# Another Topological Proof of the Infinitude of Prime Numbers

Jhixon Macías  Jhixon Macías
University of Puerto Rico at Mayaguez, Mayaguez, PR, USA
United States of America<EMAIL_ADDRESS>
###### Abstract.

We present a new topological proof of the infinitude of prime numbers with a
new topology. Furthermore, in this topology, we characterize the infinitude of
any non-empty subset of prime numbers.

###### Key words and phrases:

Primes, Topology, Greatest Common Divisor

###### 2020 Mathematics Subject Classification:

11A41; 54G05; 54H11

In [2], we introduce a new topology $\tau$ on the set of positive integers
$\mathbf{N}$ generated by the base

$\beta:=\\{\sigma_{n}:n\in\mathbf{N}\\},\text{ where
}\sigma_{n}:=\\{m\in\mathbf{N}:\gcd(n,m)=1\\}.$

Indeed,

######  Theorem 0.1 ([2]).

$\beta$ is a base for some topology on $\mathbf{N}$.

###### Proof.

It is clear that
$\displaystyle\bigcup_{n\in\mathbf{N}}\sigma_{n}=\sigma_{1}=\mathbf{N}$. On
the other hand, note that for every $n,m\in\mathbf{N}$, we have
$\sigma_{nm}=\sigma_{n}\cap\sigma_{m}$111Note that for all
$x,n,m\in\mathbf{N}$, we have $\gcd(x,n)=\gcd(x,m)=1$ if and only if
$\gcd(x,nm)=1$.. Therefore $\beta$ is a base for some topology on
$\mathbf{N}$. ∎

######  Remark 0.2.

Note that

$\begin{split}\sigma_{n}:&=\left\\{\begin{array}[]{lcc}\ \ \ \ \
\mathbf{N}&\text{if}&n=1\\\ \displaystyle\bigcup_{\begin{subarray}{c}1\leq
m<n\\\
\gcd(m,n)=1\end{subarray}}n\left(\mathbf{N}\cup\\{0\\}\right)+m&\text{if}&n>1\end{array}\right.\\\
\end{split}$

since for every integer $x$, it holds that $\gcd(n,m)=\gcd(n,nx+m)$, see [12,
Theorem 1.9]. It is easily deduced from here that $\sigma_{n}$ is infinite for
every positive integer $n$. Furthermore, it is deduced that the topology
$\tau$ is strictly coarser than Golomb’s topology.

The topological space $\mathbf{X}:=(\mathbf{N},\tau)$ does not satisfy the
$\mathrm{T}_{0}$ axiom, is hyperconnected, and ultraconnected. Among other
properties, one that will be useful is that for each integer $n>1$, we have:

(0.1)
$\mathbf{cl}_{\mathbf{X}}(\\{n\\})=\bigcap_{\begin{subarray}{c}p|n\end{subarray}}\mathbf{M}_{p}$

Here, $\mathbf{cl}_{\mathbf{X}}(\\{n\\})$ is the closure in $\mathbf{X}$ of
the singleton set $\\{n\\}$, $p$ is a prime number, and $\mathbf{M}_{p}$ is
the set of all multiples of $p$.

To prove that equation (0.1) holds, we require the following Lemma.

######  Lemma 0.3 ([2]).

If $p$ is a prime number, then
$\mathbf{cl}_{\mathbf{X}}(\\{p\\})=\mathbf{M}_{p}$.

###### Proof.

Let $x\in\mathbf{cl}_{\mathbf{X}}(\\{p\\})$. Now, if $x\notin\mathbf{M}_{p}$,
then $\gcd(x,p)=1$. Consequently, $x\in\sigma_{p}$, which implies that
$p\in\sigma_{p}$, a contradiction. Therefore, $x\in\mathbf{M}_{p}$. On the
other hand, suppose $x\in\mathbf{M}_{p}$. Then, $x=np$ for some positive
integer $n$. Now, take $\sigma_{k}\in\beta$ such that $x\in\sigma_{k}$. This
implies that $\gcd(np,k)=\gcd(x,k)=1$, so $\gcd(p,k)=1$, and thus,
$p\in\sigma_{k}$. Hence, $x\in\mathbf{cl}_{\mathbf{X}}(\\{p\\})$. ∎

######  Theorem 0.4 ([2]).

Equation (0.1) holds.

###### Proof.

Let $n>1$ a integer. Take
$x\in\displaystyle\bigcap_{\begin{subarray}{c}p|n\end{subarray}}\mathbf{M}_{p}$.
By Lemma 0.3, we have that $x\in\mathbf{cl}_{\mathbf{X}}(\\{p\\})$ for every
$p$ such that $p|n$. Therefore, for every $\sigma_{k}\in\beta$ with
$x\in\sigma_{k}$, it holds that $p\in\sigma_{k}$ for all $p$ such that $p|n$.
This implies $\gcd(k,n)=1$, and thus, $n\in\sigma_{k}$. Consequently,
$x\in\mathbf{cl}_{\mathbf{X}}(\\{n\\})$. On the other hand, suppose
$x\in\mathbf{cl}_{\mathbf{X}}(\\{n\\})$. Consider $\sigma_{k}\in\beta$ such
that $x\in\sigma_{k}$. Then, $n\in\sigma_{k}$, and therefore $\gcd(n,k)=1$.
This implies that $\gcd(p,k)=1$ for every $p|n$, and so $p\in\sigma_{k}$ for
all $p$ such that $p|n$. Thus, by Lemma 0.3, we conclude that
$x\in\displaystyle\bigcap_{\begin{subarray}{c}p|n\end{subarray}}\mathbf{M}_{p}$.
∎

######  Remark 0.5.

If $n=1$, then $\mathbf{cl_{X}}(\\{n\\})=\mathbf{N}$. Moreover note that, for
every positive integer $n>1$, we have $1\notin\mathbf{cl_{X}}(\\{n\\})$ since
$1\in\sigma_{n}$.

The objective of this short note is to provide a new topological proof of the
infinitude of prime numbers, distinct from the topological proofs presented by
Fürstenberg [6] and Golomb [7], which, in fact, are similar except for the
topology they use.

Let $\mathbf{P}$ denote the set of prime numbers. Additionally, for any set
$A$, $\\#A$ denotes the cardinality of $A$, and $\aleph_{0}:=\\#\mathbf{N}$.
Consider the following Proposition.

######  Theorem 0.6.

$\\#\mathbf{P}=\aleph_{0}$ if and only if $\mathbf{P}$ is dense in
$\mathbf{X}$.

###### Proof.

Suppose there are infinitely many prime numbers. Then, for any positive
integer $n>1$, we can choose a prime $p$ such that $p>n$, and consequently,
$p\in\sigma_{n}$ since $\gcd(n,p)=1$. Therefore, $\mathbf{P}$ is dense in
$\mathbf{X}$. On the other hand, assume that $\mathbf{P}$ is dense in
$\mathbf{X}$. Let $\\{p_{1},p_{2},\dots,p_{k}\\}$ be a finite collection of
prime numbers and consider the non-empty basic element $\sigma_{x}$ where
$x=p_{1}\cdot p_{2}\cdots p_{k}$. Note that none of the $p_{i}$ belong to
$\sigma_{x}$, but since $\mathbf{P}$ is dense in $\mathbf{X}$, there must be
another prime number $q$, different from each $p_{i}$, such that
$q\in\sigma_{x}$. Consequently, there are infinitely many prime numbers. ∎

Theorem 0.6 indicates that we only need to prove the density of $\mathbf{P}$
in $\mathbf{X}$ to establish the infinitude of prime numbers. Precisely, that
is what we will demonstrate.

To achieve our goal, consider the set
$\mathbf{N_{1}}:=\mathbf{N}\setminus\\{1\\}$ and the subspace topology

${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbf{\tau_{1}}:=\\{\mathbf{N_{1}}\cap\mathcal{O}:\mathcal{O}\in\tau\\}\
\ \text{generated by the base}\ \
\mathbf{\beta_{1}}:=\\{\mathbf{N_{1}}\cap\sigma_{n}:\sigma_{n}\in\beta\\}.}$

Also, consider the topological subspace
$\mathbf{X_{1}}:=(\mathbf{N_{1}},\mathbf{\tau_{1}})$ and the following Lemma:

######  Lemma 0.7.

$\mathbf{P}$ is dense in $\mathbf{X_{1}}$ if and only if $\mathbf{P}$ is dense
in $\mathbf{X}$.

###### Proof.

It is clear that if $\mathbf{P}$ is dense in $\mathbf{X}$, then $\mathbf{P}$
is dense in $\mathbf{X_{1}}$. On the other hand, it is clear that
$\mathbf{X_{1}}$ is dense in $\mathbf{X}$. So, by transitive property of
density, if $\mathbf{P}$ is dense in $\mathbf{X_{1}}$, then $\mathbf{P}$ is
dense in $\mathbf{X}$. ∎

Now, let’s prove that $\mathbf{P}$ is dense in $\mathbf{X_{1}}$.

######  Theorem 0.8.

$\mathbf{P}$ is dense in $\mathbf{X_{1}}$.

###### Proof.

In any topological space, it holds that the union of closures of subsets of
that space is contained in the closure of the union of those sets. Therefore,

(0.2)
$\bigcup_{p\in\mathbf{P}}\mathbf{cl}_{\mathbf{X_{1}}}(\\{p\\})\subset\mathbf{cl}_{\mathbf{X_{1}}}\left(\bigcup_{p\in\mathbf{P}}\\{p\\}\right)=\mathbf{cl}_{\mathbf{X_{1}}}(\mathbf{P})\subset\mathbf{N_{1}}$

On the other hand, from equation (0.1), it follows that

(0.3)
$\bigcup_{p\in\mathbf{P}}\mathbf{cl}_{\mathbf{X_{1}}}(\\{p\\})=\bigcup_{p\in\mathbf{P}}\left(\mathbf{cl}_{\mathbf{X}}(\\{p\\})\cap\mathbf{N_{1}}\right)=\bigcup_{p\in\mathbf{P}}\mathbf{cl}_{\mathbf{X}}(\\{p\\})=\bigcup_{p\in\mathbf{P}}\mathbf{M}_{p}$

However, using the fundamental theorem of arithmetic, it can be easily shown
that

(0.4) $\bigcup_{p\in\mathbf{P}}\mathbf{M}_{p}=\mathbf{N_{1}}$

Then, by using equations (0.2), (0.3), and (0.4), we conclude that
$\mathbf{cl}_{\mathbf{X_{1}}}(\mathbf{P})=\mathbf{N_{1}}$, i.e., $\mathbf{P}$
is dense in $\mathbf{X_{1}}$. ∎

From Theorem 0.6, Lemma 0.7 and Theorem 0.8, we can deduce that

######  Theorem 0.9.

There are infinitely many prime numbers.

