## 4 Discussion  
Using the UMAP algorithm, here we projected 20,530 gene expression data onto three different latent metric spaces in order to evaluate how different curvatures of the UMAP embedding could affect breast cancer stratification. The evaluation focused on clustering performances in terms of ARI, homogeneity, and number of estimated clusters returned by HDBSCAN, DBSCAN, agglomerative clustering, and OPTICS. We also addressed UMAP reproducibility by proposing an iterative approach to yield more stable clustering outputs.  
UMAP has been widely implemented for transcriptomic analyses [13, 14, 42, 43]. In particular, starting from single-cell RNA-seq data, [PERSON] et al. [42] and [PERSON] et al. [43] visualized tumor cell heterogeneity in triple-negative breast cancer and glioblastoma, respectively. [PERSON] et al. [13] and [PERSON] et al. [14] performed transcriptomic analyses, applying the HDBSCAN algorithm to UMAP Euclidean embedding. Moreover, [PERSON] et al. [13] also addressed how the choice of the input metric in UMAP (i.e., how we measure the distance between samples in high-dimensional input data) could influence the visualization of clustering structures.  
Nevertheless, none of the above approaches raised concerns about the reproducibility and the geometry of the UMAP embedding and, to the best of our knowledge, no study has addressed these issues.  
Our work focused on the choice of the output metric in UMAP, showing that the selection of the latent space can affect downstream clustering results. However, by exploring different metric spaces, we encountered UMAP reproducibility issues: some pairwise points were inconsistently projected onto latent spaces and so were not always clustered together over several runs, leading to unstable clustering results (see Figures 3 and 4). Since this work was the first attempt to investigate different curvatures of UMAP embedding, this  
Figure 6: BC subtype distribution over estimated clusters. Relative (vertical axis) and absolute (top of the bars) frequency of BC subtypes within each cluster, identified on spherical UMAP embedding with \(\text{NN}=65\). On the horizontal axis, we report the cluster labels. NC stands for Not Clusterable (see Methods section).  
reproducibility issue spurred us to propose a reproducible pipeline that returns consistent results, regardless of the selected output metrics.  
As reported in the Appendix (Tables A2 and A3), clustering performances were improved by applying HDBSCAN to spherical and hyperbolic metric spaces. On non-Euclidean embedding, the ARI score increased by 14 percentage points (from 0.33 to 0.47) with respect to the Euclidean latent space. Hence, in this analysis the Euclidean output metric, when applied on HDBSCAN, performs poorly compared to the other metrics. On the contrary, Euclidean and hyperbolic latent spaces showed similar ARI scores that were higher than the spherical one for three clustering algorithms out of four.  
The above results are dataset-dependent. Therefore, to validate the generalizability of our approach and the potentiality of our proposed pipeline, more datasets (single-cell RNA-seq, multi-omics, methylation data) should be tested.