GAN | AC-VAEGAN
---|---|---|---
Traditional Evaluation | GAN-train | 60.74% | 63.07%
GAN-test | 66.40% | 69.28%

###  IV-D Data Augmentation Experiment Based on AC-VAEGAN and AC-GAN

####  IV-D1 Data Augmentation Experiment of sea-land clutter dataset Based on
AC-VAEGAN and AC-GAN

Finally, the impact of data augmentation based on AC-VAEGAN/AC-GAN on the
performance improvement of ResNet18 is evaluated in the cases of imbalanced
and scarce sea-land clutter samples. We conduct six groups of comparative
experiments. The first three groups are imbalanced sample experiments, which
are based on the imbalanced datasets of NO.1-NO.3 in Table III. The last three
groups are scarce sample experiments, which are based on the scarce datasets
of NO.4-NO.6 in Table III. The steps of the experiments are as follows: (1)
Select the data augmentation dataset NO.k, k$\in\\{1,2,...6\\}$; (2) Train AC-
VAEGAN/AC-GAN based on the current dataset NO.k; (3) Use the trained AC-
VAEGAN/AC-GAN to perform data augmentation on minority class samples to obtain
a balanced sea-land clutter dataset, denoted as NO.k++/NO.k+; (4) Treat NO.k
and NO.k++/NO.k+ as training data, and original sea-land clutter test data as
test data. Then, the average classification accuracy after training ResNet18
to a steady state is recorded, and the best one after 100 repetitions of
training is taken as the final test result.

Figure 15: Data augmentation evaluation results of the sea-land clutter
dataset based on AC-VAEGAN and AC-GAN.

Fig. 15 plots the classification accuracy of NO.0-NO.6, NO.1+-NO.6+ and
NO.1++-NO.6++ tested on ResNet18, where NO.0 represents the classification
accuracy of the original dataset (98.87%). Comparing with the classification
accuracy of NO.0, the classification performance of NO.1-NO.6 is reduced.  It
is seen from the comparison results that the classification accuracy of sea-
land clutter samples after data augmentation has been improved. NO.1-NO.3
represent the imbalanced dataset of sea-land boundary, land, and sea clutter
samples as the minority, respectively. It is seen that AC-VAEGAN data
augmentation method can improve the classification performance in the case of
imbalanced samples. NO.4-NO.6 represent sea-land clutter datasets from light
to heavy scarcity, respectively. It is seen that AC-VAEGAN data augmentation
method can improve the classification performance in the case of scarce
samples, and as the sample scarcity increases, the improvement on the
classification accuracy is more obvious. Besides, the data augmentation
performance of AC-VAEGAN is generally superior to that of AC-GAN. In
conclusion, the proposed AC-VAEGAN is able to serve as an effective tool for
data augmentation in sea-land clutter classification.

In this paper, we focus on the classification of land/sea clutter. The purpose
is to match the derived classification results with a prior geographic
information, and then provide coordinate registration parameters for target
localization. Since, in a typical OTHR, the land/sea clutter is much stronger
(typically 30–50 dB) than the signal of targets, we treat the targets’ signal
as noise or interference in the classification of land/sea clutter, no matter
how close the targets’ Doppler to the clutter. In our experiments, we do not
differentiate the clutter samples with or without targets’ signal. Note that
target detection is another important issue in OTHR. We leave the extension of
our proposed method to target detection while target’s Doppler are close to
the sea/land clutter as future work.

####  IV-D2 Data Augmentation Experiment of MSTAR Dataset Based on AC-VAEGAN
and AC-GAN

Similarly, following the steps described above for data augmentation of sea-
land clutter, the impact of data augmentation based on AC-VAEGAN/AC-GAN on the
performance improvement of ResNet18 is evaluated in the cases of imbalanced
and scarce MSTAR samples in Table V. The data augmentation evaluation results
are shown in Fig. 16, which further verifies the data augmentation performance
of the proposed AC-VAEGAN.

Figure 16: Data augmentation evaluation results of the MSTAR dataset based on
AC-VAEGAN and AC-GAN.

##  V Conclusions and Future Work

A novel network, namely AC-VAEGAN, was proposed to act as a data augmentation
method for sea-land clutter classification of OTHR. An evaluation method
combining traditional evaluation of GAN domain and statistical evaluation of
signal domain was proposed to evaluate the quality of synthetic samples.
Experimental results demonstrated that AC-VAEGAN can synthesis higher quality
samples than AC-GAN. Using the samples synthesized by AC-VAEGAN as the
supplement of imbalanced and scarce datasets can improve the performance of
sea-land clutter classification model.

In future work, there are two issues that need to be addressed: (1) The loss
function of AC-VAEGAN does not contain an interpretable indicator to guide the
training process like that of classifier. Therefore, it is necessary to
improve the loss function of AC-VAEGAN by combining Wasserstein distance and
other loss functions; (2) GAN also has high requirements on the quality of
training data. In fact, there are more complex sea/land clutter samples from
OTHR. Therefore, we intend to study how to reduce the high quality
requirements of GAN for training data.  Other potential areas for future
research include the classification of more complicated sea/land clutter and
extension of the proposed network to target detection.

