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DMT-Dens: Density-preserving manifold visualization for biological data
Ruizhe Wang, Yixuan Dong, Bolin Yang, Bingo Wing-Kuen Ling, Fuji Yang, Zelin Zang
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Summary
The paper introduces DMT-Dens, a parametric manifold-visualization method designed to preserve local neighborhood structures and sampling density in high-dimensional biological data, such as single-cell RNA sequencing. Built on a latent-token Transformer encoder, DMT-Dens optimizes a combined loss function comprising rank-based manifold alignment with hard-pair aggregation and a density-preservation term based on the Pearson correlation of k-nearest-neighbor log-radius estimates. Evaluations demonstrate that DMT-Dens achieves superior density fidelity compared to baselines like t-SNE, UMAP, den-SNE, densMAP, PaCMAP, and PHATE, while maintaining competitive label separability.
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Relation Signals (7)
DMT-Dens → isappliedto → single-cell RNA sequencing
confidence 95% · Low-dimensional embeddings are widely used to explore cell-state heterogeneity in single-cell and other high-dimensional biological data.
DMT-Dens → optimizes → density preservation
confidence 95% · To preserve density, it optimizes a loss based on the Pearson correlation between k-nearest-neighbor log-radius estimates...
DMT-Dens → uses → latent-token Transformer encoder
confidence 95% · We present DMT-Dens, a parametric manifold-visualization method built on a latent-token Transformer encoder.
DMT-Dens → optimizes → manifold alignment
confidence 90% · The model integrates rank-based manifold alignment with hard-pair aggregation.
DMT-Dens → outperforms → t-SNE
confidence 85% · Benchmark evaluations demonstrate strong density preservation... DMT-Dens achieves the highest density correlation on all four biological datasets...
DMT-Dens → outperforms → UMAP
confidence 85% · Benchmark evaluations demonstrate strong density preservation... DMT-Dens achieves the highest density correlation on all four biological datasets...
DMT-Dens → outperforms → densMAP
confidence 85% · Benchmark evaluations demonstrate strong density preservation... DMT-Dens achieves the highest density correlation on all four biological datasets...
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Abstract
Abstract:Motivation: Low-dimensional embeddings are widely used to explore cell-state heterogeneity in single-cell and other high-dimensional biological data. Although many methods preserve local neighborhoods, they may distort the apparent sampling density of processed observations, altering the visual contrast between dense and sparse regions and complicating the interpretation of rare, transitional, or continuous cell-state populations. Results: We present DMT-Dens, a parametric manifold-visualization method built on a latent-token Transformer encoder. The model integrates rank-based manifold alignment with hard-pair aggregation. To preserve density, it optimizes a loss based on the Pearson correlation between k-nearest-neighbor log-radius estimates in the processed input and two-dimensional embedding spaces. Benchmark evaluations demonstrate strong density preservation, particularly on biological datasets, while retaining competitive label separability. Availability: Source code, data-processing scripts, and resolved experiment configurations are available at this https URL.
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- Source: https://arxiv.org/abs/2608.17571v1
- Canonical: https://arxiv.org/abs/2608.17571v1
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X ∗ ]Corresponding author. zangzelin@gmail.com 02026 02026 02026 Key Messages • DMT-Dens targets biological visualization settings in which density fidelity and label separability are evaluated together. • The method combines a parametric mapping and rank-based manifold alignment with an explicit kNN density-preservation loss. • The evaluation focuses on density correlation and local density correlation; SVC accuracy is reported as a descriptive label-separability diagnostic, with neighborhood and trajectory diagnostics provided as Supplementary Material. DMT-Dens: Density-preserving manifold visualization for biological dataIssue: x Ruizhe Wang Yixuan Dong Bolin Yang Bingo Wing-Kuen Ling Fuji Yang Zelin Zang Abstract Motivation: Low-dimensional embeddings are widely used to explore