There are many proofs of the infinitude of prime numbers, such as Goldbach’s
Proof [1, p.3], Elsholtz’s Proof [3], Erdos’s Proof [4], Euler’s Proof [5],
and more recent ones, see , [13], [8], [10] and [9]. Moreover, more than 200
proofs of the infinitude of primes can be found in [11]. However,
Fürstenberg’s and Golomb’s proofs are the only known a priori topological
proofs, which, in essence, as mentioned earlier, are based on the same idea,
except for the topology used. Despite being able to present a topological
proof using the same idea with the topology $\tau$ (left as an exercise to the
reader), we present a completely different proof, not only because of the
topology used but also due to the underlying idea—proving that $\mathbf{P}$ is
dense in $\mathbf{X}$.

Finally, we want to leave the reader with the following interesting Theorem.

######  Theorem 0.10.

Let $A\subset\mathbf{P}$ non-empty. Then, $A$ is dense in $\mathbf{X}$, if and
only if, $\\#A=\aleph_{0}$.

###### Proof.

Replace $\mathbf{P}$ with $A$ in the proof of Theorem 0.6. ∎

Theorem 0.10 implies a new relationship between number theory and topology, at
least we hope so. Indeed, to answer questions such as: are there infinitely
many even perfect numbers? or its equivalent, are there infinitely many
Mersenne primes? it suffices to check the density of these sets on
$\mathbf{X}$. Certainly, it may not be easy, but it is possible. The advantage
of working with $\mathbf{X}$ is that this space is hyperconnected, so any
subset is either dense or nowhere dense.

## References

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# Bayesian Layer Graph Convolutional Network for Hyperspectral Image
Classification

Mingyang Zhang, , Ziqi Di, Maoguo Gong, ,
Yue Wu, , Hao Li, , Xiangming Jiang M. Zhang, Z. Di, M. Gong, H. Li and X.
Jiang are with the School of Electronic Engineering, and the Key Laboratory of
Intelligent Perception and Image Understanding of Ministry of Education,
Xidian University, No. 2 South TaiBai Road, Xi’an 710071, China (e-mail:
<EMAIL_ADDRESS>gong@ieee.org). Y. Wu is with the School of Computer
Science and Technology, Xidian University, Xi’an 710071, China.

###### Abstract

In recent years, research on hyperspectral image (HSI) classification has
continuous progress on introducing deep network models, and recently the graph
convolutional network (GCN) based models have shown impressive performance.
However, these deep learning frameworks are based on point estimation
suffering from low generalization and inability to quantify the classification
results uncertainty. On the other hand, simply applying the distribution
estimation based Bayesian Neural Network (BNN) to classify the HSI is unable
to achieve high classification efficiency due to the large amount of
parameters. In this paper, we design a Bayesian layer as an insertion layer
into point estimation based neural networks, and propose a Bayesian layer
graph convolutional network (BLGCN) model by combining graph convolution
operations, which can effectively extract graph information and estimate the
uncertainty of classification results. Moreover, a Generative Adversarial
Network (GAN) is built to solve the sample imbalance problem of HSI dataset.
Finally, we design a dynamic control training strategy based on the confidence
interval of the classification results, which will terminate the training
early when the confidence interval reaches the presented threshold. The
experimental results show that our model achieves a balance between high
classification accuracy and strong generalization. In addition, it can
quantify the uncertainty of outputs.

###### Index Terms:

Bayesian layer; graph convolutional network; hyperspectral image
classification; generative adversarial network.

##  I Introduction

With the rapid development of remote sensing sensors, hyperspectral images
have drawn more and more attentions in academic research and various
applications [1, 2], such as mineral detection, soil testing, medical image,
and urban planning. Compared with conventional optical remote sensing image,
hyperspectral images contain abundant spectral information, which are consists
of hundreds of contiguous spectral channels [3, 4]. Due to this character,
besides spatial context information, hyperspectral images possess unique
spectral signatures for identifying materials and ground truth. For the last
few years, spatial-spectral classification for hyperspectral images has become
a hot research issue in remote sensing society.

In early researches, hyperspectral image classification focuses on designing
various feature extraction methods manually. Feature extraction is designed to
mapping the original data into a new feature space where the spatial and
spectral features are more distinguished and better for classification. For
spectral feature extraction, there are two main methods: linear transformation
based methods and nonlinear transformation based methods. Principal component
analysis (PCA) [5] and independent component analysis (ICA) [6] are two
representative methods for linear transformation based methods. They use
matrix factorization to obtain a new feature space. For nonlinear
transformation based methods, locally linear embedding (LLE) [7] and locality
preserving projections (LPP) [8] based methods are two conventional feature
extraction methods in pattern recognition, which are based on manifold
learning for seeking new feature space. However, researchers found that only
extracting spectral features has limited improvement in classification
accuracy. Since spatially adjacent ground truth is more relevant [9], spatial
context features are also considered for feature extraction combined with
spectral features. Random field based methods are classic spatial feature
extraction methods. Thereinto, Markov random field (MRF) and conditional
random field (CRF) are two representative ones. MRF can build models according
to the relevance of adjacent pixels, which can be used in multiple ground
truth scenes and multiple distribution models. Many works have been proposed
based on MRF. For example, an adaptive MRF was proposed for hyperspectral
feature extraction [10]. Further, Sun proposed a spatially adaptive Markov
random field (MRF) prior in the hidden field [11], which makes the spatial
term more smooth. CRF is a probabilistic model for labels and structural data,
which shows potentials for spatial feature extraction. Various CRF-based
methods have been proposed for spatial feature extraction, such as CRF
combined with sparse representation [12], CRF combined with ensemble learning
[13], and CRF combined with sub-pixel techniques [14]. Besides, morphological
feature based methods are also effective ways for spatial feature extraction,
among which, extended morphological attribute profiles (EMAP) is a
representative one [15]. Based on EMAP, some solid works have been proposed,
such as multiple feature learning [16] and histogram-based attribute profiles
methods [17].

The conventional manually designed feature extraction methods consist of
monolayer structure generally, which can be considered as shallow models.
However, hyperspectral images have shown the characteristics of nonlinear,
high dimensions and redundancy. Due to the capacity limitation of shallow
models, it is always not ideal for shallow models when processing this type of
data. To address this issue, the deep learning models have been introduced to
hyperspectral image classification and some inspiring works have been proposed
to implement spatial-spectral classification in recent years. Chen first
proposed to introduce a deep learning model to hyperspectral classification
[18], which using a stacked auto-encoder structure to extract high level
features. Based on this work, a deep belief network was proposed to extract
spatial-spectral features for hyperspectral images [19]. After this, various
convolutional neural network (CNN) based methods started to make their marks
in hyperspectral classification, due to their great potentials for extracting
spatial features. A 2D and 3D CNN framework were proposed to implement
classification for hyperspectral images with a supervised style [20], which
further improves the classification performance. Lee proposed a contextual CNN
with a deeper structure to extract more subtle spatial features for
classification [21]. A diverse region-based CNN was proposed to obtain
promising features with semantic context-aware representation [22]. Recently,
a CNN with multiscale convolution and diversified metric framework was
proposed, which combines multiscale learning and determinantal point process
priors to improve the diversity of features for classification [23]. However,
CNN based methods bring another challenge for hyperspectral images
classification. The absence of labeled samples can not sufficiently support
the training of CNN based models. To address this issue, data augmentation and
unsupervised pre-training are introduced to CNN based methods. For data
augmentation, Li proposed a pixel-pair feature extraction strategy to enlarge
the labeled samples [24]. With the enlarged labeled samples, the training of
deep CNN model can be improved effectively. For unsupervised pre-training
strategy, Romero proposed a greedy layerwise training framework to implement
unsupervised pre-training [25]. Further, Mou proposed an encoder-decoder
structure to train a residual CNN block in an unsupervised style [26].
Moreover, a generative model based on Wasserstein generative adversarial
network was proposed [27], which can pre-train a CNN subnetwork without
labeled samples.

Recently, besides Euclidean-space based deep learning methods, the graph-space
based deep learning methods have attracted more attention for the research of
hyperspectral image classification, which maps the original data from
Euclidean space to graph space. Due to the concepts of nodes and edges,
compared with traditional Euclidean space, the graph space can better explore
and exploit the relationship among the whole pixels of a hyperspectral image
[28]. Based on this feature, some graph neural network methods have been
proposed for the hyperspectral image classification task. Wan transformed
hyperspectral data to a graph style using image segmentation techniques, and
proposed a multiscale dynamic graph convolutional network (GCN) to implement
node classification [29]. Based on [29], a context-aware dynamic graph
convolutional network was proposed, which can flexibly explore the relations
among graph-style data [30]. Meanwhile, due to the ability of handling the
whole graph i.e. the whole data, some semi-supervised graph-based methods such
as GCNs have been proposed, which can utilize the labeled and unlabeled
samples simultaneously to reduce the requirement of labeled training samples
[31, 28, 32]. Moreover, some effective variants of GCNs have been further
designed. In [33] and [34], GCNs have been designed in dual channel style to
explore multiscale spatial information and label distribution respectively.
For better update the graph structure, in [35] and [36], an adaptive sampling
strategy and an attention mechanism are combined with the GCNs-based methods
respectively. Liu combined the advantages of convolution neural networks with
GCNs to better utilize pixel and superpixel level features [37]. From the view
of training efficiency, Hong proposed a minibatch GCN which can reduce
computational cost in large scale remote sensing data analysis [38].

