Graph-Based Few-Shot Learning for Synthetic Aperture Radar Automatic Target Recognition with Alternating Direction Method of Multipliers

Abstract

Synthetic aperture radar (SAR) automatic target recognition (ATR) underpins various remote sensing tasks, such as defense surveillance, environmental monitoring, and disaster management. However, the scarcity of annotated SAR data significantly limits the performance of conventional data-driven methods. To address this challenge, we propose a novel few-shot learning (FSL) framework: the alternating direction method of multipliers–graph convolutional network (ADMM-GCN) framework. ADMM-GCN integrates a GCN with ADMM to enhance SAR ATR under limited data conditions, effectively capturing both global and local structural information from SAR samples. Additionally, it leverages a mixed regularized loss to mitigate overfitting and employs an ADMM-based optimization strategy to improve training efficiency and model stability. Extensive experiments conducted on the Moving and Stationary Target Acquisition and Recognition (MSTAR) dataset demonstrate the superiority of ADMM-GCN, achieving an impressive accuracy of 92.18% on the challenging three-way 10-shot task and outperforming the benchmarks by 3.25%. Beyond SAR ATR, the proposed approach also advances FSL for real-world applications in remote sensing and geospatial analysis, where learning from scarce data is essential.

1. Introduction

Synthetic aperture radar (SAR) provides high-resolution imaging regardless of lighting or weather conditions [1], making it essential for defense surveillance, environmental monitoring, and disaster management [2,3,4,5]. However, achieving robust performance in these critical applications remains challenging due to the scarcity of annotated SAR data.

Historically, SAR ATR methods have relied on manually designed features, such as statistical, spectral, and structural descriptors, processed through complex signal processing frameworks [3,6,7,8,9]. Techniques such as template matching [10] and model-based approaches [11] have been instrumental in advancing the field. However, these methods depend heavily on handcrafted features, requiring significant human effort and limiting scalability. Additionally, their reliance on scenario-specific designs hinders generalization to diverse detection contexts [12,13].

Deep learning has demonstrated remarkable success in various tasks, including emotion recognition [14], healthcare applications [15], and intelligent surveillance [16]. These advancements highlight its strong capability in feature extraction and classification, offering valuable insights for SAR ATR. Deep learning has significantly improved SAR ATR performance by learning multi-level features from raw data [17,18,19]. However, its reliance on large, well-annotated datasets poses a limitation, as SAR data acquisition is costly and requires specialized expertise [20,21,22].

Few-shot learning (FSL), which allows efficient classification with a small number of labeled samples per class, presents a viable solution to the data scarcity issue [23,24]. This capability makes FSL a possible method for SAR ATR, where the specialized nature of SAR targets and the high cost of data collection often render large-scale datasets infeasible. Leveraging task-specific knowledge, FSL generalizes across new tasks, addressing the critical challenges of SAR ATR. Figure 1 provides an overview of the FSL paradigm, illustrating both training and test tasks. It illustrates a five-way one-shot FSL classification task, where the support set provides one labeled sample per class, while the query set consists of unlabeled samples for prediction. The training tasks (top of Figure 1) involve classes encountered during training. In these tasks, the model learns feature representations from the support set and uses them to classify the samples in the query set. The test tasks (bottom of Figure 1) contain novel classes that were not seen during training. The model generalizes to these unseen classes using the learned feature representations, demonstrating its ability to adapt to new SAR target categories.

To address data scarcity challenges, we propose the alternating direction method of multipliers–graph convolutional network (ADMM-GCN) framework, a novel framework for few-shot SAR ATR that integrates a GCN with the ADMM algorithm. By leveraging the representational capabilities of GCNs, ADMM-GCN effectively captures both global and local characteristics of SAR samples while mitigating overfitting through a mixed regularized loss function. Moreover, the ADMM algorithm is introduced to ensure efficient optimization, achieving consistent convergence even with limited data.

Figure 2 visualizes the ADMM-GCN framework, which processes SAR data through graph construction and iterative updates of node and edge features. In Figure 2, the workflow begins with SAR data represented as a graph, where samples correspond to nodes and their relationships to edges. Through iterative updates, node features are refined to encapsulate key characteristics of SAR targets, while edge features capture nuanced interactions between nodes. This iterative process, highlighted in the circular flow, continuously enhances feature representation and improves classification accuracy, making ADMM-GCN well suited for scenarios where data are scarce.

To summarize, this study’s main contributions to the literature are as follows:

• We propose an innovative framework termed ADMM-GCN for few-shot SAR ATR, which effectively combines global context with local feature analysis by constructing a relational graph among features, thereby enhancing the overall feature representation under limited-data scenarios.
• A mixed regularized loss function is designed to mitigate the common challenge of overfitting in FSL, enhancing the model’s stability and generalizability across diverse scenarios without relying on extensive data augmentation.
• The ADMM algorithm is integrated into few-shot SAR ATR to ensure consistent convergence to the global optimum while avoiding local optima, simplifying optimization by decomposing complex problems into tractable subproblems.
• Extensive experiments conducted on the Moving and Stationary Target Acquisition and Recognition (MSTAR) dataset verify the superiority of the proposed ADMM-GCN, achieving an impressive accuracy of 92.18% on the challenging three-way 10-shot task, outperforming the benchmarks by 3.25%.

The paper is structured as follows: Section 2 discusses related works. Section 3 explains the details of the ADMM-GCN framework. Section 4 presents the experimental validation, followed by a detailed analysis and discussions in Section 5. The paper concludes with Section 6.

2. Related Works

2.1. Few-Shot SAR Target Recognition

Few-shot SAR target recognition focuses on accurately identifying objects in SAR images with limited labeled samples. This subsection explores the development of this field through two main aspects: data-augmentation-based methods and model-optimization-based methods.

2.1.1. Data-Augmentation-Based Methods

Data augmentation enlarges the available dataset by modifying original SAR data to create synthetic samples resembling the real data, thereby enhancing model learning under data constraints. Common techniques in few-shot SAR target recognition include basic transformations such as rotation and scaling [4,25,26], as well as advanced approaches leveraging deep generative models to produce more complex variations [27,28]. For example, Ding et al. applied data augmentation strategies including target position shifts, speckle noise variations, and pose changes to improve model robustness [4]. Similarly, Song et al. proposed an adversarial autoencoder to generate SAR images with different target views, introducing valuable variability into the training data [27].

While data augmentation expands training sets, it often incurs high computational costs and requires expertise to ensure meaningful transformations. Additionally, deep generative models rely on large datasets, which is an obstacle to their effectiveness in FSL settings.

2.1.2. Model-Optimization-Based Methods

Three primary approaches can be distinguished among the current trends in model-optimization-based techniques for few-shot SAR target recognition: transfer learning [29,30], metric learning [31,32], and meta-learning [33].