## References

  * [1] H. Lan, Z. Wang, X. Bai, Q. Pan, and K. Lu, “Measurement-level target tracking fusion for over-the-horizon radar network using message passing,” _IEEE Transactions on Aerospace and Electronic Systems_ , vol. 57, no. 3, pp. 1600–1623, 2020.
  * [2] L. Yang, H. Gao, Y. Ling, and B. Li, “Localization method of wide-area distribution multistatic sky-wave over-the-horizon radar,” _IEEE Geoscience and Remote Sensing Letters_ , vol. 19, pp. 3 500 305–1–3 500 305–5, 2022.
  * [3] T. Thayaparan, D. Dupont, Y. Ibrahim, and R. Riddolls, “High-frequency ionospheric monitoring system for over-the-horizon radar in Canada,” _IEEE Transactions on Geoscience and Remote Sensing_ , vol. 57, no. 9, pp. 6372–6384, 2019.
  * [4] J. Hu, C. Jian, C. Zhuo, H. Li, and J. Xie, “Knowledge-aided ocean clutter suppression method for sky-wave over-the-horizon radar,” _IEEE Geoscience and Remote Sensing Letters_ , vol. 15, no. 3, pp. 355–358, 2018.
  * [5] Z. Guo, Z. Wang, Y. Hao, H. Lan, and Q. Pan, “An improved coordinate registration for over-the-horizon radar using reference sources,” _Electronics_ , vol. 10, no. 24, pp. 3086–1–3086–20, 2021.
  * [6] Y. Han, Z. Yang, and X. Chu, “Research on the correction of ionospheric distortion for ship detection in OTHR,” _Modern Radar_ , vol. 25, no. 10, pp. 5–8, 2003.
  * [7] M. Turley, R. Gardiner-Garden, and D. Holdsworth, “High-resolution wide area remote sensing for HF radar track registration,” in _Proceedings of 2013 International Conference on Radar_. IEEE, 2013, pp. 128–133.
  * [8] Z. Jin, Q. Pan, C. Zhao, and W. Zhou, “SVM based land/sea clutter classification algorithm,” _Applied Mechanics and Materials_ , vol. 236-237, pp. 1156–1162, 2012.
  * [9] C. Li, Z. Wang, Z. Zhang, H. Lan, and K. Lu, “Sea/land clutter recognition for over-the-horizon radar via deep CNN,” in _Proceedings of 2019 International Conference on Control, Automation and Information Sciences_. IEEE, 2019, pp. 1–5.
  * [10] Y. Zhang, Z. Wang, K. Lu, and Q. Pan, “A Bif-ResNet with transfer learning mechanism for multi-scale land/sea clutter classification,” _Submitted for Publication_.
  * [11] C. Li, Z. Wang, K. Lu, and Q. Pan, “Cross-scale land/sea clutter identification method for OTHR based on algebraic multigrid,” _Submitted for Publication_.
  * [12] A. Mikołajczyk and M. Grochowski, “Data augmentation for improving deep learning in image classification problem,” in _Proceedings of 2018 international interdisciplinary PhD workshop_. IEEE, 2018, pp. 117–122.
  * [13] C. Shorten and T. M. Khoshgoftaar, “A survey on image data augmentation for deep learning,” _Journal of Big Data_ , vol. 6, no. 60, pp. 1–48, 2019.
  * [14] W. Li, C. Chen, M. Zhang, H. Li, and Q. Du, “Data augmentation for hyperspectral image classification with deep CNN,” _IEEE Geoscience and Remote Sensing Letters_ , vol. 16, no. 4, pp. 593–597, 2018.
  * [15] J. M. Johnson and T. M. Khoshgoftaar, “Survey on deep learning with class imbalance,” _Journal of Big Data_ , vol. 6, no. 27, pp. 1–54, 2019.
  * [16] M. Buda, A. Maki, and M. A. Mazurowski, “A systematic study of the class imbalance problem in convolutional neural networks,” _Neural Networks_ , vol. 106, pp. 249–259, 2018.
  * [17] X. Zhang, Y. Wang, N. Zhang, D. Xu, H. Luo, B. Chen, and G. Ben, “Spectral–spatial fractal residual convolutional neural network with data balance augmentation for hyperspectral classification,” _IEEE Transactions on Geoscience and Remote Sensing_ , vol. 59, no. 12, pp. 10 473–10 487, 2021.
  * [18] A. Antoniou, A. Storkey, and H. Edwards, “Data augmentation generative adversarial networks,” _arXiv preprint arXiv:1711.04340_ , 2017.
  * [19] K. Bousmalis, N. Silberman, D. Dohan, D. Erhan, and D. Krishnan, “Unsupervised pixel-level domain adaptation with generative adversarial networks,” in _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2017, pp. 3722–3731.
  * [20] I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio, “Generative adversarial nets,” _Advances in neural information processing systems_ , vol. 27, pp. 1–9, 2014.
  * [21] A. Radford, L. Metz, and S. Chintala, “Unsupervised representation learning with deep convolutional generative adversarial networks,” in _Proceedings of International Conference on Machine Learning_ , 2016, pp. 1–16.
  * [22] A. B. L. Larsen, S. K. Sønderby, H. Larochelle, and O. Winther, “Autoencoding beyond pixels using a learned similarity metric,” in _Proceedings of International Conference on Machine Learning_ , 2016, pp. 1558–1566.
  * [23] M. Mirza and S. Osindero, “Conditional generative adversarial nets,” _arXiv preprint arXiv:1411.1784_ , 2014.
  * [24] A. Odena, “Semi-supervised learning with generative adversarial networks,” in _Proceedings of Data Efficient Machine Learning workshop at International Conference on Machine Learning_ , 2016, pp. 1–3.
  * [25] A. Odena, C. Olah, and J. Shlens, “Conditional image synthesis with auxiliary classifier GANs,” in _Proceedings of International Conference on Machine Learning_ , 2017, pp. 2642–2651.
  * [26] G. Mariani, F. Scheidegger, R. Istrate, C. Bekas, and C. Malossi, “BAGAN: Data augmentation with balancing GAN,” _arXiv preprint arXiv:1803.09655_ , 2018.
  * [27] K. Shmelkov, C. Schmid, and K. Alahari, “How good is my GAN?” in _Proceedings of the European Conference on Computer Vision_ , 2018, pp. 213–229.
  * [28] D. P. Kingma and M. Welling, “Auto-encoding variational Bayes,” in _Proceedings of the International Conference on Learning Representation_ , 2014.
  * [29] T. Salimans, I. Goodfellow, W. Zaremba, V. Cheung, A. Radford, and X. Chen, “Improved techniques for training gans,” _Advances in neural information processing systems_ , vol. 29, pp. 1–9, 2016.
  * [30] T. Che, Y. Li, A. P. Jacob, Y. Bengio, and W. Li, “Mode regularized generative adversarial networks,” in _Proceedings of the 5th International Conference on Learning Representations_ , 2019, pp. 1–13.
  * [31] M. Heusel, H. Ramsauer, T. Unterthiner, B. Nessler, and S. Hochreiter, “GANs trained by a two time-scale update rule converge to a local Nash equilibrium,” _Advances in neural information processing systems_ , vol. 30, pp. 1–38, 2017.
  * [32] G. K. Dziugaite, D. M. Roy, and Z. Ghahramani, “Training generative neural networks via maximum mean discrepancy optimization,” in _Proceedings of the 31st Conference on Uncertainty in Artificial Intelligence_ , 2015, pp. 258–267.
  * [33] M. Arjovsky, S. Chintala, and L. Bottou, “Wasserstein generative adversarial networks,” in _Proceedings of International conference on Machine Learning_ , 2017, pp. 214–223.
  * [34] I. Gulrajani, F. Ahmed, M. Arjovsky, V. Dumoulin, and A. C. Courville, “Improved training of Wasserstein GANs,” _Advances in neural information processing systems_ , vol. 30, pp. 1–20, 2017.
  * [35] S. Chen, H. Wang, F. Xu, and Y.-Q. Jin, “Target classification using the deep convolutional networks for SAR images,” _IEEE transactions on geoscience and remote sensing_ , vol. 54, no. 8, pp. 4806–4817, 2016.
  * [36] E. R. Keydel, S. W. Lee, and J. T. Moore, “Mstar extended operating conditions: A tutorial,” _Algorithms for Synthetic Aperture Radar Imagery III_ , vol. 2757, pp. 228–242, 1996.
  * [37] K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” in _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2016, pp. 770–778.


#  A Simple Discriminative Training Method for Machine Translation with Large-
Scale Features

Tian Xia
Wright State University
<EMAIL_ADDRESS>
&Shaodan Zhai
Wright State University
<EMAIL_ADDRESS>
&Shaojun Wang
Wright State University
<EMAIL_ADDRESS>

###### Abstract

Margin infused relaxed algorithms (MIRAs) dominate model tuning in statistical
machine translation in the case of large scale features, but also they are
famous for the complexity in implementation. We introduce a new method, which
regards an N-best list as a permutation and minimizes the Plackett-Luce loss
of ground-truth permutations. Experiments with large-scale features
demonstrate that, the new method is more robust than MERT; though it is only
matchable with MIRAs, it has a comparatively advantage, easier to implement.

##  1 Introduction

Since Och [26] proposed minimum error rate training (MERT) to exactly optimize
objective evaluation measures, MERT has become a standard model tuning
technique in statistical machine translation (SMT). Though MERT performs
better by improving its searching algorithm [23, 6, 13, 25], it does not work
reasonably when there are lots of features111The regularized MERT seems
promising from Galley et al. [14] at the cost of model complexity.. As a
result, margin infused relaxed algorithms (MIRA) dominate in this case [24,
35, 10, 31, 7].

In SMT, MIRAs consider margin losses related to sentence-level BLEUs. However,
since the BLEU is not decomposable into each sentence, these MIRA algorithms
use some heuristics to compute the exact losses, e.g., pseudo-document [10],
and document-level loss [31].

Recently, another successful work in large-scale feature tuning include force
decoding based[36], classification based [15].

We aim to provide a simpler tuning method for large-scale features than MIRAs.
Out motivation derives from an observation on MERT. As MERT considers the
quality of only top$1$ hypothesis set, there might have more-than-one set of
parameters, which have similar top1 performances in tuning, but have very
different topN hypotheses. Empirically, we expect an ideal model to benefit
the total N-best list. That is, better hypotheses should be assigned with
higher ranks, and this might decrease the error risk of top1 result on unseen
data.

Plackett[29] offered an easy-to-understand theory of modeling a permutation.
An N-best list is assumedly generated by sampling without replacement. The
$i$th hypothesis to sample relies on those ranked after it, instead of on the
whole list. This model also supports a partial permutation which accounts for
top $k$ positions in a list, regardless of the remaining. When taking $k$ as
1, this model reduces to a standard conditional probabilistic training, whose
dual problem is actual the maximum entropy based [27]. Although Och [26]
substituted direct error optimization for a maximum entropy based training,
probabilistic models correlate with BLEU well when features are rich enough.
The similar claim also appears in [37]. This also make the new method be
applicable in large-scale features.

##  2 Plackett-Luce Model

Plackett-Luce was firstly proposed to predict ranks of horses in gambling
[29]. Let $\mathbf{r}=(r_{1},r_{2}\ldots r_{N})$ be $N$ horses with a
probability distribution $\mathcal{P}$ on their abilities to win a game, and a
rank $\mathbf{\pi}=(\pi(1),\pi(2)\ldots\pi(|\mathbf{\pi}|))$ of horses can be
understood as a generative procedure, where $\pi(j)$ denotes the index of the
horse in the $j$th position.

In the 1st position, there are $N$ horses as candidates, each of which $r_{j}$
has a probability $p(r_{j})$ to be selected. Regarding the rank $\pi$, the
probability of generating the champion is $p(r_{\pi(1)})$. Then the horse
$r_{\pi(1)}$ is removed from the candidate pool.

In the 2nd position, there are only $N-1$ horses, and their probabilities to
be selected become $p(r_{j})/Z_{2}$, where $Z_{2}=1-p(r_{\pi(1)})$ is the
normalization. Then the runner-up in the rank $\pi$, the $\pi(2)$th horse, is
chosen at the probability $p(r_{\pi(2)})/Z_{2}$. We use a consistent
terminology $Z_{1}$ in selecting the champion, though $Z_{1}$ equals $1$
trivially.