However, the aforementioned deep learning-based methods, including CNN and
GCN, use point estimation for data features, which makes the network easy to
get overfitted with poor generalization ability during the learning process.
At the same time, since each forward propagation is performed with a single
sampling of the weight space, the convergence speed and learning efficiency
are greatly limited, and it is incapable for the network to estimate the
credibility of its output. In addition, in the face of the imbalanced sample
distribution of hyperspectral remote sensing datasets, it is difficult for
deep learning methods to maintain high classification accuracy on minority
classes, which increases the risks of miss alarming corresponding to high-
value samples.

To address the issues discussed above, we propose a Bayesian layer graph
convolutional network (BLGCN), in which we design the Bayesian layer.
Different from the traditional neural network, Bayesian strategy represents
weight matrices in distribution form, which transforms the training process of
networks from point estimation to distribution estimation. The weight matrices
we get from sampling are combined with input features to obtain the network
output, i.e., the classification results.

We apply the variational inference method to calculate the posterior
distribution of the weight matrices and classification loss. Consequently,
based on the given prior distribution and multiple sampling in each forward
propagation, the overfitting problem is effectively solved, and the
generalization ability of the neural network is significantly improved.

While adapting the Bayesian neural network, we give the definition of Bayesian
layer and design its structure. The participation of Bayesian layer enables
researchers to fine-tune the network backbone to satisfy the specific task.
Compared with the traditional Bayesian neural network, our method can reach
the balance of high classification accuracy and low time cost.

Besides, in order to tackle the task of minority class recognition, generative
adversarial methods [39] are introduced to perform feature learning and data
generation for minority class. By expanding the training set with generation
data, the recognition accuracy of minority class is effectively increased.

At the same time, for the purpose of using the uncertainty information of
classification results to guide the training process, we design a dynamic
control strategy with Bayesian method. By setting the threshold on the
validation set classification accuracy, it is stipulated that the training
process will be terminated when the upper limit of the confidence interval
reaches the threshold. This strategy shortens the time cost of training
process and increases the training efficiency.

To sum up, our main contributions in this paper are listed as follows:

(1) We propose the BLGCN framework, which can reach the balance of high
classification accuracy and low time cost.

(2) The Bayesian method carried by Bayesian layer enables the network to
quantify the uncertainty of the classification results.

(3) The generative adversarial method we import solve the minority class
recognition problem efficiently.

(4) A dynamic control strategy based on Bayesian method is involved in the
training process which further reduces the time cost of model learning.

The remainder of our paper is organized as follows. Background and motivations
of the methods we used are introduced in Section II. In Section III we
formulate the theoretical basis of BLGCN and relevant image processing
methods, including data-augmentation with generative adversarial methods and
our training strategy. We conduct the experiments in Section IV. Finally, this
article is concluded in Section V.

##  II Background and Motivation

###  II-A Graph Convolutional Network

Since the graph convolutional networks presented by Kipf&Welling in 2017 [40],
it has been introduced into a wide range of applications, including
recommendation system, graph embedding and node classification. As an
important part of remote sensing images analysis, the application of GCN on
hyperspectral image classification can also be seen in several works [41],
[42], [43]. To begin with, the GCN method assumes that the given settings can
be simplified into a graph $G_{obs}=(V,\varepsilon)$, where $V$ represents the
set of $N$ nodes and $\varepsilon$ is the set of their edges. Each node $i$
has a set of feature vectors $x_{i}\in\mathbbm{R}^{d\times 1}$ in $d$
dimensions, and the label set related to a subset of nodes $L\subset V$ can be
expressed as $y_{i}$. In the image classification task, the value of label set
identifies categories.

The core layer of a GCN can be expressed as follows:

$\displaystyle\mathbb{H}^{(1)}$
$\displaystyle=\sigma(\hat{\mathbb{A}}_{G}\mathbb{X}\mathbb{W}^{(0)}),$ (1)
$\displaystyle\mathbb{H}^{(l+1)}$
$\displaystyle=\sigma(\hat{\mathbb{A}}_{G}\mathbb{H}^{(l)}\mathbb{W}^{(l)}).$

Here $\mathbb{W}^{(l)}$ denotes the weights of the $l$th layer in a GCN.
$\mathbb{H}^{(l)}$ represents the output features from $l$th layer, and the
function $\sigma$ is a nonlinear activation function. The normalized adjacency
matrix $\hat{\mathbb{A}}_{G}$ is derived from a given graph and determines the
mixing situation of the output features across the graph at each convolutional
layer.

For an $L$-layers GCN, the final output is $Z=\mathbb{H}^{(L)}$. The model
learns the weights of network through the backpropagation with the target of
minimizing the error metric function between the given labels $Y_{L}$ and the
network prediction outputs $Z_{L}$.

###  II-B Bayesian Method

With the development of convolutional neural networks and Transformers in
recent years, mainstream deep learning frameworks have become more and more
complex, and the network depth can reach hundreds or even thousands of layers.
Although these large-scale neural networks are capable of information
perception and feature extraction, there still remains hidden dangers of easy
overfitting and lack of the ability to estimate credibility of the model
output. Hence, Bayesian deep learning method [44] has been widely used in
research and gets a significant effect since it can capture the cognitive
uncertainty inherent in data-driven models as well as maintaining a high
accuracy[45].

For the given dataset $\bm{X}$ and label set $\bm{Y}$, when predicting the
probability distribution of the test data pair $x^{*}$ and $y^{*}$, according
to the marginal probability calculation, we have

$p(y^{*}\mid x^{*},\bm{Y},\bm{X})=\int p(y^{*}\mid
x^{*},\omega)p(\omega\mid\bm{Y},\bm{X})$ (2)

where $\omega$ is the model parameter, and the problem is transformed into
finding the maximum posterior distribution of the parameters on the training
dataset $\bm{X}$,$\bm{Y}$. According to the Bayesian formula, we have

$p(\omega\mid\bm{Y},\bm{X})=\frac{p(\bm{Y}\mid\bm{X},\omega)p(\omega)}{p(\bm{Y}\mid\bm{X})}.$
(3)

The integral in Eq. 2 is intractable for most situations, and there are
various integral methods proposed to infer the $p(\omega\mid\bm{Y},\bm{X})$
including variational inference [46] and Markov Chain Monte Carlo (MCMC)
method [44].

The general Bayesian deep learning is defined slightly differently in various
articles, but it usually refers to the Bayesian neural network. Based on the
framework of the traditional convolutional neural network, Bayesian neural
network inherits the method of Bayesian deep learning. Assuming the weight
$\omega$ subjects to a specific distribution, the network optimizing goal is
to maximizing the posterior distribution $p(\omega\mid\bm{Y},\bm{X})$. The
backpropagation process remains the same with traditional neural network.

By structurally combining probabilistic model and neural network, Bayesian
neural network also keeps their advantages. While retaining the ability of
neural network perception and feature extraction, Bayesian method applies
distribution estimation instead of point estimation by representing the weight
space with probability distribution. It provides theoretical premise for
quantifying the confidence level of the learning results. By measuring the
uncertainty of learning results, we can design evaluation models, which
dynamically evaluate the training progress and determine whether to proceed.
At the same time, the training method based on distribution estimation speeds
up the learning convergence process and ensures the robustness of the training
results by sampling the weight distribution multiple times in a forward
propagation process.

###  II-C Generative Adversarial Network

Nowadays, GAN [39] is becoming a popular model with strong generative
capabilities and high efficiency. Generally, it consists of a discriminative
network $\bm{D}$ and a generative network $\bm{G}$. The generative network
$\bm{G}$ needs to generate features that match the training features $\bm{x}$
as much as possible based on the given random features $\bm{z}$, which
subjects to a specific distribution (e.g., uniform distribution or the
Gaussian distribution). On the contrary, the aim of the discriminative network
$\bm{D}$ is to distinguish whether the sample data is generated features or
training features $\bm{x}$. The training processes of the two networks are
arranged to perform alternately, which can be considered as a two-player mini-
max game and has a value function as follows:

$\displaystyle\min_{\bm{G}}\max_{\bm{D}}V(\bm{D},\bm{G})$
$\displaystyle=\mathbbm{E}_{\bm{x}\sim p_{data}(\bm{x})}[\log\bm{D}(\bm{x})]$
(4) $\displaystyle+\mathbbm{E}_{\bm{z}\sim
p_{\bm{z}}(\bm{z})}[\log(1-\bm{D}(\bm{G}(\bm{z})))].$

It is assumed that the training features $\bm{x}$ subject to the distribution
of $p_{data}(\bm{x})$, and the random features $\bm{z}$ subject to an
arbitrary distribution $p_{\bm{z}}(\bm{z})$. As training processes, the
discriminator network and the generative network tend to converge and reach a
point where the features generated are so real that the discriminator is
unable to judge whether they are real features.

When processing hyperspectral remote sensing image data, researchers are often
troubled by the problem of sample imbalance. In the Indian Pines dataset, the
sample size of the smallest class is only equals to eight thousandths of the
sample size of the largest class. The resulting model will easily have poor
classification performance on the minority class. If we simply replicate the
minority class data, we can achieve high recognition accuracy on the training
dataset, but the model we got will lack generalization ability when dealing
with unknown datasets. Therefore, we need to design the model from the data
generation point of view. The traditional generative adversarial network has
strong generative ability and adaptability, but when dealing with the pixel
data with long-sequence features in hyperspectral images, the feature
extraction ability will be greatly reduced, and become difficult to generate
new features.

Over the past years, GAN methods including 1D-GAN [47], 3D-GAN [47] and other
networks [48], [49], [50] have demonstrated their superior ability in
unsupervised or semi-supervised learning, pioneers have started to apply them
in dealing with hyperspectral remote sensing problems. However, we find that
these models have limited feature expression capabilities and overfitting
problem. The feature extraction method also loses part of the original
information.

Therefore, we make some improvements to the generative adversarial network to
solve the problem. By means of feature vector interleaved combination and unit
matrix feature extraction, the newly designed generative adversarial network
achieves better feature expression ability and strong generalization at the
same time.

##  III Methodology

This section elaborates our proposed BLGCN model and the corresponding HSI
processing framework. First, we preprocess the input HSI raw data with the
help of simple linear iterative cluster (SLIC) algorithm [51], and construct
feature matrix and adjacency matrix. After that, we train our GAN network for
the minority classes, augment the minority classes data, and update the
adjacency matrix. We input the obtained features into the BLGCN model, and
dynamically adjust the training strategy by quantifying the uncertainty of the
output.