Transfer learning enables pre-trained models to adapt to new tasks by transferring knowledge across domains. For instance, Rostami et al. [29] address SAR-EO domain transfer using a sliced Wasserstein distance (SWD)-based invariant embedding space, and Tai et al. [30] propose a few-shot transfer learning strategy incorporating selective feature transfer and a Bayesian CNN for SAR image classification; while transfer learning offers high efficiency and minimal fine-tuning, its effectiveness relies on similarity between the source and target domains. Significant domain shifts can degrade performance, requiring specialized adaptation strategies. Additionally, the availability of sufficiently large and well-curated source datasets remains a practical constraint.

Metric learning maps samples into an optimal metric space, emphasizing similarity-based representations. For instance, Wang et al. [31] introduced an attribute-guided multi-scale prototypical network with subband decomposition to enhance feature extraction from limited SAR data. Similarly, Ren et al. [32] proposed a transductive prototype reasoning approach that refines class prototypes, improving target identity reasoning with few labeled samples. However, metric learning often prioritizes local similarity patterns, potentially neglecting broader structural features that are critical for tasks such as object recognition. To improve SAR target classification, an effective balance between local and global feature representations is needed.

Meta-learning trains models to quickly adapt to new tasks, improving few-shot performance. However, it requires extensive meta-training and risks overfitting to small datasets. Meta-learning enhances model adaptability by leveraging cross-task learning, enabling efficient generalization from limited samples. For instance, Fu et al. [33] proposed MSAR, a meta-learning framework that optimizes initialization and update strategies for better adaptation to new tasks. However, meta-learning involves high computational complexity and is prone to overfitting, particularly in data-scarce scenarios.

In general, although prior research has significantly advanced the field of SAR target recognition under few-shot scenarios, challenges such as model complexity, susceptibility to overfitting, and effective handling of highly limited data remain unresolved, warranting further exploration.

2.2. Alternating Direction Method of Multipliers

Gradient-based optimization techniques, such as Adam [34] and stochastic gradient descent (SGD) [35], are widely used in deep learning. However, these methods face several limitations:

• They often converge to local optima, making it difficult to reach the global optimum and hindering the overall training process;
• Their effectiveness is highly sensitive to input data quality, requiring meticulous preprocessing to ensure convergence, which complicates training and affects model performance.

These challenges become even more pronounced in FSL, where the extremely limited number of labeled samples exacerbates issues such as overfitting and unstable convergence. Additionally, gradient-based optimizers rely on well-defined loss landscapes and proper initialization, making them vulnerable to poor convergence in low-data regimes.

To address these challenges, alternative optimization approaches have been explored. One such alternative is the alternating direction method of multipliers (ADMM) [36], which has shown significant promise for constrained and structured optimization problems. ADMM has been widely applied in deep learning due to its advantages [37]:

• ADMM decomposes the optimization problem into smaller, more manageable subproblems, each of which can be solved optimally with theoretical guarantees of convergence. This decomposition is particularly beneficial in FSL, where limited data necessitates a stable and structured training process.
• Unlike gradient-based methods, ADMM is inherently robust to parameter initialization, ensuring stable convergence even when training data are scarce.
• By introducing an auxiliary variable, ADMM enforces constraints during optimization, which not only stabilizes training but also enhances generalization, making it well suited for FSL applications.

In few-shot SAR target recognition, these properties make ADMM particularly advantageous. First, the ability of ADMM to break down complex optimization problems helps mitigate the impact of gradient vanishing, a common issue in low-data regimes. Additionally, its built-in regularization mechanisms help prevent overfitting, which is crucial when training data are extremely limited.

Several prior studies have highlighted the effectiveness of ADMM in handling complex regularization functions and improving generalization performance in deep learning applications [38,39]. These works demonstrate that ADMM achieves superior convergence properties compared to standard gradient-based optimizers such as Adam, particularly in scenarios where optimization constraints and structured learning play a crucial role.

Despite its demonstrated success in various deep learning tasks, ADMM remains underexplored in few-shot SAR target recognition. This presents an opportunity to investigate its potential to improve model interpretability, generalization, and stability. The proposed approach aims to bridge this gap by leveraging the properties of ADMM to enhance few-shot SAR target recognition performance.

3. Methodology

3.1. Framework of ADMM-GCN

Figure 3 presents the overall architecture of the ADMM-GCN framework, which consists of three main components: the embedding module (EM), the graph convolutional module (GCM), and the ADMM optimization process.

The EM extracts initial feature representations from SAR images, providing a foundational understanding of the target characteristics. These features are subsequently processed by the GCM, which encodes and propagates global contextual dependencies using graph-based algorithms. By aggregating and refining features, this module enables the model to effectively capture intricate patterns and interdependencies, which are essential for accurate target classification in FSL scenarios.

Furthermore, the ADMM optimization process is integrated into the framework to iteratively refine the model parameters by decomposing the original optimization problem into a sequence of subproblems. In this process, ADMM follows three key steps: updating the primal variable $\theta$ (network parameters) via Equation (18a), updating the auxiliary variable z via Equation (18b), and updating the dual variable $\alpha$ via Equation (18c). Figure 3 visually illustrates these updates, demonstrating the interaction between ADMM, the GCM, and the EM. The detailed ADMM methodology and optimization steps are provided in Section 3.4, while further descriptions of each component and their integration within the framework are discussed in subsequent sections.

3.2. Network Architecture

3.2.1. Embedding Module

The EM in the ADMM-GCN framework acts as the feature extractor, transforming SAR images into latent feature representations, as illustrated in Figure 4. This module is a critical preparatory stage, generating feature representations that serve as input for the subsequent GCM, enabling effective contextual information propagation and classification.

For the ith sample $X_i$, the associated embedding vector ${\mathrm{\Xi }}_{\mathit{i}}$ can be expressed as

$${\mathrm{\Xi }}_{\mathit{i}}=f_{\theta }\left(X_i\right)$$

(1)

where $f_{\theta }:{\mathbb{R}}^{H\times W}\to {\mathbb{R}}^P$ is the embedding function, $\theta$ represents the learnable parameters, H and W indicate the dimensions of the input image, and P specifies the embedding vector’s dimensionality.

Starting with a 3 × 3 convolutional layer, the EM extracts fundamental features from the SAR input. Subsequently, batch normalization ensures stability, while the ReLU activation function introduces non-linearity, enabling effective feature extraction. A maxpooling layer then downsamples the feature map, highlighting dominant features while reducing computational complexity.

As the network progresses, two additional convolutional layers with filter sizes increasing from 32 to 128 are employed to capture intricate features. These layers, paired with batch normalization and ReLU activation, ensure high-quality feature representation. Intermittent maxpooling layers further abstract the feature maps, reducing their size while preserving critical information. In the final stage, a 12 × 12 convolutional kernel operates on a 128-channel feature map, synthesizing high-level features into a 64-channel output. This design balances computational efficiency with analytical depth, enabling the module to generate rich, compact feature representations.