This procedure iterates to the last rank in $\pi$. The key idea for the
Plackett-Luce model is the choice in the $i$th position in a rank
$\mathbf{\pi}$ only depends on the candidates not chosen at previous stages.
The probability of generating a rank $\pi$ is given as follows

$p(\mathbf{\pi})=\prod_{j=1}^{|\boldsymbol{\pi}|}\frac{p(r_{\pi(j)})}{Z_{j}}$
(1)

where $Z_{j}=1-\sum_{t=1}^{j-1}p(r_{\pi(t)})$.

We offer a toy example (Table 1) to demonstrate this procedure.

$\mathbf{r}$ | $r_{1}$ | $r_{2}$ | $r_{3}$
---|---|---|---
$\mathbf{\pi}$ | 2 | 3 | 1
$Z$ | 1 | 1-$p(r_{2})$ | 1-($p(r_{2})+p(r_{3}))$
$p(\pi)$ | $\frac{p(r_{2})}{Z_{1}}$ | $\frac{p(r_{3})}{Z_{2}}$ | $\frac{p(r_{1})}{Z_{3}}$
Table 1: The probability of the rank $\boldsymbol{\pi}=(2,3,1)$ is
$p(r_{2})\cdot p(r_{3})/(1-p(r_{2}))$ in a simplified form, as
$\frac{p(r_{2})}{Z_{1}}=p(r_{2})$ and $\frac{p(r_{1})}{Z_{3}}=1$.

######  Theorem 1.

The permutation probabilities $p(\mathbf{\pi})$ form a probability
distribution over a set of permutations $\Omega_{\pi}$. For example, for each
$\mathbf{\pi}\in\Omega_{\pi}$, we have $p(\mathbf{\pi})>0$, and
$\sum_{\pi\in\Omega_{\pi}}p(\mathbf{\pi})=1$.

We have to note that, $\Omega_{\pi}$ is not necessarily required to be
completely ranked permutations in theory and in practice, since gamblers might
be interested in only the champion and runner-up, and thus $|\mathbf{\pi}|\leq
N$. In experiments, we would examine the effects on different length of
permutations, systems being termed $PL(|\pi|)$.

######  Theorem 2.

Given any two permutations $\mathbf{\pi}$ and $\mathbf{\pi}\prime$, and they
are different only in two positions $p$ and $q$, $p<q$, with
$\pi(p)=\mathbf{\pi}\prime(q)$ and $\pi(q)=\mathbf{\pi}\prime(p)$. If
$p(\pi(p))>p(\pi(q))$, then $p(\pi)>p(\pi\prime)$.

In other words, exchanging two positions in a permutation where the horse more
likely to win is not ranked before the other would lead to an increase of the
permutation probability.

This suggests the ground-truth permutation, ranked decreasingly by their
probabilities, owns the maximum permutation probability on a given
distribution. In SMT, we are motivated to optimize parameters to maximize the
likelihood of ground-truth permutation of an N-best hypotheses.

Due to the limitation of space, see [29, 5] for the proofs of the theorems.

##  3 Plackett-Luce Model in Statistical Machine Translation

In SMT, let $\mathbf{f}=(f_{1},f_{2}\ldots)$ denote source sentences, and
$\mathbf{e}=(\\{e_{1,1},\ldots\\},\\{e_{2,1},\ldots\\}\ldots)$ denote target
hypotheses. A set of features are defined on both source and target side. We
refer to $h(e_{i,*})$ as a feature vector of a hypothesis from the $i$th
source sentence, and its score from a ranking function is defined as the inner
product $h(e_{i,*})^{T}\boldsymbol{w}$ of the weight vector $\boldsymbol{w}$
and the feature vector.

We first follow the popular exponential style to define a parameterized
probability distribution over a list of hypotheses.

$p(e_{i,j})=\frac{\exp\\{h(e_{i,j})^{T}\boldsymbol{w}\\}}{\sum_{k}\exp\\{h(e_{i,k})^{T}\boldsymbol{w}\\}}$
(2)

The ground-truth permutation of an $n$best list is simply obtained after
ranking by their sentence-level BLEUs. Here we only concentrate on their
relative ranks which are straightforward to compute in practice, e.g. add 1
smoothing. Let $\pi_{i}^{*}$ be the ground-truth permutation of hypotheses
from the $i$th source sentences, and our optimization objective is maximizing
the log-likelihood of the ground-truth permutations and penalized using a
zero-mean and unit-variance Gaussian prior. This results in the following
objective and gradient:

$\mathcal{L}=\log\\{\prod_{i}p(\pi_{i}^{*},\mathcal{P}(\mathbf{\boldsymbol{w}}))\\}-\frac{1}{2}\boldsymbol{w}^{T}\boldsymbol{w}$
(3)

$\frac{\partial\mathcal{L}}{\partial\boldsymbol{w}}=\sum_{i}\sum_{j}\\{h(e_{i,\pi_{i}^{*}(j)})-\sum_{t=j}(h(e_{i,\pi_{i}^{*}(t)})\cdot\frac{p(e_{i,\pi_{i}^{*}(t)})}{Z_{i,j}})\\}-\boldsymbol{w}$
(4)

where $Z_{i,j}$ is defined as the $Z_{j}$ in Formula (1) of the $i$th source
sentence.

The log-likelihood function is smooth, differentiable, and concave with the
weight vector $\boldsymbol{w}$, and its local maximal solution is also a
global maximum. Iteratively selecting one parameter in $\alpha$ for tuning in
a line search style (or MERT style) could also converge into the global global
maximum [2]. In practice, we use more fast limited-memory BFGS (L-BFGS)
algorithm [4].

### N-best Hypotheses Resample

The log-likelihood of a Plackett-Luce model is not a strict upper bound of the
BLEU score, however, it correlates with BLEU well in the case of rich
features. The concept of “rich” is actually qualitative, and obscure to define
in different applications. We empirically provide a formula to measure the
richness in the scenario of machine translation.

$r=\frac{\textrm{the size of features }}{\textrm{the average size of N-best
lists}}$ (5)

The greater, the richer. In practice, we find a rough threshold of r is 5.

In engineering, the size of an N-best list with unique hypotheses is usually
less than several thousands. This suggests that, if features are up to
thousands or more, the Plackett-Luce model is quite suitable here. Otherwise,
we could reduce the size of N-best lists by sampling to make $r$ beyond the
threshold.

Their may be other efficient sampling methods, and here we adopt a simple one.
If we want to $m$ samples from a list of hypotheses $\mathbf{e}$, first, the
$\frac{m}{3}$ best hypotheses and the $\frac{m}{3}$ worst hypotheses are taken
by their sentence-level BLEUs. Second, we sample the remaining hypotheses on
distribution $p(e_{i})\propto\exp(h(e_{i})^{T}\boldsymbol{w})$, where
$\mathbf{\boldsymbol{w}}$ is an initial weight from last iteration.

##  4 Evaluation

| MT02(dev) | MT04(%) | MT05(%)
---|---|---|---
MERT | 34.61 | 31.76 | 28.85
MIRA | 35.31 | 32.25 | 29.37
PL(1) | 34.20 | 31.70 | 28.90
PL(2) | 34.31 | 31.83 | 29.10
PL(3) | 34.39 | 32.05 | 29.20
PL(4) | 34.40 | 32.13 | 29.46+
PL(5) | 34.46 | 32.19+ | 29.42+
PL(6) | 34.37 | 32.16 | 29.30
PL(7) | 34.39 | 32.20+ | 29.32
PL(8) | 34.70 | 32.19+ | 29.10
PL(9) | 34.30 | 32.07 | 29.22
PL(10) | 34.30 | 32.14 | 29.19
Table 2: PL($k$): Plackett-Luce model optimizing the ground-truth permutation
with length $k$. The significant symbols (+ at 0.05 level) are compared with
MERT. The bold font numbers signifies better results compared to M(1) system.

We compare our method with MERT and MIRA222MIRA is from the open-source Moses
[16] in two tasks, iterative training, and N-best list rerank. We do not list
PRO [15] as our baseline, as Cherry et al.[7] have compared PRO with MIRA and
MERT massively.

In the first task, we align the FBIS data (about 230K sentence pairs) with
GIZA++, and train a 4-gram language model on the Xinhua portion of Gigaword
corpus. A hierarchical phrase-based (HPB) model (Chiang, 2007) is tuned on
NIST MT 2002, and tested on MT 2004 and 2005. All features are eight basic
ones [9] and extra 220 group features. We design such feature templates to
group grammars by the length of source side and target side, (feat-
type,a$\leq$src-side$\leq$b,c$\leq$tgt-side$\leq$d), where the feat-type
denotes any of the relative frequency, reversed relative frequency, lexical
probability and reversed lexical probability, and [a, b], [c, d] enumerate all
possible subranges of [1, 10], as the maximum length on both sides of a
hierarchical grammar is limited to 10. There are 4 $\times$ 55 extra group
features.