###  III-A Data Preprocessing

In order to meet the input requirements of our model, we need to perform
preprocessing operations on the original image. We first apply the SLIC
algorithm to divide the original image into a number of superpixel blocks, the
pixels in each block have strong spatial-spectral similarity with each other.
The essence of the SLIC algorithm is $K$-means clustering, which starts to
evolve from several initial points randomly placed on the image, and
continuously clusters pixels with similar spatial-spectral characteristics.
When the segmentation is completed, the original data can be simplified to a
set of superpixel blocks.

For each superpixel block, we calculate the category distribution of pixels in
the block and select the most occurring category as the label of the
superpixel block. To improve the classification efficiency, we filter out all
superpixel patches annotated as background. In order to extract the spatial
information in the original image, we save the adjacency relationship between
the superpixels in adjacency matrix and mark it as $\mathbb{A}$ based on the
original spatial adjacency relationship between pixels.

We calculate the mean value of the pixel features contained over each
dimension in superpixel block and save it as the texture feature $\bm{t}$, we
sum the squares of the feature vector mean value and save it as the spectral
feature $\bm{s}$. After concatenating the texture feature vector and the
spectral feature vector we can get the feature matrix
$\bm{F}=(\bm{t}\parallel\bm{s})$.

###  III-B Data Augmentation with GAN

Some of the preprocessed hyperspectral images still have a serious problem of
sample imbalance. To solve it, we design a data augment framework based on the
characteristics of feature matrix with the help of generative adversarial
method, the framework is shown in Fig. 1, the integral settings in the
framework are based on the Indian Pines HSI dataset.

####  III-B1 Formulation

We design the generative adversarial data augmentation framework by answering
two questions. The first one is how to increase the amounts of the minority
class training samples inputted into the generative adversarial network while
ensuring the quality of data, and the second task is how to learn and generate
high-dimensional feature vectors without losing the original information.

Figure 1: GAN-framework

####  III-B2 Feature Extraction

After the image preprocessing process, for the sample class $m_{i}$, we mark
the number of superpixel blocks in the class as $b_{m}$, and the feature
vector of superpixel block as $f_{m}^{j}\in\mathbbm{R}^{(1\times
d)},j\in[0,b_{m})$, $d$ represents the dimension of the feature vector after
preprocessing. We filter out the maximum amounts of superpixel blocks in the
classes as $(b_{m})_{max}$. For the class $m_{i}$ which meets the conditions
of $b_{m_{i}}<0.02\times(b_{m})_{max}$, we define it as minority class,
extract its feature vectors $f_{m_{i}}^{j}$ and splice it vertically into a
minority class feature matrix $\mathbb{F}_{i}\in\mathbbm{R}^{b_{m_{i}}\times
d}.$

Due to the high spectral dimension of the feature matrix, the existing
mainstream spectral feature extraction methods and minority class data
generation strategies are usually performed by applying PCA method [5], which
leads to the loss of effective information. As a result, it reduces the
classification accuracy of the model.

In order to achieve a balance between generation efficiency and data
diversity, we design a transition matrix optimization strategy based on
adversarial methods, we use the original feature vector as the input of the
discriminator and train the discriminator’s ability to identify minority class
feature vectors. The feature matrix perturbed by the transition matrix is
imported into the generator, then we input the generated result into the
discriminator. The generator and discriminator are trained respectively. We
assume that the weight matrix of the generator is $\mathbb{W}_{\bm{G}}$, in
order to realize the perturbation operation on the original feature matrix, it
is necessary to initialize the $mean(\mathbb{W}_{\bm{G}})=1$, and set a small
variance to adapt to the data characteristic of the preprocessed feature
matrix.

####  III-B3 Feature Enhancement

For the purpose of cooperating with the optimization strategy designed, we
construct a feature enhancement algorithm for minority class sequence data.
For the minority class feature matrix $\mathbb{F}_{i}$ generated from
preprocessing, we replicate it longitudinally to $d$-dimension and get
$\mathbb{F}_{i}^{\prime}$, we import it into the discriminator as real data
for training. We perform the operation
$\mathbb{F}_{i}^{\prime\prime}=\mathbb{F}_{i}^{\prime}\times\mathbb{I}_{d}$
and do the perturbation to matrix $\mathbb{F}_{i}^{\prime\prime}$ with
generator. Through this process, we construct a diagonal feature matrix that
can be input into the generator for convolution, and the output of the
generator has the characteristic that each row is a slice of new generated
feature vector. By mixing in order, we effectively increase the data diversity
of the training set. At the same time, if the replication order of the
original feature matrix $\mathbb{F}_{i}$ to generate the feature matrix is
adjusted, we will get different forms of enhanced features vector, which
greatly reduces the overfitting probability of the model.

####  III-B4 Loss Function

We design the loss function of the data augmentation model by inheriting the
form of the GAN network loss function. To set an example, we only perform the
data enhancement operation to a single class and ignore the mark of $i$, we
define $\bm{y}_{real}$ and $\bm{y}_{fake}$ which have all elements equal to 1
and 0 with a proper size. First we calculate the training loss $\bm{G}_{loss}$
of the generator

$\bm{G}_{loss}=BCE\\_loss(\bm{D}(\bm{G}(\mathbb{F}^{\prime}\otimes\mathbb{I}),\bm{y}_{real}),$
(5)

in which the $BCE\\_loss$ refers to the binary cross entropy loss function,
then input the original data and the generated data into the discriminator.
Training loss $\bm{D}_{loss}$ with following equations:

$\displaystyle\bm{D}_{real\\_loss}$
$\displaystyle=BCE\\_loss(\bm{D}(\mathbb{F}),\bm{y}_{real})$ (6)
$\displaystyle\bm{D}_{fake\\_loss}$
$\displaystyle=BCE\\_loss(\bm{D}(\bm{G}(\mathbb{F}^{\prime}\otimes\mathbb{I}),\bm{y}_{fake})$
(7) $\displaystyle\bm{D}_{loss}$
$\displaystyle=\bm{D}_{real\\_loss}+\bm{D}_{fake\\_loss}.$ (8)

We define the final loss function as follows:

$\min_{\bm{G}}\max_{\bm{D}}V(\bm{D},\bm{G})=\bm{G}_{loss}-\bm{D}_{loss}.$ (9)

After the training process, we take the output of the generator filling into
the original feature matrix according to the model learning requirements, so
as to realize the data enhancement for the minority class.

###  III-C Bayesian Layer and BLGCN

####  III-C1 Bayesian Deep Learning

In general, for a Bayesian deep neural network, its predicted value can be
calculated by Eq. 2, and the calculation depends on the approximate inference
of the posterior distribution $p(y^{*}\mid x^{*},\bm{Y},\bm{X})$ expressed in
Eq. 3. We take the approach of variational inference and use a simple
distribution $q(\omega)$ to approximate the posterior distribution
$p(\omega\mid\bm{Y},\bm{X}).$

In order to make $q(\omega)$ fit $p(\omega\mid\bm{Y},\bm{X})$ as closely as
possible, we only need to minimize Kullback–Leibler divergence
$KL[q(\omega)\parallel p(\omega\mid\bm{Y},\bm{X})]$ [44].

To present uncertainty, we assume that both the approximate posterior and the
prior follow Gaussian distribution, i.e.,

$\displaystyle q(\omega)=q(\omega\mid\theta),\theta=(\mu,\sigma)$ (10)
$\displaystyle
p(\omega_{i}\mid\bm{Y},\bm{X})\sim\mathbbm{N}(\omega_{i}\mid\mu_{i},{\sigma_{i}}^{2}).$
(11)

The original problem can be transformed into optimizing the parameter

$\displaystyle\theta^{*}$
$\displaystyle=\mathop{\arg\min}_{\theta}KL[q(\omega\mid\theta)\parallel
p(\omega\mid\bm{Y},\bm{X}))]$ (12)
$\displaystyle=\mathop{\arg\min}_{\theta}\mathbbm{E}_{q(\omega\mid\theta)}[\log\frac{q(\omega\mid\theta)}{p(\omega\mid\bm{Y},\bm{X})}]$
$\displaystyle=\mathop{\arg\min}_{\theta}\mathbbm{E}_{q(\omega\mid\theta)}[\log\frac{q(\omega\mid\theta)p(\bm{Y}\mid\bm{X})}{p(\omega\mid\bm{Y},\bm{X})p(\omega)}]$
$\displaystyle=\mathop{\arg\min}_{\theta}\mathbbm{E}_{q(\omega\mid\theta)}[\log\frac{q(\omega\mid\theta)}{p(\bm{Y}\mid\bm{X},\omega)p(\omega)}]$

and expressed as minimizing the loss function

$\displaystyle Loss=KL[q(\omega\mid\theta)\parallel
p(\omega)]-\mathbbm{E}_{q(\omega)}\log(p(\bm{Y}\mid\bm{X},\omega)).$ (13)

We apply the reparameter trick here [45], for the weights
$\omega_{i}\sim\mathbbm{N}(\mu_{i},{\theta_{i}}^{2})$, we have
$\omega_{i}=\mu_{i}+\theta_{i}\odot\epsilon_{i}$, where
$\epsilon_{i}\sim\mathbbm{N}(0,1)$, and $\odot$ represents Hadamard product.
Then for the backpropagation of the loss function Eq. 13, we have

$\displaystyle\frac{\partial}{\partial\theta}\mathbbm{E}_{q(\epsilon)}\\{\log[\frac{q(\omega\mid\theta)}{p(\bm{Y}\mid\bm{X},\omega)p(\omega)}]\\}$
(14)
$\displaystyle=\mathbbm{E}_{q(\epsilon)}\\{\frac{\partial}{\partial\theta}\log[\frac{q(\omega\mid\theta)}{p(\bm{Y}\mid\bm{X},\omega)p(\omega)}]\\}.$

In order to ensure that the range of parameter $\theta$ includes the entire
real number axis, we need to perform the reparameter trick on it, with
$\sigma=\log(1+e^{\rho})$ we have $\theta=(\mu,\rho).$

Figure 2: Bayesian Layer

####  III-C2 Bayesian Layer

The Bayesian neural network used in the current research field is relatively
fixed in form, and has not been effectively combined with the convolutional
neural network [45]. Researchers usually simply migrate the Bayesian forward
propagation method and loss function to the traditional neural networks, and
ignore adjusting the network structure in combination with specific tasks.
Since the weight matrix of Bayesian neural network obeys a specific
distribution, whether it is dealt with variable inference method or the MCMC
method, we need to perform multiple sampling in the forward propagation with
approximate calculation, which will take a great time cost. And when dealing
with the hyperspectral image classification task, directly applying Bayesian
theory to completely transform graph convolutional neural networks limits the
room to fine-tune the network framework and improve the classification
results.