The extracted features are seamlessly relayed to the GCM, where the GCN interprets the data structure and relationships for higher-level contextual analysis. This integrated workflow transforms localized features into a global perspective, enabling comprehensive interpretation and analysis in FSL scenarios.

3.2.2. Graph Convolutional Module

GCNs [40] stand at the forefront of graph-based deep learning, building upon the core concepts of CNNs. Mimicking CNNs, modern GCNs learn the common local and global structural patterns of graphs through designed convolution functions [41]. The embedding of a node is generated by collecting features from its neighbors and then enhancing this information through a series of linear transformations and non-linear activations.

Figure 5 illustrates feature aggregation and the edge information interaction within a GCN. Initially, edges AB and CD, represented as dashed lines, are not directly connected. Through the graph convolution process and successive feature aggregation, nodes progressively update their features, exemplified by the path $A\to B\to C\to D$, and then edge AB and edge CD have feature fusion as well as interaction through edge BC. Blue nodes signify the initial stages of feature aggregation, while red nodes represent the outcome after several iterations of feature aggregation and edge information interaction.

As the process unfolds across layers, the resulting embeddings reflect not only individual node characteristics but also their interconnections within the graph. Such depth of information, combined with the network’s end-to-end trainability, is essential for effectively tackling few-shot SAR target recognition tasks. The intricate interconnectedness and information exchange between nodes as well as edges underscore the effectiveness of GCNs in capturing the relational inferences crucial for advanced recognition tasks.

Let $\mathcal{G}=(\mathcal{V},\mathcal{E})$ represent a graph, where $\mathcal{V}$ is a collection of N nodes and $\mathcal{E}$ defines the edges connecting these nodes. In this undirected graph, a node $v_i\in \mathcal{V}$ is linked to another node $v_j$ via an edge $e_{i,j}=\{v_i,v_j\}\in \mathcal{E}$, which reflects a similarity relationship between the two. The topology of the graph is encoded in the adjacency matrix $\mathit{A}\in {\mathbb{R}}^{N\times N}$, where $a_{i,j}=e_{i,j}$.

Each node $v_i$ is assigned a feature vector ${\mathit{x}}_{\mathit{i}}$, and all node features are stored in the matrix $\mathit{X}\in {\mathbb{R}}^{N\times M}$, where M indicates the dimension of the feature space. The similarity $e_{i,j}$ between nodes i and j is determined using a function $f_d$, expressed as

$$e_{i,j}=f_d({\mathit{x}}_{\mathit{i}},{\mathit{x}}_{\mathit{j}})=MLP\left(|{\mathit{x}}_{\mathit{i}}-{\mathit{x}}_{\mathit{j}}|\right),$$

where $f_d$ measures the similarity by leveraging a multilayer perceptron (MLP) trained on the absolute difference between the feature vectors of nodes $v_i$ and $v_j$.

Figure 6 illustrates a graphical representation of data characterized by both node and edge features. Nodes $v_i$ in the graph are each associated with an M-dimensional feature vector, represented as $[x_1,x_2,\dots ,x_M]$. Edges $e_{i,j}$ between nodes $v_i$ and $v_j$ are represented by their own attribute vectors, capturing the similarities as well as the interactions between the connected nodes.

GCNs extend traditional CNNs to the graph domain. Utilizing the simplification introduced in [40], the convolution operation in a GCN is defined as

$${\mathit{g}}_{\upsilon }★\mathit{s}=\upsilon \left(\mathit{I}+{\mathit{D}}^{-{\displaystyle \frac{1}{2}}}\mathit{A}{\mathit{D}}^{-{\displaystyle \frac{1}{2}}}\right)\mathit{s},$$

(2)

where $\mathit{s}$ denotes the input signal, ${\mathit{g}}_{\upsilon }$ the spectral filter, $\mathit{I}$ the identity matrix, ★ the convolution operator, and $\upsilon$ the Chebyshev coefficient.

The degree matrix $\mathit{D}$ of $\mathit{A}$ is defined as

$$\mathit{D}=\mathrm{diag}(d_1,d_2,\dots ,d_n),$$

(3)

where each $d_i={\sum }_j{a}_{ij}$ represents the degree of vertex i.

Using the normalization approach introduced by Kipf and Welling [40], the convolution matrix is transformed as

$$\mathit{I}+{\mathit{D}}^{-{\displaystyle \frac{1}{2}}}\mathit{A}{\mathit{D}}^{-{\displaystyle \frac{1}{2}}}\to {\tilde{\mathit{D}}}^{-{\displaystyle \frac{1}{2}}}\tilde{\mathit{A}}{\tilde{\mathit{D}}}^{-{\displaystyle \frac{1}{2}}},$$

(4)

where $\tilde{\mathit{A}}=\mathit{A}+\mathit{I}$ and $\tilde{\mathit{D}}={\sum }_j{\tilde{\mathit{A}}}_{ij}$.

Extending the convolution definition to cater to a signal with M input channels, that is, $\mathit{X}\in {\mathbb{R}}^{N\times M}$ where each vertex is affiliated with an M-dimensional feature vector, the propagation rule for this model is given by:

$${\mathit{H}}^{(l+1)}=\sigma \left({\tilde{\mathit{D}}}^{-{\displaystyle \frac{1}{2}}}\tilde{\mathit{A}}{\tilde{\mathit{D}}}^{-{\displaystyle \frac{1}{2}}}{\mathit{H}}^{\left(l\right)}{\mathrm{\Theta }}^{\left(l\right)}\right),$$

(5)

The activation function $\sigma$ used in the proposed GCN module is the ReLU function, formulated as $\mathrm{ReLU}(·)=max(0,·)$. Considering the balance between model complexity and computational efficiency, a three-layer architecture is adopted. This configuration ensures efficient data processing while retaining key data features. The output $\mathit{Z}$ of the third layer is derived using the established propagation rule:

$$\mathit{Z}={\mathit{H}}^{\left(3\right)}=\widehat{\mathit{A}}\mathrm{ReLU}\left(\widehat{\mathit{A}}\mathrm{ReLU}\left(\widehat{\mathit{A}}\mathit{X}{\mathrm{\Theta }}^{\left(0\right)}\right){\mathrm{\Theta }}^{\left(1\right)}\right){\mathrm{\Theta }}^{\left(2\right)},$$

(6)

where $\widehat{\mathit{A}}$ represents the normalized adjacency matrix, defined as

$$\widehat{\mathit{A}}={\tilde{\mathit{D}}}^{-{\displaystyle \frac{1}{2}}}\tilde{\mathit{A}}{\tilde{\mathit{D}}}^{-{\displaystyle \frac{1}{2}}}.$$

(7)

To accentuate intra-class similarity while maintaining clear inter-class divergence, this method computes cluster centers for each category. This approach highlights the nuances of each category, ensuring that similar instances within a category are closely aligned, while distinctly separating different categories. The cluster center ${\mathit{c}}_{\mathit{n}}$ for the nth category is the mean of the embedding vectors in its support set:

$${\mathit{c}}_{\mathit{n}}={\displaystyle \frac{1}{K}}\sum _{{\mathit{z}}_{\mathit{i}}\in {\mathcal{S}}^n}{\mathit{z}}_{\mathit{i}},$$

(8)

where K denotes the shot number, ${\mathcal{S}}^n$ the support set for the nth category, and ${\mathit{z}}_{\mathit{i}}$ the corresponding feature vector.

