Gamma Distribution PCA-Enhanced Feature Learning for Angle-Robust SAR Target Recognition
Authors: Chong Zhang
Peng Zhang
Mengke Li
Venue: ICML 2025 poster (2025)
Keywords: Synthetic aperture radar; Automatic target recognition
Primary area: applications->computer_vision

Abstract: Scattering characteristics of synthetic aperture radar (SAR) targets are typically related to observed azimuth and depression angles. 
However, in practice, it is difficult to obtain adequate training samples at all observation angles, which probably leads to poor robustness of deep networks. In this paper, we first propose a Gamma-Distribution Principal Component Analysis ($\Gamma$PCA) model that fully accounts for the statistical characteristics of SAR data. The $\Gamma$PCA derives consistent convolution kernels to effectively capture the angle-invariant features of the same target at various attitude angles, thus alleviating deep models' sensitivity to angle changes in SAR target recognition task. We validate $\Gamma$PCA model based on two commonly used backbones, ResNet and ViT, and conduct multiple robustness experiments on the MSTAR benchmark dataset. The experimental results demonstrate that $\Gamma$PCA effectively enables the model to withstand substantial distributional discrepancy caused by angle changes. Additionally, $\Gamma$PCA convolution kernel is designed to require no parameter updates, introducing no extra computational burden to the network. The source code is available at \href{https://github.com/ChGrey/GammaPCA}{https://github.com/ChGrey/GammaPCA}.

Scattering characteristics of synthetic aperture radar (SAR) targets are typically related to ob- served azimuth and depression angles. However, in practice, it is difficult to obtain adequate train- ing samples at all observation angles, which prob- ably leads to poor robustness of deep networks. In this paper, first propose Gamma- Distiaaa-Dttiittiinn Principal Component Analysis (TPCA) model that fully accounts for the statistical character- istics of SAR data. The IPCA derives consis- tent convolution kernels to effectively capture the angle-invariant features of the same target various attitude angles, thus alleviating deep models' sensitivity to angle changes in SAR tar- get recognition task. We validate ПPCA model based on two commonly used backbones, ResNet and ViT, and conduct multiple robustness exper- iments on the MSTAR benchmark dataset. The experimental results demonstrate that TPCA ef- fectively enables the model withstand sub- stantial distributional discrepancy caused by an- gle changes. Additionally. IPCA convolution kernel is designed to require no parameter up- dates, introducing no extra computational burden to the network. The source code available at Gamma Distribution PCA-Enhanced Feature Learning for Angle-Robel
SAR Target Recognition
Chong Zhang'
Peng Zhang"
Mengke Li
Abstract
Figure 1: SAR images of the same target captured at differ- ent angles. (Left) Azimuth 09-90", Depression 17;; (Middle) Azimuth 270--360", Depression 17;; (Right) Azimuth 0 90% Depression 30%
Zhang et al., 2023a: Yang et al., 2022; Zhang et al., 2021) and military (Song et al., 2022; L.v et al., 2023; Kechagias- Stamatis & Aouf, 2021) fields. Automatic target recognition (ATR) is fundamental observation mission of SAR. In the SAR ATR field, numerous studies have attempted to develop general and robust feature extractors by adapting deep net- works initially designed for optical images (Pei et al., 2023; Zeng et al., 2022; Guo et al., 2020; Pei et al., 2023; Li et al., 2020; Wang et al., 2021). While these approaches have demonstrated satisfactory performance, they often struggle maintain robustness when recognizing the same target at different azimuth depression angles. This is primar- ily because the electromagnetic scattering characteristics of the same SAR target will vary depending on the angle of incidence of the electromagnetic wave.
In recent years, numerous studies focusing on SAR ATR have been proposed from various perspectives, including electromagnetic scattering feature extraction (L.i et al., 2022; Feng al., 2023; Zhang et al., 2020; Li al., 2021; Chen et al., 2024; Huang et al., 2024b), phase information (Wang et al., 2022; Liu & Lang, 2021; Zeng et al., 2022), shadow information (Guo et al., 2024) attitude information (Guo et al., 2024; Wang et al., 2024; Oh et al., 2020) as well as methods based on transfer learning (Zhou et al.. 2024b; Shi et al., 2024: Zhang al., 2023c), name few. These methods have empirically demonstrated superior perfor- mance, thereby emphasizing the importance of accounting for the unique characteristics of SAR data in the SAR ATR task. However, most of physical-basee based methods rely on extra physical prior knowledge, which limits their gener- alizability to a wider range of task scenarios. While deep
1. Introduction
Synthetic aperture radar (SAR), with its all-weather, all- time observation capabilities, plays an important role in various remote sensing observation missions. has found widespread application in both civil (Yang et al., 2023:
'Corresponding author. 'National Key Laboratory of Radar Signal Processing, School of Electronic Engineering. Xidian Uni- versity, Xi' an. China. *College of Computer Science and Software Engineering, Shenzhen University, Shenzhen, China. Correspon- dence to: Peng Zhang <pzhang@ xidian.cdu..commmmm edu.en>>, Mengke Li <esmengkeli@ saanennkeli@gmail.coom gmail.com>.
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Gamma Distribution PCA-Enhanced Feature Learning for Angle-Robust SAR Target Recognition
learning-based approaches mainly consider the problem of data seareity, and the data shift problem caused by imaging angle variation is usually ignored.
The proposed IPCA can be seamlessly integrated into various deep models without introducing additional pa- rameters, effectively enhancing the robustness of target recognition across different azimuth and depression angles for SAR data.
In reality, the distribution of strong scattering points and shadow areas in SAR imagery is highly sensitive to the attitude of target, that is, the azimuth and depression an- gle of the target (Dong Liu, 2022). The same target observed from different angles often exhibits distinct vi- sual appearances, scattering centers, geometric contours, etc., as shown in Figure 1. Therefore, the variation in the scattering characteristies of the same target with azimuth and depression angle is a critical factor affecting the perfor- mance of existing methods, yet it has not been explicitly or comprehensively addressed. Although prior studies have attempted to solve this issue based on multiview fusion and azimuth estimation (Liao et al., 2024; Ge et al., 2022; He et al., 2021), these methods are still unable to maintain the robust representation of different inputs from the same class when training samples from multiple angles are lacking, As a result, the sensitivity of target scattering properties to the viewing angle remains challenging in SAR ATR.
2. Related Work
2.1. Attributed Scattering Center-Based Approaches
The scarcity and attitude sensitivity of SAR data pose sig- nificant challenges for achieving robust target recognition (TRR) under limited data conditions. Physical features, known for their robustness and explainability (Datcu et al., 2023), have been widely utilized in addressing these chal- lenges. Among various physical cues, attributed scatter- ing centers (ASC) have emerged as prominent approach. Early works (Potter & Moses, 1997) demonstrated the ef- fectiveness of ASC in describing local scattering character- istics, leading to the development of robust feature extrac- tors (Ding et al., 2017; 2018) for SAR TRR tasks.
Recent advancements have seen the integration of ASC with deep neural networks (DNNs), evolving from simple feature fusion (Zhang et al., 2020) to more sophisticated ASC-driven models (Zhao et al., 2024; Feng et al., 2022; 2021). These models have shown resilience against signifi- cant variations in depression angles (Huang et al., 2024b). Additionally, other physical information such as azimuth, phase, and shadow characteristics have been leveraged to enhance SAR TRR. For example, Zhang et al. (2024) utilize azimuth information to filter and learn salient representa- tions, thereby making the model more robust azimuth variations. Choi et al. (2022) and Guo al. (2024) focus shadow characteristics to guide deep models in learning target structures. Complex-valued networks, integrating am- plitude and phase information, have also shown promise in addressing SAR TRR challenges, as demonstrated by Deng et al. (2022); Zeng et al. (2022); Zhou et al. (2023).
To address the aforementioned issue, we propose feature extractor based on generalized Cammaadistribution princi- pal component analysis (denoted as IPCA) to obtain angle- insensitive features. The Gamma distribution is well-suited to model the statistics of multiple independent scattering points, making it particularly effective for SAR targets. This characteristic allows for the extraction of robust low-rank in- formation from targets observed multiple viewing angles. We extend the Pearson mean square error optimization ob- jective of principal component analysis (PCA) based on the Gamma distribution, resulting in IPCA. We further derive the consistency projection matrix, enabling the extraction of low-rank information across different target angles. This ap- proach unifies the intrinsic characteristics of a target across various attitudes, significantly enhancing the ability of mod- els to discriminate and generalize unseen attitude angles. The proposed I'PCA is a simple, general, and efficient fea- ture extractor that can be seamlessly integrated with various backbone architectures. We demonstrate the effectiveness of our method by incorporating IPCA into both CNN- based backbones. The experimental results
Most physics-driven models are inherently task-specific and rely heavily on complex expert knowledge, which limits their generalizability to more refined downstream tasks. Ad- ditionally, experiments involving these methods typically address only one specific scenario, such as missing depres- sion or azimuth information, leaving their robustness under more complex and challenging conditions untested.
and Transformer-base demonstrate that the proposed I I'PCA effectively improves the performance of existing models. The main contributions of this paper are follows:
We fully consider the non-Gaussian statistics of SAR targets and derive novel TPCA that can extract low- rank features of SAR targets.
2.2. Deep Model-Driven Approaches
In contrast to physics-driven methods, deep model-driven approaches leverage established deep learning frameworks to address the challenges in SAR TRR. On one hand, a series
With SAR target samples observed at finite angles, the proposed TPCA derives a projection matrix to construct convolutional kernels, effectively capturing angle-insensitive information.
of studies have developed task-specific deep networks for SAR TRR by utilizing well-established architectures such as ResNet (Shang et al., 2023) and Transformer (Sun et al.,
2