cell-state heterogeneity in single-cell and other high-dimensional biological data. Although many methods preserve local neighborhoods, they may distort the apparent sampling density of processed observations, altering the visual contrast between dense and sparse regions and complicating the interpretation of rare, transitional, or continuous cell-state populations. Results: We present DMT-Dens, a parametric manifold-visualization method built on a latent-token Transformer encoder. The model integrates rank-based manifold alignment with hard-pair aggregation. To preserve density, it optimizes a loss based on the Pearson correlation between k-nearest-neighbor log-radius estimates in the processed input and two-dimensional embedding spaces. Benchmark evaluations demonstrate strong density preservation, particularly on biological datasets, while retaining competitive label separability. Availability: Source code, data-processing scripts, and resolved experiment configurations are available at https://github.com/Ruizhe-wang/DMT-Dens. Contact: zangzelin@gmail.com Supplementary information: Supplementary data are included after the main article in this arXiv version. †journal: Bioinformatics†doi: DOI added during production†copyright-year: 2026†year: 2026†access: Advance Access Publication Date: to be assigned†appnotes: Original Paper†firstpage: 1†address: Tsientang Institute for Advanced Study, , Zhejiang, China†corresponding: [†received: Date†revised: Date†accepted: Date 1 Introduction Figure 1: Motivation for density-preserving biological visualization. a, Ground-truth synthetic three-dimensional branching tree with a dense root, sparse transitions, and multiple branch points, colored by branch identity. b, Two-dimensional embeddings of the same data. The displayed baseline embeddings differ in local density, apparent continuity, and adjacency. Adding a density objective to t-SNE and UMAP yields the density-aware baselines den-SNE and densMAP. The red outlines mark apparent disconnections in this representative UMAP embedding and their partial reconnection in densMAP. In the displayed DMT-Dens embedding, density variation and continuous paths are visually retained. c, Density preservation versus label separability across the benchmark datasets. Each point shows the mean rank of one method over the eight datasets with complete baseline coverage (density-correlation rank versus SVC-accuracy rank), and whiskers show the standard error across datasets. The dashed curve marks the Pareto frontier of the baseline methods. Low-dimensional visualization is widely used to explore high-dimensional data. In single-cell RNA sequencing (scRNA-seq), two-dimensional embeddings support cell-type annotation, quality control, and exploratory analysis of developmental continua and heterogeneous cell states (1; 9). Deep manifold-learning methods have also been developed for structure-preserving visualization and data integration in single-cell and spatial transcriptomics (26; 25; 27). These methods map sparse, noisy gene-expression profiles from high-dimensional space to two or three dimensions for visual inspection. Common visualization methods do not preserve every property relevant to biological interpretation. Principal Component Analysis (PCA) provides a linear view of global variation. Nonlinear neighbor-embedding methods such as t-Distributed Stochastic Neighbor Embedding (t-SNE) and Uniform Manifold Approximation and Projection (UMAP) instead emphasize local structure and often produce clearly separated groups (21; 12; 1). However, cluster area and spread in these embeddings do not reliably reflect cell-state abundance or transcriptional variability (15; 6; 9). t-SNE may expand densely sampled regions and contract sparse parts of continuous trajectories (14; 9), whereas UMAP may exaggerate population separation and alter local density (6; 10). Here, density refers specifically to the sample density of the processed input representation. Experimental sampling determines which cells enter that representation, and tissue dissociation can induce transcriptional changes before density is calculated (20). Feature selection, normalization, and dimensionality-reduction steps such as PCA further define the geometry in which density is measured. Although processed-input density is not a calibrated measure of biological abundance, it is an empirical structural property presented to the visualization method. Preserving this relative