Subsequently, the feature output of a query sample ${\mathit{z}}_{\mathit{i}}$ is processed through a $Softmax$ classifier to determine the class probability distribution. The predicted label $\widehat{y_i}$ for the query sample is calculated using the $Softmax$ function:

$$p(\widehat{y_i}=n|{\mathit{z}}_{\mathit{i}})={\displaystyle \frac{e^{-S({\mathit{c}}_{\mathit{n}},{\mathit{z}}_{\mathit{i}})}}{{\sum }_{j=1}^N{e}^{-S({\mathit{c}}_{\mathit{n}},{\mathit{z}}_{\mathit{i}})}}},$$

(9)

where S is a metric function adhering to the cosine similarity, defined as

$$S({\mathit{c}}_{\mathit{n}},{\mathit{z}}_{\mathit{i}})={\displaystyle \frac{{\mathit{c}}_{\mathit{n}}·{\mathit{z}}_{\mathit{i}}}{\parallel {\mathit{c}}_{\mathit{n}}\parallel \times \parallel {\mathit{z}}_{\mathit{i}}\parallel }}.$$

(10)

3.3. Construction of Regularized Mixed Loss

To address the prevalent challenge of overfitting in FSL, a specialized regularized mixed loss function has been designed within the ADMM-GCN framework $\mathcal{F}$. This method not only streamlines the model architecture but also alleviates overfitting, thereby improving both stability and generalizability. The neural network’s output in the ADMM-GCN framework is formulated as

$$y_i={\mathcal{F}}_{\theta }^i\left(x\right),$$

(11)

where $y_i$ represents the ith output, ${\mathcal{F}}_{\theta }^i\left(x\right)$ represents the function corresponding to the ith output’s response to input x, and $\theta$ symbolizes the weight parameters.

The cross-entropy loss $L\left(\theta \right)$ is defined as

$$L\left(\theta \right)=-\sum \widehat{y_i}log{y}_i=-\sum \widehat{y_i}log{f}_{\theta }^i\left(x\right),$$

(12)

where $\widehat{y_i}$ is the actual label and $y_i$ denotes the neural network’s output.

In ADMM-GCN, a loss function has been devised that extends the original loss function by incorporating a regularization term. For parameters $\theta$, the expression for the $L_2$ regularization term is formulated as

$$R\left(\theta \right)={\displaystyle \frac{\lambda }{2}}{\parallel \theta \parallel }_2={\displaystyle \frac{\lambda }{2}}\sqrt{{\theta }_1^2+{\theta }_2^2+\dots +{\theta }_n^2},$$

(13)

where $\lambda$ represents the regularization parameter that balances the strength of the regularization, while ${\parallel \theta \parallel }_2$ denotes the $L_2$ norm of the weight parameter.

Consequently, the mixed regularized loss is given as

$$\begin{array}{cc}\hfill L_R& =L\left(\theta \right)+R\left(\theta \right)=-\sum \widehat{y_i}log{y}_i+{\displaystyle \frac{\lambda }{2}}{\parallel \theta \parallel }_2=-\sum \widehat{y_i}log{f}_{\theta }^i\left(x\right)+{\displaystyle \frac{\lambda }{2}}{\parallel \theta \parallel }_2.\hfill \end{array}$$

(14)

3.4. ADMM Optimizer

Having established the loss function, we now shift our attention to its optimization. The steps of the ADMM algorithm for regularized loss optimization are summarized in Algorithm 1.

Consider the following minimization problem:

$$\underset{\theta }{\arg \min }-\sum \widehat{y_i}log{f}_{\theta }^i\left(x\right)+{\displaystyle \frac{\lambda }{2}}{\parallel \theta \parallel }_2.$$

(15)

Following this, this method examines the constrained version, defined as follows:

$$\begin{array}{cc}\hfill \underset{\theta ,z}{\arg \min }& -\sum \widehat{y_i}log{f}_{\theta }^i\left(x\right)+{\displaystyle \frac{\lambda }{2}}{\parallel z\parallel }_2\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& z=\theta \hfill \end{array}$$

(16)

According to the typical neural network framework, an iterative procedure solution is typically required to solve (15). The ADMM technique, gaining significant attention for its efficacy in non-convex deep learning frameworks [42,43,44], is therefore used in an attempt to address the minimization problem (16). Therefore, the augmented Lagrangian function for the problem is given as

$$\begin{array}{cc}\hfill \mathcal{L}=& -\sum \widehat{y_i}log{f}_{\theta }^i\left(x\right)+{\displaystyle \frac{\lambda }{2}}{\parallel \theta \parallel }_2+\langle \alpha ,\theta -z\rangle +{\displaystyle \frac{\rho }{2}}{\parallel \theta -z\parallel }_2^2,\hfill \end{array}$$

(17)

where $\alpha$ denotes the Lagrangian multiplier corresponding to the constraint $z=\theta$, while $\rho >0$ is the penalty parameter.

Following the ADMM paradigm, the optimal point is sought by maximizing over the dual variable $\alpha$ along with minimizing over the primal variables $\theta$ and z, respectively. The ADMM algorithm proceeds by alternately updating $\{\theta ,z,\alpha \}$ through the resolution of the following subproblems:

$$\{\begin{array}{}\mathrm{(18a)}& \underset{\mathit{\theta }}{\arg \min }-\sum \widehat{y_i}log{f}_{\mathit{\theta }}^i\left(x\right)+{\displaystyle \frac{\rho }{2}}{\parallel \mathit{\theta }-z+{\displaystyle \frac{\mathit{\alpha }}{\mathit{\rho }}}\parallel }_2^2\hfill \mathrm{(18b)}& \underset{z}{\arg \min }-\sum \widehat{y_i}log{f}_{\mathit{\theta }}^i\left(x\right)+{\displaystyle \frac{\lambda }{2}}{\parallel z\parallel }_2\hfill \mathrm{(18c)}& {\alpha }_{t+1}\leftarrow {\alpha }_t+\rho (\mathit{\theta }-z)\hfill \end{array}$$

An approximate solution to the first problem (18a) can be derived by utilizing a gradient-based strategy over a specified number of iterations. The Adam iterative approach [34] is adopted for this purpose. Additionally, PyTorch’s automatic differentiation functionality could be leveraged for variable-related numerical gradient computations [45].