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Gamma Distribution PCA-Enhanced Feature Learning for Angle- Robust SAR Target Recognition
2024). For instance, Lv et al. (2024) propose Transformer- based multi-view joint model that addresses challenges lated small target deformation and depression variation by denoising SAR images using low- fwwfrrequeney prior informa- tion and integrating multi-view features. On the other hand, various deep learning methods have been applied to SAR TRR tasks. Notably, recent studies have framed the limi- tations of SAR data as a few-shot learning (FSL) problem and have sought address it by incorporating diverse FSL methodologies (Zhou et al., 2024b; Zhang et al., 2023c). For example, Zhang et al. (2023b) establish a unified bench- mark for few-shot SAR image classification to evaluate the performance of various classical SAR ATR methods. Sim- ilarly, Shi et al. (2024) propose an unsupervised domain adaptation approach that employs contrastive learning to model and align the distributions of simulated and real data, thereby mitigating the issue of SAR data scarcity. Deep model-driven approaches exhibit superior generalizability compared to physics-driven approaches. Furthermore, some hybrid methods have emerged that integrate physical infor- mation with deep learning frameworks, achieving a balance between the two paradigms (Zhou et al., 2024b; Sun al., 2024). However, few studies have explored scenarios involv- ing simultaneous variations in both azimuth and depression angles, limiting their applicability in real world.
Angle
Azimuth 0'-90"
Azimuth 90'-180*
251
ZIL131
ZSU234
Figure 2: Histogram distribution of three categories of SAR targets at different azimuths.
3.1. Gamma-Distributer Principal Component Analysis
To extract low-rank information of targets under varying target angles, we follow the minimum mean square error criterion of generalized PCA (Landgraf & Lee, 2020), where optimal projection of original d dimensional data to k dimensional subspace (k: < d) is identified with squared error loss. This approach ensures that the projected data retains essential structural information, thereby facilitating feature extraction. Specifically, this projection involves seeking a center l E R2d and a rank-k matrix U E R where U I and I is the k-dimensional unit matrix, such that the following objective function is minimized:
3. Proposed Method
The scattering characteristics of SAR targets are inherently related the imaging angle. However, acquiring training samples that cover the full range of possible angles is often impractical. As result, enhancing the robustness of recog- nition against imaging angle variations remains significant challenge. Fortunately, we observe that, for a given tar- get, the statistical properties of SAR images exhibit notable consistency despite variations in imaging angles. Figure 2 illustrates this situation and Li al. (2011) have empirically demonstrated that the distribution of SAR targets follows gamma distribution. This statistical characteristic forms the basis for our approach to enhance the recognition robustness of targets at unseen angles.
x 14-4-UU UU (x, (x]1]]| s.t. U = I.-
min
(I)
Traditionally, optimal data representation is derived under the assumption that follows Gaussian distribution. How- ever, as discussed above, the SAR target typically follows Gamma distribution rather than Gaussian distribution. In the following, we will discuss how to derive TPCA in detail.
In this section, we propose a cuimma-distriuttdd Principal Component Analysis (PCA) model, named TPCA, which can extract low-rank information from SAR targets across multiple viewing angles. TPCA extends Pearson's mean square eтror optimization objective of PCA to the gamma distribution and derives a consistent projection matrix ca- pable of effectively capturing low-rank information under varying target angles, thus achieving robust and accurate target recognition even in scenarios where training samples from different viewing angles are limited. On the other hand, TPCA can be seamlessly integrated with various deep learning backbones.
TPCA Formulation. The Gamma distribution is specific instance of the exponential family of distributions. Gen- erally, the probability density function of the exponential family for random variable is expressed as:
x - b(0)
+ c(x.,)) (2)
f(r 0) = =exp
all)
where 0 is the canonical natural parameter, a(v)) is the scale parameter, and a(sp), 6(0), c(æ, ip) are all determined by specific distribution.
We express the probability density function of Gamma dis-
3

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Gamma Distribution PCA-Enhanced Feature Learning for Angle-Robus, SAR Target Recognition
tribution into the form of exponential family distribution:
Then, how to derive the rank-k matrix U R and the center ja IRd ? To derive them, we minimize the deviance:
_ [(t) e
u, v) [allu,
ux
O(X,))= = 1=l
(log [xx [x] (0) log J [x (0,x)) v)).
=expfv-log u -ouuuuua log T(u)+ + (u-)+((11)) 1v- 1ogg log x u
+ log u
(10)
eexp
log (vg (0)+ + (v- log 2
where the data matrix X [may] and the natural parameter
(3)
matrix ê = [0,y]-
In the above equation, r(v)= So x." dd. If we as- sume that- - = 0, v = =2² Equation (3) can be refor- mulated as
Substituting Equation (4) into Equation (10), and further
considering 0= ---r we derive the following formula:
a0+1og(-0) 1pt
+ 1 log 1 g2
Dy Dylong 015)
Ni 0,g -log( (0i(-
[xx = exp {
82
log r
1 log x
62
(11)
(4)
By comparing Equation (2) with Equation (4), the undeter- mined functions of Gamma distribution are as follows:
This formula can also be seen as minimizing the deviance of natural parameters in the k-dimensional subspace. Sub- stituting Equation (9) into Equation (11), the optimization objective function can be written as
((p) = x²
b(0) = - logo 0).
cxx,pp) = x² log x² log r + 82 -1) log
Jij (LV0" (o) + 1)); -15) - llog (ns [UV" (0: + m)),))
Σ 1.j
(5)
max MER U-
According to the property of exponential family distribu- tion (Landgraf& & Lee, 2020), the mean of Gamma distribu- tion can be directly derived from:
(12)
Hence, by minimizing the the objective function in Equa- tion (12), IPCA finds center μ E and rank-k matrix U Rd such that U U- Is.
(X (0)) -11
(6)
In addition, the saturated model is considered (Landgraf & Lee, 2020), and we have a canonical link function g(-) which satisfics g (0)) = and the mean 8'(0)= = x. This also implies that g()( = g(z) = 0. Therefore, we can approximate the natural parameter in IPCA by:
The above objective function is generally non-convex, mak- ing it challenging to find the global optimal solution. There- fore, we iteratively optimize the objective function using the Majorization Minimization Majorizaton-MMnimnztttoon (MM) algorithm (Mairal, 2013;
Sun al., 2016; Landgraf & Lee, 2020). Then, U and μ are estimated iteratively by:
x. 8 ( --, i.e. 0 0=---
(U -p(:) T 1' "
1 [1+1) = (1% (1) In)
Optimal data representation in IPCA is achieved when the estimated mean vector jJ equals the sample mean x and the estimated subspace Ü consists of the first & eigenvectors of the sample covariance matrix. Therefore, the approximation of x, by Equation (1) can be viewed equivalently the approximation of 0.g by
(13)
FOO) H 11 (1 T
pp(t) (p (i)
C ( (1()))
(14)
H (H+II()()))
( 14
., μ1 + UUT (-0.1 -14),,
(8)
where U((+1) is the first k eigenvectors of F(!).
where 0, is the natural parameter for the i-th case of sat- urated model. For similarity, we denote that (0) =
The detailed derivation of Equation (13) and Equation (14) are provided in Appendix B. In the MM algorithm, μ is
[ 0/221 _..2......6... 1T. and the estimation of natural param- eter d in d-dimensional subspace can be defined as
initialized as the column-wise mean vector of θ, and ( is initialized using the first right singular vectors of Ô. The procedure of TPCA is summarized in Algorithm 1.
0.j (LV0" (7:(0) +1) 14)
(9)
3.2. IPCA Attitude- Robust Convolution Kernel
IPCA Objective. By Equation (9), we can derive the ap- proximation form 0,y of the natural parameter Ozy IPCA.
In the preprocessing of original SAR images, following the approach of PCANet (Chan et al., 2015), we divide each
4

