## 5 Conclusions  
We consider our results a warning for future UMAP implementations applied upstream of other analyses, such as clustering. For the case study of tumor stratification, we showed that, with HDBSCAN, keeping default UMAP parameters, i.e., Euclidean output metric, would have impoverished the downstream results. Furthermore, we observed high variability in estimating the number of clusters and in clustering performances. For datasets with well-separated groups, it might not be necessary to worry about the UMAP reproducibility issue and investigate various topological UMAP embeddings. However, for more complex data such as RNA-seq, different geometries of output spaces should be explored if one cannot be assumed a priori. UMAP has mostly been implemented with a Euclidean metric. We hope our work will encourage further research on non-Euclidean embeddings for the analysis of clustering or tumor stratification, even with other metric spaces to investigate further samples' relationships. Future works should validate different UMAP output metrics on several datasets, analogous to [PERSON] et al. [13] who evaluated the importance of the UMAP input metric. Moreover, as suggested by [44], supervised or semi-supervised dimensionality reduction can provide more informative latent space representations. Therefore, future efforts can integrate our pipeline with a supervised version of UMAP. Finally, even multi-omics data should be exploited [45] within our approach to enhance medical and biological interpretation of the final clustering results. [PERSON] and [PERSON] conducted data analysis and drafted the first version of the paper. [PERSON], [PERSON], [PERSON] ([PERSON]), [PERSON], [PERSON], [PERSON] and [PERSON] participated in the development. The overall work was supervised by [PERSON] and A.C. ([PERSON]). All the authors performed a critical review and approved the final version. All authors have read and agreed to the published version of the manuscript. The research center CMP\({}^{3}\)VdA and the Project 5000 genomie@VdA are co-founded by \"Fondo Europeo di Sviluppo Regionale (FESR) Programma Investimenti per la crescita e l'occupazione 2014/20\" (European Social Fund, ESF, and European Regional Development Fund, ERDF), the Autonomous Region of the Aosta Valley, and the Italian Ministry of Labour and Social Policy (CUP B68H19005520007). [PERSON], M.A., [PERSON] ([PERSON]) and S.M. have carried out this work supported by a grant of the EU-ESF, the Autonomous Region of the Aosta Valley, and the Italian Ministry of Labour and Social Policy. The Astronomical Observatory of the Autonomous Region of the Aosta Valley (OAVdA) is managed by the Fondazione Clement Fililietroz-ONLUS, which is supported by the Regional Government of the Aosta Valley, the Town Municipality of Nus, and the \"Unite des Communes validoaties Mont-Emilius\". **Informed Consent Statement**: Not applicable. **Data Availability Statement:** The breast cancer RNA-seq dataset (Illumina HiSeq 2000 RNA Sequencing platform, level 3 transcription, log2(x + 1) transformed RSEM-normalized count) and the related clinical variables were downloaded from the University of California Santa Cruz (UCSC) cancer browser website ([[https://xenabrowser.net/datapages](https://xenabrowser.net/datapages)]([https://xenabrowser.net/datapages](https://xenabrowser.net/datapages)), accessed on 20 April 2022). The breast cancer subtypes are available in a GitHub repository: [[https://github.com/yxchspring/GOEGCN_BRCA_Subtypes](https://github.com/yxchspring/GOEGCN_BRCA_Subtypes)]([https://github.com/yxchspring/GOEGCN_BRCA_Subtypes](https://github.com/yxchspring/GOEGCN_BRCA_Subtypes)), accessed on 20 April 2022. **Acknowledgments:** The Project 5000 genomi@VdA ([https://5000 genomivda.it/en/](https://5000 genomivda.it/en/), accessed on 20 April 2022) is a scientific project that has enabled the creation of a new research center dedicated to personalized, preventive, and predictive medicine (CMP3 VdA) for neurodevelopmental, neurodegenerative, and oncological diseases.