The second problem, (18b), can be tackled in a manner akin to that of [46], and the exact solution of z is formulated as

$$z=Soft(\theta +{\displaystyle \frac{\alpha }{\rho }}),$$

(19)

where $Soft\left(\right)$ denotes the soft threshold operator. The corresponding formulation for $Soft\left(\right)$ is

$$Sof{t}_{{\displaystyle \frac{\mu }{2}}}\left(y\right)=\underset{x}{\arg \min }{(y-x)}^2+\mu \left|x\right|.$$

(20)

Algorithm 1 ADMM Algorithm for Optimization with $L_2$ Regularization

1: Input: $\lambda$; $\rho$; initial values of $\theta$, z, and $\alpha$ 2: Output: Optimal $\theta$ and z 3: repeat 4:    Update $\theta$ by solving $$\begin{array}{cc}\hfill \theta & \leftarrow \underset{\theta }{\arg \min }-\sum \widehat{y_i}log{f}_{\theta }^i\left(x\right)+{\displaystyle \frac{\rho }{2}}{\parallel \theta -z+{\displaystyle \frac{\alpha }{\rho }}\parallel }_2^2\hfill \end{array}$$ 5:    Update z by solving $$\begin{array}{cc}\hfill z& \leftarrow \underset{z}{\arg \min }-\sum \widehat{y_i}log{f}_{\theta }^i\left(x\right)+{\displaystyle \frac{\lambda }{2}}{\parallel z\parallel }_2\hfill \end{array}$$ 6:    Update Lagrangian multiplier $\alpha$: $$\begin{array}{cc}\hfill \alpha & \leftarrow \alpha +\rho (\theta -z)\hfill \end{array}$$ 7: until Convergence criteria are met

4. Experiment

4.1. Dataset

The Moving and Stationary Target Acquisition and Recognition (MSTAR) dataset [47] serves as a widely recognized benchmark in the field of SAR image analysis and target recognition. This dataset features X-band imaging radar operating in HH polarization, producing images with dimensions of 128 × 128 pixels and a spatial resolution of 0.3 × 0.3 m, including ten distinct military vehicle targets. In this study, the training and testing sets are constructed using targets imaged at depression angles of 17° and 15°, respectively, following standard experimental protocols.

Table 1. Numbers of samples for each target category in the MSTAR dataset.

Target Category	BRDM2	BMP2	BTR60	BTR70	D7	T62	T72	ZIL131	ZSU234	2S1
Number	572	428	451	429	573	572	428	573	573	573

summarizes the number of samples for each target category. Figure 7 illustrates optical and SAR image samples of these military vehicles within the MSTAR dataset.

To create the training task set ${\mathcal{T}}_{train}$ and the test task set ${\mathcal{T}}_{test}$, the initial dataset is partitioned into two non-overlapping subsets: the training set ${\mathcal{D}}_{tr}$ and the test set ${\mathcal{D}}_{te}$, such that ${\mathcal{D}}_{tr}\cap {\mathcal{D}}_{te}=\varnothing$. To ensure methodological rigor and experimental consistency, the categories in the ${\mathcal{D}}_{tr}$ and ${\mathcal{D}}_{te}$ remain fixed throughout the experiment.

For a five-way classification task, the test set ${\mathcal{D}}_{te}$ includes the following categories: BTR60, BRDM2, T72, 2S1, and D7. The training set ${\mathcal{D}}_{tr}$ consists of the remaining five categories. In a three-way classification scenario, the test set ${\mathcal{D}}_{te}$ comprises BTR60, BRDM2, and T72, while the training set ${\mathcal{D}}_{tr}$ includes the remaining seven categories.

Table 2. Categorization of the MSTAR dataset into disjoint training (${\mathcal{D}}_{tr}$) and test (${\mathcal{D}}_{te}$) datasets under different N-way K-shot experimental settings.

N-Way K-Shot Task	${\mathcal{D}}_{\mathit{tr}}$ Categories	${\mathcal{D}}_{\mathit{te}}$ Categories
5-way K-shot	ZIL131, BMP2, T62, BTR70, ZSU234	BTR60, BRDM2, T72, 2S1, D7
3-way K-shot	D7, T62, 2S1, ZIL131, BMP2, ZSU234, BTR70	BTR60, BRDM2, T72

summarizes the dataset categorization under different experimental configurations.

4.2. N-Way K-Shot Task

In FSL, the concept of N-way K-shot is essential for assessing a model’s ability to generalize and adapt to novel target classes with limited labeled samples. Here, N-way specifies the amount of target classes, while K-shot indicates the amount of labeled samples available for each category.

In this study, we consider two main experimental configurations: three-way K-shot tasks and five-way K-shot tasks. For each configuration, the model’s performance is systematically evaluated under varying K-shot settings, including one-, five-, and ten-shot. The N-way configurations involve either three or five distinct target classes, and the K-shot settings determine the training samples per category. These diverse experimental setups provide a comprehensive evaluation of the proposed method.

4.3. Implementation Details

To achieve the experimental objectives, hyperparameters have been carefully selected. Specifically, a batch size of eight is used, with a learning rate of 0.01 for five-way K-shot tasks and 0.001 for three-way classification tasks. To prevent overfitting, the training process is capped at a maximum of 35,000 iterations. Additionally, an early-stopping mechanism is employed, which halts training if the loss function does not decrease for 20 consecutive iterations, thereby mitigating the risk of overfitting. The experiments are implemented using the PyTorch framework (version 2.4.1) with Python 3.8.

In FSL, the episode-based training method is widely employed, as extensively discussed in prior works [48,49,50]. This approach simulates the testing phase during training by aligning the configurations of training tasks ${\mathcal{T}}_{train}$ and test tasks ${\mathcal{T}}_{test}$, both adhering to specific N-way K-shot setups. The detailed procedure for episode training in ADMM-GCN is presented in Algorithm 2.

Algorithm 2 Episode Training for ADMM-GCN

1: Input: Training dataset ${\mathcal{D}}_{train}$, Number of ways N, Number of shots K, Number of episodes P  2: Output: Trained model  3: for $p\leftarrow 1$ to P do  4:    Randomly select N categories from ${\mathcal{D}}_{\mathrm{train}}$  5:    Initialize empty support set $\mathcal{S}$ and query set $\mathcal{Q}$  6:    for each category C in the selected N categories do  7:      Randomly select K samples from C and add to $\mathcal{S}$  8:      Randomly select 1 sample from C and add to $\mathcal{Q}$  9:    end for 10:    while not converged do 11:      Train the model using the support set $\mathcal{S}$ 12:      Compute the loss $\mathcal{L}$ on $\mathcal{S}$ 13:      Backpropagate gradients and update model parameters $\mathrm{\Theta }$ 14:      Evaluate the model on the query set $\mathcal{Q}$ 15:      Check convergence criteria 16:    end while 17: end for 18: Return: Trained model

The fundamental principle of episode training lies in organizing training data into discrete episodes, each consisting of two key components: a support set $\mathcal{S}$ and a query set $\mathcal{Q}$. The support set $\mathcal{S}$ contains a limited number of classes and serves as the foundation for model learning, while the query set $\mathcal{Q}$ includes samples from the same classes and acts as the evaluation set. To construct ${\mathcal{T}}_{train}$, N random categories are first selected from the training set ${\mathcal{D}}_{tr}$. Then, K samples are randomly drawn from each of these categories to form the support set $\mathcal{S}$. Additionally, one sample from the remaining data in each category is randomly chosen to form the query set $\mathcal{Q}$. This process is repeated P times, resulting in P distinct tasks within the training set ${\mathcal{T}}_{train}$.