density prevents the projection from introducing additional distortion and allows concentration and dispersion in the embedding to reflect the analyzed representation more faithfully (14). den-SNE and densMAP address density distortion by adding density-preservation terms to the t-SNE and UMAP objectives, respectively (14). These methods improve density fidelity, but remain non-parametric and inherit the neighborhood-affinity objectives of their parent methods: pairwise probabilistic affinities in t-SNE/den-SNE and a fuzzy neighbor graph in UMAP/densMAP. Consequently, manifold organization is still governed by the corresponding parent attraction–repulsion formulation, with density preservation added to the inherited structural objective. This leaves a more specific methodological gap: an explicit parametric model that jointly learns neighborhood-rank structure and the relative density profile of the processed observations. Such joint optimization is relevant because biological data are often modeled as lying on a low-dimensional, nonlinear manifold whose observation density is nonuniform (4; 3; 14). Within this scope, faithful sample-density visualization provides a controlled basis for examining heterogeneous and transitional regions while distinguishing projection-induced distortion from variation already present in the analyzed data. We developed DMT-Dens to address this gap. Its primary contribution is an explicit density loss that aligns k-nearest-neighbor log-radius profiles between the processed input and the two-dimensional embedding. This loss is optimized within a parametric deep manifold-transformation model (26). The accompanying manifold objective represents neighborhood relations by rank affinity and emphasizes hard pairs during optimization. These mechanisms support neighborhood organization while the density term targets relative concentration and dispersion. Bidirectional affinity construction and two-scale density aggregation are additional refinements of the full objective. To determine whether DMT-Dens improves density fidelity while retaining useful label and neighborhood structure, we compare its two-dimensional representations with those produced by t-SNE (21), UMAP (12), den-SNE and densMAP (14), PaCMAP (23), and PHATE (13). The evaluation metrics, baseline settings, and hyperparameter-search protocol are described in the Methods and Supplementary Material. Under this common protocol, DMT-Dens achieves the highest density correlation on all four biological datasets and on six of the nine datasets overall, while its linear SVC accuracy ranks among the top two methods on seven datasets. Ablation experiments identify the density loss as the main source of improved density fidelity. Supplementary analyses of synthetic dyngen trajectories (5) and a Caenorhabditis elegans developmental time course (17) provide an exploratory assessment of topology- and time-related diagnostics. 2 Material and methods 2.1 Problem definition Let =[1⊤,…,N⊤]⊤∈ℝN×DX=[x_1 ,…,x_N ] ^N× D denote an input matrix with N observations and D features, where i∈ℝDx_i ^D is observation i. For single-cell data, an observation is a cell represented by processed gene-expression measurements or a derived feature vector. The formulation also covers the image, text, and sensor data used in our experiments. The encoder eϕ:ℝD→ℝ40e_φ:R^D ^40 maps ix_i to a 40-dimensional latent representation i=eϕ(i)z_i=e_φ(x_i). The projection module pψ:ℝ40→ℝ2p_ψ:R^40 ^2 maps iz_i to a two-dimensional coordinate i=pψ(i)y_i=p_ψ(z_i). We denote the full mapping by fθ=pψ∘eϕf_θ=p_ψ e_φ, with θ=(ϕ,ψ)θ=(φ,ψ), and collect the coordinates in =[1⊤,…,N⊤]⊤∈ℝN×2Y=[y_1 ,…,y_N ] ^N× 2. We optimize the map toward two objectives. (i) Manifold preservation: the manifold-alignment loss encourages the embedding to retain relative neighborhood orderings from feature space. If ix_i is closer to jx_j than to kx_k, the loss favors the corresponding ordering between iy_i, jy_j, and ky_k. (i) Density preservation: the density loss encourages agreement between local-density profiles in feature and embedding spaces, helping to maintain the visual contrast between dense and sparse regions. We express these objectives through a manifold-preservation loss ℒmanifoldL_manifold and a density-preservation loss ℒdensL_dens. The model parameters are estimated by minimizing their weighted sum, ℒ(θ)=ℒmanifold(θ)+λdℒdens(θ),L(θ)=L_manifold(θ)+ _d\,L_dens(θ), (1) where λd≥0 _d≥ 0 controls the contribution of the density term. Figure 2: DMT-Dens architecture and training objective. (A) A latent-token Transformer encoder and projection module map the input matrix X to the embedding Y. For each observation, the encoder forms a fixed set of latent tokens, applies self-attention within the token set, and pools the result into a 40-dimensional representation for two-dimensional projection. (B) The manifold-preservation loss derives bidirectional rank affinities PijP_ij from paired input-space views. Binary cross-entropy matches these targets to cross-view Student t-kernel affinities QijQ_ij over selected hard pairs. (C) The density-preservation loss compares kNN log-density profiles gsXg_s^X and gsYg_s^Y across anchor observations at scales s∈⌊k/2⌋,ks∈\ k/2 ,k\. Stop-gradient is applied to the input-space profile. (D) The final objective is ℒ(θ)=ℒmanifold(θ)+λdℒdens(θ)L(θ)=L_manifold(θ)+ _dL_dens(θ); its gradients update the encoder and projection module. 2.2 Latent-token Transformer encoder The encoder eϕe_φ represents each observation with a fixed number of latent tokens. Self-attention operates within this token set and is not applied directly across input features or observations. The attention sequence length is independent of the input dimension D, and the embedding of an observation does not depend on other observations in its inference batch. We first normalize each input using ¯i=BNin(i). x_i=BN_in(x_i). (2) A learned linear map compresses the normalized feature vector into M latent vectors of width r, i=reshape(c¯i,M,r)∈ℝM×r,S_i=reshape\! (W_c x_i,\,M,\,r ) ^M× r, (3) where c∈ℝMr×DW_c ^Mr× D. Each vector is independently expanded to the token width dtd_t and assigned a learned latent-identity embedding, (0)i,m=i,m+m,m=1,…,M,T^(0)_i,m=S_i,mE_m+a_m, m=1,…,M, (4) where m∈ℝr×dtE_m ^r× d_t and m∈ℝdta_m ^d_t. The resulting token matrix i(0)∈ℝM×dtT^(0)_i ^M× d_t has the same size for datasets with different feature dimensions. The tokens are processed by L pre-normalized Transformer blocks (22). For block ℓ , the updates are i(ℓ) ^( )_i =i(ℓ−1)+MSAℓ(LNℓ,1(i(ℓ−1))), =T^( -1)_i+MSA_ \! (LN_ ,1(T^( -1)_i) ), (5) i(ℓ) ^( )_i =i(ℓ)+FFNℓ(LNℓ,2(i(ℓ))), =U^( )_i+FFN_ \! (LN_ ,2(U^( )_i) ), (6) where MSAMSA denotes multi-head self-attention over the M tokens of one observation, and FFNFFN is a two-layer feed-forward network with GELU activation. The encoder uses neither labels nor cross-observation attention. After the final block, we average the layer-normalized tokens and map the pooled vector to 40 dimensions, i _i =1M∑m=1MLNout(i,m(L)), = 1M _m=1^MLN_out\! (T^(L)_i,m ), (7) i _i =BNout(oi+o). =BN_out\! (W_oh_i+b_o ). (8) The projection module produces i=pψ(i)y_i=p_ψ(z_i). Our final configuration uses M=32M=32, r=16r=16, dt=224d_t=224, L=2L=2, and four attention heads. The feed-forward hidden width is 4dt4d_t; attention and residual dropout are set to zero. Attention cost is determined by the fixed token count M and does not scale quadratically with the original feature dimension D. 2.3 Manifold-preservation loss Each mini-batch yields two paired views, ℬ(a)=i(a)i=1B,ℬ(b)=i(b)i=1B,B^(a)=\x^(a)_i\_i=1^B, ^(b)=\x^(b)_i\_i=1^B, (9) where i(a)=ix^(a)_i=x_i is the original observation and i(b)x^(b)_i is a neighborhood-augmented version. To form the augmented view, we project the full preprocessed dataset onto DpcaD_pca principal components and build an approximate KaugK_aug-nearest-neighbor graph after excluding self-matches. These PCA coordinates are used only for neighbor search. For each sampled observation, we draw an index jiaugj_i^aug uniformly from its KaugK_aug graph neighbors and interpolate in the original preprocessed feature space, i(b)=αii+(1−αi)jiaug,αi∼Uniform(αmin,αmax).x^(b)_i= _i\,x_i+(1- _i)\,x_j_i^aug, _i ( _ , _ ). (10) The vectors i(a)x^(a)_i, jiaugx_j_i^aug, and i(b)x^(b)_i all lie in the original D-dimensional preprocessed feature space. PCA does not alter the vectors used in the distance calculations below. Each observation contributes one augmented vector, giving B vectors in each view. For the nine-dataset benchmark, Kaug=200K_aug=200, Dpca=64D_pca=64, αmin=0.05 _ =0.05, and αmax=1 _ =1. Parameter settings that differ in the case studies are reported in the Supplementary Material. The shared mapping fθf_θ processes both views to produce paired two-dimensional embeddings, i(a)=fθ(i(a)),i(b)=fθ(i(b)).y^(a)_i=f_θ(x^(a)_i), ^(b)_i=f_θ(x^(b)_i). (11) Before