Supplementary datasets (large images, interactive 3D visualizations, spreadsheets) for Kočí et al. "Sparse sampling and rare-variant depletion distort PCA visualizations of population structure: recovery with objective-guided manifold learning"

Supplementary Dataset 1: Comprehensive summary of manifold-learning-derived LDEs across the full parameter space of the objective-guided embedding-optimization protocol. Each point in the PC-space-dimensionality versus k-NN biplots is colored by the median Δr² or Δρ² between an manifold-learning-derived embedding and the corresponding drift-normalized classic PC1–PC2–PC3 space. Each figure in this dataset consists of a collection of such biplots spanning different PC-space types, PC-space distance metrics, and manifold-learning algorithms. Individual figures are defined by the combination of LDE-ranking objective, landscape-sampling approach, either random or clustered, and MAF-filtering threshold, either no filtering or MAF = 1%. Separate sets of figures show the PC-space-dimensionality versus k-NN parameter space using either log₁₀ or linear axis scales.

Supplementary Dataset 2: An interactive R application for visualizing any selected PC1–PC2–PC3 space derived from the IBR-LDM simulations and for generating UMAP, UMAP/densMAP, densMAP, or PHATE 3D LDEs from these data using user-defined parameters. For each embedding, we display Pearson’s correlation with geographic distances and Spearman’s correlation with GRM-derived dissimilarities, the best-performing LDE-ranking objective.

Supplementary Dataset 3: Parameters and fits of the top 100 LDEs for each objective function are shown for each of the four case studies in which simple, non-layered LDE ranking was used.

Supplementary Dataset 4: Comprehensive summary of manifold-learning-derived LDEs across the full parameter space of the objective-guided embedding-optimization protocol in the Orang Asli case study. Each point in the PC-space-dimensionality versus k-NN biplots is colored by the Δr² or Δρ² between a manifold-learning-derived embedding and the corresponding drift-normalized classic PC1–PC2 space (MAC = 1, no missing data). Each figure in this dataset consists of a collection of such biplots spanning different PC-space types, PC-space distance metrics, and manifold-learning algorithms. Individual figures are defined by the LDE-ranking objective: F_ST, Hamming distance (default), GRM-derived dissimilarity, or outgroup f₃-statistics, each evaluated using either Pearson’s or Spearman’s correlation. Geography-based Pearson and Spearman objectives are shown for comparison. Separate sets of figures show the PC-space-dimensionality versus k-NN parameter space using either log₁₀ or linear axis scales.

Supplementary Dataset 5: Alternative PC1–PC2–PC3 spaces (a–h) and the top-ranked manifold-learning-derived LDE (i) from the Cervus elaphus case study, shown as interactive 3D plots. For the PCA algorithm, SNP-filtering scheme, and missing-data removal regime used for each analysis, see the file names and Supplementary Table 1. Corresponding goodness-of-fit statistics are provided in Supplementary Table 1 and Supplementary Dataset 3. Individuals are colored by the geographic regions defined by de Jong et al. (2025).

Supplementary Dataset 6: Comprehensive summary of manifold-learning-derived LDEs across the full parameter space of the objective-guided embedding-optimization protocol in the Cervus elaphus case study. Each point in the PC-space-dimensionality versus k-NN biplots is colored by the Δr² or Δρ² between a manifold-learning-derived embedding and the corresponding drift-normalized classic PC1–PC2–PC3 space (MAC = 1, no missing data). Each figure in this dataset consists of a collection of such biplots spanning different PC-space types, PC-space distance metrics, and manifold-learning algorithms. Individual figures are defined by the LDE-ranking objective: F_ST, Hamming distance (default), GRM-derived dissimilarity, or outgroup f₃-statistics, each evaluated using either Pearson’s or Spearman’s correlation. Geography-based Pearson and Spearman objectives are shown for comparison. Separate sets of figures show the PC-space-dimensionality versus k-NN parameter space using either log₁₀ or linear axis scales.

Supplementary Dataset 7: Alternative PC1–PC2–PC3 spaces (a–h) and the top-ranked manifold-learning-derived LDE (i) from the PNG case study, shown as interactive 3D plots. For the PCA algorithm, SNP-filtering scheme, and missing-data removal regime used for each analysis, see the file names and Supplementary Table 1. Corresponding goodness-of-fit statistics are provided in Supplementary Table 1 and Supplementary Dataset 3. Following Bergström et al. (2017), individuals whose parents were born in the same province are colored according to their province of parental origin, defined by Papua New Guinea provincial boundaries.