Therefore, we propose the concept of Bayesian layer. We define the Bayesian
layer as a convolutional layer which all weight parameters and biases are
represented in distributed form as shown in Fig. 2. In the process of forward
propagation, the input features matrix will be divided into several row
vectors $\overrightarrow{x_{n}}$, and the Bayesian matrix can be seen as a set
of column vectors $\overrightarrow{h_{n}}$, which is assembled by weight
parameters of $\mu$ and $\theta$. In the matrix multiplication operation, we
have
$\overrightarrow{y_{nm}}=\overrightarrow{x_{n}}\times\overrightarrow{h_{m}}$.

At the same time, we stipulate that the loss function must have the form of KL
divergence which can be optimized by the variational inference method. By
analyzing the form of the Bayesian convolutional neural network loss function,
we find that the likelihood function term can be replaced by a multi-
classification loss. The reason is, given the model output $y$, the likelihood
function $L(\omega\mid y)$ about the parameter $\omega$ is numerically equals
to the probability $p(Y=y\mid\omega)$ of the output $y$ given the parameter
$\omega$. In the process of each forward propagation, the weight parameters of
the non-Bayesian layer remain unchanged when the Bayesian layer is sampled
multiple times, so each time the gradient is updated in the backpropagation,
the existence of non-Bayesian layer will not influence the derivation
operation towards the Bayesian layer.

As a result, if we use the classification loss function of traditional neural
network such as negative log-likelihood loss (nll loss) to represent the
likelihood function, it can provide the theoretical basis for the realization
of the Bayesian layer. Here we can rewrite Eq. 13 as follows:

$\displaystyle\begin{split}Loss&=\sum_{i}\log{q(\omega_{i}\mid\theta_{i})}-\sum_{i}\log{p(\omega_{i})}\\\
&-\sum_{j}\log{p(y_{j}\mid\omega,x_{j})}\\\
&=\sum_{i}\log{q(\omega_{i}\mid\theta_{i})}-\sum_{i}\log{p(\omega_{i})}-nll\\_loss.\end{split}$
(15)

According to the former demonstration, when upgrading the weight parameters of
a Bayesian layers convolutional neural network in gradient calculation and
back propagation, we know that the prior distribution loss
$\sum_{i}\log{p(\omega_{i})}$ and posterior distribution loss
$\sum_{i}\log{q(\omega_{i}\mid\theta_{i})}$ only relate to the weights of
Bayesian layer, and the multi-classification loss relates to all the weight
parameters of the neural network. Therefore, the parameter updates of the
Bayesian layer and the non-Bayesian layer can be performed sequentially under
the same process without interfering with each other.

Meanwhile, it also gives us more choices in diversification. Facing various
tasks, we can choose different insertion methods of the Bayesian layer, which
can achieve the uncertainty quantization output while ensuring that the loss
function structure of the original neural network is not damaged.

####  III-C3 BLGCN

We combine the idea of graph convolution with Bayesian layers, where the
output of each Bayesian layer is multiplied with the preprocessed adjacency
matrix to fuse the spatial features of hyperspectral images. More
specifically, the adjacency matrix $\mathbb{A}$ got from the preprocessing is
renormalized with the method proposed by Kipf&Welling [40]. To generalize the
formulation of $\hat{\mathbb{A}}_{G}$, we perform the trick

$\displaystyle\hat{\mathbb{A}}_{G}=\hat{\mathbb{D}}^{-\frac{1}{2}}\hat{\mathbb{A}}\hat{\mathbb{D}}^{-\frac{1}{2}}$
(16)

with $\hat{\mathbb{A}}=\mathbb{A}+\mathbb{I}$ and
$\hat{\mathbb{D}}_{ii}=\sum\nolimits_{j}\hat{\mathbb{A}}_{ij}$. Here,
$\mathbb{I}$ denotes the identity matrix which has proper size to calculate.
Hence, for each Bayesian layer, the output can be expressed with Eq. 1 after
sampling on the weight parameter matrix.

The main framework of BLGCN is shown in Fig. 3, we set a feature extraction
module before the two Bayesian layer modules, which consists of two full-
connected convolutional layers and a ReLU layer, it can roughly extract the
data features and provide a solid foundation for the Bayesian processing.

It should be noted that for the data-enhanced hyperspectral data, in order to
maintain the consistency of the spatial information, the adjacency matrix
needs to be expanded. We assume that the newly generated superpixels have
similar spatial characteristics with the original superpixels of the same
class. We randomly match the generated ones with those initial superpixels,
and give them the same adjacency relationship with the matched ones, thereby
regenerating a new adjacency matrix.

Figure 3: Bayesian layer graph convolutional network

###  III-D Training Strategy based on Bayesian Method

The prediction results given by deep learning models are not always reliable,
and some application fields of hyperspectral image classification have high
requirements for confidence. By modeling uncertainty with output variance, we
dynamically evaluate the confidence level of the training results and make
decisions on whether terminate the training process.

From the central limit theorem, we know that if the sample size is large
enough, the sampling distribution can be approximated as a normal
distribution. Due to the existence of the Bayesian layer, we can consider that
outputs of the multiple times forward propagation after fixing the model are
independent from each other. Therefore, by calculating the mean and variance
of the multiple forward propagation output, we can obtain an interval estimate
of the accuracy and a confidence interval under the premise of given
confidence.

To enable dynamic evaluation of the training process, we set two accuracy
thresholds and calculate the accuracy of the validation set after each
training process. When it reaches the first threshold, the model weights are
fixed after each training batch and the forward propagation is performed 30
times. We record the output results and calculate whether the confidence
interval upper bound of the validation set accuracy reaches the second
threshold we set under 95% confidence level, and if it does, the training will
be stopped immediately. For training tasks with specific sample classification
accuracy requirements, we only need to change the two judgment operations on
the validation set accuracy to the specific class accuracy.

We applied statistical knowledge to calculate the confidence interval of the
accuracy. For the case of large sample size, we used the Z-test to determine
whether the sample mean was significantly different from the population mean.
We first calculate the sample mean $\mu$ and standard error $SE$, then
determine the confidence level and get the standard score $z$. When the
confidence level is set to 95%, the standard score $z$ is numerically equal to
1.96, and the bounds of the confidence interval $[a,b]$ are shown in below:

$\displaystyle a=\mu-\left|z\right|\times SE$ (17) $\displaystyle
b=\mu+\left|z\right|\times SE.$

We combine the proposed model with dynamic control strategy, and formulate the
whole classification process in forms of flow chart. The implementation
details of the process are shown in Algorithm 1.

Algorithm 1 Proposed BLGCN with Dynamic Control Strategy for HSI
Classification

$\mathbb{Input:}$ Input image; number of iterations $i$; Threshold
$\bm{T_{1}}$,$\bm{T_{2}}$
$\mathbb{Output:}$ Classification Result

1:// Preprocessing the dataset

2:Segment the input image into superpixels with SLIC;

3:Label the superpixels and exclude the background;

4:Construct the adjacency matrices $\bm{A}$;

5:Extract the image information into the feature matrix $\bm{F}$;

6:if The sample imbalance problem is serious then

7: Train the GAN network on preprocessed dataset with Eq. 9;

8: Expand the feature matrix $\bm{F}$ with generated data;

9: Update the adjacency matrices $\bm{A}$;

10:// Training process with dynamic control

11:for epoch in range $i$ do

12: Train the model with BLGCN with Eq. 15;

13: if validation set accuracy reaches $\bm{T_{1}}$ then

14: Fix the model;

15: Perform the forward propagation 30 times;

16: Calculate the confidence interval with Eq. 17;

17: if confidence interval reaches $\bm{T_{2}}$ then

18: Break

19:return result.

##  IV Experimental Analysis

###  IV-A Dataset Description and Evaluation Criteria

In our experiment, we select four mainstream hyperspectral remote sensing
datasets, including Indian Pines, Salinas, Pavia University and Houston
University to verify the advantages of the model we proposed.

Indian Pines dataset was obtained by AirBorne Visible&Infrared Imaging
Spectrometer sensor over the Indian Pines region in 1992. It contains
145$\times$145 pixels and consists of 220 spectral bands covering the range
from 0.4 $\mu$m to 2.5 $\mu$m. 20 bands absorbed by water vapor are removed
and the remaining 200 spectral bands are used for model classification. This
dataset contains 16 kinds of landscapes. After the image preprocessing, the
pixels in the dataset are aggregated into several superpixels. The number of
superpixels in each landscape class and their corresponding training and
testing samples are listed in Table I. The ground-truth map is shown in Fig.
4.

Figure 4: Indian Pines. (a) False color image. (b) Ground truth map. TABLE I:
NUMBERS OF LABELED AND UNLABELED SUPERPIXELS USED IN THE INDIAN PINES DATASET ID |  | Class | Labeled | Unlabeled
---|---|---|---|---
1 |  | Alfalfa | 5 | 41
2 |  | Corn-notill | 143 | 1285
3 |  | Corn-mintill | 83 | 747
4 |  | Corn | 24 | 213
5 |  | Grass-pasture | 48 | 435
6 |  | Grass-trees | 73 | 657
7 |  | Grass-pasture-mowed | 3 | 25
8 |  | Hay-windrowed | 48 | 430
9 |  | Oats | 2 | 18
10 |  | Soybean-notill | 97 | 875
11 |  | Soybean-mintill | 246 | 2209
12 |  | Soybean-clean | 59 | 534
13 |  | Wheat | 21 | 185
14 |  | Woods | 127 | 1138
15 |  | Buildings-grass-trees-drives | 39 | 347
16 |  | Stone-steel-towers | 9 | 84

The University of Pavia dataset was acquired over the Pavia University in
Italy in 2001. It contains 610$\times$340 pixels in 103 spectral bands and has
a wavelength range from 0.43 $\mu$m to 0.86 $\mu$m after removing the noisy
bands. Table III lists the details of the superpixels and the amounts of
training and testing samples. Fig. 6 shows the ground-truth map of the
dataset.