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Gamma Distribution PCA-Enhanced Feature Learning for Rogle-Robustteo SAR Target Recognition
Algorithm 1 IPCA Algorithm.
Class
251
251
ZSU234
2SU234
Image
(Depression (5*) (Depression 30
(Depression 15")
(Depression 30")
Require: Input data X= X2,, X2, .NN)
number of ex- tracted principal components I., size of projection ma- trix k. number of iterations for MM algorithm T'
Original SAR image
Ensure:
Output orthonormal optimal projection matrix V 1: Initialize: (=()) to the column means of é and U(0) to the first right singular vectors of θ, where θ is calculated by Equation (7).
Principal Component O
2: for 0 to T do
3: Update the mean vector μ (++1) by 1 ((tt))
In- 4: Calculate and record the deviance M(O (()) by
(1 ( 1.,) (HU (U pll) T
2 Principal
Component O
(oe0)
(H+1, (I)))) u (Ui)))
Figure 3: IPCA feature visualization.
M
$ plo In
where matxx (V.,)) denotes that mapping the /-th column of matrix V to a new matrix W E R
5:
if M S min[mnne (+) k=1 then
V- V-U
6:
The optimal projection matrix can be regarded as a conven- tional convolution kernel to convolve with the original SAR images:
7:
end if
Update the mean vector ((++1) ( (by Equation (29) in Appendix B.2), where ((++1) is the first ht eigenvec-
8:
tors of F(())
O = W E m , 1,2, ,N,1 1,2, (17)
9: end for
where O denotes the I-th principal component of i-th im- age, which has a size of i x m. Figure 3 illustrates the feature extraction performance of IPCA convolution ker- nels trained with limited angles while applied to samples under unseen angles.
SAR image into multiple patches, which are then vectorized for further processing. Suppose TPCA model receives a data set with N images, where the optimal projection matrix has dimensions k k, and the goal is to extract L princi- pal components from each picture. Initially, input images should be reshaped to uniform size, and we denoted the re- shaped images as I= [ E mmm Then, all the images are padded with blank pixels and divided into m patches of size k kt For the i-th image, we vectorize its all patches and concatenate them to a new matrix P, E R xmn? where i [(m m k + 2p)/ />] + , is is the number of blank pixels padded around a image, and is the stride size of dividing patches.
TPCA functions as filter within deep networks, facilitat- ing more effective preservation of angle-invariant features for the same SAR targets across different attitude angles, thereby enhancing the robustness of the networks. The implementation process of TPCA is illustrated in Figure 4.
In this paper, each SAR image is decomposed into two prin- cipal components, denoted as O and O.. These principal component feature maps retain the same spatial dimensions as the original SAR image, enabling them to be concate- nated with the single-channel image along the channel di- mension. The resulting three-channel image, comprising the two feature maps and the original image, serves as input for subsequent deep network processing.
Subsequently, we calculate the mean matrix over [ and derive its sccond-order statistics, which serve as input to principal component analysis:
X
2 PPT
(15)
Notably, in IPCA, we employ Min-Max Scaling for image normalization instead of conventional normalization tech- niques such as LayerNorm, BatchNorm, or InstanceNorm. It motivated by two key considerations: Preservation of principal component independence: In SAR images, each feature map corresponds to a single principal component of the target, and these components exhibit significant diver- sity. To maintain the independence between principal com- ponents of different styles, it is crucial to avoid normaliza- tion methods that might introduce unintended dependencies. 2) Distributional compatibility: LayerNorm, BatchNorm,
where P= P=
1 P,. Additionally, we divide the result by 2 to ensure that each matrix element falls within a range of 0 to 1, thereby facilitating the calculation of IPCA.
By the processing shown in Algorithm 1, we can finally get an orthonormal optimal projection matrix V R xI. Then can be reshaped into I. convolution kernels of size l x k.
Wi = ma (V.), 1,,,
(16)
5

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Gamma Distribution PCA-Enhanced Feature Learning for SAR Target Recognition
ПРСА Model
Training set
Gamma Principal
reshape padding
reshape
Compoment Analysis
N
NX
P-
V
Divided into
Vectorize and align patches matrices
Sum and culculate
Obtaining optimal projection matrix
Convolation
Input in batches
patches
statistics
kernell: kernel2:
Channel
Inpat
Convolution Layer
Output 2
Output
Normalization
Channel Concat
Max- Min Scale
Channel
Deep Networks
Predict results
Channel
Connecting Layer
Batch size
Figure 4: Implementation of Gamma-Distribution Principal Analysis ([PCA)..
and InstanceNorm are designed under the assumption that the data follows Gaussian distribution. However, SAR data inherently follow Gamma distribution, and applying Gaussian-based normalization methods could distort the in- trinsic distributional characteristics of the data. Min-Max scaling normalizes data by scaling each channel of an image to the range [-1, based on its maximum and minimum values. This approach distribution-correlaeed preserv- ing the inherent structure of SAR data without imposing Gaussian assumptions.
eral SOTA approaches, including VisionLSTM-T (Alkin et al., 2024), MSNet-PIHA (Huang et al., 2024b), and WTConvNeXt-T (Finder et al., 2025).
Settings. For the dataset, in previous studies, experimental settings typically considered variations in either azimuth or depression angle individually. However, real-world scenarios, limited azimuth coverage and changes depres- sion angle often occur simultaneously. To better evaluate model robustness under such realistic conditions, design more challenging experimental setting that incorporates both azimuth insufficiency and depression variation. Our experimental design is as follows.
4. Empirical Study
Azimuth Robustness Test. To simulate the scenario that a wide range of azimuth angles are missing, all models are trained and validated by data from only one azimuth quad- at depression 17 (e.g., Azimuth 00-900, Depression 17%), which means these data only consist of 25 % azimuth information at that depression angle. For testing, well- trained model will be tested in the full-azimuth (09-3609) testing set at depression 15"
4.1. Description of MSTAR Dataset
The data set used in this paper is the static ground military target data set, MSTAR (Keydel al., 1996). MSTAR collected by a high resolution beam stacking SAR. It consists of various vehicle images different azimuth and depression angles. The detailed composition of MSTAR dataset is illustrated in Appendix C.I.
Azimuth Depression Robustness Test. The processes of model training and validation are the same as Azimuth Ro- bustness Test. However, all trained models will be tested in testing set at depression 30,, full-azimuth.
4.2. Comparison Methods and Experimental Settings
Comparison Methods. To demonstrate the effectiveness and generalizability of the proposed method, we integrate TPCA into two representative backbones, R0sNet101 (He et al., 2016) and ViT-B/16 (Dosovitskiy, 2020), referred to as TPCA-ResNet and IPCC- respectively. In addition to the baseline modeisFurthermore, w namely ResNet101 and ViT-B/16, We compare method with three recently proposed backbones: Swin Transformer (Liu et al., 2021), EfficientNet (Tan & Le, 2019), and VisionMamba (Zhu et al., 2024). Furthermore, we evaluate our method against sev-
To reduce the randomness, we conducted the above ex- periments on two separate training sets, which come from opposite azimuth quadrants: 0--90" and -809- is generally believed that two opposite azimuth quadrants can provide complementary azimuthal information, therefore. we would like to repeatedly evaluate model's stability by using two complementary training sets. A detailed descrip- tion of the experimental strategy is provided in Appendix A.
6

