Pulsar-candidate Selection Using a Generative Adversarial Network and ResNeXt

Radio pulsars, first discovered in 1967 (Hewish et al. 1968), are fast-spinning and highly magnetized compact neutron stars, capable of generating beamed radiation (Mitra 2017). The rotation period of a pulsar is equal to its pulse period, which is short and stable. The search for pulsars is crucial in physics and astronomy. Pulsars are formed via the core-collapse supernovae of massive stars, thus they can assist researchers in learning more about what happens when a star collapses. Moreover, the combination of the neutron stars' extreme compactness and their clock-like emission characteristics allow for unique tests of general relativity (Hulse & Taylor 1975; Lyne et al. 2004; Antoniadis et al. 2013). For studying matter at densities above nuclear saturation density, pulsars offer a distinctive environment (Backer et al. 1982; Raaijmakers et al. 2019, 2020). Pulsars also play an important role as interstellar-medium detectors (Armstrong et al. 1981).

A total of 3320 pulsars have been detected to date (Manchester et al. 2005), ³ with the majority discovered using modern radio telescopes. This includes the discovery of 201 pulsars with the Galactic Plane Pulsar Snapshot survey (Han et al. 2021) through an L-band 19-beam receiver of the Five-hundred-meter Aperture Spherical Telescope (FAST). The survey's goal is to discover pulsars within the Galactic latitude of 10° from the Galactic plane. Other modern pulsar surveys include the High-Time Resolution Universe (HTRU) survey (Sengar et al. 2022), the Green Bank North Celestial Cap survey (Lynch et al. 2021), and the Low-Frequency Array Tied-Array All-Sky Survey (Sanidas et al. 2019). Finally, the high-sensitivity radio searches of unassociated γ-ray sources has proven to be an effective way of finding new pulsars (Abdo et al. 2013; Wang et al. 2021).

Signals observed by radio telescopes are processed and folded into diagnostic plots, which are subsequently analyzed to identify pulsar candidates. Through drift scanning, modern radio observatories such as FAST (Jiang et al. 2018) generate more than a million pulsar-candidate samples per night (Wang et al. 2019). Besides, targeted searches looking for radio periodicity at a certain location or blind searches looking for pulses in a patch of the sky are also common pulsar-search techniques. However, due to radio-frequency interference (RFI) and noise, only a small percentage of these candidate samples are real pulsars. Experts need to perform further analysis to confirm whether these candidates are real pulsars. The process of manually screening pulsar candidates from these enormous samples is time consuming, hence a machine-based solution to the problem of pulsar-candidate identification is required.

In recent years, machine learning and artificial intelligence have gradually matured and been applied to pulsar-candidate identification. Lyon et al. (2016) designed eight statistical features for describing candidates and selected the Gaussian Hellinger Very Fast Decision Tree (GH-VFDT) machine-learning classifier. Eatough et al. (2010) used 12 candidate features extracted artificially to train single hidden-layer artificial neural networks (ANNs) for automatic pulsar-candidate identification. Bates et al. (2012) combined the feature extraction method of Eatough et al. (2010) to design 22 features as inputs of an ANN model for classification tasks. Morello et al. (2014) proposed the Straightforward Pulsar Identification using Neural Networks method. The proposed method used six empirical features and a larger training data set in order to achieve identification tasks with simpler rules compared to the methods of Eatough et al. (2010) and Bates et al. (2012) and improve the performance of machine-learning methods. As the capabilities of deep-learning models gradually became stronger, researchers began to use the network or the combination of the network and manual design to replace the method of manual extraction of a large number of complex features. For example, Zhu et al. (2014) directly used four pulsar diagnostic plots (frequency-phase plot (FPP), time-phase plot (TPP), summed-profile histogram (SPH), and dispersion measure (DM) curve, see Section 3.1 for details) as inputs of a pulsar image-based classification system (PICS). On the basis of PICS, Wang et al. (2019) used a convolutional neural network (CNN) as a feature extractor and used a residual model based on a graphics-processing unit for acceleration to perform a classification task. Lin et al. (2020b) proposed a multi-input CNN (MICNN). To obtain more information, the network structure combined the deep-image features extracted by a CNN from FPP and TPP with the statistical features, which include the mean, the standard deviation, the kurtosis, and the skewness from the SPH and DM curve designed manually.