Supplementary Dataset 8: Comprehensive summary of manifold-learning-derived LDEs across the full parameter space of the objective-guided embedding-optimization protocol in the PNG case study. Each point in the PC-space-dimensionality versus k-NN biplots is colored by the Δr² or Δρ² between a manifold-learning-derived embedding and the corresponding drift-normalized classic PC1–PC2–PC3 space (MAC = 1, no missing data). Each figure in this dataset consists of a collection of such biplots spanning different PC-space types, PC-space distance metrics, and manifold-learning algorithms. Individual figures are defined by the LDE-ranking objective: F_ST, Hamming distance (default), GRM-derived dissimilarity, or outgroup f₃-statistics, each evaluated using either Pearson’s or Spearman’s correlation. Geography-based Pearson and Spearman objectives are shown for comparison. Separate sets of figures show the PC-space-dimensionality versus k-NN parameter space using either log₁₀ or linear axis scales.

Supplementary Dataset 9: Alternative PC1–PC2–PC3 spaces (a–h) and the top-ranked manifold-learning-derived LDE (i) from the SEA case study, shown as interactive 3D plots. For the PCA algorithm, SNP-filtering scheme, and missing-data removal regime used for each analysis, see the file names and Supplementary Table 1. Corresponding goodness-of-fit statistics are provided in Supplementary Table 1 and Supplementary Dataset 3. Individuals are colored according to their linguistic affiliation (language family).

Supplementary Dataset 10: Comprehensive summary of manifold-learning-derived LDEs across the full parameter space of the objective-guided embedding-optimization protocol in the SEA case study. Each point in the PC-space-dimensionality versus k-NN biplots is colored by the Δr² or Δρ² between a manifold-learning-derived embedding and the corresponding drift-normalized classic PC1–PC2–PC3 space (MAC = 1, no missing data). Each figure in this dataset consists of a collection of such biplots spanning different PC-space types, PC-space distance metrics, and manifold-learning algorithms. Individual figures are defined by the LDE-ranking objective: F_ST, Hamming distance (default), GRM-derived dissimilarity, or outgroup f₃-statistics, each evaluated using either Pearson’s or Spearman’s correlation. Geography-based Pearson and Spearman objectives are shown for comparison. Separate sets of figures show the PC-space-dimensionality versus k-NN parameter space using either log₁₀ or linear axis scales.

Supplementary Dataset 11: PCA LDEs visualizing the projected (ancient Eurasian; a–c) and basis (present-day Eurasian; d–f) individuals separately. The PC1–PC2–PC3 spaces are presented as interactive 3D plots. A separate series of interactive plots displays the projected and basis individuals jointly (g–i). Shown are results from three PCA approaches: classic PC scores on data that were centered and normalized by genetic-drift dispersion prior to eigendecomposition (a, d, g); classic PC scores on centered data (b, e, h); and eigenvectors on data centered and normalized by genetic-drift dispersion (c, f, i). Ancient Eurasian individuals are colored according to the 23 largest IBD-sharing communities including 20 or more individuals (Supplementary Table 2) in all plots; present-day Eurasian individuals are colored according to the 41 largest IBD-sharing communities including 10 or more individuals (Supplementary Table 2) in plots d–f and shown as grey dots in plots g–i. All fidelity metrics for these 3D PC spaces are provided in Supplementary Table 4.

Supplementary Dataset 12: Three prioritized results from the multi-step LDE-ranking procedure applied to 5,133 ancient Eurasian individuals: the final LDE (a, d) and two additional LDEs (b, c) produced under the highest k-NN settings among the top 20 LDEs from step 3. Each LDE is shown as an interactive 3D plot, with individuals colored by the 23 largest IBD-sharing communities (a–c) or by 10 geographic regions (d). In plot d, members of the CEE IBD-sharing community are shown as medium-sized points, while CEE individuals from the Roman and Migration period (eCEE) are emphasized as the largest points. The parameter settings used to generate these LDEs, along with all associated fidelity metrics, are listed in Supplementary Table 4.

Supplementary Dataset 13: Comparison of manifold-learning optimization outcomes (using 5,133 ancient Eurasian individuals) obtained under different objectives. For each of the following key objectives, the best-performing 3D LDE is shown:  (a),  (b),  with the default missingness correction (c),  (d), median Mahalanobis distance among the 23 largest IBD-sharing communities (e), neighborhood-preservation metric at k = 500 (f) or k = 50 (g), and median Mahalanobis distance among the 21 Y-lineages with intra-lineage coalescence around 2,800 ya (h). The LDEs are displayed as interactive 3D plots, with individuals colored by the 23 largest IBD-sharing communities. Settings used to generate each LDE and all fidelity metrics are provided in Supplementary Table 4.