In the second task, we rerank an N-best list from a HPB system with 7491
features from a third party. The system uses six million parallel sentence
pairs available to the DARPA BOLT Chinese-English task. This system includes
51 dense features (translation probabilities, provenance features, etc.) and
up to 7440 sparse features (mostly lexical and fertility-based). The language
model is a 6-gram model trained on a 10 billion words, including the English
side of our parallel corpora plus other corpora such as Gigaword (LDC2011T07)
and Google News. For the tuning and test sets, we use 1275 and 1239 sentences
respectively from the LDC2010E30 corpus.

Figure 1: PL($k$) with 500 L-BFGS iterations, k=1,3,5,7,9,12,15 compared with
MIRA in reranking.

###  4.1 Plackett-Luce Model for SMT Tuning

We conduct a full training of machine translation models. By default, a
decoder is invoked for at most 40 times, and each time it outputs $200$
hypotheses to be combined with those from previous iterations and sent into
tuning algorithms.

In getting the ground-truth permutations, there are many ties with the same
sentence-level BLEU, and we just take one randomly. In this section, all
systems have only around two hundred features, hence in Plackett-Luce based
training, we sample 30 hypotheses in an accumulative $n$best list in each
round of training.

All results are shown in Table 2, we can see that all PL($k$) systems does not
perform well as MERT or MIRA in the development data, this maybe due to that
PL($k$) systems do not optimize BLEU and the features here are relatively not
enough compared to the size of N-best lists (empirical Formula 5). However,
PL($k$) systems are better than MERT in testing. PL($k$) systems consider the
quality of hypotheses from the $2$th to the $k$th, which is guessed to act the
role of the margin like SVM in classification . Interestingly, MIRA wins first
in training, and still performs quite well in testing.

The PL(1) system is equivalent to a max-entropy based algorithm [27] whose
dual problem is actually maximizing the conditional probability of one oracle
hypothesis. When we increase the $k$, the performances improve at first. After
reaching a maximum around $k=5$, they decrease slowly. We explain this
phenomenon as this, when features are rich enough, higher BLEU scores could be
easily fitted, then longer ground-truth permutations include more useful
information.

###  4.2 Plackett-Luce Model for SMT Reranking

After being de-duplicated, the N-best list has an average size of around 300,
and with 7491 features. Refer to Formula 5, this is ideal to use the Plackett-
Luce model. Results are shown in Figure 1. We observe some interesting
phenomena.

First, the Plackett-Luce models boost the training BLEU very greatly, even up
to 2.5 points higher than MIRA. This verifies our assumption, richer features
benefit BLEU, though they are optimized towards a different objective.

Second, the over-fitting problem of the Plackett-Luce models PL($k$) is
alleviated with moderately large $k$. In PL(1), the over-fitting is quite
obvious, the portion in which the curve overpasses MIRA is the smallest
compared to other $k$, and its convergent performance is below the baseline.
When $k$ is not smaller than 5, the curves are almost above the MIRA line.
After 500 L-BFGS iterations, their performances are no less than the baseline,
though only by a small margin.

This experiment displays, in large-scale features, the Plackett-Luce model
correlates with BLEU score very well, and alleviates overfitting in some
degree.

## References

  * [1] Adam L Berger, Vincent J Della Pietra, and Stephen A Della Pietra.  A maximum entropy approach to natural language processing.  Computational linguistics, 22(1), 1996.
  * [2] Dimitri P Bertsekas.  Nonlinear programming.  1999\.
  * [3] Phil Blunsom, Trevor Cohn, and Miles Osborne.  A discriminative latent variable model for statistical machine translation.  In Proc. of ACL, 2008.
  * [4] Richard H Byrd, Peihuang Lu, Jorge Nocedal, and Ciyou Zhu.  A limited memory algorithm for bound constrained optimization.  SIAM Journal on Scientific Computing, 16(5), 1995.
  * [5] Zhe Cao, Tao Qin, Tie-Yan Liu, Ming-Feng Tsai, and Hang Li.  Learning to rank: from pairwise approach to listwise approach.  In Proc. of ICML, 2007.
  * [6] Daniel Cer, Daniel Jurafsky, and Christopher D Manning.  Regularization and search for minimum error rate training.  In Proc. of WSMT, 2008.
  * [7] Colin Cherry and George Foster.  Batch tuning strategies for statistical machine translation.  In Proceedings of the 2012 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pages 427–436. Association for Computational Linguistics, 2012\.
  * [8] David Chiang.  A hierarchical phrase-based model for statistical machine translation.  In Proc. of ACL, 2005.
  * [9] David Chiang.  Hierarchical phrase-based translation.  computational linguistics, 33(2), 2007.
  * [10] David Chiang, Yuval Marton, and Philip Resnik.  Online large-margin training of syntactic and structural translation features.  In Proc. of EMNLP, 2008.
  * [11] Michael Collins.  Discriminative training methods for hidden markov models: Theory and experiments with perceptron algorithms.  In Proc. of EMNLP, 2002.
  * [12] Kevin Duh and Katrin Kirchhoff.  Beyond log-linear models: boosted minimum error rate training for n-best re-ranking.  In Proc. of ACL: Short Papers, 2008.
  * [13] Michel Galley and Chris Quirk.  Optimal search for minimum error rate training.  In Proc. of EMNLP, 2011.
  * [14] Michel Galley, Chris Quirk, Colin Cherry, and Kristina Toutanova.  Regularized minimum error rate training.  In EMNLP, pages 1948–1959, 2013.
  * [15] Mark Hopkins and Jonathan May.  Tuning as ranking.  In Proc. of EMNLP, 2011.
  * [16] Philipp Koehn, Hieu Hoang, Alexandra Birch, Chris Callison-Burch, Marcello Federico, Nicola Bertoldi, Brooke Cowan, Wade Shen, Christine Moran, Richard Zens, et al.  Moses: Open source toolkit for statistical machine translation.  In Proc. of ACL: Poster, 2007.
  * [17] Philipp Koehn, Franz Josef Och, and Daniel Marcu.  Statistical phrase-based translation.  In Proc. of NAACL, 2003.
  * [18] Shankar Kumar and William J Byrne.  Minimum bayes-risk decoding for statistical machine translation.  In Proc. of HLT-NAACL, 2004.
  * [19] Shankar Kumar, Wolfgang Macherey, Chris Dyer, and Franz Och.  Efficient minimum error rate training and minimum bayes-risk decoding for translation hypergraphs and lattices.  In Proc. of Joint ACL and AFNLP, 2009.
  * [20] John Lafferty, Andrew McCallum, and Fernando CN Pereira.  Conditional random fields: Probabilistic models for segmenting and labeling sequence data.  In Prof. of ICML, 2001.
  * [21] Zhifei Li and Jason Eisner.  First-and second-order expectation semirings with applications to minimum-risk training on translation forests.  In Proc. of EMNLP, 2009.
  * [22] Percy Liang, Alexandre Bouchard-Côté, Dan Klein, and Ben Taskar.  An end-to-end discriminative approach to machine translation.  In Proc. of ACL, 2006.
  * [23] Wolfgang Macherey, Franz Josef Och, Ignacio Thayer, and Jakob Uszkoreit.  Lattice-based minimum error rate training for statistical machine translation.  In Proc. of EMNLP, 2008.
  * [24] Ryan McDonald, Koby Crammer, and Fernando Pereira.  Online large-margin training of dependency parsers.  In Proc. of ACL, 2005.
  * [25] Robert C Moore and Chris Quirk.  Random restarts in minimum error rate training for statistical machine translation.  In Proceedings of the 22nd International Conference on Computational Linguistics-Volume 1, pages 585–592. Association for Computational Linguistics, 2008.
  * [26] Franz Josef Och.  Minimum error rate training in statistical machine translation.  In Proc. of ACL, 2003.
  * [27] Franz Josef Och and Hermann Ney.  Discriminative training and maximum entropy models for statistical machine translation.  In Proc. of ACL, 2002.
  * [28] Adam Pauls, John DeNero, and Dan Klein.  Consensus training for consensus decoding in machine translation.  In Proc. of EMNLP, 2009.
  * [29] Robin L Plackett.  The analysis of permutations.  Applied Statistics, 1975.
  * [30] David A Smith and Jason Eisner.  Minimum risk annealing for training log-linear models.  In Proc. of COLING/ACL: Poster, 2006.
  * [31] Ming Tan, Tian Xia, Shaojun Wang, and Bowen Zhou.  A corpus level mira tuning strategy for machine translation.  In EMNLP, pages 851–856, 2013.
  * [32] Ben Taskar, Dan Klein, Michael Collins, Daphne Koller, and Christopher D Manning.  Max-margin parsing.  In Proc. of EMNLP, volume 1, 2004.
  * [33] Christoph Tillmann and Tong Zhang.  A discriminative global training algorithm for statistical mt.  In Proc. of ACL, 2006.
  * [34] Roy W Tromble, Shankar Kumar, Franz Och, and Wolfgang Macherey.  Lattice minimum bayes-risk decoding for statistical machine translation.  In Proc. of EMNLP, 2008.
  * [35] Taro Watanabe, Jun Suzuki, Hajime Tsukada, and Hideki Isozaki.  Online large-margin training for statistical machine translation.  In Proc. of EMNLP, 2007.
  * [36] Heng Yu, Liang Huang, Haitao Mi, and Kai Zhao.  Max-violation perceptron and forced decoding for scalable mt training.  In EMNLP, pages 1112–1123, 2013.
  * [37] Ji Zhu and Trevor Hastie.  Kernel logistic regression and the import vector machine.  In Proc. of NIPS, 2001.