However, in actual large-scale pulsar detection projects, the candidates that can be confirmed as real pulsars are far less common than the nonpulsar candidates (those that are RFI). For example, in the data set of HTRU Medlat (Morello et al. 2014), the ratio of positive to negative samples is 1:75. The model fitting is affected by this sample imbalance. Moreover, in the study of pulsar classification using currently available algorithms, Lyon et al. (2014) found that the imbalance problem caused the model to have a low recall for positive samples. There are two common methods to solve sample imbalance: oversampling (downsampling) and data enhancement. To oversample the minority-category data, Liu et al. (2021) copied the samples that were randomly extracted from the positive samples until the required number was obtained and designed a 14-layer deep residual network (ResNet) for pulsar-candidate identification. In 2002, Chawla et al. (2002) proposed the synthetic minority oversampling technique (SMOTE) algorithm, which was widely used in academia and industry (Wang et al. 2006; Jeatrakul et al. 2010; Ramentol et al. 2012; Prusty et al. 2017; Douzas & Bacao 2019; Maldonado et al. 2019), and achieved good results in different fields. Lin et al. (2020a) applied SMOTE to pulsar identification, effectively improving recall of the model compared with the approach without the oversampling technique. Wang et al. (2019) designed a simple and effective oversampling method: For the FPP or TPP of a confirmed pulsar, retain its maximum pulse. Randomly select three other positive samples, adding up their corresponding FPP or TPP with random coefficients into one plot. Then add this plot and the retained maximum pulse to a synthesized sample. Finally, the synthesized sample was used to train the CNN. Lin et al. (2020b) carried out data enhancement by transforming images and adding Gaussian noise (TIAGN), which helped to avoid the problem of model overfitting caused by simple oversampling in Lin et al. (2020a). For instance, in an FPP, bins of the same number are cut off from each subband and concatenated to the end of its corresponding subband to generate a new image. Finally, the Gaussian noise is added into the transformed image. In view of the excellent performance of the generative adversarial network (GAN) proposed by Goodfellow et al. (2016) in data enhancement (GANs are explained in Section 2.1 in detail), Guo et al. (2019) used a deep convolutional GAN (DCGAN) to solve the problem of insufficient positive samples and generated reliable positive pulsar samples through the adversarial learning between the generator and discriminator. Then the support vector machine (SVM) is adopted as the classifier for predicting pulsar candidates. In the practical scenario, there is typically a small amount of labeled data along with a large amount of unlabeled data. Semisupervised learning addresses this imbalance (Balakrishnan et al. 2021). Considering that a semisupervised generative adversarial network (SGAN) can learn from unlabeled data of pulsar surveys, Balakrishnan et al. (2021) employed an SGAN for pulsar-candidate identification with limited labeled data in the early stages of pulsar surveys on new instruments. The main advantage of the network is the capacity to leverage readily available unlabeled candidates for achieving excellent results.

In this paper, we introduce a framework for identifying pulsar candidates. To begin, this study uses the data enhancement method to enlarge the data set in order to balance the positive and negative examples. A DCGAN is utilized to create positive samples that look like real pulsar candidates which is based on Guo et al. (2019), and the generated positive samples are subsequently fused into the unbalanced data set. Second, based on the fact that ResNeXt can reduce network complexity while increasing network accuracy, this research offers a ResNeXt-based pulsar-candidate identification approach using a balanced data set which includes original data from the HTRU Medlat data set (Morello et al. 2014) and generated data from DCGAN. Experiments show that the suggested pulsar-candidate identification framework is effective.