Supplementary Dataset 14: Illustrating the LDE ranking procedure applied to the ancient Eurasian dataset: fits, Mahalanobis and Bhattacharya distances between IBD-sharing communities, and four metrics assessing how well the geometry of the (s – 1)-dimensional PC space (Fischer & Ma 2024) is preserved in an LDE are shown for the broader optimal region of the parameter space: 85,188 PHATE embeddings on classic PC scores. The results are displayed on logarithmic (a) or linear (b) PC space dimensionality and k-NN axes. In each row, the results are grouped by: (i) PCA variant – classic PCA with centering only, and classic PCA with centering plus normalization by genetic-drift dispersion; (ii) distance/affinity in the input PC space – Euclidean (ℓ₂), Manhattan (ℓ₁), and cosine.

Supplementary Dataset 15: Illustrating the LDE ranking procedure applied to the present-day Eurasian dataset: fits, Mahalanobis and Bhattacharya distances between IBD-sharing communities, and four metrics assessing how well the geometry of the (s – 1)-dimensional PC space (Fischer & Ma 2024) is preserved in an LDE are shown for the region of the parameter space we explored: 83,544 PHATE embeddings on classic PC scores. The results are displayed on logarithmic (a) or linear (b) PC space dimensionality and k-NN axes. In each row, the results are grouped by: (i) PCA variant – classic PCA with centering only, and classic PCA with centering plus normalization by genetic-drift dispersion; (ii) distance/affinity in the input PC space – Euclidean (ℓ₂), Manhattan (ℓ₁), and cosine.

Supplementary Dataset 16: Comparison of manifold-learning optimization outcomes (using 2,425 present-day Eurasian individuals) obtained under different objectives. For each of the following key objectives, the best-performing 3D LDE is shown:  (a),  (b),  with the default missingness correction (c),  (d),  (e),  (f),  (g),  (h), median Mahalanobis distance among the 41 largest IBD-sharing communities (i), neighborhood-preservation metric at k = 250 (j) or k = 50 (k). The LDEs are displayed as interactive 3D plots, with individuals colored by the 41 largest IBD-sharing communities. Settings used to generate each LDE and all fidelity metrics are provided in Supplementary Table 4.

Supplementary Dataset 17: Five prioritized results from the multi-step LDE-ranking procedure applied to 2,425 present-day Eurasian individuals: the final LDE (a, f) and the four next highest-scoring LDEs (b–e). Each LDE is shown as an interactive 3D plot, with individuals colored by the 41 largest IBD-sharing communities or by groups used for the HapNe-IBD analysis (f). The parameter settings used to generate these LDEs, along with all associated fidelity metrics, are listed in Supplementary Table 4.

Supplementary Dataset 18: Optimization landscape from objective-guided manifold-learning search in the ancient Eurasian case study: fits of 809,286 unique 3D LDE configurations to genetic and geographic distances. Values shown are differences in squared correlation relative to a baseline of classic 3D PC space (with data normalized by genetic‐drift dispersion): or Panels a–c use Spearman’s rank correlation, and panels d–f use Pearson’s linear correlation. Each panel reports fits to pairwise distances among 5,133 individuals: (a, d) Hamming genetic distances; (b, e) GRM genetic distances; (c, f) great-circle geographic distances. The color scales use a power (warp-top) transform  with to enhance contrast near the best-fitting regions of the landscape. In each panel, the results are grouped by: (i) PCA variant – smartPCA eigenvectors, classic PC scores with centering only, and classic PC scores with centering plus normalization by genetic-drift dispersion; (ii) distance/affinity in the input PC space – Euclidean (ℓ₂), Manhattan (ℓ₁), cosine, and correlation; (iii) manifold-learning algorithm and its settings – UMAP (two min_dist values), a UMAP–densMAP hybrid (dens_scale = 0.5), densMAP (dens_scale = 1), and PHATE. Both the PC space dimensionality and k-NN axes use a logarithmic scale.

Supplementary Dataset 19: The optimization landscapes from Supplementary Dataset 18, displayed on linear (non-logarithmic) PC space dimensionality and k-NN axes.

Supplementary Dataset 20: Most frequent Y-lineages in the ancient Eurasian dataset at selected intra-lineage coalescence cut-offs, 2,800 ya (a) and 4,600 ya (b), visualized on the 2D isometric unfolding of the best-fitting 3D LDE. Interactive 2D plots are shown. The sets of 21 (a) and 31 (b) Y-chromosomal lineages, each represented by at least 10 individuals, were defined using a reduced subset of individuals with SNP counts above the 25th percentile; however, the plots themselves were not filtered by this SNP-count threshold. Males carrying other Y-lineages are shown as grey dots and labeled “zzz”.

Supplementary Dataset 21: Quantifying Y-lineage separation in PHATE embeddings based on classic PC spaces in the ancient Eurasian case study.