On the Vertex-Degree-Function Indices of Connected
$(n,m)$-Graphs of Maximum Degree at Most Four

Abeer M. Albalahi1, Igor Ž. Milovanović2, Zahid Raza3,
Akbar Ali1,***Corresponding author, Amjad E. Hamza1

1Department of Mathematics, College of Science,
University of Ha’il, Ha’il, Saudi Arabia
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
2Faculty of Electronic Engineering,
University of Niš, Niš, Serbia
<EMAIL_ADDRESS>
3Department of Mathematics, College of Sciences,
University of Sharjah, Sharjah, UAE
<EMAIL_ADDRESS>
Abstract

Consider a graph $G$ and a real-valued function $f$ defined on the degree set
of $G$. The sum of the outputs $f(d_{v})$ over all vertices $v\in V(G)$ of $G$
is usually known as the vertex-degree-function indices and is denoted by
$H_{f}(G)$, where $d_{v}$ represents the degree of a vertex $v$ of $G$. This
paper gives sharp bounds on the index $H_{f}(G)$ in terms of order and size of
$G$ when $G$ is connected and has the maximum degree at most $4$. All the
graphs achieving the derived bounds are also determined by their degree sets.
Bounds involving several existing indices – including the general zeroth-order
Randić index and coindex, the general multiplicative first/second Zagreb
index, the variable sum lodeg index, the variable sum exdeg index – are
deduced as the special cases of the obtained ones.
Keywords: chemical graph theory; topological index; vertex-degree-function
indices; degree of a vertex.
AMS Subject Classification: 05C07, 05C09, 05C92.

##  1 Introduction

This study is concerned with only connected and finite graphs. The
(chemical-)graph theoretical concepts used in the this paper without providing
their definitions can be found in the related books like [26, 22, 6, 5].

A topological index is a function defined on the set of all graphs with the
condition that it remains the same under the graph isomorphism. The degree set
of a graph $G$ is the set consisting of all distinct elements of the degree
sequence of $G$. Consider a graph $G$ and a real-valued function $f$ defined
on the degree set of $G$. The sum of the outputs $f(d_{v})$ over all vertices
$v\in V(G)$ of $G$ is usually known as the vertex-degree-function indices and
is denoted by $H_{f}(G)$, where $d_{v}$ represents the degree of a vertex $v$
of $G$. Thus,

$H_{f}(G)=\sum_{v\in V(G)}f(d_{v}).$ (1)

Although the terminology and notation of the topological index $H_{f}$ that is
being used by several researchers was coined by Yao et al. [27], to the best
of authors’ knowledge such indices were studied first by Linial and Rozenman
in [14]. These indices have been the subject of several recent papers; see for
example the recent articles [20, 10, 21], recent review paper [12], and
related publications cited therein.

If vertices $u$ and $v$ are adjacent in $G$, we write $u\sim v$, otherwise we
write $u\nsim v$. Let $TI(G)$ be a vertex–degree–based topological index of
the form:

$TI(G)=\sum_{u\sim v}(f(d_{u})+f(d_{v}))=\sum_{u\in V(G)}d_{u}f(d_{u})\,;$

the right-handed identity is a special case of a more general identity
reported in [7]. Then, the corresponding coindex, $\overline{TI}(G)$ can be
defined [9, 15] as:

$\overline{TI}(G)=\sum_{u\nsim v}(f(d_{u})+f(d_{v}))=\sum_{u\in
V(G)}(n-1-d_{u})f(d_{u})\,.$

The following identity is valid

$TI(G)+\overline{TI}(G)=(n-1)\sum_{u\in V(G)}f(d_{u})=(n-1)H_{f}(G)\,.$ (2)

In what follows, some existing indices are given that are special cases of
Equation (1).

  * •

Equation (1) gives the general zeroth-order Randić index if $f(x)=x^{\alpha}$
(see for example [17, 2, 13, 16, 3]), where $\alpha$ is a real number.

  * •

The general zeroth–order Randić coindex is obtained from equation (1)
corresponding to the choice $f(x)=(n-1-x)x^{\alpha-1}$, where $n$ is the order
of the graph under consideration and $\alpha$ is a real number (see e.g. [19,
18]). Particularly, if $\alpha=3$, then the forgotten topological coindex
$\overline{F}(G)=\sum_{u\in V(G)}(n-1-d_{u})d_{u}^{2}$ is obtained (see for
example [4, 8]); the forgotten topological coindex is same as the Lanzhou
index [25].

  * •

One gets the natural logarithm of the general multiplicative first Zagreb
index (general multiplicative second Zagreb index, respectively) [23] by
taking $f(x)=\ln x^{a}$ ($f(x)=\ln x^{ax}$, respectively), where
$a\in\mathbb{R}$ (that is the set of real numbers).

  * •

The substitution $f(x)=x(\ln x)^{a}$ in Equation (1) yields the variable sum
lodeg index [24], where $a\in\mathbb{R}$

  * •

If $f(x)=xa^{x}$ then Equation (1) gives the variable sum exdeg index (see for
example [24, 1]), where $a>0$ with $a\neq 1$.

A graph with $n$ vertices and $m$ edges is known as an $(n,m)$-graph. A
chemical graph is the one with the maximum degree at most four. This paper
gives sharp bounds on the index $H_{f}(G)$ for chemical $(n,m)$-graphs in
terms of $m$ and $n$. All the graphs achieving the derived bounds are also
identified. Bounds involving the above-mentioned existing indices (that is,
the general zeroth-order Randić index and coindex, the general multiplicative
first/second Zagreb index, the variable sum lodeg index, the variable sum
exdeg index) are deduced as the special cases of the obtained bounds.

##  2 Main Results

For a graph $G$, its number of vertices having the degree $r$ is denoted by
$n_{r}$. If $G$ is a chemical $(n,m)$-graph, then

$H_{f}(G)=\sum_{i=1}^{4}n_{i}\,f(i),$ (3) $\sum_{i=1}^{4}n_{i}=n,$ (4)
$\sum_{i=1}^{4}i\,n_{i}=2m,$ (5)

We solve Equations (4) and (5) for the quantities $n_{1},n_{4}$, and then
substitute their values in Equation (3):

$\displaystyle H_{f}(G)$
$\displaystyle=\frac{1}{3}\Big{(}4f(1)-f(4)\Big{)}n+\frac{2}{3}\Big{(}f(4)-f(1)\Big{)}m$
$\displaystyle\quad\
+\left(f(2)-\frac{2}{3}f(1)-\frac{1}{3}f(4)\right)n_{2}+\left(f(3)-\frac{1}{3}f(1)-\frac{2}{3}f(4)\right)n_{3}\,.$
(6)

Let us take

$\displaystyle\Gamma_{\\!\\!f}(G)$
$\displaystyle=\left(f(2)-\frac{2}{3}f(1)-\frac{1}{3}f(4)\right)n_{2}+\left(f(3)-\frac{1}{3}f(1)-\frac{2}{3}f(4)\right)n_{3}\,.$
(7)

Now, Equation (2) yields

$H_{f}(G)=\frac{1}{3}\Big{(}4f(1)-f(4)\Big{)}n+\frac{2}{3}\Big{(}f(4)-f(1)\Big{)}m+\Gamma_{\\!\\!f}(G)\,.$
(8)

Let

$\xi_{1}=f(2)-\frac{2}{3}f(1)-\frac{1}{3}f(4)\quad\text{and}\quad\xi_{2}=f(3)-\frac{1}{3}f(1)-\frac{2}{3}f(4)$
(9)

be the coefficients of $n_{2}$ and $n_{3}$, respectively, in (7). From
Equation (8), it is evident that if one wants to establish a bound on $H_{f}$
for chemical $(n,m)$-graphs in terms of $m$ and $n$, it is enough to determine
the least or greatest $\Gamma_{\\!\\!f}$-value for chemical $(n,m)$-graphs.
Thence, in the next lemma, we derive a bound on $\Gamma_{\\!\\!f}$ for
chemical $(n,m)$-graphs.