This section delves into the details of the pulsar-candidate recognition framework proposed in this paper. How the DCGAN model generates positive pulsar images is discussed in Section 2.1 and the design of our ResNeXt structure is explained in Section 2.2. We first run two training sessions on the identical DCGAN structure to generate FPP and TPP. The original data set and the generated images are combined to create a balanced data set, which is then used to train ResNeXt. For both FPP and TPP, we separately train the identification model ResNeXt.

2.1. DCGAN-based Pulsar-candidate Data Enhancement

Goodfellow et al. (2016) introduced a deep-learning system called the GAN. Through an adversarial-learning process, the system captures the distribution of training data and generates new data from it. GAN is made up of two models: the generator and the discriminator. The generator's purpose is to generate data that is as close to the positive sample as possible, whereas the discriminator's goal is to properly assess whether the input data is real or generated. DCGAN (Radford et al. 2015) combines CNNs and the GAN to improve the GAN training stability and extract more broad and effective image features, which are commonly used in image-generating applications (Suarez et al. 2017; Lu et al. 2019; Wang & Liu 2020; Wu et al. 2020; Dewi et al. 2021).

Following the design requirements of the DCGAN (Radford et al. 2015), the discriminator in this paper is a network model containing a four-level convolution structure, which uses strided convolution to replace space pooling in a CNN. To increase model stability, batch normalization is applied to all layers except the input layer. The LeakyReLU activation function is used for all layers but the last fully connected layer. The specific discriminator structure is shown in Figure 1. The input of this model is a gray-scale pulsar image with the size of 48 × 48. After four convolution operations with the step size of 2 and the kernel size of 4 × 4, which is smaller than that in Guo et al. (2019) and reduces the number of parameters and computational complexity, the model outputs the predicted probability through the sigmoid neuron finally. After each convolution operation, the size of the corresponding feature map is the tensor of 24 × 24 × 64, 12 × 12 × 128, 6 × 6 × 256, and 3 × 3 × 512 respectively. The height and width are halved, and the number of channels is doubled.

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The generator designed in this paper is a network model with four layers of transpose convolution structure and applies batch normalization to all layers except the output layer. Except for the output layer which uses a tanh activation function, the other layers use ReLU as the activation function. The specific generator structure is shown in Figure 2. This model takes random noise with a uniform distribution in 1 × 100 dimensions as the input, and obtains a tensor with the size of 3 × 3 × 512 by adjusting dimensions. After four layers of a convolution kernel with the size of 4 × 4 and the step size of 2 using a transpose convolution operation, finally the generator outputs a gray image with a size of 48 × 48. After each transpose convolution operation, the size of the corresponding feature map is 6 × 6 × 256, 12 × 12 × 128, and 24 × 24 × 64, respectively. The height and width are doubled, and the number of channels is halved.

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The DCGAN model is executed based on the framework of Python 3.6 and Pytorch 1.10.0. In the training process, the model adopts a batch size of 32 and Adam Optimizer (Kingma & Ba 2014) with a learning rate of 0.0002 and β₁ = 0.5.

2.2. ResNeXt-based Pulsar-candidate Identification

To improve the accuracy of the model, classic methods employ ways to deepen or widen the network (Lecun et al. 1998; Krizhevsky et al. 2012; Simonyan & Zisserman 2014; Zeiler & Fergus 2014; Szegedy et al. 2014; Huang et al. 2016). The difficulty and computational overhead of network design grows as the number of hyperparameters (for example, the number of channels) increases. As a result, Xie et al. (2016) introduced the aggregated residual transformation (ResNeXt) for deep neural networks in order to improve network accuracy while maintaining (reducing) network complexity.