5. Discussion and Analysis

5.1. Comparison Experiments

The experimental results of ADMM-GCN in comparison with various FSL methods on the MSTAR dataset under different N-way K-shot settings are summarized in

Table 3. Accuracy (%) results of ADMM-GCN compared with other FSL methods on the MSTAR dataset across different N-way K-shot settings, with 95% confidence intervals.

Methods	3-Way			5-Way
	1-Shot	5-Shot	10-Shot	1-Shot	5-Shot	10-Shot
ProtoNet [50]	71.24 ± 0.45	80.79 ± 0.38	82.37 ± 0.33	50.42 ± 0.89	63.74 ± 0.78	67.95 ± 0.70
RelationNet [51]	75.32 ± 0.49	84.29 ± 0.43	86.76 ± 0.35	53.81 ± 0.91	66.52 ± 0.84	72.20 ± 0.62
TPN [52]	80.45 ± 0.48	87.32 ± 0.46	88.93 ± 0.42	57.44 ± 0.92	65.70 ± 0.83	70.37 ± 0.73
MSAR [33]	69.23 ± 0.51	84.71 ± 0.59	87.96 ± 0.49	53.50 ± 1.00	60.50 ± 0.90	64.72 ± 0.71
DeepEMD [53]	76.01 ± 0.42	83.23 ± 0.39	86.24 ± 0.34	55.61 ± 0.82	65.17 ± 0.75	69.66 ± 0.60
BSCapNet [54]	73.01 ± 0.47	86.62 ± 0.48	84.60 ± 0.42	64.81 ± 0.87	67.50 ± 0.79	73.55 ± 0.56
ADMM-GCN (ours)	84.31 ± 0.39	89.70 ± 0.39	92.18 ± 0.38	61.79 ± 0.56	68.75 ± 0.52	74.01 ± 0.53

. To ensure a comprehensive and fair evaluation, we incorporate widely adopted methods in FSL and SAR ATR, including ProtoNet [50], RelationNet [51], TPN [52], MSAR [33], DeepEMD [53], and BSCapNet [54].

In the three-way classification task, ADMM-GCN achieves the highest accuracy across all K-shot settings. Specifically, in the one-shot scenario, ADMM-GCN attains an accuracy of 84.31% ± 0.39, significantly outperforming ProtoNet (71.24% ± 0.45) and RelationNet (75.32% ± 0.49). Furthermore, ADMM-GCN surpasses advanced methods such as DeepEMD (76.01% ± 0.42) and BSCapNet (73.01% ± 0.47), demonstrating its superior feature extraction and classification capability under extreme sample scarcity. As the number of shots increases, all methods exhibit improved classification accuracy. In the 10-shot setting, ADMM-GCN achieves 92.18% ± 0.38, outperforming all other baselines, including TPN (88.93% ± 0.42) and RelationNet (86.76% ± 0.35).

The five-way classification task poses greater challenges due to an increased number of classes, leading to lower overall accuracy compared to the three-way task. Despite this, ADMM-GCN consistently outperforms existing methods. In the one-shot setting, ADMM-GCN achieves 61.79% ± 0.56, surpassing ProtoNet (50.42% ± 0.89) and RelationNet (53.81% ± 0.91). The performance gap remains noticeable in the 10-shot setting, where ADMM-GCN attains 74.01% ± 0.53, compared to TPN (70.37% ± 0.73) and DeepEMD (69.66% ± 0.60). The results suggest that ADMM-GCN maintains robust classification performance in complex scenarios with a higher number of categories. The model’s superior feature learning capability enables it to extract more discriminative representations, even under conditions with limited training samples. Moreover, the inclusion of confidence intervals in the results ensures statistical reliability, further validating the effectiveness of ADMM-GCN in few-shot SAR ATR.

5.2. Performance Assessment Under Varying Conditions

To evaluate the performance of ADMM-GCN in real-world SAR ATR scenarios, we conduct additional experiments under varying conditions, including noise injection, random cropping, and rotation operations. These perturbations simulate common challenges encountered in practical SAR target recognition, such as sensor-induced noise, partial occlusions due to environmental obstacles, and variations in target orientation.

Noise is a common issue in SAR imaging, often resulting from electronic interference, clutter, and sensor limitations. To replicate these effects, Gaussian noise is injected into SAR images, introducing pixel-level distortions to evaluate the model’s ability to extract discriminative features despite signal degradation. In addition to noise, real-world SAR images frequently suffer from partial occlusion due to obstacles such as buildings and vegetation. To simulate this, random cropping is applied, removing portions of the target object and assessing the model’s ability to generalize with incomplete feature representations. Furthermore, SAR target images are often captured from varying aspect angles due to changes in sensor positions and target orientations. To evaluate the model’s robustness to such variations, random rotations are applied, evaluating whether ADMM-GCN can maintain classification accuracy across different orientations.

The experimental results, summarized in

Table 4. Experimental results of ADMM-GCN under varying conditions across different N-way K-shot settings (Accuracy, %).

Perturbation Settings			3-Way			5-Way
Noise Injection	Random Cropping	Rotation	1-Shot	5-Shot	10-Shot	1-Shot	5-Shot	10-Shot
			84.31	89.70	92.18	61.79	68.75	74.01
✓			81.38	89.62	90.26	60.12	66.04	73.74
	✓		80.32	88.70	90.02	61.46	64.62	73.80
		✓	82.32	89.06	90.50	58.44	65.82	72.16
✓	✓	✓	81.10	89.58	90.88	56.98	65.46	71.10

, indicate that ADMM-GCN maintains stable classification performance under various perturbations, with a modest decline in accuracy observed under more challenging conditions. In the table, checkmarks (✓) indicate that the corresponding perturbation was applied. The introduction of noise injection slightly reduces accuracy, reflecting the increased difficulty in feature extraction due to signal distortions. For instance, in the three-way one-shot setting, accuracy decreases from 84.31% to 81.38%, and in the five-way one-shot setting, it drops from 61.79% to 60.12%. Random cropping also leads to minor performance degradation, suggesting that ADMM-GCN can still effectively classify targets even when partial information is missing. In the three-way one-shot scenario, accuracy drops to 80.32%, and in the five-way one-shot scenario, it remains relatively stable at 61.46%. Rotation perturbations introduce variations in accuracy, highlighting the impact of aspect angle differences on SAR image representation. The accuracy under this setting is 82.32% for three-way one-shot and 58.44% for five-way one-shot, indicating that angular variations can affect feature representation but the model still generalizes well. When all three perturbations are applied simultaneously, the model exhibits a moderate performance drop, yet it still maintains acceptable classification accuracy. The most significant decrease is observed in the five-way one-shot scenario, where accuracy drops to 56.98%, while in the three-way 5-shot setting, it stabilizes at 89.58%. Despite these variations in environmental conditions, ADMM-GCN maintains acceptable classification performance, demonstrating its potential for deployment in operational environments where noisy, partially obscured, or variably oriented targets are encountered.