###### Lemma 1

Let $G$ be a chemical $(n,m)$-graph such that $n_{2}+n_{3}\geq 2$.

(i).

If both $\xi_{1}$ and $\xi_{2}$ are negative such that
$2\xi_{2}<\xi_{1}<\xi_{2}/2$, then

$\Gamma_{\\!\\!f}(G)<\min\left\\{\xi_{1},\xi_{2}\right\\}.$

(ii).

If both $\xi_{1}$ and $\xi_{2}$ are positive such that
$\xi_{2}/2<\xi_{1}<2\xi_{2}$, then

$\Gamma_{\\!\\!f}(G)>\max\left\\{\xi_{1},\xi_{2}\right\\}.$

Proof:  (i) Take $\max\left\\{\xi_{1},\xi_{2}\right\\}=\xi_{max}$. Note that

$\Gamma_{\\!\\!f}(G)=\xi_{1}n_{2}+\xi_{2}n_{3}\leq(n_{2}+n_{3})\xi_{max}\leq
2\xi_{max}<\min\left\\{\xi_{1},\xi_{2}\right\\}.$

(ii) Let $\min\left\\{\xi_{1},\xi_{2}\right\\}=\xi_{min}$. Then

$\Gamma_{\\!\\!f}(G)=\xi_{1}n_{2}+\xi_{2}n_{3}\geq(n_{2}+n_{3})\xi_{min}\geq
2\xi_{min}>\max\left\\{\xi_{1},\xi_{2}\right\\}.$

$\square$

Recall that the degree set of a graph $G$ is the set of all unequal degrees of
vertices of $G$.

###### Theorem 2

Let $G$ be a chemical $(n,m)$-graph, where $n\geq 5$. Let $\xi_{1}$ and
$\xi_{2}$ be the numbers defined in (9).

(i).

If both $\xi_{1}$ and $\xi_{2}$ are negative such that
$2\xi_{2}<\xi_{1}<\xi_{2}/2$, then

$\displaystyle H_{f}(G)$
$\displaystyle\leq\frac{1}{3}\Big{(}4f(1)-f(4)\Big{)}n+\frac{2}{3}\Big{(}f(4)-f(1)\Big{)}m$
$\displaystyle\quad\ +\begin{cases}\displaystyle
f(2)-\frac{2}{3}f(1)-\frac{1}{3}f(4)&\text{if~{} $2m-n\equiv
1\pmod{3}$}\\\\[17.07164pt] \displaystyle
f(3)-\frac{1}{3}f(1)-\frac{2}{3}f(4)&\text{if~{} $2m-n\equiv
2\pmod{3}$}\\\\[17.07164pt] 0&\text{if~{} $2m-n\equiv 0\pmod{3}$}\end{cases}$

with equality if and only if the degree set of $G$ is
$\bullet$ $\\{1,2,4\\}$ and $G$ contains only one vertex of degree $2$
whenever $2m-n\equiv 1\pmod{3}$;
$\bullet$ $\\{1,3,4\\}$ and $G$ contains only one vertex of degree $3$
whenever $2m-n\equiv 2\pmod{3}$;
$\bullet$ $\\{1,4\\}$ whenever $2m-n\equiv 0\pmod{3}$.

(ii)

If both $\xi_{1}$ and $\xi_{2}$ are positive such that
$\xi_{2}/2<\xi_{1}<2\xi_{2}$, then

$\displaystyle H_{f}(G)$
$\displaystyle\geq\frac{1}{3}\Big{(}4f(1)-f(4)\Big{)}n+\frac{2}{3}\Big{(}f(4)-f(1)\Big{)}m$
$\displaystyle\quad\ +\begin{cases}\displaystyle
f(2)-\frac{2}{3}f(1)-\frac{1}{3}f(4)&\text{if~{} $2m-n\equiv
1\pmod{3}$}\\\\[17.07164pt] \displaystyle
f(3)-\frac{1}{3}f(1)-\frac{2}{3}f(4)&\text{if~{} $2m-n\equiv
2\pmod{3}$}\\\\[17.07164pt] 0&\text{if~{} $2m-n\equiv 0\pmod{3}$}\end{cases}$

with equality if and only if the degree set of $G$ is
$\bullet$ $\\{1,2,4\\}$ and $G$ contains only one vertex of degree $2$
whenever $2m-n\equiv 1\pmod{3}$;
$\bullet$ $\\{1,3,4\\}$ and $G$ contains only one vertex of degree $3$
whenever $2m-n\equiv 2\pmod{3}$;
$\bullet$ $\\{1,4\\}$ whenever $2m-n\equiv 0\pmod{3}$.

Proof:  Because the proofs of the both parts are similar to each other, we
prove only Part (i). If the inequality $n_{2}+n_{3}\geq 2$ holds, then by
using Lemma 1 and Equation (8), one has

$\displaystyle H_{f}(G)$
$\displaystyle<\frac{1}{3}\Big{(}4f(1)-f(4)\Big{)}n+\frac{2}{3}\Big{(}f(4)-f(1)\Big{)}m$
$\displaystyle\quad\
+\min\left\\{f(2)-\frac{2}{3}f(1)-\frac{1}{3}f(4),f(3)-\frac{1}{3}f(1)-\frac{2}{3}f(4)\right\\}$
$\displaystyle<\frac{1}{3}\Big{(}4f(1)-f(4)\Big{)}n+\frac{2}{3}\Big{(}f(4)-f(1)\Big{)}m$

as desired.
In the remaining proof, assume that $n_{2}+n_{3}\leq 1$. Then,
$(n_{2},n_{3})\in\\{(0,0),(1,0),(0,1)\\}$. From Equations (4) and (5), it
follows that $n_{2}+2n_{3}\equiv 2m-n\pmod{3}$ (see for example [11]), which
gives

$(n_{2},n_{3})=\begin{cases}(1,0)&\text{if\,~{} $2m-n\equiv
1\pmod{3}$,}\\\\[5.69054pt] (0,1)&\text{if\,~{} $2m-n\equiv
2\pmod{3}$,}\\\\[5.69054pt] (0,0)&\text{if\,~{} $2m-n\equiv
0\pmod{3}$.}\end{cases}$

The required result follows now from Equation (2). $\square$

In what follows, we consider some well-known topological indices that satisfy
the assumptions of Theorem 2 and hence yield different corollaries of Theorem
2.

First, we take $f(x)=x^{\alpha}$. Then $H_{f}$ is the general zeroth-order
Randić index ${}^{0}\\!R_{\alpha}$. Here, we have

$\xi_{1}=f(2)-\frac{2}{3}f(1)-\frac{1}{3}f(4)=\begin{cases}\displaystyle-\frac{(2^{\alpha}-2)(2^{\alpha}-1)}{3}<0&\text{
if either $\alpha>1$ or $\alpha<0$,}\\\\[14.22636pt]
\displaystyle-\frac{(2^{\alpha}-2)(2^{\alpha}-1)}{3}>0&\text{ if
$0<\alpha<1$,}\end{cases}$

and

$\xi_{2}=f(3)-\frac{1}{3}f(1)-\frac{2}{3}f(4)=\begin{cases}\displaystyle\frac{3^{\alpha+1}-2^{2\alpha+1}-1}{3}<0&\text{
if either $\alpha>1$ or $\alpha<0$,}\\\\[14.22636pt]
\displaystyle\frac{3^{\alpha+1}-2^{2\alpha+1}-1}{3}>0&\text{ if
$0<\alpha<1$.}\end{cases}$

Also,

$2\xi_{2}=\frac{2(3^{\alpha+1}-2^{2\alpha+1}-1)}{3}<\xi_{1}=-\frac{(2^{\alpha}-2)(2^{\alpha}-1)}{3}<\frac{\xi_{2}}{2}=\frac{3^{\alpha+1}-2^{2\alpha+1}-1}{6}$
(10)

holds if either $\alpha>1$ or $\alpha<0$. If each inequality sign “$<$” of
(10) is replaced with “$>$” then the resulting inequality holds for
$0<\alpha<1$. Thus, we have the following known [11] result as a direct
consequence of Theorem 2.

