# Semi-Supervised Deep Subspace Embedding for Binary Classification of Sella Turcica
## 2 Materials and Methods
### Proposed Semi-Supervised Deep Subspace Embedding (SSLDSE) Architecture
The process begins with extracting deep features from each image, followed by converting these features into probability distributions. To estimate these distributions for a pair of images, a binning technique is employed, where features are binned into fixed intervals. The number of bins as well as the binning strategy (e.g., equal width or equal frequency) are optimized for the dataset. To ensure numerical stability and to avoid zero probabilities in the binning process, a smoothing constant \(\varepsilon\) is added to each bin count prior to normalization. Given two sets of features \(P\) (from image 1) and \(Q\) (from image 2), the KL divergence between the distributions of their CNN outputs is calculated as  
\[D_{\text{KL}}(P\parallel Q)=\sum_{i}P(Y_{i}\mid X)\ln\left(\frac{P(Y_{i}\mid X )}{Q(Y_{i})}\right), \tag{2}\]  
Here, \(i\) represents the bin index for the estimated feature distributions. This approach enables a fine-grained comparison between the image pairs, capturing the inherent differences in ST shapes. **[**Efficient Computation and Manifold Learning Integration**]** Computing the KL divergence for all image pairs in large datasets can be computationally prohibitive. Therefore, an optimized sampling method is applied to select a representative subset of image pairs for divergence calculation. This reduces the computational load significantly while maintaining analytical precision. Parallel processing across multiple cores further enhances the efficiency of these calculations, allowing for the rapid processing of large datasets without sacrificing detail. To capture the complex, non-linear relationships inherent in these features, manifold learning techniques, particularly t-distributed Stochastic Neighbor Embedding (t-SNE), are incorporated [63; 64]. t-SNE helps embed high-dimensional divergence values into a lower-dimensional space while preserving both the local and global structures of the data. This allows the model to generalize effectively across variations in ST. The embedding reveals clusters and patterns that are not immediately obvious in the original high-dimensional feature space. Let \(I_{1}\) and \(I_{2}\) denote the sets of images represented by bridging and non-bridging ST forms, respectively. For each image \(I\) in \(I_{1}\) and \(I_{2}\), a feature representation \(f(I)\) is extracted. The KL divergence \(D(f(I_{1}),(I_{2}))\) is computed for each image pair \(I_{1}\) in \(I_{1}\) and \(I_{2}\) in \(I_{2}\), resulting in a matrix of divergence values. A non-linear embedding technique is then used to visualize these relationships in a lower-dimensional space. \(I_{1}\) and \(I_{2}\) are the sets of feature representations for the bridging and non-bridging ST images to formalize this process. The embedding of \(D\) in a lower-dimensional space is given by  
\[D=\text{Embed}(D(I_{1},I_{2})), \tag{3}\]  
where \(D(I_{1},I_{2})\) is the matrix of KL divergence values between all pairs of feature representations from \(I_{1}\) and \(I_{1}\), and \(Embed()\) is the embedding function t-SNE. Because both \(D(f_{1},f_{2})\) and \(D(f_{2},f_{1})\) are non-negative and unbounded, this visualization highlights the divergence patterns, elucidating the morphological differences between bridging and non-bridging ST images. The embedding further enhances the model's ability to distinguish between classes by preserving the non-linear relationships inherent in the data. [Normalized KL Divergence and Information Distinguishability] While the KL divergence offers valuable insight into the separability of feature distributions, its unbounded nature can make comparisons across different datasets or conditions challenging. To address this, the Information Distinguishability (\(ID\)) measure proposed by [PERSON] et al.