5.3. Ablation Study

This section conducts a series of experiments to validate the effectiveness of each essential part of the suggested ADMM-GCN. The configuration settings listed in Table 2 are used for all ablation assessments.

5.3.1. Effectiveness Assessment of the EM

First, experiments are conducted to assess the effectiveness of the EM within the proposed ADMM-GCN framework. The experimental results are demonstrated in

Table 5. Ablation study results for assessing the effectiveness of the EM on 3-way and 5-way K-shot tasks (Accuracy, %).

N-Way	EM Settings	K-Shot
		1-Shot	5-Shot	10-Shot
3-way	w/o EM	72.61	75.43	78.25
	w EM	84.31 (11.70 ↑)	89.70 (14.27 ↑)	92.18 (13.93 ↑)
5-way	w/o EM	58.48	68.69	72.34
	w EM	61.79 (3.31 ↑)	68.75 (0.06↑)	74.01 (1.67 ↑)

, where “w” indicates that the EM is included, while “w/o” indicates that the EM is excluded.

As shown in Table 5, the classification accuracy of ADMM-GCN with the EM is consistently higher than that of the model without the EM. These results demonstrate that the EM effectively extracts embedding features, significantly enhancing the performance of few-shot SAR target classification.

5.3.2. Effectiveness Assessment of the GCM

Next, experiments are carried out to verify the effectiveness of the GCM in enhancing the performance.

Table 6. Ablation study results for assessing the effectiveness of the GCM on 3-way and 5-way K-shot tasks (Accuracy, %).

N-Way	GCM Settings	K-Shot
		1-Shot	5-Shot	10-Shot
3-way	w/o GCM	70.63	87.99	90.16
	w GCM	84.31 (13.68 ↑)	89.70 (1.71 ↑)	92.18 (2.02 ↑)
5-way	w/o GCM	56.85	65.68	72.70
	w GCM	61.79 (4.94 ↑)	68.75 (3.07 ↑)	74.01 (1.31 ↑)

shows the corresponding experimental results, where “w” indicates that the GCM is included, while “w/o” denotes that it is excluded.

As shown in Table 6, the ADMM-GCN, which incorporates the GCM, consistently outperforms the model without the GCM across six different experimental settings. Notably, in the five-way one-shot task, the classification accuracy improves by approximately 5% when the GCM is employed. These experimental results demonstrate that the GCM effectively enhances the model’s classification performance. Specifically, the graph structure and convolutional operations improve the discriminative ability of the extracted features and strengthen the overall feature representation by constructing a relational graph among features. This, in turn, significantly boosts the few-shot SAR target classification performance.

5.3.3. Impact Evaluation of Mixed Regularized Loss

Several experiments were conducted to evaluate the impact of the mixed regularized loss. The results, as shown in

Table 7. Experimental results of 3-way K-shot tasks: comparison between classical loss and mixed regularized loss in terms of accuracy (%).

Configurations			Accuracy (%)
Loss Settings	$\mathit{K}$-Shot		Min	Max	Mean	Standard Deviation
Classical Loss	1-shot		78.64	82.57	80.30	0.7684
	5-shot		81.46	85.26	83.78	0.7745
	10-shot		82.00	86.79	84.40	0.7697
Mixed Regularized Loss	1-shot		82.67	86.79	84.31 (4.01 ↑)	0.5467 (0.2217 ↓)
	5-shot		88.50	90.86	89.70 (5.92 ↑)	0.5440 (0.2305 ↓)
	10-shot		90.50	93.43	92.18 (7.78 ↑)	0.5374 (0.2323 ↓)

and

Table 8. Experimental results of 5-way K-shot tasks: comparison between classical loss and mixed regularized loss in terms of accuracy (%).

Configurations			Accuracy (%)
Loss Settings	$\mathit{K}$-Shot		Min	Max	Mean	Standard Deviation
Classical Loss	1-shot		37.14	43.07	39.94	1.1498
	5-shot		62.07	66.50	64.27	0.9465
	10-shot		64.56	65.82	65.40	0.9453
Mixed Regularized Loss	1-shot		59.14	65.21	61.79 (21.85 ↑)	0.7856 (0.3642 ↓)
	5-shot		65.21	69.57	68.75 (4.48 ↑)	0.7198 (0.2267 ↓)
	10-shot		71.43	76.29	74.01 (8.61 ↑)	0.7384 (0.2069 ↓)

, and Figure 8, highlight the performance of models in three-way and five-way few-shot SAR target classification tasks. These tables and figures provide comprehensive insights by presenting average accuracy and standard deviations, capturing both the precision and stability of the models.

In the three-way K-shot tasks, the mixed regularized term significantly enhances both accuracy and stability. For instance, in the one-shot task, the accuracy increases from 80.30% to 84.31%, while the standard deviation decreases from 0.7684 to 0.5467. This demonstrates the mixed regularized term’s effectiveness in improving model generalization and reducing variability, thus ensuring more consistent performance across diverse settings.

Similarly, for five-way K-shot tasks, the introduction of the regularized term leads to consistent improvements. In the one-shot task, accuracy rises from 39.94% to 61.79%, with a corresponding decrease in standard deviation from 1.1498 to 0.7856. A similar trend is observed in 5-shot and 10-shot tasks, where the term contributes to both accuracy enhancements and variability reductions. By boosting accuracy and reducing fluctuations, the regularized term enables more reliable and stable performance, which is essential for practical applications of few-shot SAR target classification.

To further investigate the effectiveness of addressing overfitting challenges, we have conducted additional experiments comparing the proposed mixed regularized loss with two widely used regularization techniques: L1 regularization [55] and ElasticNet [56].

Table 9. Comparison of N-way K-shot accuracy (%) between the mixed regularized loss and other regularization techniques.

Regularization Settings	3-Way			5-Way
	1-Shot	5-Shot	10-Shot	1-Shot	5-Shot	10-Shot
L1 Regularization [55]	67.06	85.48	85.84	57.96	63.18	67.64
ElasticNet [56]	80.84	87.28	91.44	58.04	67.40	66.12
Mixed Regularized Loss	84.31	89.70	92.18	61.79	68.75	74.01

presents the accuracy differences across various N-way K-shot settings.

