Exploring gene expression profiles using density-based dimensionality reduction methods

Abstract

Traditional statistical methods, such as principal component analysis, often fail to capture complex dependencies in gene expression data. To address this limitation, we propose a functional framework combining multidimensional scaling with a density-based version of principal component analysis. By representing gene expression profiles through estimated distributions, the method captures both distributional shape and variability across individuals. Using artificial datasets, we show that clustering performed on density-based scores accurately recovers the original class structure. We also compare our approach with two nonlinear dimensionality reduction techniques, uniform manifold approximation and projection and diffusion maps, as dimensionality increases, highlighting the importance of $L_2$ normalization in preserving discriminative power. For moderate dimensions, densities are estimated using a multivariate gamma kernel, well suited to the non-negative and asymmetric nature of transcriptomic data. Finally, we establish convergence results for the estimated inner products and prove the spectral consistency of the resulting eigenvalues and eigenvectors. The extracted principal components effectively capture both lower-order statistical moments and complex gene interaction patterns that are often inaccessible to classical linear methods.

Keywords

• Clustering
• gene-gene interactions
• multidimensional scaling
• nonparametric gamma kernel
• principal component.

Thanks

The author wishes to express his deepest gratitude to Professor R. Boumaza, whose passion for statistics and scientific rigor has profoundly influenced this work. His wise counsel, high intellectual standards, and unwavering support have been essential throughout the development of this research. The author is also grateful to the Centre de Recherche sur l’Information Scientifique et Technique (CERIST) for providing access to the IBN BADIS High-Performance Computing (HPC) infrastructure, which made the extensive computations in this study possible. Additionally, the author sincerely thanks the anonymous reviewers for their constructive comments and insightful feedback, which significantly improved the scientific quality and clarity of this paper. Finally, this manuscript was translated with the assistance of DeepL and was subsequently reviewed and refined by the author to ensure technical accuracy.

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