# The Paradigm Shift in Hyperspectral Image Compression: A Neural Video Representation Methodology
## 3 Proposed Methods
### The Necessity of a Paradigm Shift in Hyperspectral Image Compression

Table 1: _Cont._It has been provided that the expected loss \(L(\Theta_{1}+\Theta_{2})\) has the following relationship with the empirical loss: \(\hat{L}(\Theta_{1}+\Theta_{2})\)[28; 29], so we will not provide further discussion. \[L(\Theta_{1}+\Theta_{2})\leq\hat{L}(\Theta_{1}+\Theta_{2})+2\mathcal{R}_{N}( \cdot)+\mathcal{O}\bigg{(}\frac{1}{\sqrt{N}}\bigg{)} \tag{4}\]  
\(\mathcal{R}_{N}(\cdot)\) reflects the maximum potential variability of the loss through random variables in the parameter space. From a heuristic perspective, the feasible domain of \(\mathbf{Y}\) cannot reflect the pattern of the compression task when input images are limited, so this transforms it into a simple pattern matching task. For Problem (2):  
Under the condition of a fixed generalization error, the sample complexity of the model, i.e., the minimum number of required samples, is exponentially related to the dimensionality of the feature space. To enable the model to learn similar statistical information from the data, the number of data points must be increased. For example, to achieve the distortion of \(\epsilon\), the number of training items required is as follows [29; 30; 31]:  
\[N^{*}=\mathcal{O}\left(\frac{\mathcal{O}\big{(}k^{2}CI\big{)}+\log\Big{(} \frac{1}{\delta}\Big{)}}{\epsilon^{2}}\right) \tag{5}\]  
\(\delta\) is a probability tolerance parameter; specifically, it is the acceptable probability of generalization failure, i.e., the proportion of likelihood by which our expected generalization performance may deviate. \(k\) denotes the kernel size of the CNN, \(C\) denotes the number of CNN kernels, and \(I\) denotes the input channel. Typically, \(C\) is several times larger than \(I\); here, we assume that \(C=kI\). Then, we have  
\[N^{*}=\mathcal{O}\left(\frac{\mathcal{O}\big{(}k^{3}I^{2}\big{)}+\log\Big{(} \frac{1}{\delta}\Big{)}}{\epsilon^{2}}\right) \tag{6}\]  
It can be observed that as the number of input channels increases, the required size of the training dataset increases quadratically, and the existing open-source hyperspectral datasets fall far short of the training requirements. In this example, by using an RGB dataset, we randomly selected a certain number of data points and trained the model multiple times to obtain an average, resulting in the curve shown in Figure 1. (The testing images are not included in any training set). Figure 1: The variance in the PSNR on different number of training images using the method of [PERSON] et al. [32] (the bpp setting is over 0.5 bpp). For Problem (3):  
As discussed in Problem (1) and Problem (2), the learning-based method is not realistic for the hyperspectral image compression scenario; therefore, we consider employing the index to replace the learning process towards latent features. In other words, we cut down the constraints of Formulation (3) and transform the feature domain learning task into an index-to-pixel learning task. Instead of employing significant computational resources to learn feature dependencies in high-dimensional hyperspectral image data, we prefer to learn a simpler task at a much lower cost, even though this task may not yet possess generalizability. The formulation of the index-to-pixel learning task is as follows:  
\[ \begin{split}&\min_{\Theta_{2}}\bigl{|}\bigl{|}\mathbf{x}- \mathcal{F}_{\Theta_{2}}(\Xi)\bigr{|}\bigr{|}\\ &\text{s.t.}\ \Theta_{2}\in\mathbf{Size}(\Theta_{2})\end{split}  \tag{7}\]  
This is easier to train because the term of \(\mathbf{Size}(\Theta_{2})\) and the index \(\Xi\) can be a constant when the input and the compression ratio are deterministic. We compare the complexity of Formulations (3) and (7) when compressing \(N\) images (assume the required images are satisfied and each image is with the shape of \((I,W,H)\)):  
\[\underbrace{\mathcal{O}\left(\underbrace{ \begin{subarray}{c}\mathcal{k}^{2}C^{ 2}I^{2}WH\\ \text{contained in network itself contained in training}\end{subarray}