ResNeXt (Xie et al. 2016) is a hybrid of ResNet (He et al. 2015) and Inception (Szegedy et al. 2014). Residuals, which indicate the discrepancy between the predicted and observed values, are ResNet's primary contribution. The network degradation issue caused by the rise in network depth can be partially resolved by adding residual connections to the neural network because the input can be directly transferred from any low layer to a high layer. The gradient dispersion issue can be somewhat reduced in the case of back propagation since the gradient at the high layer can be directly transported to the low layer through residual connection without any intermediate weight matrix change. The main idea of Inception is grouping convolution. For example, in a classic Inception module, four sets of convolution operations are performed on one input, and the convolution kernels are 1 × 1, 3 × 3, 5 × 5, and 1 × 1 respectively. Inception enhances (reduces) feature dimensions with convolution operations with the kernel size of 1 × 1. Convolution kernels of a different size are used for grouping convolution to extract features at different scales. Finally, the results of the four groups of operations are added as output. ResNeXt differs from Inception in that it does not require the manual design of complicated Inception structural features; instead, each branch uses the same structures and residuals. Xie et al. (2016) proposed the following formula for the aggregation transformation which means adding up the results of an input that is processed by several sets of the same structure:

$$\begin{eqnarray}&&R(X)=\sum _{i=1}^{C}{T}_{i}(X).\end{eqnarray} \tag{ 1 }$$

Here X is the input of the neural network, T_i (X) is the neural network function such as a series of convolution operations, and C is the number of groups of the neural network, which is called cardinality. Every neural network function has the same structure, and the output is the sum of the results of all groups. Considering that the residual connection solves the problem of gradient disappearance in the deep neural network model, the residual connection is added to the formula of the aggregation transform, which is expressed as

$$\begin{eqnarray}&&R(X)=X+\sum _{i=1}^{C}{T}_{i}(X).\end{eqnarray} \tag{ 2 }$$

Figure 3 depicts the structure of the network model proposed in this paper, which is inspired by ResNeXt50 (Xie et al. 2016). The model's overall training process is depicted on the left side of the diagram, with the input being 48 × 48 gray-scale images of the pulsar candidates including FPP or TPP respectively (see Section 3.1 for details) and Conv being the convolution operation. The three sequentials have the same structure, and each one is made up of three consecutive BottleneckBlock residual blocks with a total of 64 groupings, as seen in the middle of Figure 3. The construction of BottleneckBlock residual blocks is shown on the right side of Figure 3. The extracted pulsar features enter the Adaptive_avg_pool2 layer after a series of group convolution processes, which seeks to minimize the dimensionality of the features while retaining the background information. The features pass through the Softmax layer to acquire the final output of the prediction probability. FC stands for the fully connected layer, and the features travel through the Softmax layer to obtain the final output of the prediction probability.

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The ResNeXt model proposed in this paper is executed based on the framework of Python 3.6 and Pytorch 1.10.0. In order to make the model converge, we adopt multistep learning-rate decay. Different decay rates are used in different epochs. The initial learning rate is 0.001 and the decay rate is 0.1. The batch size is set as 64. The weight decay of (L₂ regularization) 0.0004 is employed to avoid model overfitting.

3.1. Data Sets

In this paper, the HTRU Medlat data set is used for model training and testing. Morello et al. (2014) processed RFI cancellation, dispersion cancellation, time-series analysis, and candidate classification for periodic signals collected by radio telescopes to obtain a set of pulsar-candidate diagnostic plots with labels. The pulsar diagnostic plots include an FPP, TPP, SPH, and a DM curve. The TPP can be obtained by summating the data after time-series analysis on the frequency, while the FPP can be generated by summating the data after time-series analysis over the full observing interval. The presence of one or more vertical lines in the plot indicates the presence of a real pulsar (vertical lines in the frequency-versus-phase plot can also be generated by periodic broadband RFI). When different DM values are used for dedispersion, the DM curve displays how the signal-to-noise ratio of the pulse curve changes. The DM curve for actual pulsars usually peaks at a nonzero value. By folding and summing all frequency and time-domain data, the ideal pulsar-data summed-profile histogram displays one or more wave peaks. Meanwhile, RFI can produce multiple peaks in the profile histogram making it more difficult to detect pulsars with complex pulse profiles. The DM curve is typically employed as an extra indicator in these conditions because RFI would peak at (or close to) a DM of 0.