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Gamma Distribution PCA-Enhanced Feature Learning for Angle-Robus, SAR Target Recognition
Table I: Recognition results of different networks. Training set: Depression 17", Azimuth -90 Testing set: Depres- sion 15", Full-Azimuth & Depression 30,, Full-Azimuth. OA is short for Overall Accuracy.
251
D7 T52
251
127
172
BROMZ BTRRO
2TT.131 25U334
23
BERTO
BTR20
234:234
Methods
A-R Test
AD-R Test
OA (%)
Resnet101
72.57=0.25
43.95=0.01
63.6240.14
EfficientNNet-33
67.2240.02
45.1520.07
60.27+0.01
VIT-B/16
78.12=0.04
60.11+0.24
72,45+0.04
Swin-B
61.58=0.03
#0.01
66.75+0.02
(a) VIT-B/16
(b) TPCA-ViT
VisionLSTM--T
55.25+0.21
76.54+0.24
61.95+0.22
VisionMamba-T
62.37=0.00 06
66.67=0.03
63. 68.004
251
D7
D7
BMF2
T62
MSNet-PIHA
68.08=0.03
55.75+0.06
64.20+0.03
BROM2
172
BRIM2
BTT.00
23L131
T72
WTConvNeXI-T
72.55=0.01
57.69=0.05
67.87+0.01
BTRTO
BTR70
25U234
IPCA-Resnet (Ours)
73.7920.02
52.96=0.05
67.23+0.01
IPCA-VIT (Ours)
78.48=0.01
70.6720.16
76.02+0.01
Table 2: Recognition results of different networks. Train-
ing set: Depression 17,, Azimuth 1809-270"; Testing set: Depression 15" Full-Azimuth & Depression 30, Full- Azimuth. OA is short for Overall Accuracy.
(c) R0snet101
(d) RPCA-Resnet
Methods
A-R Test
AD-R Test
OA (%)
Figure 5: t-SNE visualizations of feature distributions for different backbones in A-R Test. Training set: Depression 17", Azimuth -9-90%
Resnet101
75.30=0.01
48.12+0.68
66.74+0.11
EfficientNet-B3
69.2420.09
45.69+0.07
61,83+0.07
ViT-B/16
79.92=0.00 04
58.5620.12
73. 1940.01
For the backbone, due to the small size our dataset, all models are pre-trained on ImageNet and then fine-tuned on dataset. All networks only use Resize and CenterCrop to preprocess the input data. For the hyperparamcters, the TPCA part uses L = 2 kernels with kernel size of k 17.
Swin-B
70.17=0.08
65.53+0.05
68.71+0.06
VisionLSTM
55.4920.13 13
74.03+1.26
61.33+0.16
VisionMamba-T
66.21=0.02
72.98+0.02
68.3440.01
MSNet-PIHA
65.3620.02
56.5520.11
62.5940.03
WTComyNeXI-T
76.4620.07
63.29+0.51
72.32+0.16
TPCA-Resnel (Ours)
71.3420.04
66.21+0.02
69.7240.03
4.3. Results on MSTAR dataset
IPCA-VIT (Ours)
80.15+0.01
64.48+0.05
75.2240.01
We report the test accuracy in Table 1 and Table 2. Through an overall comparison, TPCA-VIT achieves an overall ac- curacy of 76.02% and 75.22% on the two separate training sets, respectively, with both being the best performance among all models. The results demonstrate the superiority and stability of TPCA- VIT. In terms of individual tests, PPCA-VIT consistently main- tains leading performances on A-R Test even with different training sets. On AD-R Test, it is noticeable that the perfor- mance of most models has dropped significantly, which is consistent with the previous discussion of the attitude sensi- tivity effect. Depression angle a crucial factor that affects the target's is reflection intensity and shadow shape. Thus. its variations generally cause structure distribution shift of the dataset. Although the performances of ПPCA-ViT on AD-R Test are not leading, we can find that models with high accu- racy on AD-R Test always perform poorly on A-R Test, for example, Vision-LSTM and VisionMamba, which are not considered to robust to various attitude variations. Our comparative analysis indicates that ViT performs robustly
and maintains stable performance across differrnnt scenarios.
Furthermore, vertical comparisons show substantial im- provements in model robustness when integrated with I'CCA. The overall accuracy of I'PCA-ViT increases by 3.57% and 2.03% over the original ViT-B/16 on the training set and 1809-270", respectively. Similarly, NPCA- ResNet improves the original ResNet101 by 3.61% and 2.98% the training set -0-90% and -20". In particu- lar, the improvements on the AD-R Test for both IPCCA ViT and TPCA-ResNet are particularly significant. Under more severe depression variation, the enhanced representations of the target extracted by TPCA help the models better adapt their learned patterns to the distribution shift the test set, These results collectively demonstrate the effectiveness and general applicability of our method across different back- hone architectures.
In order to visually see the impact adding IPCA on the feature extraction ability of models, we report the t-SNE visualizations of feature distributions for ViT-B/16, TPCA-

---

Gamma Distribution PCA-Enhanced Feature Learning for Angle-Robuse SAR Target Recognition
Table 3: Recognition results of different networks. Training set: Depression 17", Azimuth 900 Testing set: Depres- sion 15,, Full-Azimuth Depression 30", Full-Azimuth.
Table 4: Recognition results of different networks on SAR- AlReraft-1.0 dataset.
Methods
Overall Acc. (%)
Methods
A-R Test
AD-R Test
Overall Acc. (%)
Resnet101
97.33=0.02
Kernel size =
TPCA+Resnet101
98.67=0.01
VTT-B/16 (Baseline)
77.98
54.58
70.61
ViT-B/16
97.05=0.03
Conv-ViT
69.69
63.73
67.82
IPCA+VVT-B/16
98.12=0.02
WT-VI
71.63
57.27
67.11
Swin-B
97.41=0.01
PCA-VIT
78.35
52.33
70.16
IPCA+Swin-B
98.32=0.02
PPCA-VIT (Ours)
76,58
63.82
72.56
Kernel size
11
VTT-B/16 (Baseline)
77.98
54.58
70.61
torize each patch as a dimension for dimensionality reduc- tion. We expect IPCA to consider more target information rather than background information. This processing ef- fectively avoids I PCA from capturing the variance of the entire dimension of the image, and leads to a more accurate estimation of the distribution parameters of a target during dimension reduction. In addition, this is also the reason why a larger kernel has better effect for both IPCA and stan- dard PCA. Because a smaller kernel can only make patches cover local parts of a target, which makes the subsequent PCA processing has large deviation for the estimation of target distribution characteristics.
Conv-ViT
65.61
68.3
66.46
WT-VIT
69.03
68.04
68.72
PCA-VIT
76,08
61.67
71.55
TPCA-VIT (Ours)
79.13
61.40
73.55
Kemel size
17
VTT-B/16 (Baseline)
77.98
54.58
70.61
Conv-ViT
66.39
53.50
62.33
WT-VIT
71.51
68.22
70.47
PIA-VIT
78.06
63.64
73.52
PPCA-VIT (Ours)
79.88
67.59
76.01
ViT, Resnet and IPCA-ViT in A-R Test, as shown in Fig- ure It is evident that both PPCA-ViT and IPCA-Resnet have more compact feature distributions than their original counterparts, consistently across nearly all categories. The clustering result verifies that IPCA can retain targets' main characteristics and extract effective features to discriminate different categories of targets.
To further observe the difference between standard PCA and TPCA, we have removed the subsequent deep network and only used a PCA convolutional layer and a I'PCA con- volutional layer to extract features of original SAR images respectively, and then obtained their feature visualizations by t-SNE. The results are shown in Figure 6. Obviously, standard PCA is far less effective in characterizing the fea- tures of SAR compared to I PCA. This is largely because the Gaussian assumption of standard PCA does not fit the distribution characteristics the SAR data, leading to inad- equate separability between classes.
Furthermore, we use different kinds of convolutional layers to replace our TPCA layer in TPCA- -PTA-VIT to compare their performances, including a conventional convolutional layer, standard PCA convolutional layer and Wavelet convo- lutional layer (Finder et al., 2025). We temporarily denote them as Conv-ViT, PiA-ViT, and WTT, respectively, for clarity in the following discussions. These convolutional layers all maintain the same formulation of outputs as I'PCA model. In particular, the standard PCA convolution kernel built by using the same framework as our IPCA model, but its weights are derived through conventional PCA pro- cessing. The experimental results are reported in Table 3.
4.4. Results on Additional Dataset
MSTAR is a typical dataset for SAR ATR task, and most studies (Huang al., 2024b; Zhang et al., 2024; Wang et al., 2024; Li et al., 2023) rely solely on MSTAR their experimental dataset. To further evaluate the generality of our method, we construct new dataset from the widely used SAR aircraft target detection dataset, SAR-AIRcraft- 1.0. We crop every single target from the original SAR image and obtain an available SAR ATR dataset. detailed description of this dataset is provided in Appendix C.2.
The comparison in Table 3 shows that PPCA-ViL consis- tently achieves prominent performance across different ker- nel sizes. Besides, adding an extra convolutional layer tends to degrade the performance of ViT in most cases, regardless of whether small or large kernels are used. However, this trend does not hold to the standard PCA and TPCA variants. In fact, both PCA-VIT and TPCA-ViT exhibit improved performance as the kernel size increases, which aligns with the design principles underlying their convolutional kernels.
Since SAR-AIReraft-I. lacks diverse and separable imag- ing angles (Zhirui et al., 2023; Zhou et al., 2024a; Huang et al., 2024a), we employ standard target recognition strat- egy and conduct experiments accordingly. Specifically, 80% of the dataset is allocated for training and the remaining 20% for testing. The corresponding results are reported in Table 4. Unlike the results on the MSTAR dataset, several
Instead of directly performing IPCA on the whole SAR image, we split the image into overlapping patches and vec-
8

