## 5 Algorithm Design
### Multi-Modal Feature Modeling  
We address the proactive push task for remote sensing data by categorizing input data into two modalities--image and attributes--for multi-modal feature modeling. For remote sensing image data, we consider visible-light images and SAR images. To address the heterogeneity of multi-source remote sensing data and onboard computing constraints in the integrated satellite-terrestrial networks, we propose dual-modal adaptation optimizations based on the lightweight MobileNetV3-Small architecture:  
**Heterogeneous input layer adaptation:** To address the data discrepancy between single-channel SAR images and three-channel visible-light images, the input layer's convolution kernel dimensions are restructured--adjusting the original \(3\times 3\times 3\) kernel to a \(3\times 3\times 1\) configuration. This reduces the number of parameters in the initial layer by66.7% while retaining the lightweight characteristics of depthwise separable convolutions, ensuring channel compatibility for SAR data feature extraction.  
**Pre-training strategies:** Distinct approaches are adopted for visible-light and SAR modalities. For visible-light image processing, ImageNet pre-trained weights are directly loaded, with the first three convolutional layers frozen to leverage the transfer learning for inheriting general visual semantic representations, while only the deep feature extraction layers (Layers 4-16) and the classification head are fine-tuned to accelerate convergence. For single-channel SAR images, a three-stage adaptation is implemented: (1) the input layer adaptation averages the weights of the original three-channel convolution kernel along the channel dimension to align with the SAR's single-channel input; (2) Intermediate layers reuse pre-trained weights, exploiting the cross-modal generalization capability of depthwise separable convolutions; and (3) Kaiming normal initialization is applied to the classification head to mitigate the channel dimension mismatch and gradient vanishing issues. This strategy ensures efficient cross-modal compatibility while preserving computational efficiency.  
The attribute data include remote sensing attribute metadata, satellite cache states, and user cache states from the previous \(\tau\) time slots. We employ a three-layer MLP to jointly encode the remote sensing attribute data and satellite cache states:  
\[f^{(1)}_{attr}=ReLU(W_{1}\bullet[x_{meta};x_{cache_{state}}]+b1), \tag{12}\]  
\[f^{(2)}_{attr}=ReLU(W_{1}\bullet f^{(1)}_{attr}+b_{2}), \tag{13}\]  
\[f_{attr}=W_{3}\bullet f^{(2)}_{attr}+b_{3}\in\mathbb{R}^{d}, \tag{14}\]  
where \(x_{meta}\) denotes the remote sensing attribute data, \(x_{cache_{state}}\) represents the satellite cache state, and \(W_{1}\),\(W_{2}\), \(W_{3}\) are the weight matrices of each layer, with \(b_{1}\), \(b_{2}\), \(b_{3}\) being the corresponding bias terms.  
An LSTM module is employed to model the temporal evolution of user cache states over the previous \(T\) time slots  
\[(x_{lstm},(h_{T},c_{T}))=LSTM(x_{cache_{user}},(h_{0},c_{0})), \tag{15}\]  
\[f_{cache_{user}}=x^{cache_{user}}_{t}[-1,:,:]\in\mathbb{R}^{d}, \tag{16}\]  
where \(x_{lstm}\) contains the hidden states across \(T\) time steps, \(h_{T}\) and \(c_{T}\) represent the final hidden and cell states. The last time-step hidden state is extracted as the temporal feature of user cache dynamics. The MLP-extracted attribute features \(f_{attr}\) and LSTM temporal features are fused via the residual addition  
\[f^{fused}_{attr}=f_{attr}+f_{user}\in\mathbb{R}^{d}. \tag{17}\]