There are 1196 positive candidate examples, which are sampled from 521 distinct real pulsars labeled artificially based on comparison with known pulsars, and 89,996 negative candidate examples generated from RFI in the HTRU Medlat data set. Positive samples are numbered from 0 to 1195. Negative samples are numbered from 0 to 89,995. Diagnostic plots of a real pulsar candidate are shown in Figure 4, and diagnostic plots of a nonpulsar candidate are shown in Figure 5. The subbands plot (FPP) and subintegrations plot (TPP) are used for experimental analysis in this paper.

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3.2. Evaluation Metrics

The evaluation metrics used for the pulsar-candidate identification task include precision, recall, and F1-score. Table 1 shows the confusion matrix for the binary classification problem, including True Positive (TP, a real pulsar candidate and the prediction is positive), True Negative (TN, a nonpulsar candidate and the prediction is negative), False Positive (FP, a nonpulsar candidate and the prediction is positive) and False Negative (a real pulsar candidate and the prediction is negative). The precision, recall and F1-score are expressed as follows:

$$\begin{eqnarray}&&\mathrm{Precision}=\displaystyle \frac{{\rm{TP}}}{{\rm{TP}}+{\rm{FP}}},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\mathrm{Recall}=\displaystyle \frac{{\rm{TP}}}{{\rm{TP}}+{\rm{FN}}},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&\text{F1-score}\,=\,\displaystyle \frac{2\times {\rm{P}}{\rm{r}}{\rm{e}}{\rm{c}}{\rm{i}}{\rm{s}}{\rm{i}}{\rm{o}}{\rm{n}}\times {\rm{R}}{\rm{e}}{\rm{c}}{\rm{a}}{\rm{l}}{\rm{l}}}{{\rm{P}}{\rm{r}}{\rm{e}}{\rm{c}}{\rm{i}}{\rm{s}}{\rm{i}}{\rm{o}}{\rm{n}}+{\rm{R}}{\rm{e}}{\rm{c}}{\rm{a}}{\rm{l}}{\rm{l}}}.\end{eqnarray} \tag{ 5 }$$

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Different evaluation metrics have different meanings: precision denotes the proportion of true positive samples among all positive predictions, recall denotes the proportion of true correct predictions among all positive samples, and the harmonized average of the two is used as a comprehensive metric to consider the balance between them, that is, the F1-score. All three measures have a range of [0, 1], with a score closer to 1 indicating that the model is more effective.

3.3. Data Partitioning

Figure 6 shows six real pulsar candidates from the HTRU Medlat data set, all of which are 48 × 48 pixels wide. Figure 7 displays six positive pulsar sample images generated by DCGAN; the generated images are the same size as the original data. The comparison reveals that the DCGAN-generated positive sample images including FPP and TPP are similar to real positive images, with real pulsar-like features and decreased image noise. The inception distance (FID) is used to evaluate the performance of the DCGAN at image generation (Heusel et al. 2017), with a smaller number indicating that the generated positive samples are closer to real pulsar candidates. Through calculations, FID is 45.18 between two types of images. As a result, the created positive sample images can be utilized to supplement the original HTRU Medlat data set as recognition model input data.

Table 1. Confusion Matrix for Binary Classification Problems

	Predicted Class: Negative	Predicted Class: Positive
Actual Class: Negative	TN	FP
Actual Class: Positive	FN	TP

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In the experiments of this paper, the specific experimental data are divided as shown in Table 2. There are two parts of the experimental data set: training and testing (every data set includes FPP and TPP). In HTRU Medlat, there are 1,196 positive samples, and 500 of them are chosen at random as test data, and the remaining 696 are used as training data. There are 89,996 negative samples in total, with 500 being chosen at random as test data. The original data in HTRU Medlat for the positive and negative samples in training set 1 is 696 positive samples and 10,000 negative samples that are randomly chosen, with a positive to negative sample ratio of 7:100. In our work, we randomly select 10,000 samples instead of more for the experiment, which is enough to train the identification network (Du et al. 2018) and reduces computation and the requirement for computer hardware to make our work easier to follow. To ensure the reliability of the experiment, we repeat for different sets of 10,000 samples. The positive and negative samples in training set 2 are balanced because training set 2 includes the samples from training set 1 and adds 9304 examples created by the generative model to the positive samples. The experiments are evaluated using the same test set for both training sets.