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Gamma Distribution PCA-Enhanced Feature Learning for Angle-Robust Robust SAR Target Recognition
S0
281
74.40%
76.01%
BMP? 102
BRIDM
172
BRDM2
162
70
69.60%
*70.6
BIRM
BIRAC
ZIL:31 25U234
BTR40
23L
2IL131
Oveall, Accuney (%)
50
62
62.19%
56.15%
58.24%
+63.04%
51.51%
52.16%
so/is
0018.08%
47.22%
52.22%
55.30%
44 44.4554
13%%%%
40
33.20%
32.47%
GPCA-VT (Nused))
-G#CA.Reasel (ooiee))
30
27.10%
AGPTAA-VIT (Nmoslos)
23.34%
(Notied)
RRenet (Naaad
(Noiseles))
attor (Noissked)
(a) Standard PCA
(b) IPCA
10
20
30
50)
Equivalout Number of Looks (ENL)
Figure 6: Standard PCA vs. 1PCA for feature extraction.
Figure 7: Anti-noise performance comparison.
methods achieve satisfying performance on the dataset. This is because SAR-AIReraft-1.. does not include imaging angle annotations, making the task not angle-limited and inherently less complicated. Nevertheless, methods inte- grated with TPCA still outperform their original backbones, further validating the effectiveness and generalizability of our approach across different datasets and conditions.
5. Conclusion
In the practical SAR target recognition task, data scarcity poses significant challenge. To address this issue, we have proposed L'PCA, which effectively improves the robustness of recognition models when the angle information of train- ing data is insufficient. Moreover, TPCA is designed as seg-andd-layy module, enabling broad applicability across various deep learning architectures. The core idea of TPCA is to provide low-rank estimates of natural parameters by projecting saturated model parameters, thereby mining ef- fective fow-dimensional features. This approach makes the principal components of the raw data more salient, allow- ing the network to directly learn essential features of the same target across different imaging angles, significantly mitigating the impact of angle variations. The effectiveness and generalizability of our method have been demonstrated through different angle-insun ingle-insufficiet experiments.
4.5. Anti-Noise Experiment
Compared to optical imagery, SAR images are considerably more susceptible to noise. Among various types of noise, speckle the most prevalent and detrimental in SAR data. It a form of inherent multiplicative noise due the coherent imaging mechanism of SAR, typically manifesting itself as random granular fluctuations across the image. Owing to its strong coupling with the underlying signal, speckle is noto- riously difficult to suppress. can frequently cover weak target information, such as dark shadows or small objects, and severely degrade critical image features like edges and textures, thereby affecting the performance of downstream processing tasks. Consequently, anti-nois, performance evaluation is indispensable for ensuring the reliability and effectiveness of SAR ATR methods.
A Acknowledgements
This work was supported in part by the Foundation of Na- tional Key Laboratory of Radar Signal Processing under Grant (KGJ202304, JKW202403), NSFC (62306181), and Guangdong Basic and Applied Basic Research Foundation (2024A1515010163)
We conduct anti-noise experiments under various speckle conditions. Noise levels are quantified using a standard criterion for multiplicative noise: the Equivalent Number of Looks (ENL), defined 212 /02, where μ denotes the mean pixel value and represents the standard deviation. A higher ENL value indicates lower noise and vice versa. The effects of varying noise levels on different models are illustrated in Figure 7. As shown in the figure. both ViT-B/16 and ResNet101 integrated with TPCA exhibit superior anti-noise performance compared their original counterparts. This improvement is primarily due to the ability of TPCA to extract principal components while suppressing subsidiary components, which are dominated by noise. Such selective retention is crucial for enhancing the dual robustness of our method against both angular variations and speckle.
Impact Statement
This paper aims to advance automatic target recognition for SAR imagery and machine learning. The proposed methods may have implications for military applications, such as target recognition, threat detection, and surveillance. While these technologies could enhance situational awareness and operational efficiency, we acknowledge their dual-use nature and emphasize the importance of responsible deployment. We encourage the adoption of ethical guidelines and policy frameworks to maximize societal benefit while minimizing unintended consequences.
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Gamma Distribution PCA-Enhanced Feature Learning for Mugle-Robust.oo SAR Target Recognition
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Gamma Distribution PCA-Enhanced Feature Learning for Angle-Robusy SAR Target Recognition
A. Supplementary Description of Experiments
The data are divided into the training set and validation set with ratio of 0.8. The visualization of the experimental strategy shown in Figure 8.
Trainting set Depression 17
Azimuth Robustness Test
2S1
BMP2
BRDM2)
HMPP
UNDNE
D Depression
Testing Set at
Classes
BTR60
Azimuth
BTR70
D7
Classos
0-360
AR Yeas
All nin-
T72
180*
Trriiinn sets
180
Azimuth
ZIL131
ZSU234
90
Training Sct1
Training Set.2
90
zlouth
270"
at Depresioon 17
Depression 17 270*
Training sets from opposite quadrants
Azimuth
ADR Todel
ADR Tou-2
2S1
Testing Set at
Classes
BRDM2
Depression
Azimuth & Depression Robustness Test
0-360 Azimuth
ZSU223
Figure Experimental strategy.
B. Supplementary Derivation
B.1. Solvable convex optimization objective
We suppose that ((e,g [0()) is the quadratic approximation of deviance D( (0 the point 0 and it is defined
O (60) = (xxii 0(
2wne (1)(0(()) - (0is (0:j-0!1))
(18)
+ (1 (B., 0100
where refers to the number of current iteration.
As discussed above, we have b(0) = -lo((-0) for data in the Gamma distribution, and the quadratic approximation O (()) can be replaced as:
( (000 = -((ros 0
(-1/0(0-0) -zis)
(0.s -00)
+ 2wns
-
(19)
tw// (0!0) 2 (0. 010)
Since 8(() is not constant, we have the following expression:
(8) max 0
(0 )
(20)
13

---

Gamma Distribution PCA-Enhanced Feature Learning for Angle-Robust Robust SAR Target Recognition
The objective function of MM algorithm expressed by:
M (8.g (0()) : (2 8 (/
+ 2wig (-1/0/) Iis)
(0n-019))
(21)
+u² (0s (t)
Obviously, the tangency condition of MM algorithm satisfied: M(O,, ==
) Daz,, (())) and the control
condition is also satisfied for 0: M (0,, (g) 2 (0,s ) because of Equation (20), similar to the case in traditional
generalized PCA (Landgraf & Lee, 2020).
Then summing M(1 ) to get (()))
e
= Σ M(O )
(22)
Σ
(1)
0.
(8
+3,s))
))
+C.
0
2
Further, we introduce the following substitution:
2jj
(2)
(1)
Wij
1
+ miz).
(23)
Equation (22) can be rewritten as:
(+))
Σ
(0)
0 1j
2j
(9)
+C
2
10)
2 1
0.
2ij
(I)
C
(24)
2
(t)
(1)
2
1
20
(t)
+
+C.
In k-dimensional subspace, the estimation of 0.g is defined in Equation (9), M( (())
can be rewritten
Σ // 13
[00 (7,(0)++ 113 -4 21g
[0]
+C
[UU [ (1.(0)+ -1y (0
20
+C
(25)
(t)
[ (n) + 1) - (pg) + p1) }=} C.
In the above equation, we introduce the following expressions for ease of notation:
(t) max max((())) (oxx((O))))) p. = =!f /0
(26)
Let p(t) 1 H | (1)) ... nn (0)) and q; is the diagonal element of the diagonal matrix ( M( (0))
can be further written the matrix form:
(0()) (H + 1, (1())) u(e (wO)) -Q(10) (p + (1))
M
+ +C,
(27)
14

---

Gamma Distribution PCA-Enhanced Feature Learning for Angle-Robust SAR Target Recognition
where ||-1|= is the Frobenius Norm. We minimize M by iteration to find the projection matrix U and the mean vector pa:
(0r))² (H + I. (2)))))) u (ui))
2
arg min U-U-I
+ C.
(28)
& (p()) - (00)))
(2)))
B.2. The modification of U and
Holding y fixed, the minimum of Equation (28) can be found by updating U. According to the property of Frobenius Norm, it can be rewritten in the form of a trace of matrices. Therefore, Equation (27) can be rewritten as follows:
M(
1/2 (=)))) (O
1
F
tr H (101))12) (1 ()) u (
(HH10))))))))))))) (( (1/)) (10)++1((e0))) {
(1o (anrnoo) (0 (1l0)o
H H (1+I,())))7) ffo
(29)
(id) (pl) (10 +1n(11000 (elloo
tr (w0o) H (H+In((u00)) 900
(pto) In(ml0)))) Qt0) H H (n(11)))2 (
(H+II( u ( 1)((()))) (oo)
Denote that:
H In (1(0)) 4' Q10)) (p()) + (1((0)))) )
F(t) (1), (()) p(())
(pl) + (n (e0) H) (0))
(30)
H (H+ ot) [ 11+ (1 (9)))
The minimization of (M (+) is equivalent to:
arg max M( (()) = arg max tr
(U()) p(0)
(H, (H, (1) p()) 0
(31)
(U)r
where the trace is maximized with respect orthonormal by the first & eigenvectors of F()) (H, 1l)) p(() (()).
On the other hand, holding U fixed, the optimal be found by minimizing (()))
M(e ())
|(00)) 1/2 H (1( u (Q))) (p()) + (, ())) 1
1/2
HU + (0 (0) (")0
p(()) |
1/2
= 11)0) 1/2 HU (U p()) + (Q 11 (10) -
11 ||
(32)
Let A (t)
(U (U(0) p(o) E Rad and ( = (110 -1)
-1a) E Rdxd, and we can convert the computation
15