For the three-way one-shot setting, the mixed regularized loss attains 84.31%, outperforming L1 regularization (67.06%) and ElasticNet (80.84%). Similarly, in the 10-shot setting, the mixed regularized loss achieves 92.18%, surpassing ElasticNet (91.44%) and L1 (85.84%).

For the five-way K-shot tasks, the improvements are also evident. The mixed regularized loss achieves 61.79% accuracy in the one-shot setting, compared to 57.96% and 58.04% for L1 and ElasticNet regularization, respectively. As the number of shots increases, the mixed regularized loss maintains its advantage, reaching 74.01% in the 10-shot setting, while L1 and ElasticNet achieve 67.64% and 66.12%, respectively.

These results suggest that the mixed regularized loss provides superior generalization and increased accuracy compared to L1 regularization and ElasticNet, while L1 regularization encourages sparsity, it can overly penalize model parameters in FSL scenarios, leading to suboptimal feature utilization. ElasticNet, which combines L1 and L2 regularization, improves performance but struggles to maintain model expressiveness. In contrast, the mixed regularized loss balances regularization and model complexity, effectively reducing overfitting while preserving informative features, making it suitable for few-shot SAR ATR tasks.

5.3.4. Performance of ADMM Optimizer

To comprehensively evaluate the effectiveness of the ADMM optimizer, we compare its performance with three widely used optimization methods: SGD [35], RMSprop [57], and Adam [34]. The experimental results, presented in

Table 10. Comparison of three-way K-shot accuracy (%) between ADMM optimizer and other optimizers.

Optimizer Settings	K-Shot
	1-Shot	5-Shot	10-Shot
SGD [35]	75.54	86.46	86.76
RMSprop [57]	76.66	85.32	87.42
Adam [34]	82.98	88.46	90.62
ADMM (Ours)	84.31	89.70	92.18

and

Table 11. Comparison of 5-way K-shot accuracy (%) between ADMM optimizer and other optimizers.

Optimizer Settings	K-Shot
	1-Shot	5-Shot	10-Shot
SGD [35]	54.92	60.26	61.40
RMSprop [57]	57.34	63.34	65.28
Adam [34]	60.73	67.19	72.67
ADMM (Ours)	61.67	68.75	74.01

, as well as Figure 9, demonstrate the accuracy improvements of the ADMM optimizer across different task settings.

Table 10 presents the results for three-way K-shot tasks. For the one-, five-, and ten-shot settings, the ADMM optimizer achieves accuracies of 84.31%, 89.70%, and 92.18%, respectively. In comparison, the Adam optimizer attains 82.98%, 88.46%, and 90.62%. The RMSprop and SGD optimizers exhibit lower performance, with SGD achieving 75.54% in the one-shot setting and RMSprop reaching 87.42% in the 10-shot setting. Among all tested optimizers, ADMM consistently yields the highest accuracy, with relative improvements of 1.33%, 1.24%, and 1.56% over Adam. The performance gap is even more pronounced when compared to RMSprop and SGD.

Similarly, Table 11 presents the results for five-way K-shot tasks. In the one-, five-, and ten-shot settings, the ADMM optimizer achieves accuracies of 61.67%, 68.75%, and 74.01%, respectively, outperforming the Adam optimizer, which attains 60.73%, 67.19%, and 72.67%. The accuracy improvements over Adam are 0.94%, 1.56%, and 1.34%, respectively. Additionally, RMSprop and SGD exhibit lower performance, with SGD achieving 54.92% in the one-shot setting and RMSprop reaching 65.28% in the ten-shot setting.

These trends are further illustrated in Figure 9, which compares the test accuracy achieved by the ADMM optimizer and three other optimizers (Adam, RMSprop, and SGD) across three-way and five-way K-shot tasks. Figure 9a shows the test accuracy for three-way K-shot tasks, while Figure 9b presents the results for five-way K-shot tasks. As shown in Figure 9, the ADMM optimizer (red solid line) consistently outperforms all baseline optimizers across different shot numbers. In the three-way one-shot task, ADMM achieves 84.31%, while Adam attains 82.98%, RMSprop reaches 76.66%, and SGD performs the worst at 75.54%. Similarly, in the five-way 10-shot setting, ADMM achieves the highest accuracy of 74.01%, surpassing Adam (72.67%), RMSprop (65.28%), and SGD (61.40%). These visualizations further validate the effectiveness of the ADMM optimizer in few-shot SAR target recognition, demonstrating its capability to achieve higher accuracy and enhance model generalization across different task configurations.

5.4. Hyperparameter $\lambda$ Analysis

$\lambda$ is a key parameter that balances the strength of regularization in the proposed method. To evaluate its impact, a series of experiments were conducted across a wide range of $\lambda$ values for both three-way and five-way tasks, under one-shot, five-shot, and ten-shot settings.

Figure 10 presents the classification accuracy for different $\lambda$ values in N-way K-shot tasks. In each subplot, the horizontal axis represents the value of $\lambda$, while the vertical axis denotes the average classification accuracy (%). For instance, in the three-way one-shot task (subplot a), the classification accuracy remains stable across a wide range of $\lambda$ values, from $1\times {10}^{-6}$ to $1\times {10}^{-5}$. However, a slight decline in performance is observed when $\lambda$ increases to $1\times {10}^{-4}$, where accuracy decreases from approximately 85% to 82%. The observed trend in the three-way one-shot task is consistent across all other settings. Such consistent behavior across diverse N-way K-shot tasks indicates the method’s robustness to a wide range of $\lambda$ settings.

6. Conclusions

In this paper, we propose ADMM-GCN, a novel graph-based framework for few-shot SAR target recognition, designed to address the critical challenge of data scarcity in SAR ATR. By leveraging the learning capabilities of GCNs, ADMM-GCN proficiently extracts both global and local features from few-shot SAR samples. To further enhance performance in FSL scenarios, this study designs a mixed regularized loss function to mitigate the risk of overfitting during training. Additionally, an ADMM-based algorithm is developed to address the optimization model. Extensive experiments conducted on the MSTAR dataset validate the effectiveness of ADMM-GCN, which achieves superior performance across all evaluated settings. Notably, the method achieves an accuracy of 92.18% on the challenging three-way 10-shot task, outperforming benchmarks by 3.25%. While ADMM-GCN offers advantages, its optimization process involves iterative updates, leading to a modest increase in computational cost.

To further improve efficiency, future research will explore several strategies. One promising direction is to develop a hybrid ADMM–Adam optimizer for few-shot SAR target recognition, leveraging ADMM’s constrained optimization capabilities alongside Adam’s fast convergence properties to improve computational efficiency while maintaining robust optimization. Additionally, exploring lightweight variants of ADMM-GCN could facilitate its deployment in real-world SAR processing systems with constrained computational resources.