Figure 5: Pavia University. (a) False color image. (b) Ground truth map. TABLE
II:
NUMBERS OF LABELED AND UNLABELED SUPERPIXELS USED IN THE PAVIA UNIVERSITY ID |  | Class | Labeled | Unlabeled
---|---|---|---|---
1 |  | Asphalt | 663 | 5968
2 |  | Meadows | 1854 | 16795
3 |  | Gravel | 210 | 1889
4 |  | Trees | 306 | 2758
5 |  | Painted metal sheets | 135 | 1210
6 |  | Bare soil | 503 | 4526
7 |  | Bitumem | 133 | 1197
8 |  | Self-blocking bricks | 368 | 3314
9 |  | Shadows | 95 | 852

Salinas dataset captured the Salinas Valley in California. This dataset
contains 512$\times$127 pixels in 224 bands. 20 bands absorbed by water vapor
are removed and the remaining 204 spectral bands are used for classification.
Its ground-truth map is shown in Fig. 5 and the number of superpixels with
their information are listed in Table II.

Figure 6: Salinas. (a) False color image. (b) Ground truth map. TABLE III:
NUMBERS OF LABELED AND UNLABELED SUPERPIXELS USED IN THE SALINAS DATASET ID |  | Class | Labeled | Unlabeled
---|---|---|---|---
1 |  | Brocoli-green-weeds-1 | 201 | 1808
2 |  | Brocoli-green-weeds-2 | 373 | 3353
3 |  | Fallow | 198 | 1778
4 |  | Fallow-rough-plow | 139 | 1255
5 |  | Fallow-smooth | 268 | 2410
6 |  | Stubble | 393 | 3536
7 |  | Celery | 358 | 3221
8 |  | Grapes-untrained | 1127 | 10144
9 |  | Soil-vinyard-develope | 620 | 5583
10 |  | Corn-senesced-green-weeds | 328 | 2950
11 |  | Lettuce-romaine-4wk | 107 | 961
12 |  | Lettuce-romaine-5wk | 193 | 1734
13 |  | Lettuce-romaine-6wk | 92 | 824
14 |  | Lettuce-romaine-7wk | 107 | 963
15 |  | Vinyard-untrained | 727 | 6541
16 |  | Vinyard-vertical-trellis | 181 | 1626

The University of Houston dataset is provided by the IEEE GRSS Data Fusion
Contest in 2013. This dataset consists of 349$\times$1905 pixels and contains
144 spectral bands covering the range from 364 nm to 1046 nm. The number of
superpixels in each landscape class and their corresponding training and
testing samples are given in Table IV with its ground-truth map shown in Fig.
7.

Figure 7: Houston University. (a) False color image. (b) Ground truth map.
TABLE IV:
NUMBERS OF LABELED AND UNLABELED SUPERPIXELS USED IN THE HOUSTON UNIVERSITY ID |  | Class | Labeled | Unlabeled
---|---|---|---|---
1 |  | Healthy grass | 125 | 1126
2 |  | Stressed grass | 125 | 1129
3 |  | Synthetic grass | 70 | 627
4 |  | Trees | 124 | 1120
5 |  | Soil | 124 | 1118
6 |  | Water | 33 | 292
7 |  | Residential | 127 | 1141
8 |  | Commercial | 124 | 1120
9 |  | Road | 125 | 1127
10 |  | Highway | 123 | 1104
11 |  | Railway | 124 | 1111
12 |  | Parking Lot1 | 123 | 1110
13 |  | Parking Lot2 | 47 | 422
14 |  | Tennis Court | 43 | 385
15 |  | Running track | 66 | 594

In our experiment, four metrics are adopted to quantificationally evaluate the
classification performance which are per-class accuracy, overall accuracy
(OA), average accuracy (AA), and kappa coefficient (Kappa).

###  IV-B Experimental Settings

In our experiment, the proposed model is implemented via Pytorch with the Adam
optimizer. We train our model with 0.2 dropout rate, 5$\times 10^{-4}$ weight
decay rate. We adopt the learning rate decay strategy in the training process.
We set the initial learning rate as 1$\times 10^{-3}$ and use the multistep
learning rate strategy with gamma fixed to 0.9, and three milestones are 1500,
2500 and 3500. The threshold parameter of assigning the pseudo label to the
unlabeled nodes is set to 0.9, which means that besides the training sample,
all the rest samples are distributed to the validation set.

In the SLIC algorithm in image preprocessing, we set the compactness parameter
to 0.08 for all experiments. And in our GAN algorithm in data augment process,
the initial standard deviation of the generator weight matrix and the
discriminator weight matrix are set to 1$\times 10^{-5}$ and 0.01,
respectively. Because of the data characteristics after the normalization
operation in data preprocessing, the learning rate in our GAN network is set
to 1${\times 10^{-7}}$. For the landscape class which has the superpixels
sample size less than 0.02 times of the superpixels sample size of the largest
class sample, we define it as minority class and perform the data augment
process towards it, the generated samples are filled into the original dataset
until the sample size of the minority class reaches the threshold of 0.05
times of the superpixels sample size of the largest class sample.

###  IV-C Classification Results

In order to evaluate the effectiveness of our proposed BLGCN model, we conduct
comparative experiments with various comparison algorithms on the selected
dataset. To make the experiment results reliable, various SOTA algorithms are
selected as the competitors. Among the algorithms based on traditional machine
learning, we selected the support vector machine algorithm (SVM) [52]. The
commonly used SVM kernel includes linear kernel, polynomial kernel, Gaussian
kernel and Sigmoid kernel. We make the SVM network search for optimization in
the given parameter set, and then select the appropriate hyperparameters for
the classification task. For convolutional neural network based on deep
learning methods, we select the 3D-CNN convolutional neural network [53] which
performs well over the HSI datasets. For graph convolutional neural network
based model, we select several algorithms that have been widely used in recent
years, such as GCN [40], spectral-spatial graph convolutional network (S2GCN)
[31], and multiscale dynamic GCN (MDGCN) [29], etc. They are powered by the
GCN idea to improve the network in a targeted manner, and have achieved better
classification results on the HSI datasets. We perform 30 independent
experiments on each dataset based on the algorithm chosen based on the
application and recorded the mean and standard deviation of the results. It is
worth noting that since the classification accuracies are quite poor when the
3D-CNN model is performed on the HSI datasets with 10% training set ratio, we
only show the classification results with 30% training set ratio on the last
three datasets.

The classification results on the Indian Pines dataset show the effectiveness
of our algorithm. The traditional algorithm (SVM) and the original GCN have
serious misclassification problem in the face of minority class categories
classification task, and the accuracy of the 3D-CNN method based on full-
supervised classification is also relatively low. When the proportion of
training set samples is increased from 10% to 30%, the classification effect
has been significantly improved. Compared with other GCN methods, our method
has obvious advantages in the task of distinguishing minority class
categories, i.e., Class 7 and Class 9, while maintaining high classification
accuracy on other sample categories. The relevant results are listed in Table
V.

TABLE V:
CLASSIFICATION RESULTS ON INDIAN PINES DATASET

Class |  | SVM | 3D-CNN(0.1) | 3D-CNN(0.3) | GCN | S2GCN | MDGCN | BLGCN
---|---|---|---|---|---|---|---|---
1 |  | 59.81$\pm$11.20 | 78.93$\pm$16.75 | 95.55$\pm$3.99 | 73.27$\pm$16.47 | 92.05$\pm$5.64 | 90.02$\pm$4.62 | 92.30$\pm$3.63
2 |  | 73.53$\pm$1.70 | 82.19$\pm$1.75 | 94.42$\pm$1.43 | 81.03$\pm$1.66 | 82.84$\pm$2.32 | 77.83$\pm$1.87 | 96.80$\pm$1.19
3 |  | 65.63$\pm$2.81 | 77.25$\pm$2.98 | 89.73$\pm$1.49 | 81.90$\pm$0.79 | 80.34$\pm$3.15 | 95.39$\pm$2.56 | 95.94$\pm$3.27
4 |  | 54.66$\pm$5.29 | 76.59$\pm$3.90 | 92.81$\pm$0.99 | 69.76$\pm$0.64 | 90.10$\pm$1.85 | 95.69$\pm$1.49 | 96.92$\pm$6.19
5 |  | 88.67$\pm$1.77 | 92.41$\pm$1.82 | 95.69$\pm$0.87 | 95.44$\pm$0.42 | 94.10$\pm$0.1 | 92.36$\pm$0.04 | 93.25$\pm$4.50
6 |  | 92.48$\pm$1.07 | 97.41$\pm$0.82 | 99.09$\pm$0.23 | 100.00$\pm$0.00 | 94.25$\pm$0.24 | 94.32$\pm$0.16 | 99.40$\pm$0.80
7 |  | 75.42$\pm$7.90 | 80.53$\pm$12.10 | 92.75$\pm$3.29 | 0.00$\pm$0.00 | 67.49$\pm$6.22 | 82.03$\pm$5.09 | 93.30$\pm$7.90
8 |  | 96.22$\pm$0.90 | 98.41$\pm$0.89 | 99.86$\pm$0.14 | 100.00$\pm$0.00 | 95.93$\pm$0.03 | 96.00$\pm$0.00 | 100.00$\pm$0.00
9 |  | 36.98$\pm$13.95 | 73.95$\pm$15.89 | 95.62$\pm$3.80 | 0.00$\pm$0.00 | 75.13$\pm$7.85 | 79.02$\pm$6.44 | 100.00$\pm$0.00
10 |  | 69.44$\pm$1.92 | 83.21$\pm$2.03 | 93.02$\pm$1.27 | 79.26$\pm$0.94 | 90.86$\pm$0.95 | 87.39$\pm$0.74 | 92.19$\pm$4.00
11 |  | 77.65$\pm$1.43 | 85.97$\pm$1.61 | 94.69$\pm$0.76 | 91.73$\pm$0.08 | 82.38$\pm$2.14 | 90.62$\pm$1.72 | 98.08$\pm$1.21
12 |  | 70.43$\pm$4.76 | 80.17$\pm$3.01 | 93.77$\pm$1.11 | 86.66$\pm$1.20 | 77.62$\pm$4.79 | 90.33$\pm$3.91 | 97.94$\pm$3.29
13 |  | 94.12$\pm$2.11 | 97.57$\pm$1.51 | 99.29$\pm$0.44 | 93.14$\pm$2.80 | 96.32$\pm$0.72 | 97.08$\pm$0.55 | 99.86$\pm$0.33
14 |  | 93.85$\pm$0.71 | 96.15$\pm$0.95 | 98.73$\pm$0.34 | 97.37$\pm$0.98 | 93.71$\pm$1.45 | 96.81$\pm$1.15 | 99.72$\pm$0.82
15 |  | 59.32$\pm$3.54 | 73.52$\pm$3.76 | 87.91$\pm$1.44 | 86.86$\pm$0.60 | 84.13$\pm$1.18 | 96.81$\pm$0.93 | 95.07$\pm$3.12
16 |  | 94.34$\pm$2.33 | 96.64$\pm$2.13 | 99.00$\pm$0.62 | 96.14$\pm$1.70 | 96.87$\pm$1.18 | 92.91$\pm$0.00 | 100.00$\pm$0.00
OA |  | 78.78$\pm$1.03 | 86.08$\pm$0.77 | 94.24$\pm$0.58 | 88.80$\pm$0.53 | 84.91$\pm$1.79 | 87.09$\pm$1.33 | 97.22$\pm$0.81
AA |  | 75.09$\pm$3.93 | 85.62$\pm$2.38 | 95.26$\pm$0.44 | 75.97$\pm$0.45 | 87.08$\pm$2.23 | 90.25$\pm$1.98 | 97.02$\pm$1.15
Kappa |  | 75.70$\pm$1.20 | 84.14$\pm$0.87 | 93.46$\pm$0.68 | 87.21$\pm$0.57 | 83.02$\pm$0.98 | 86.53$\pm$0.86 | 96.81$\pm$0.94

