MANIFOLD-BASED SHAPLEY FOR SAR RECOGNIZATION NETWORK EXPLANATION

We denote the network as $f$, with the sample to be explained represented as $I\in\mathbb{R}^{C\times W\times H}$, and the data manifold as $U\in\mathbb{R}^{1\times L}$. We leverage the StyleGAN2 [13] framework to train the generator, denoted as $G$, which learns the underlying manifold structure and decodes the mapping, resulting in $I=G(U)$. Subsequently, we proceed to train the reconstructor, denoted as $R$, using Image2StyleGAN [14] to obtain the encoding mapping: $U=R(I)$.

Now, we have successfully established a mutual mapping relationship between high-dimensional data and low-dimensional manifolds, denoted as $I\Leftrightarrow U$.

In section 2.1, we have successfully mapped the high-dimensional feature to the low-dimensional manifold. This mapping allows us to compute the shapley value on the manifold directly. For a manifold comprising $L$ features, we can express formula (LABEL:eq1) as follows:

In this equation, $S_{U}$ represents a subset $S_{U}\subseteq L=\{1,2,\ldots,l\}$ of the feature set $L$, while $S_{U}\subseteq L\backslash\{p\}$ indicates that $S_{U}$ is a subset of $L$ and does not include feature $p$. The symbol $v$ denotes the value function. For a specific sample denoted as $u^{\prime}$, the function $v$ can be expressed as:

We have successfully established a mapping between the high-dimensional feature, $I$, and the low-dimensional manifold, $U$. However, the shapley value represents feature importance and the challenge now lies in determining how to mapping this importance effectively.

We adopte a gradient-based approach to assess this importance mapping. First, define $U_{p}$ as an element within the position-encoded $p$-index manifold. It is intuitive to understand that slight perturbations in $U_{p}$ will induce changes in the high-dimensional features represented by $I$. We define this mapping as $\gamma$, unit-importance-mapping, which can be expressed mathematically as follows:

In this equation, $I$ represents an image, a coalition of pixels. Formula (3) can be interpreted as mapping the shapley value from the low-dimensional manifold to the high-dimensional feature space for each feature:

To preserve the shapley characteristics within the shapley methods, we express the shapley value $\phi_{v}\left(U_{p}\right)$ of $U_{p}$ as a weighted sum of the individual shapley values $\phi_{v}\left(I_{c,w,h}^{p}\right)$ within the high-dimensional feature space $I$:

Combining Formulas (4) and (5):

For all elements in manifold $U$:

Now, we have obtained the saliency map denoted as $M_{\text{manifold }}=\phi_{v}^{M}\left(I_{c,w,h}\right)$.

Manifold-based successfully addresses the issue of traditional Shapely method’s overlooking on feature interdependence. Nevertheless, the manifold dimension somewhat constrains this method, and altering the manifold dimension necessitates relearning the mapping relationship between high-dimensional features and low-dimensional manifold—a process that incurs substantial computational costs. To tackle this problem, we propose a hybrid approach by reintegrating the feature-independent traditional shapley method into the manifold shapley method:

The fusion coefficient $\alpha\in[0,1]$ is determined with the objective of minimizing average drop in confidence, which can be expressed mathematically as:

among them, $\otimes$ stands for Hadamard product. The set $t=\{1,2\ldots,C\}$ signifies that the network encompasses a total of $C$ confidence categories, with $t$ representing the Top-1 confidence category assigned to the input $I$. Utilizing equation (8), we can derive the fusion coefficient $\alpha$ and subsequently compute Fusion-Shap, denoted as $M_{fusion}$, elements in saliency map are Fusion-shapley value $\phi_{v}^{F}$.

In contrast to optical images and human intuitive perception, it is more difficult to understand DNNs’ decision-making logic on SAR images due to the intricate nature of SAR imaging mechanism. SAR recognition network’s decision-making is not solely contingent upon the target area; interference spots and shadow regions also holds significance in network decision-making. Figure 3 illustrates visualization results of Grad CAM [17], LRP [18], IG [19], SG [20], SHAP [3] and Fusion-SHAP. Table 1 shows the results of subjective evaluation (explanation validity from lowest to highest: 1-10).

Taylor Interactions [15] considers various explanation methods by aggregating the individual effects $\varphi(k)$ and the interaction $I(k)$ of characteristic effects, each governed by distinct rules. Subsequently, it introduces three criteria for assessing the attribution method’s reliability: Low Approximation Error (LAE), No Unrelated Allocation (NUA) and Complete Allocation (CA). The results of the attribution validation are presented in Table 1.

(1) compute the shapley values for high-dimensional features and low-dimensional manifolds. Both of these components notably satisfy the three computed aforementioned characteristics. The contribution of each feature within manifold can be decomposed as follows:

Fusion-Shap can be viewed as the redistribution of shapley values through manifold information. Since $\sum\phi_{v}(i)=\sum\phi_{v}(p)$, this distribution still adheres to the shapley value properties. To illustrate this more intuitively, the shapley value of feature $U_{p}$ within manifold $U$ is reassigned to the high-dimensional space $I$, corresponding to the cumulative shapley values of multiple high-dimensional features. We reasonably hypothesize that the shapley value of feature $U_{p}$ in manifold $U$ is mapped to the high-dimensional space and allocated to two or more features. Thus, we can express Fusion-Shap in the form of Taylor interaction:

This Taylor interaction formulation satisfies three axioms, substantiating the effectiveness of Fusion Shap.

We introduce fidelity and sensitivity [16] for evaluating model performance. For a given method $\psi(f,I)$, we consider a given meaningful perturbation $P$ with probability distribution $\mu_{P}$. Infidelity of $\psi$ can be defined as:

and the sensitivity index can be expressed as:

among them, parameter $r$ denotes the radius of the domain or field under consideration.

Table 1 presents the results of infidelity and sensitivity. F-shap meets the axiom verification criteria and achieves the best infidelity, sensitivity, and subjective evaluation metrics.

We have examined several manifold dimension configurations and calculated the fidelity and sensitivity of network explanations. Figure 4 showcase the manifold dimensions and model reliability. It is evident that increasing the manifold dimension reduces model’s infidelity and sensitivity.