---

Gamma Distribution PCA-Enhanced Feature Learning for Angle-Robuse SAR Target Recognition
of Frobenius Norm to the computation of trace of matrices:
O ( =tr (010)//211) (I, [ (0")))))
- (A11))1)) (110)112) ((
[ (019)112) ( (01))1/2110)
+ tr
(33)
(111)100) 101))1121111 (0)
= =t (A
ir (B(O)T n (1,,)² (10A
+ t (B
-2tr 2A (1)
+ tr (1())(((())) (1,)² (0
Then we take the partial derivative of two parts of (() with respect to (t) The partial derivative of former part in Equation (33) is as follows:
0plt)
② )um to
(34)
(O "1 I.
1, is essentially the sum of all the elements of As for the partial derivative of latter part in Equation (33), because (1 (t) the partial derivative of this part can be expressed as:
a
tr (B(O) (1.""
pp(()
1 In d (1m))
(B(O) (M(O)
Julcy
(35)
10 (1,)
tr (H (B())² mm)
Opli
(1,)² I
1, B B())
Therefore, get:
0 O (0))
0p = (B OA 1, +2,,) 1 (B())
(36)
Then let Equation (36) equals to 0 to find its extreme point:
B (BO0 = (1,))²
(37)
16

---

Gamma Distribution PCA-Enhanced Feature Learning for Angle-Robuse SAR Target Recognition
In the above equation, ( (B can be simplified:
B ( (())
[a U ((()))) 1 (00
- (00 -0000
(38)
=La (U0)7
=-
B
Further substitute A (0) = HUO (UlO) p(() and ((t) = (T")( Ia) into Equation (37), and (z finally
derived as follows:
R (+) (1 ( "00000
(39)
# (11)) (U - -p(t) 1
C. Detailed Composition of Experimental Dataset.
C.1. Description of MSTAR dataset
There are three imaging depression angles in MSTAR: 15,, 17 and 30", each depression consists of data from complete azimuth angles (0--360)). MSTAR has divided all images into four quadrants according to their azimuth angles, 09-90", -0--180" 1809-270%, 27000 2709-360", respectively. While dataset at depression 15 and 17 both include categories: BMP2, BTR70, T72, 2S1, BRDM2, D7, BTR60, T62, ZIL131, ZSU234, dataset at depression 30 only includes categories: 2S1, BRDM2 and ZSU34.
Table 5: Description of MSTAR dataset.
Depression
17
15a
30
Azimuth
0"-90"
90--180"
180"-270"
270--360"
0"-360"
0"-360"
2S1
71
74
79
75
274
288
BMP2
61
60
51
61
195
N/A
BRDM2
72
73
274
420
BTR60
68
68
60
60
195
N/A
BTR70
56
52
61
64
196
N/A
D7
70
79
274
N/A
T62
73
74
80
72
273
N/A
T72
62
56
51
63
196
N/A
ZIL131
72
274
N/A
ZSU234
72
73
80
74
274
406
C.2. Description of SAR-ATReraft-1.0 dataset
The images this dataset are obtained from satellite Gaofen-3 with single polarization, a spatial resolution of meter and Spotlight imaging mode. This dataset contains 4,368 images and covers aircraft categories, namely A220, A320/321, A330, ARJ21, Boeing737, Boeing787 and other, where "other" represents aircraft instances that do not belong to the mentioned six categories.
Table Deseription of SAR-AIRcraft-LO0 dataset.
Class
A220
A320/321
A330
ARJ21
Boeing737
Boeing787
other
all
Number
2065
939
290
713
1495
1677
2041
9220
17