Table VI shows the classification effects of different algorithms on the Pavia
University dataset. Among these methods, the algorithm we proposed still
maintains the highest classification accuracy, and has achieved 100%
recognition and classification accuracy in several categories. It is worth
noting that due to the relatively balanced sample size of the PU dataset, the
3D-CNN model also reaches a good classification effect after we increase the
training set ratio to 30%, and gets the highest classification accuracy in
some categories. For the samples class that are scattered in spatial such as
Class 4, the classification methods based on GCN fail to achieve good results.
The main reason is that its spatial adjacency relationship is relatively
sparse, and makes it easy to be misclassified when using the GCN methods.

TABLE VI:
CLASSIFICATION RESULTS ON PAVIA UNIVERSITY DATASET

Class |  | SVM | 3D-CNN(0.3) | GCN |  S2GCN | MDGCN | BLGCN
---|---|---|---|---|---|---|---
1 |  |  92.40$\pm$1.71 | 97.94$\pm$0.10 |  70.97$\pm$0.57 |  90.69$\pm$1.84 |  91.09$\pm$0.45 |  96.45$\pm$4.34
2 |  |  95.54$\pm$0.84 |  97.44$\pm$0.05 |  96.55$\pm$0.28 |  85.02$\pm$3.22 |  96.64$\pm$0.41 | 99.12$\pm$0.75
3 |  |  79.92$\pm$1.23 | 94.56$\pm$0.26 |  24.74$\pm$0.55 |  85.91$\pm$3.18 |  89.61$\pm$0.29 |  83.25$\pm$11.57
4 |  |  93.70$\pm$2.17 | 98.02$\pm$0.13 |  47.25$\pm$1.61 |  88.72$\pm$1.24 |  81.57$\pm$1.97 |  66.76$\pm$1.49
5 |  |  99.55$\pm$0.22 |  99.88$\pm$0.04 |  100.00$\pm$0.00 |  97.66$\pm$0.98 |  96.86$\pm$0.21 | 100.00$\pm$0.00
6 |  |  85.61$\pm$2.29 |  98.26$\pm$0.19 |  99.95$\pm$0.15 |  86.52$\pm$1.59 |  92.75$\pm$0.48 | 100.00$\pm$0.00
7 |  |  76.53$\pm$8.90 | 98.36$\pm$0.09 |  94.24$\pm$1.39 |  96.57$\pm$0.76 |  96.32$\pm$1.26 |  98.00$\pm$2.58
8 |  |  86.22$\pm$1.10 |  96.52$\pm$0.37 | 100.00$\pm$0.00 |  87.86$\pm$3.01 |  92.49$\pm$1.00 |  91.71$\pm$1.83
9 |  |  99.90$\pm$1.41 |  99.68$\pm$0.11 |  0.00$\pm$0.00 |  96.58$\pm$0.79 |  78.89$\pm$0.78 | 100.00$\pm$0.00
OA |  |  91.96$\pm$1.41 |  96.58$\pm$0.04 |  90.84$\pm$0.19 |  87.64$\pm$1.82 |  93.16$\pm$0.26 | 97.56$\pm$0.38
AA |  |  89.82$\pm$0.28 | 97.80$\pm$0.05 |  80.37$\pm$0.33 |  90.70$\pm$0.65 |  90.70$\pm$0.74 |  91.42$\pm$2.21
Kappa |  |  89.20$\pm$1.80 |  95.44$\pm$0.05 |  85.40$\pm$0.27 |  84.66$\pm$1.97 |  91.77$\pm$0.39 | 96.15$\pm$0.73

The classification results we get on the Salinas dataset are listed in Table
VII. Compared with other algorithms, BLGCN basically achieves the highest
classification accuracy in all categories, while maintaining high stability.
We find that the SVM method also has higher classification accuracy than the
deep learning algorithm, which shows that there is also room for the
application of traditional classification methods in datasets with sufficient
and balanced sample data such as Salinas.

TABLE VII:
CLASSIFICATION RESULTS ON SALINAS DATASET

Class |  | SVM | 3D-CNN(0.3) | GCN |  S2GCN | MDGCN | BLGCN
---|---|---|---|---|---|---|---
1 |  |  98.80$\pm$0.10 |  98.38$\pm$0.04 |  97.44$\pm$0.55 |  97.90$\pm$0.43 |  98.26$\pm$0.02 | 100.00$\pm$0.00
2 |  |  99.80$\pm$0.11 |  98.88$\pm$0.06 |  99.94$\pm$0.13 |  98.07$\pm$0.29 |  98.18$\pm$0.13 | 100.00$\pm$0.00
3 |  |  98.94$\pm$0.51 |  99.19$\pm$0.32 |  96.88$\pm$2.42 |  96.06$\pm$0.55 |  98.08$\pm$0.15 | 100.00$\pm$0.00
4 |  |  99.25$\pm$0.16 |  99.38$\pm$0.17 |  97.42$\pm$0.62 |  97.99$\pm$0.51 |  95.81$\pm$0.87 | 100.00$\pm$0.00
5 |  |  99.00$\pm$0.00 |  99.11$\pm$0.38 |  89.53$\pm$0.17 |  96.46$\pm$0.60 |  96.27$\pm$1.32 | 99.11$\pm$1.04
6 |  |  99.91$\pm$0.12 |  99.55$\pm$0.05 |  100.00$\pm$0.00 |  98.21$\pm$0.19 |  97.40$\pm$1.02 | 100.00$\pm$0.00
7 |  |  99.88$\pm$0.15 |  99.50$\pm$0.11 |  99.72$\pm$0.27 |  97.95$\pm$0.24 |  96.49$\pm$1.56 | 100.00$\pm$0.00
8 |  |  84.77$\pm$0.24 |  77.78$\pm$11.49 |  97.68$\pm$0.43 |  69.88$\pm$5.89 |  91.18$\pm$2.96 | 99.19$\pm$1.04
9 |  |  99.56$\pm$0.10 |  99.75$\pm$0.05 |  99.42$\pm$0.01 |  97.22$\pm$0.97 |  98.28$\pm$0.79 | 100.00$\pm$0.00
10 |  |  96.23$\pm$0.63 |  97.57$\pm$0.05 |  95.84$\pm$0.49 |  89.95$\pm$2.41 |  96.62$\pm$1.69 | 97.67$\pm$1.93
11 |  |  97.81$\pm$1.24 |  97.66$\pm$0.50 |  99.08$\pm$1.94 |  96.90$\pm$1.66 |  97.68$\pm$2.04 | 99.55$\pm$0.63
12 |  |  97.41$\pm$0.61 |  98.69$\pm$0.12 |  100.00$\pm$0.00 |  98.44$\pm$0.96 |  97.39$\pm$2.57 | 100.00$\pm$0.00
13 |  |  99.31$\pm$0.60 |  98.61$\pm$0.68 |  100.00$\pm$0.00 |  96.73$\pm$1.02 |  95.91$\pm$3.19 | 100.00$\pm$0.00
14 |  |  97.41$\pm$0.64 |  97.86$\pm$0.93 |  95.72$\pm$4.65 |  94.67$\pm$1.87 |  96.23$\pm$0.89 | 98.97$\pm$0.00
15 |  |  71.80$\pm$0.81 |  76.68$\pm$5.87 |  75.94$\pm$9.67 |  69.56$\pm$3.58 | 94.06$\pm$2.09 |  91.79$\pm$1.22
16 |  | 99.21$\pm$0.21 |  94.47$\pm$0.18 |  93.69$\pm$3.79 |  95.81$\pm$1.03 |  95.58$\pm$1.45 |  94.32$\pm$8.00
OA |  |  92.63$\pm$1.71 |  91.03$\pm$3.10 |  94.75$\pm$1.66 |  87.40$\pm$1.24 |  96.52$\pm$2.81 | 98.30$\pm$0.22
AA |  |  95.90$\pm$0.03 |  95.88$\pm$1.09 |  96.22$\pm$1.33 |  93.14$\pm$0.10 |  96.52$\pm$0.92 | 99.02$\pm$0.24
Kappa |  |  91.80$\pm$0.20 |  90.05$\pm$3.39 |  94.13$\pm$1.86 |  86.12$\pm$2.01 |  95.27$\pm$0.96 | 98.10$\pm$0.25

We also apply BLGCN on the Houston University dataset to compare the
classification efficiency. As we can see from Table VIII, the algorithm
proposed in this paper greatly improves the classification accuracy, has
obvious advantages over other algorithms, and has high classification
performance on various sample classes. Besides, the comparison algorithms
based on GCN have the problem that the classification accuracies on the
majority samples classes such as Class 7 $\sim$ 12 are relatively low due to
the minor ratio of training set samples. Because of the strong generalization
ability in Bayesian layer, our model alleviates this problem successfully.