Table 2. Partitioning of Data Sets Based on HTRU Medlat

Data Sets	Positive	Negative	Total
Original samples	1196	89,996	91,192
Training set 1	696	10,000	10,696
Training set 2	696 + 9304	10,000	20,000
Test set	500	500	1000

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3.4. Evaluation Results and Analysis

Table 4 displays the results of several methods for the pulsar-candidate identification on the HTRU Medlat data set, which contains subband and subintegration plots. Lyon et al. (2016) only use eight features based on the experience of experts and the training data is unbalanced, resulting in poor performance. Wang et al. (2019) employ an oversampling strategy to expand positive samples and feed them into DCNN-S for training, which increases the model's generalization ability and improves the recall by 3.4% over GH-VFDT. In the DCGAN-L2-SVM approach, DCGAN is used for training to create pulsar samples, and the discriminators in the trained DCGAN are utilized to extract pulsar features, before an SVM is used for classification. When compared to the GH-VFDT method, the F1-score of this method improves by 3.8% and 3.5%, respectively, indicating that the features extracted by the discriminator of the GAN are slightly more accurate than the statistical features and more consistent with the pulsar's latent feature distribution. Lin et al. (2020b) devised the TIAGN technique, and use four diagnostic plots of the pulsars as an input to the identification network, fully taking into account the features of the subband and subintegration plots, as well as the statistical features of the other two diagnostic plots, with 96.7% and 96.8% accuracy. To tackle the pulsar identification problem, Liu et al. (2021) use oversampling and a residual network, and the accuracy, recall, and F1-score all increase dramatically, with the recall rate reaching 100%, indicating that the residual structure has a stronger ability to fit the data. According to these findings, the pulsar-candidate recognition framework proposed in this paper has a superior classification performance, with all three evaluation metrics reaching 100%, indicating that the DCGAN model designed in this paper expands the data set with high quality and the designed ResNeXt structure has a satisfactory classification performance. Grouping convolutions in ResNeXt have strong and effective feature extraction capabilities and improve the accuracy of the network. Therefore ResNeXt extracts the latent features of the FPP and TPP effectively and utilizes them to identify the most promising candidates for follow-up without DM curve or pulse-profile information.

Identifying pulsar candidates based on the experience of experts usually requires some additional observations, which can be very time consuming. In this research, a methodology based on DCGAN and ResNeXt is proposed for automatically identifying pulsar candidates. DCGAN is used to generate real pulsar candidates (positive samples) since the number of positive samples in the pulsar-candidate identification task is substantially lower than the number of nonpulsar candidates (negative samples), which has a significant impact on the performance of the identification model. The converged model, which is trained on a data set of only positive samples, is used to generate a sequence of high-quality samples to increase the data set. Only the actual samples in the FPP and TPP are used for data production in the four diagnostic plots of the HTRU Medlat data set. After data augmentation, a balanced training set and an unbalanced training set are utilized to train the ResNeXt-based pulsar recognition model, and a test set of 500 positive and negative samples is used to evaluate the model. Experiments of our work and methods suggested by other studies are also listed to evaluate the performance of our framework. On the HTRU Medlat data set, the method described in this study is compared to methods proposed by researchers in recent years, and the trials show that the suggested framework gets the best results in three evaluation metrics: accuracy, recall, and F1-score.

The research work described in this paper was supported by the Joint Research Fund in Astronomy (U2031136) under cooperative agreement between the NSFC and CAS and the National Key Research and Development Program of China (No. 2018AAA0100203).