Reviews:
[{"review_id": "dxdI5H4ePZ", "reviewer": "Reviewer_kgn7", "summary": "This paper addresses a key challenge in SAR target recognition: the sensitivity of deep learning models to variations in azimuth and depression angles, which cause significant shifts in scattering characteristics. The authors propose a Gamma-Distribution Principal Component Analysis (ΓPCA) model to extract angle-invariant features by leveraging the statistical properties of SAR data. ΓPCA derives consistency convolution kernels without requiring parameter updates, thus adding no computational burden. The method is evaluated on the MSTAR dataset using ResNet and ViT backbones, demonstrating improved robustness to angle-induced distributional discrepancies. The key contributions include: (1) a novel ΓPCA framework tailored to SAR’s non-Gaussian statistics, (2) angle-insensitive feature extraction via a fixed projection matrix, and (3) seamless integration into existing architectures. While the approach shows promise, its generalizability to more diverse SAR targets and complex clutter scenarios remains to be further validated. The work stands out for its principled statistical modeling and parameter-free design, offering a potential advance over ad-hoc multiview fusion or data augmentation techniques.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> Comment1. Dependence on Gamma distribution assumptions (may not hold for all targets/clutter). \n\n**Answer1:** \nWe appreciate this insightful observation regarding the distributional assumption. To provide clarification:\n1. Theoretical and empirical validation: The applicability of Gamma distribution to SAR image statistics is theoretically grounded and empirically validated, as evidenced by [R1, R2]. These works demonstrate its effectiveness in capturing the multiplicative speckle noise and scattering characteristics inherent to SAR systems.\n2. Scope of the Gamma assumption: Our method adopts the Gamma distribution to model the reflection intensity of the entire SAR image (encompassing targets and clutter collectively). Of course, statistical distributions such as generalized Gaussian, fisher, etc. can be used, but these distributions are complex, and the corresponding parameter estimation and derivation of matrix $U$  are challenging to realize in practice.\n3. Robustness of approach: By operating at this holistic statistical level, our method maintains robustness across diverse imaging scenarios, even when local target/clutter distributions vary. This aligns with established practices in SAR image analysis, where system-level statistical modeling often supersedes component-specific assumptions.\n\n**Reference:**\n\n[R1] Li, H. C., et al. On the empirical-statistical modeling of SAR images with generalized gamma distribution. *IEEE Journal of selected topics in signal processing*, 2011.  \n[R2] Nascimento, A. D. C., et al. Compound truncated Poisson gamma distribution for understanding multimodal SAR intensities. *Journal of Applied Statistics*, 2023.\n\n----\n\n> Comment2. Limited discussion on computational efficiency vs. performance trade-offs. \n\n**Answer2:** \nThanks for your detailed comment. Below, we analyze the computational complexity of our proposed ΓPCA method and compare it with standard PCA. \n\nThe primary computational cost of ΓPCA mainly arises from the derivation using the MM.\n\n1) Update the mean vector $\\mu^{(t+1)}$ by equation (13) in our manuscript:\n\nThe update involves matrix multiplications and a linear system solve. Dominant operations include:\n\n- $O(d^2k)$ for $U^{(t)}(U^{(t)})^T$ \n\n- $O(nd^2)$ for matrix multiplication of $H(·)(·)^T$\n\nHere, $n$ is the number of samples, $d$ is the original feature dimension and $k$ is the reduced dimension. Therefore, the total complexity for $\\mu^{(t+1)}$ is $O(nd^2+d^2k)$.\n\n\n2) Calculate the deviance $M(\\boldsymbol{\\Theta}|\\boldsymbol{\\Theta}^{(t)})$ by equation (27):\n\nComputes the Frobenius norm of an $n \\times d$ matrix after rank-$k$ projections. Dominate terms:\n\n- $O(ndk)$ for matrix products with $\\mathrm{H}+1_n\\mu^{(t)}$ with $U^{(t)}$.\n\n- $O(d^{2}k)$ for $U^{(t)}(U^{(t)})^T$ operations.\n\n The total complexity for $M(\\boldsymbol{\\Theta}|\\boldsymbol{\\Theta}^{(t)})$ is $O(ndk+d^2k)$.\n\n3) Calculate the matrix $F^{(t)}$ by equation (30) and update $U^{(t+1)}$ to the first $k$ eigenvectors of $F^{(t)}$:\n\n- $O(nd^{2})$ for the matrix conjunction of $(·)^{T}Q^{(t)}(·)$.\n- $O(d^{3})$ for singular value decomposition (SVD).\n\nTherefore, the total complexity for $U^{(t+1)}$ is $O(nd^{2}+d^3)$.\n\nFrom the above discussion, the overall computational complexity of ΓPCA algorithm is approximately $O(nd^2+d^2k)+O(ndk+d^2k)+O(nd^2+d^3)\\approx O(nd^2+d^2k+ndk+d^3).$\n\nIn practical applications, parameters typically exhibit the following relationships: $n \\gg d \\gg k$. Therefore, the overall computational complexity of ΓPCA algorithm is $O(nd^2+d^3)$.\n\n\nIn contrast, the computational complexity of the standard PCA method is mainly determined by two key operations:\n\n1. Computation of covariance matrix: $\\Sigma=\\frac{1}{n}\\mathbf{X}^T\\mathbf{X}$, which has a computational complexity of $O(nd^2)$.\n\n2. Eigenvalue decomposition of covariance matrix: $\\Sigma=\\mathrm{U}\\Lambda\\mathrm{U}^T$, with a computational complexity of $O(d^3)$, equivalent to the SVD operation in ΓPCA . \n\nHere, $\\text{X}\\in\\mathbb{R}^{n\\times d}$ is the data matrix, $U$ is the eigenvector matrix, and $\\Lambda$ is the eigenvalue diagonal matrix. Consequently, the overall computational complexity of standard PCA is approximately $O(nd^2+d^3)$.\n\nThe computational complexity is identical to that of standard PCA under the condition of $n \\gg d \\gg k$. The overall computational cost of our method is on the same order of magnitude as standard PCA, making the new algorithm computationally acceptable.", "claims_and_evidence": "The paper makes several strong claims, most of which are supported by empirical evidence, but some aspects could benefit from further clarification or validation.", "methods_and_evaluation_criteria": "To verify the effectiveness of the proposed method, the authors conducted a variety of robustness experiments on the MSTAR benchmark dataset. Experimental results show that the ΓPCA model can significantly improve the performance of existing models when facing angle changes. In addition, the ΓPCA model does not require parameter updates, so it does not bring additional computational burden to the network.\n\nThe proposed ΓPCA method and evaluation criteria (i.e., the MSTAR benchmark dataset) are meaningful for addressing the angle variation problem in SAR target recognition. The MSTAR dataset contains SAR images at different angles, which provides a suitable testing environment for evaluating the performance of the model at different observation angles. Therefore, it can be considered that the proposed model and evaluation method are designed for this problem or application and are reasonable.", "theoretical_claims": "N/A, there is no theoretical claims in the paper.", "experimental_designs_or_analyses": "The experimental analyses are sound.", "relation_to_broader_scientific_literature": "The contribution of this paper is to propose a new feature extraction method that is particularly suitable for SAR data and can improve the robustness of the model in the face of angle changes. These contributions provide new perspectives and solutions compared to existing methods in the literature, especially in dealing with the non-Gaussian properties and angle changes of SAR data.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "Strengths:\n\nStrong motivation: Tackles a well-known but underexplored problem (angle sensitivity) in SAR ATR.\n\nNovelty: ΓPCA extends PCA to Gamma distributions, aligning with SAR physics.\n\nPracticality: No added parameters or training overhead.\n\nWeaknesses:\n\nDependence on Gamma distribution assumptions (may not hold for all targets/clutter).\n\nLimited discussion on computational efficiency vs. performance trade-offs.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "Please refer to the entries above.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "OratfEqP7z", "reviewer": "Reviewer_8Si4", "summary": "This paper proposes Gamma-PCA, a Gamma-distribution Principal Component Analysis model, to enhance angle-robust SAR target recognition by leveraging SAR's non-Gaussian statistics. The method derives consistent projection kernels to capture angle-invariant features without parameter updates, seamlessly integrating with CNN/Transformer backbones. Experiments on MSTAR demonstrate its effectiveness in mitigating performance degradation under significant azimuth/depression angle variations while maintaining computational efficiency.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely appreciate your positive feedback on our work and are truly grateful for the time and effort you have dedicated to reviewing our manuscript. Should you have any additional comments or suggestions, we would be delighted to engage in further discussion and improvement.", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "Yes.", "experimental_designs_or_analyses": "Yes.", "relation_to_broader_scientific_literature": "NA", "essential_references_not_discussed": "NA", "other_strengths_and_weaknesses": "see the comments.", "other_comments_or_suggestions": "NA", "questions_for_authors": "NA", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "L9ztO0LXJH", "reviewer": "Reviewer_otus", "summary": "The paper proposes a Gamma-Distribution Principal Component Analysis ($\\Gamma$PCA) model to enhance the robustness of Synthetic Aperture Radar (SAR) target recognition against angle variations. The key idea is to leverage the Gamma distribution to extract low-rank features that are invariant to changes in azimuth and depression angles. The authors derive a consistency projection matrix from $\\Gamma$PCA to construct convolutional kernels that capture angle-insensitive information. The proposed method is integrated into deep learning backbones (ResNet and ViT) and validated on the MSTAR dataset, demonstrating improved robustness and performance compared to baseline models.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> Comment1. The experimental validation is limited to a single dataset (MSTAR). \n\n**Answer1:** \nThanks for pointing out this important concern. We acknowledge this limitation. Unlike optical imagery, SAR data is challenging to obtain at scale due to sensor costs, operational constraints, and the factor of security sensitivities. Therefore, most studies [R3-R6] rely solely on MSTAR for evaluation. To enhance experimental validation, we construct a dataset based on SAR-AIRcraft-1.0, a widely used public SAR aircraft detection dataset. Compared to MSTAR, SAR-AIRcraft-1.0 offers richer data diversity and more complex operational scenarios. We extract individual targets from the original imagery to create a dedicated SAR ATR dataset. Below, we provide a detailed description of its composition.\n\n| Class  | A220 | A320/321 | A330 | ARJ21 | Boeing737 | Boeing787 | other | all  |\n| ------ | ---- | -------- | ---- | ----- | --------- | --------- | ----- | ---- |\n| Number | 2065 | 939      | 290  | 713   | 1495      | 1677      | 2041  | 9220 |\n\n\nSince SAR-AIRcraft-1.0 lacks diverse and separable imaging angles [R7-R9], a standard target recognition strategy and conducted experiments are implemented. Specifically,  80% of the data was used for training, and the remaining 20% for testing. The experimental results are outlined below:\n\n\n| Methods        | Overall Acc. (%) |\n| -------------- | ---------------- |\n| Resnet101      | 97.33±0.02       |\n| ΓPCA+Resnet101 | _**98.67±0.01**_ |\n| ViT-B/16       | 97.05±0.03       |\n| ΓPCA+ViT-B/16  | _**98.12±0.02**_ |\n| Swin-B         | 97.41±0.01       |\n| ΓPCA+Swin-B    | _**98.32±0.01**_ |\n\n\nCompared to the MSTAR dataset, SAR-AIRcraft-1.0 is lack of imaging angle annotations, and thus the angle-related constraints are eliminated in fact.  