TABLE VIII:
CLASSIFICATION RESULTS ON HOUSTON UNIVERSITY DATASET

Class |  | SVM | 3D-CNN(0.3) | GCN |  S2GCN | MDGCN | BLGCN
---|---|---|---|---|---|---|---
1 |  |  95.80$\pm$0.60 |  98.88$\pm$0.44 |  81.07$\pm$0.00 |  95.41$\pm$2.03 |  92.72$\pm$0.97 | 99.58$\pm$0.70
2 |  |  97.41$\pm$0.51 |  98.84$\pm$0.38 | 100.00$\pm$0.00 |  97.66$\pm$1.08 |  92.97$\pm$0.78 |  99.34$\pm$0.06
3 |  |  99.25$\pm$0.48 |  99.42$\pm$0.44 |  74.89$\pm$2.21 |  97.97$\pm$1.35 |  97.39$\pm$0.60 | 100.00$\pm$0.00
4 |  |  96.83$\pm$0.75 |  99.30$\pm$0.35 |  89.24$\pm$0.00 |  96.78$\pm$1.14 |  94.87$\pm$1.05 | 100.00$\pm$0.00
5 |  |  95.44$\pm$1.41 |  99.38$\pm$0.29 |  92.34$\pm$3.97 |  96.76$\pm$0.97 |  98.27$\pm$0.32 | 99.97$\pm$0.06
6 |  |  96.16$\pm$2.59 |  96.44$\pm$2.26 |  86.90$\pm$0.00 |  95.95$\pm$1.45 |  92.58$\pm$1.14 | 98.23$\pm$1.71
7 |  |  83.21$\pm$1.84 |  93.26$\pm$0.89 |  51.54$\pm$1.43 |  82.71$\pm$3.54 |  87.01$\pm$2.28 | 97.91$\pm$0.96
8 |  |  72.74$\pm$1.72 |  94.56$\pm$2.89 |  90.73$\pm$0.00 |  75.44$\pm$4.11 |  79.83$\pm$4.07 | 94.82$\pm$1.71
9 |  |  76.51$\pm$1.21 |  90.02$\pm$0.76 |  32.06$\pm$7.07 |  81.41$\pm$2.69 |  88.96$\pm$0.19 | 92.97$\pm$2.56
10 |  |  81.32$\pm$1.40 |  91.90$\pm$0.47 |  65.99$\pm$2.96 |  86.05$\pm$2.01 |  89.38$\pm$1.70 | 99.83$\pm$0.07
11 |  |  81.74$\pm$1.26 |  92.92$\pm$0.86 |  62.56$\pm$2.42 |  87.75$\pm$2.00 |  86.07$\pm$1.64 | 98.62$\pm$0.92
12 |  |  69.19$\pm$1.82 |  91.24$\pm$2.09 |  70.63$\pm$3.68 |  77.91$\pm$4.68 |  88.76$\pm$2.97 | 99.69$\pm$0.30
13 |  |  48.51$\pm$6.95 |  98.94$\pm$4.48 |  0.00$\pm$0.00 |  74.93$\pm$4.26 |  92.08$\pm$1.02 | 96.49$\pm$1.29
14 |  |  96.25$\pm$1.47 |  98.94$\pm$0.34 |  98.88$\pm$2.51 |  98.53$\pm$0.44 |  98.69$\pm$0.39 | 100.00$\pm$0.00
15 |  |  98.08$\pm$0.51 |  98.80$\pm$0.48 |  72.44$\pm$4.62 |  97.13$\pm$0.68 |  95.55$\pm$0.44 | 100.00$\pm$0.00
OA |  |  85.67$\pm$0.43 |  95.14$\pm$0.51 |  72.93$\pm$0.44 |  88.49$\pm$1.57 |  90.71$\pm$1.00 | 98.36$\pm$0.32
AA |  |  86.28$\pm$0.96 |  95.60$\pm$0.71 |  70.44$\pm$0.54 |  89.50$\pm$1.04 |  91.68$\pm$1.07 | 98.30$\pm$0.40
Kappa |  |  84.71$\pm$0.51 |  94.74$\pm$0.53 |  71.73$\pm$0.47 |  87.62$\pm$1.46 |  90.02$\pm$0.89 | 98.23$\pm$0.35

###  IV-D Ablation Study

We design ablation experiments for multiple functional modules implemented in
BLGCN, and verify the necessity of each module in order to achieve high
accuracy in classification. It is also been validated that dynamic control
effects by processing the Indian Pines dataset.

####  IV-D1 Graph convolution operation

By using the method of graph convolution, our algorithm effectively extracts
the spatial adjacency relationship between the superpixels of hyperspectral
images, which greatly improves the classification efficiency. Among our
method, the key operation of extracting the spatial relationship is the
convolution operation of the feature matrix and the adjacency matrix.
Therefore, we verify the necessity of the graph convolution module by removing
the adjacency matrix convolution operation. The relevant comparison results
are listed in Table IX. We can find that after removing the graph convolution
operation, the overall classification results are greatly affected, and the
accuracy rates have dropped significantly.

####  IV-D2 Minority class data generation module

Facing the problem of low accuracy in minority class classification that often
occurs in practical classification tasks, this algorithm combines the
generative adversarial network method to achieve effective expansion of
minority classes. Therefore, we design a comparative experiment of removing
the minority generation module and directly input the original data into the
classification module. We list the results of the experiments in Table IX, in
which the classification accuracies of the minority class 1, 7, and class 9
have dropped significantly after the minority class data generation module is
removed, and they cannot be effectively classified at all.

####  IV-D3 Dynamic control training module guided by confidence intervals

Due to the uncertainty of the Bayesian neural network itself, we combine this
feature to quantify the output uncertainty by calculating the confidence
interval of the classification result, and dynamically control the training
process whether to proceed. We design a comparative experiment on this module
by removing the uncertainty calculation of the BLGCN classification results
and the two thresholds in the training process, and we plot the correspondence
between the accuracy of the validation set and the training batch in Fig. 8.
It is shown that when given a 95% confidence level, the model automatically
stops learning when the confidence interval reaches the second threshold,
which saves plenty of training time.

TABLE IX:
ABLATION STUDY RESULTS ON INDIAN PINES DATASET

Class |  | BLGCN without GCN | BLGCN without GAN | BLGCN
---|---|---|---|---
1 |  | 0.00$\pm$0.00 | 0.00$\pm$0.00 | 92.30$\pm$3.63
2 |  | 72.82$\pm$18.21 | 94.05$\pm$7.43 | 96.80$\pm$1.19
3 |  | 59.58$\pm$21.11 | 92.45$\pm$3.12 | 95.94$\pm$3.27
4 |  | 49.04$\pm$15.69 | 98.44$\pm$1.00 | 96.92$\pm$6.19
5 |  | 89.76$\pm$2.87 | 96.85$\pm$1.42 | 93.26$\pm$4.50
6 |  | 94.38$\pm$2.50 | 99.77$\pm$0.50 | 99.40$\pm$0.80
7 |  | 0.82$\pm$1.73 | 0.00$\pm$0.00 | 93.30$\pm$7.91
8 |  | 95.90$\pm$3.71 | 100.00$\pm$0.00 | 100.00$\pm$0.00
9 |  | 17.50$\pm$28.09 | 0.00$\pm$0.00 | 100.00$\pm$0.00
10 |  | 57.64$\pm$30.88 | 84.42$\pm$4.68 | 92.19$\pm$4.00
11 |  | 69.82$\pm$15.07 | 94.39$\pm$9.73 | 98.08$\pm$1.21
12 |  | 69.54$\pm$5.62 | 94.67$\pm$6.39 | 97.94$\pm$3.28
13 |  | 96.32$\pm$2.86 | 100.00$\pm$0.00 | 99.86$\pm$0.33
14 |  | 97.37$\pm$0.50 | 100.00$\pm$0.00 | 99.72$\pm$0.82
15 |  | 44.52$\pm$1.60 | 90.81$\pm$2.16 | 95.07$\pm$3.12
16 |  | 90.32$\pm$4.91 | 100.00$\pm$0.00 | 100.00$\pm$0.00
OA |  | 74.28$\pm$4.41 | 94.08$\pm$2.19 | 97.22$\pm$0.81
AA |  | 65.36$\pm$3.17 | 81.85$\pm$0.95 | 97.02$\pm$1.16
Kappa |  | 70.70$\pm$4.74 | 93.23$\pm$2.48 | 96.81$\pm$0.94

Figure 8: Ablation Study of Dynamic Control

##  V Conclusion

In this paper, we propose a novel BLGCN method for HSI classification. The
proposed Bayesian layer is applied to the GCN-based networks to quantify the
uncertainty of the output results. Since the distribution form is presented on
only a few parts of the network, the proposed method maintains high
classification accuracy and generalization ability. Meanwhile, newly designed
GANs on minority class data are trained to enlarge its capacity and solve the
sample imbalance problem in HSI dataset. To improve the training efficiency, a
dynamic control strategy is designed for an early termination of the training
process when the confidence interval reaches the threshold. The experimental
results on four open source HSI datasets demonstrate the superiority of our
proposed BLGCN. In addition, ablation studies are arranged to verify the
contribution of different modules including graph convolution operation, data
generation module and dynamic control strategy.

In the future, further research will be implemented on the various application
field of Bayesian layer, including image classification and graph node
prediction. Moreover, we will continue to improve the theoretical basis of
Bayesian layer and make it available to various neural networks.

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