###### Corollary 1

Let $G$ be a chemical $(n,m)$-graph, where $n\geq 5$. If either $\alpha>1$ or
$\alpha<0$, then

${}^{0}\\!R_{\alpha}(G)$
$\displaystyle\leq\frac{4-4^{\alpha}}{3}\,n+\frac{2(4^{\alpha}-1)}{3}\,m+\begin{cases}\displaystyle-\frac{(2^{\alpha}-2)(2^{\alpha}-1)}{3}&\text{if\,~{}
$2m-n\equiv 1\pmod{3}$}\\\\[11.38109pt]
\displaystyle\frac{3^{\alpha+1}-2^{2\alpha+1}-1}{3}&\text{if\,~{} $2m-n\equiv
2\pmod{3}$}\\\\[11.38109pt] 0&\text{if\,~{} $2m-n\equiv
0\pmod{3}$}\end{cases}$

with equality if and only if the degree set of $G$ is the same as specified in
Theorem 2. If $0<\alpha<1$ then the above inequality for
${}^{0}\\!R_{\alpha}(G)$ is reversed.

Now, we take $f(x)=xa^{x}$ with $a>0$ but $a\neq 1$. Then $H_{f}$ is the
variable sum exdeg index $SEI_{a}$. Here, we have

$\xi_{1}=f(2)-\frac{2}{3}f(1)-\frac{1}{3}f(4)=\begin{cases}\displaystyle-\frac{2a(a-1)(2a^{2}+2a-1)}{3}<0&\text{
if either $0<a<\displaystyle\frac{1}{3}$ or $a>1$}\\\\[14.22636pt]
\displaystyle-\frac{2a(a-1)(2a^{2}+2a-1)}{3}>0&\text{ if
$\displaystyle\frac{1}{2}<a<1$,}\end{cases}$

and

$\xi_{2}=f(3)-\frac{1}{3}f(1)-\frac{2}{3}f(4)=\begin{cases}\displaystyle-\frac{a(a-1)(8a^{2}-a-1)}{3}<0&\text{
if either $0<a<\displaystyle\frac{1}{3}$ or $a>1$}\\\\[14.22636pt]
\displaystyle-\frac{a(a-1)(8a^{2}-a-1)}{3}>0&\text{ if
$\displaystyle\frac{1}{2}<a<1$,}\end{cases}$

Also,

$\displaystyle 2\xi_{2}=-\frac{2a(a-1)(8a^{2}-a-1)}{3}$
$\displaystyle<\xi_{1}=-\frac{2a(a-1)(2a^{2}+2a-1)}{3}$
$\displaystyle<\frac{\xi_{2}}{2}=-\frac{a(a-1)(8a^{2}-a-1)}{6}$ (11)

holds if either $a>1$ or $0<a<\frac{1}{3}$. If each inequality sign “$<$” in
(2) is replaced with “$>$” then the resulting inequality holds for
$\frac{1}{2}<a<1$. Thus, we have the next result that follows directly from
Theorem 2.

###### Corollary 2

Let $G$ be a chemical $(n,m)$-graph, where $n\geq 5$. If either $a>1$ or
$0<a<\frac{1}{3}$, then

$\displaystyle SEI_{a}(G)$
$\displaystyle\leq\frac{4a(1-a^{3})n}{3}+\frac{2a(4a^{3}-1)m}{3}+\begin{cases}\displaystyle\frac{2a(1-a)(2a^{2}+2a-1)}{3}&\text{if\,~{}
$2m-n\equiv 1\pmod{3}$}\\\\[11.38109pt]
\displaystyle\frac{a(1-a)(8a^{2}-a-1)}{3}&\text{if\,~{} $2m-n\equiv
2\pmod{3}$}\\\\[11.38109pt] 0&\text{if\,~{} $2m-n\equiv
0\pmod{3}$}\end{cases}$

with equality if and only if the degree set of $G$ is the same as specified in
Theorem 2. If $\frac{1}{2}<a<1$ then the above inequality for $SEI_{a}(G)$ is
reversed.

Next, we take $f(x)=x(\ln x)^{a}$ with $a>0$. Then $H_{f}$ is the variable sum
lodeg index $SLI_{a}$. Here, for $a>\frac{\ln 3-\ln 4}{\ln(\ln 2)-\ln(\ln
3)}~{}(\approx 0.6246)$, we have

$\xi_{1}=f(2)-\frac{2}{3}f(1)-\frac{1}{3}f(4)=\frac{2\big{(}3(\ln 2)^{a}-2(\ln
4)^{a}\big{)}}{3}<0,$
$\xi_{2}=f(3)-\frac{1}{3}f(1)-\frac{2}{3}f(4)=\frac{9(\ln 3)^{a}-8(\ln
4)^{a}}{3}<0$

and

$\displaystyle 2\xi_{2}=\frac{2\big{(}9(\ln 3)^{a}-8(\ln 4)^{a}\big{)}}{3}$
$\displaystyle<\xi_{1}=\frac{2\big{(}3(\ln 2)^{a}-2(\ln
4)^{a}\big{)}}{3}<\frac{\xi_{2}}{2}=\frac{9(\ln 3)^{a}-8(\ln 4)^{a}}{6}.$

Hence, the following corollary is another direct consequence of Theorem 2.

###### Corollary 3

Let $G$ be a chemical $(n,m)$-graph, where $n\geq 5$. If

$a>\frac{\ln 3-\ln 4}{\ln(\ln 2)-\ln(\ln 3)}~{}~{}(\approx 0.6246),$

then

$\displaystyle SLI_{a}(G)$ $\displaystyle\leq\frac{8(\ln
4)^{a}}{3}\,m-\frac{4(\ln
4)^{a}}{3}\,n+\begin{cases}\displaystyle\frac{2\big{(}3(\ln 2)^{a}-2(\ln
4)^{a}\big{)}}{3}&\text{if\,~{} $2m-n\equiv 1\pmod{3}$}\\\\[11.38109pt]
\displaystyle\frac{9(\ln 3)^{a}-8(\ln 4)^{a}}{3}&\text{if\,~{} $2m-n\equiv
2\pmod{3}$}\\\\[11.38109pt] 0&\text{if\,~{} $2m-n\equiv
0\pmod{3}$}\end{cases}$

with equality if and only if the degree set of $G$ is the same as specified in
Theorem 2.

Finally, if we take $f(x)=(n-1-x)x^{2}$, or $f(x)=\ln x^{ax}$, or $f(x)=\ln
x^{a}$, then $H_{f}$ is the forgotten topological coindex $\overline{F}(G)$
(see [4, 8]), or the natural logarithm of the general multiplicative first
Zagreb index $\ln\Pi_{1,a}$, or the natural logarithm of the general
multiplicative second Zagreb index $\ln\Pi_{2,a}$, respectively.

  * •

If we take $f(x)=(n-1-x)x^{2}$ with $n\geq 11$, or $f(x)=\ln x^{ax}$ with
$a>0$, or $f(x)=\ln x^{a}$ with $a<0$, then $f$ satisfies the conditions of
Theorem 2(i).

  * •

If we take $f(x)=\ln x^{a}$ with $a>0$, or $f(x)=\ln x^{ax}$ with $a<0$, then
$f$ satisfies the conditions of Theorem 2(ii).

Hence, the next result follows immediately from Theorem 2.

###### Corollary 4

Let $G$ be a chemical $(n,m)$-graph, where $n\geq 5$. If $a<0$ then

$\displaystyle\Pi_{1,a}(G)$ $\displaystyle\leq\begin{cases}\displaystyle
2^{\frac{a(4m-2n+1)}{3}}&\text{if\,~{} $2m-n\equiv 1\pmod{3}$}\\\\[11.38109pt]
\displaystyle 2^{\frac{2a(2m-n-2)}{3}}3^{a}&\text{if\,~{} $2m-n\equiv
2\pmod{3}$}\\\\[11.38109pt] \displaystyle 2^{\frac{2a(2m-n)}{3}}&\text{if\,~{}
$2m-n\equiv 0\pmod{3}$,}\end{cases}$ $\displaystyle\Pi_{2,a}(G)$
$\displaystyle\geq\begin{cases}\displaystyle
2^{\frac{2a(8m-4n-1)}{3}}&\text{if\,~{} $2m-n\equiv
1\pmod{3}$}\\\\[11.38109pt] \displaystyle
2^{\frac{8a(2m-n-2)}{3}}3^{3a}&\text{if\,~{} $2m-n\equiv
2\pmod{3}$}\\\\[11.38109pt] \displaystyle 2^{\frac{8a(2m-n)}{3}}&\text{if\,~{}
$2m-n\equiv 0\pmod{3}$,}\end{cases}$

and if $n\geq 11$ then

$\overline{F}(G)\leq\begin{cases}2\Big{(}m(5n-26)-n(2n-11)+8\Big{)}&\text{if\,~{}
$2m-n\equiv 1\pmod{3}$}\\\\[11.38109pt]
2\Big{(}m(5n-26)-n(2n-11)+9\Big{)}&\text{if\,~{} $2m-n\equiv
2\pmod{3}$}\\\\[11.38109pt] 2\Big{(}m(5n-26)-2n(n-6)\Big{)}&\text{if\,~{}
$2m-n\equiv 0\pmod{3}$,}\end{cases}$

where the equality sign in any of the above inequalities involving
$\Pi_{1,a}(G)$, $\Pi_{2,a}(G)$, $\overline{F}(G)$, holds if and only if the
degree set of $G$ is the same as specified in Theorem 2. If $a>0$ then the
above inequalities involving $\Pi_{1,a}(G)$ and $\Pi_{2,a}(G)$ are reversed.

