Supplementary Material of the Manuscript : Exploring Gene Expression Profiles Using Density-based Dimensionality Reduction Methods

This Supplementary Material provides additional technical details supporting the main manuscript. It contains detailed proofs of the main results, as well as several technical lemmas together with their complete proofs. It also includes additional tables, a detailed description of the gene expression data used in the study, and a complementary analysis of these data providing further interpretation of the results.

The document further presents extended algorithmic descriptions and additional implementation details that could not be included in the main text due to space limitations. In particular, we provide a complete description of the computational complexity of the proposed framework, detailed specifications of the kernel-based affinity functions, as well as additional simulation studies used to assess the robustness of the method.

Moreover, this document includes detailed pseudocode and implementation guidelines for the proposed Gamma-kernel-based functional PCA (FPCA/MDS) approach, along with extended discussions on hyperparameter selection and numerical stability considerations. It also contains auxiliary technical results that complement the main theoretical developments and provide additional justification for certain methodological steps.

The results presented in this Supplementary Material aim to ensure full reproducibility of the proposed methodology and to facilitate its implementation in related applications involving high-dimensional gene expression data or functional data. Throughout this document, equation numbers and section references are consistent with the main manuscript unless otherwise specified.