Consequently, in this experiment on SAR-AIRcraft-1.0, most of the evaluated networks exhibit superior performance. But notably, existing mainstream networks incorporating our ΓPCA consistently outperform their original backbones, even under this less constrained evaluation scenario. These results validate the generalizability of our method across diverse data characteristics and scenarios.\n\n**Reference:**\n\n[R3] Huang, Z., et al. Physics inspired hybrid attention for SAR target recognition. *ISPRS Journal of Photogrammetry and Remote Sensing*, 2024.  \n[R4] Zhang, L., et al. Optimal azimuth angle selection for limited SAR vehicle target recognition. *International Journal of Applied Earth Observation and Geoinformation*, 2024.  \n[R5] Wang, R., et al. MIGA-Net: Multi-view image information learning based on graph attention network for SAR target recognition. *IEEE Transactions on Circuits and Systems for Video Technology*, 2024.  \n[R6] Li, W., et al. Hierarchical disentanglement-alignment network for robust SAR vehicle recognition. *IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing*, 2023.  \n[R7] Zhirui, W., et al. SAR-AIRcraft-1.0: High-resolution SAR Aircraft Detection and Recognition Dataset. *Journal of Radars*, 2023.  \n[R8] Huang, B., et al. Scattering Enhancement and Feature Fusion Network for Aircraft Detection in SAR Images. *IEEE Transactions on Circuits and Systems for Video Technology*, 2024.  \n[R9] Zhou, J., et al. DiffDet4SAR: Diffusion-based aircraft target detection network for SAR images. *IEEE Geoscience and Remote Sensing Letters*, 2024.\n\n---\n\n\n> Suggestions and writing issues: \n\n**Answer2:** \nThank you for your careful review and valuable suggestions. We will thoroughly proofread and incorporate these revisions into the final version of the paper.", "claims_and_evidence": "* The authors take the non-Gaussian nature of SAR into consideration and deveplot the $\\Gamma$PCA model accordingly.\n* The experimental results on the MSTAR dataset demonstrate that $\\Gamma$PCA can improve the robustness of ResNet and ViT models against angle variations.", "methods_and_evaluation_criteria": "The evaluation criteria look good.", "theoretical_claims": "There is no proof for theoretical claims to be checked.", "experimental_designs_or_analyses": "* The authors conduct experiments on the MSTAR dataset to evaluate the azimuth robustness and azimuth & depression robustness of the proposed method.\n* Experiments with state-of-the-art models (e.g., ViT and ResNet) further validate the effectiveness of the proposed method.", "relation_to_broader_scientific_literature": "The key contributions are not related to the broader scientific literature.", "essential_references_not_discussed": "Based on the information provided, there are no obvious essential references missing.", "other_strengths_and_weaknesses": "Strengths:\n* The paper presents a novel approach to addressing a significant challenge in SAR target recognition: robustness to angle variations.\n* The method is demonstrated to be effective across different deep learning architectures (ResNet and ViT), highlighting its generalizability.\n\nWeaknesses:\n* The experimental validation is limited to a single dataset (MSTAR), which may restrict the generalizability of the findings.", "other_comments_or_suggestions": "* Line 143, the formulation lacks ending period.\n* Figure 4 can be lifted to former pages.\n* Figures 5 and 6, the same label repeats 6 times, please simplify it. Also, the font can be larger.", "questions_for_authors": "There is no further questions for the authors.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "2TRWXXf29E", "reviewer": "Reviewer_Ub2X", "summary": "This paper proposes a Gamma-Distribution Principal Component Analysis (ΓPCA) method for angle-robust SAR target recognition. The key idea is to integrate the Gamma distribution into PCA to account for the statistical properties of SAR data, thereby enhancing feature extraction across varying observation angles. The authors claim that ΓPCA-derived convolution kernels can capture angle-invariant features without adding additional computational burden. The method is evaluated on the MSTAR dataset with ResNet and ViT backbones.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely appreciate the reviewer’s valuable comments. We will enhance the related work and include more experimental results in our final version. Below are our detailed responses to each point.\n> Q1. MSTAR is small.\n\n**A1:** We constructed a new dataset based on SAR-AIRcraft-1.0. A detailed description of the dataset and the experimental results can be found in **Answer1** for **Reviewer Otus**.\n\n> Q2. SAR-specific baselines.\n\n**A2:** We incorporat the ASC-based method MSNet-PIHA [R10] into our experiments. The comparison results are shown below.\n\n|Methods|A-R Test|AD-R Test| Overall Acc. (%)|\n|---|---|---|---|\n|MSNet-PIHA| 68.08±0.03|55.75±0.06| 64.20±0.03|\n|ΓPCA-Resnet|73.79±0.02|52.96±0.05|67.23±0.01|\n|ΓPCA-ViT|78.48±0.01|70.67±0.16|76.02±0.01|\n\nMSNet-PIHA performs well when trained on diverse azimuth angles [R10] but struggles with limited-angle data, which can lead to overfitting and suboptimal predictions. While ASC information enhances feature extraction with limited data, severe azimuth constraints remain challenging. In contrast, ΓPCA demonstrates better angle robustness.\n\n[R10] Huang, Z., et al. Physics inspired hybrid attention for SAR target recognition. *ISPRS*, 2024. \n\n> Q3. Statistical significance tests.\n\n**A3:** We conduct significance tests by repeating all experiments in our paper three times and reported the mean and variance to reduce the influence of randomness. Part of the updated experimental results are presented below.\n\nTable R1. Partially updated recognition results of Table 1.\n\n|Methods|A-R Test|AD-R Test|Overall Acc. (%)|\n|---|---|---|---|\n|Resnet|72.57±0.25|43.95±0.01|63.62±0.14|\n|ViT|78.12±0.04|60.11±0.24|72.45±0.04|\n|ΓPCA-Resnet|73.79±0.02|52.96±0.05|67.23±0.01|\n|ΓPCA-ViT|78.48±0.01|70.67±0.16|76.02±0.01|\n\nMore results can be found at: https://anonymous.4open.science/r/ICML2025-E1F8. \n\n> Q4. Theoretical proof of ΓPCA is better than PCA.\n\n**A4:** Previous work in [R11] has proven that generalized PCA based on exponential family distributions theoretically outperforms traditional PCA. As a variant, our ΓPCA, based on the Gamma distribution, inherits this theoretical advantage.\n\n[R11] Landgraf, A. J., et al. Generalized principal component analysis: Projection of saturated model parameters. *Technometrics*, 2020.\n\n> Q5. Theoretical link between angle-robustness and ΓPCA projections.\n\n**A5:** Under different orientation observation angles, the statistics of the same class of SAR targets are similar. Base on this characteristic, ΓPCA assumes that the SAR target image follows the Gamma distribution, and then derives ΓPCA projection which can extract the principal component information, i.e., the low-rank information, of these SAR targets. Then the projection matrix is used to construct the convolution kernel to extract the SAR target features, which are characterized by a low sensitivity to angle.\n> Q6. Comparison of different components.\n\n**A6:** We provide an isolated comparison of different components (see Table 3, page 8, PCA-ViT vs. ΓPCA-ViT). As demonstrated in Table 3, ΓPCA consistently outperforms conventional PCA with ViT backbone for SAR data. For example, with a kernel size of 17, PCA-ViT and ΓPCA-ViT achieve accuracies of 73.52\\% and 76.01\\%, respectively, demonstrating its effectiveness in non-Gaussian SAR data analysis. \n> Q7. Effects of different settings.\n\n**A7:** Kernel size analysis is presented in Sec. 4.3 (Page 8, Left Column, Lines 410–434), with quantitative comparisons in Table 3. To isolate the impact of ΓPCA, all experiments exploit the same training settings (optimizer and learning rate: the default configurations of the original backbones, batch size: 32), ensuring a fair comparison within the scope of the paper on fundamental ΓPCA theory for SAR data.\n> Q8. Real-World Relevance.\n\n**A8:** We conducted experiments under various noise conditions, focusing on multiplicative noise, i.e. speckle, the most common noise type in SAR image. Noise levels are quantified by the Equivalent Number of Looks (ENL), defined as: μ^2/σ^2, where μ and σ are mean and standard deviation of pixel values. A higher ENL value indicates lower noise. \n\n|ENL|ViT|ΓPCA+ViT|\n|---|---|---|\n|Noiseless|70.61|76.01|\n|20|51.51|69.60|\n|10|47.22|62.19|\n|5|45.30|56.23|\n|2|41.20|48.09|\n\nMore results are available at: https://anonymous.4open.science/r/ICML2025-E1F8. These results show the anti-noise performance of ΓPCA, attributed to its ability to extract principal components while filtering out noise-dominated ones.\n\nOur study primarily addresses the challenge of angle-induced performance degradation in SAR ATR. Extensive experiments validate the dual robustness of ΓPCA to angular variations and speckle noise. While sensor types and imaging artifacts are also important for SAR ATR, these aspects fall beyond the scope of our current work, as they pose distinct research challenges. We will further clarify the influence of these two aspects in the final version.", "claims_and_evidence": "no", "methods_and_evaluation_criteria": "The study is based entirely on MSTAR, which, while widely used, is a relatively small dataset that may not reflect real-world SAR conditions.\n The authors compare ΓPCA with ViT, ResNet, and a few SOTA models, but crucial SAR-specific baselines (e.g., UIU-Net, ASC-based models) are missing.\nNo statistical significance tests (e.g., t-tests, Wilcoxon tests) are performed to validate improvements.", "theoretical_claims": "The paper extends PCA to a Gamma-distributed setting, claiming it better represents SAR image statistics.\nThe derivation of ΓPCA is mathematically rigorous, but no theoretical proof is provided that ΓPCA is better than traditional PCA or other low-rank approximations.\nThe study assumes that angle variation can be mitigated by low-rank feature extraction, but does not establish a formal theoretical link between angle-robustness and Gamma-PCA projections.", "experimental_designs_or_analyses": "The paper compares against ViT, ResNet, and some recent models but ignores SAR-specific methods such as UIU-Net or ASC-based approaches.\n The study shows ΓPCA with different architectures but does not isolate the contributions of different components (e.g., Gamma assumption vs. PCA).\nThe study does not explore the effect of different kernel sizes, training strategies, or dataset variations.\nReal-World Relevance: No analysis is provided on how the method performs under different noise conditions, sensor types, or imaging artifacts.", "relation_to_broader_scientific_literature": "The paper references many relevant SAR and deep learning studies but fails to discuss critical related work in:\nStatistical modeling of SAR images (Weibull, K-distribution, etc.)\nDeep learning for SAR ATR (UIU-Net, few-shot SAR methods)\nAngle-invariant feature learning in computer vision", "essential_references_not_discussed": "no", "other_strengths_and_weaknesses": "no more.", "other_comments_or_suggestions": "no more", "questions_for_authors": "no more", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {"ethical_review_flag": "Flag this paper for an ethics review."}}]

Author rebuttal:

Meta-review: This paper proposes a method to  address the sensitivity of deep learning models to variations in azimuth and depression angles in SAR data, which is termed  Gamma-Distribution Principal Component Analysis. After the rebuttal and discussion, three reviewers agree on the benefit of the approach in extending PCA to Gamma distributions, aligning with SAR physics, while one of the reviewers finds the theoretical and practical merits of ΓPCA not sufficiently validated and the novelty limited. While the AC agrees that the novelty is limited to this one aspect, the paper is still sufficiently motivated, theoretically sound, and it addresses an interesting problem from a signal processing perspective.
Decision: Accept (poster)