From Theorem 2 and the identity (2), the next result follows.

###### Theorem 3

Let $G$ be a chemical $(n,m)$-graph, where $n\geq 5$. Let $\xi_{1}$ and
$\xi_{2}$ be the numbers defined in (9).

(i)

If both $\xi_{1}$ and $\xi_{2}$ are negative such that
$2\xi_{2}<\xi_{1}<\xi_{2}/2$, then

$\displaystyle TI(G)+\overline{TI}(G)$
$\displaystyle\leq(n-1)\left(\frac{1}{3}\Big{(}4f(1)-f(4)\Big{)}n+\frac{2}{3}\Big{(}f(4)-f(1)\Big{)}m\right)$
$\displaystyle\quad\
+\begin{cases}\displaystyle(n-1)\Big{(}f(2)-\frac{2}{3}f(1)-\frac{1}{3}f(4)\Big{)}&\text{if~{}
$2m-n\equiv 1\pmod{3}$}\\\\[17.07164pt]
\displaystyle(n-1)\Big{(}f(3)-\frac{1}{3}f(1)-\frac{2}{3}f(4)\Big{)}&\text{if~{}
$2m-n\equiv 2\pmod{3}$}\\\\[17.07164pt] 0&\text{if~{} $2m-n\equiv
0\pmod{3}$,}\end{cases}$

with equality if and only if the degree set of $G$ is
$\bullet$ $\\{1,2,4\\}$ and $G$ contains only one vertex of degree $2$
whenever $2m-n\equiv 1\pmod{3}$;
$\bullet$ $\\{1,3,4\\}$ and $G$ contains only one vertex of degree $3$
whenever $2m-n\equiv 2\pmod{3}$;
$\bullet$ $\\{1,4\\}$ whenever $2m-n\equiv 0\pmod{3}$.

(ii)

If both $\xi_{1}$ and $\xi_{2}$ are positive such that
$\xi_{2}/2<\xi_{1}<2\xi_{2}$, then

$\displaystyle TI(G)+\overline{TI}(G)$
$\displaystyle\geq(n-1)\left(\frac{1}{3}\Big{(}4f(1)-f(4)\Big{)}n+\frac{2}{3}\Big{(}f(4)-f(1)\Big{)}m\right)$
$\displaystyle\quad\
+\begin{cases}\displaystyle(n-1)\Big{(}f(2)-\frac{2}{3}f(1)-\frac{1}{3}f(4)\Big{)}&\text{if\,~{}
$2m-n\equiv 1\pmod{3}$}\\\\[17.07164pt]
\displaystyle(n-1)\Big{(}f(3)-\frac{1}{3}f(1)-\frac{2}{3}f(4)\Big{)}&\text{if\,~{}
$2m-n\equiv 2\pmod{3}$}\\\\[17.07164pt] 0&\text{if\,~{} $2m-n\equiv
0\pmod{3}$}\end{cases}$

with equality if and only if the degree set of $G$ is
$\bullet$ $\\{1,2,4\\}$ and $G$ contains only one vertex of degree $2$
whenever $2m-n\equiv 1\pmod{3}$;
$\bullet$ $\\{1,3,4\\}$ and $G$ contains only one vertex of degree $3$
whenever $2m-n\equiv 2\pmod{3}$;
$\bullet$ $\\{1,4\\}$ whenever $2m-n\equiv 0\pmod{3}$.

## Acknowledgements

The authors are very grateful to the handling editor and the referee for
helpful comments and suggestions. This research has been funded by the
Scientific Research Deanship, University of Ha’il, Saudi Arabia, through
project number RG-22 005.

## References

  * [1] A. Ali, D. Dimitrov, On the extremal graphs with respect to bond incident degree indices, _Discrete Appl. Math._ 238, 32–40 (2018).
  * [2] A. Ali, D. Dimitrov, Z. Du, F. Ishfaq, On the extremal graphs for general sum-connectivity index ($\chi_{\alpha}$) with given cyclomatic number when $\alpha>1$, _Discrete Appl. Math._ 257, 19–30 (2019).
  * [3] A. Ali, I. Gutman, E. Milovanović, I. Milovanović, Sum of powers of the degrees of graphs: extremal results and bounds, _MATCH Commun. Math. Comput. Chem._ 80, 5–84 (2018).
  * [4] M. Azari, F. Falahati-Nezhad, Some results on forgotten topological coindex, _Iranian J. Math. Chem._ 10, 307–318 (2019).
  * [5] J. A. Bondy, U. S. R. Murty, Graph Theory, Springer (2008).
  * [6] G. Chartrand, L. Lesniak, P. Zhang, Graphs & Digraphs, CRC Press (2016).
  * [7] T. Došlić, T. Réti, D. Vukičević, On the vertex degree indices of connected graphs, _Chem. Phys. Lett._ 512, 283–286 (2011).
  * [8] A. Ghalavand, A. R. Ashrafi, On forgotten coindex of chemical graphs, _MATCH Commun. Math. Comput. Chem._ 83, 221–232 (2020).
  * [9] I. Gutman, On coindices of graphs and their complements, _Appl. Math. Comput._ 305, 161–165 (2017).
  * [10] Z. Hu, L. Li, X. Li, D. Peng, Extremal graphs for topological index defined by a degree-based edge-weight function, _MATCH Commun. Math. Comput. Chem._ 88, 505–520 (2022).
  * [11] Y. Hu, X. Li, Y. Shi, T. Xu, I. Gutman, On molecular graphs with smallest and greatest zeroth-order general Randić index, _MATCH Commun. Math. Comput. Chem._ 54, 425–434 (2005).
  * [12] X. Li, D. Peng, Extremal problems for graphical function-indices and $f$-weighted adjacency matrix, _Discrete Math. Lett._ 9, 57–66 (2022).
  * [13] X. Li, J. Zheng, A unified approach of the extremal trees for different indices, _MATCH Commun. Math. Comput. Chem._ 54, 195–208 (2005).
  * [14] N. Linial, E. Rozenman, An extremal problem on degree sequences of graphs, _Graphs Combin._ 18, 573–582 (2002).
  * [15] M. Matejić, I. Milovanović, E. Milovanović, Some remarks on the general zeroth–order Randić coindex, _Filomat_ 36, 2083–2090 (2022).
  * [16] A. Milićević, S. Nikolić, On variable Zagreb indices, _Croat. Chem. Acta_ 27, 97–101 (2004).
  * [17] P. Milošević, I. Milovanović, E. Milovanović, M. Matejić, Some inequalities for general zeroth–order Randić index, _Filomat_ 33, 5249–5258 (2019).
  * [18] I. Milovanović, M. Matejić, E. Milovanović, A note on the general zeroth–order Randić coindex of graphs, _Contrib. Math._ 1, 17–21 (2020).
  * [19] T. Monsour, C. Song, The $a$ and $(a,b)$-analogs of Zagreb indices and coindices of graphs, _Int. J. Comb._ 2012, #909285 (2012).
  * [20] I. Tomescu, Graphs with given cyclomatic number extremal relatively to vertex degree function index for convex functions, _MATCH Commun. Math. Comput. Chem._ 87, 109–114 (2022).
  * [21] I. Tomescu, Extremal vertex-degree function index for trees and unicyclic graphs with given independence number, _Discrete Appl. Math._ 306, 83–88 (2022).
  * [22] N. Trinajstić, Chemical Graph Theory, CRC Press (1992).
  * [23] T. Vetrík, S. Balachandran, General multiplicative Zagreb indices of trees, _Discrete Appl. Math._ 247, 341–351 (2018).
  * [24] D. Vukičević, Bond additive modeling 5. mathematical properties of the variable sum exdeg index, _Croat. Chem. Acta_ 84, 93–101 (2011).
  * [25] D. Vukičević, Q. Li, J. Sedlar, T. Došlić, Lanzhou index, _MATCH Commun. Math. Comput. Chem._ 80, 863–876 (2018).
  * [26] S. Wagner, H. Wang, Introduction to Chemical Graph Theory, CRC Press (2018).
  * [27] Y. Yao, M. Liu, F. Belardo, C. Yang, Unified extremal results of topological index and graph spectrum, _Discrete Appl. Math._ 271, 218–232 (2019).