computing embedding-space affinities, we standardize each coordinate within each view. For view v∈a,bv∈\a,b\ and coordinate r∈1,2r∈\1,2\, let y~i,r(v)=(yi,r(v)−μr(v))/σr(v) y^(v)_i,r=(y^(v)_i,r-μ^(v)_r)/σ^(v)_r, where μr(v)μ^(v)_r and σr(v)σ^(v)_r are the mini-batch mean and standard deviation. These statistics are held constant during backpropagation. We write the standardized embedding as ~i(v)=(y~i,1(v),y~i,2(v))⊤ y^(v)_i=( y^(v)_i,1, y^(v)_i,2) . We define the raw cross-view squared Euclidean distance in the input space as Δijab=‖i(a)−j(b)‖22. ^ab_ij= ^(a)_i-x^(b)_j _2^2. (12) The augmentation generally makes Δiiab ^ab_i nonzero. To treat matched observations as nearest across views, we floor off-diagonal squared distances at εdist>0 _dist>0 and set the paired diagonal to zero: dijab=max(Δijab,εdist),i≠j,0,i=j.d^ab_ij= cases ( ^ab_ij, _dist),&i≠ j,\\ 0,&i=j. cases (13) For each i(a)x^(a)_i, let πia→b(j) _i^a→ b(j) be the ordinal rank of j(b)x^(b)_j after sorting dijabj=1B\d^ab_ij\_j=1^B in ascending order. For a fixed i, the index j ranges over the B augmented vectors in the mini-batch. The KaugK_aug candidates used to construct i(b)x^(b)_i are not part of this ranking. Ranks are zero-based, so πia→b(j)∈0,…,B−1 _i^a→ b(j)∈\0,…,B-1\. For tied off-diagonal distances, torch.argsort assigns distinct consecutive ranks in its returned order; average ranks are not used. Since diiab=0d^ab_i=0 and all off-diagonal entries are positive, the paired diagonal has rank 00. The reverse rank πjb→a(i) _j^b→ a(i) is defined in the same way by sorting distances from j(b)x^(b)_j to the observations in the first view. The directional rank affinities are defined as Pija→b=ηπia→b(j),Pijb→a=ηπjb→a(i),P^a→ b_ij=η^\, _i^a→ b(j), P^b→ a_ij=η^\, _j^b→ a(i), (14) where η∈(0,1)η∈(0,1) controls affinity decay with rank. The paired diagonal has rank 00 in both directions, giving Piia→b=Piib→a=1P^a→ b_i=P^b→ a_i=1. We combine the two directional affinities using their geometric mean, Pij=max(Pija→bPijb→a,εaff),P_ij= \! ( P^a→ b_ijP^b→ a_ij, _aff ), (15) where εaff>0 _aff>0 is a numerical floor. This construction gives Pii=1P_i=1 and encodes the bidirectional neighborhood ranks of cross-view pairs. The affinity matrix P is held fixed during optimization. Let δijab=‖~i(a)−~j(b)‖2δ^ab_ij= y^(a)_i- y^(b)_j _2 (16) denote the distance between the standardized embeddings of the two views. The corresponding Student t-kernel value is Q~ij=(1+(δijab)2ν)−(ν+1)/2, Q_ij= (1+ (δ^ab_ij)^2ν )^-(ν+1)/2, (17) where ν>0ν>0 controls the tail weight of the kernel. The embedding-space affinity is obtained by row normalization, Qij=Q~ij∑ℓ≠iQ~iℓ.Q_ij= Q_ij _ ≠ i Q_i . (18) Pairwise binary cross-entropy matches the input-space targets to the embedding-space affinities, ℓij=−[Pijlog(Qij+εBCE)+(1−Pij)log(1−Qij+εBCE)], _ij=- [P_ij (Q_ij+ _BCE)+(1-P_ij) (1-Q_ij+ _BCE) ], (19) where εBCE>0 _BCE>0 stabilizes the logarithms. The pointwise loss measures the mismatch between an input-space rank affinity and its embedding-space counterpart. Hard-pair selection retains the cross-view pairs with the largest pointwise losses. For each i, let τi _i be the m-th largest value in ℓijj=1B\ _ij\_j=1^B, where m=min(m0,B)m= (m_0,B), and define ℳi=j:ℓij≥τiM_i=\\,j: _ij≥ _i\,\. We use |⋅||·| for set cardinality, so |ℳi||M_i| is the number of indices retained for observation i. The threshold gives |ℳi|≥m|M_i|≥ m, with equality when no losses are tied at τi _i. The full set of retained pairs is ℳ=(i,j):j∈ℳi.M=\(i,j):j _i\. (20) The manifold-preservation loss is then defined as ℒmanifold=1|ℳ|∑(i,j)∈ℳℓij.L_manifold= 1|M| _(i,j) _ij. (21) The mini-batch loss is consequently concentrated on cross-view relationships with the largest current mismatch between PijP_ij and QijQ_ij. The manifold-preservation loss integrates bidirectional rank targets, Student t-kernel similarities, and hard-pair aggregation. It is optimized jointly with the density-preservation loss in Eq. (1). 2.4 Density-preservation loss We estimate local density using k-nearest-neighbor (kNN) radii. For a reference index set ℛR and a query a∈ℛa , define the input- and embedding-space radii as rsX(a,ℛ) r_s^X(a,R) =OSs(∥a−j∥2:j∈ℛ∖a), =OS_s\! ( \ _a-x_j _2:j \a\ \ ), (22) rsY(a,ℛ) r_s^Y(a,R) =OSs(∥a−j∥2:j∈ℛ∖a), =OS_s\! ( \ _a-y_j _2:j \a\ \ ), where OSsOS_s denotes the s-th smallest value. The query is excluded from both reference neighborhoods. In a space of ambient dimension d, let nℛ=|ℛ|−1n_R=|R|-1 be the number of eligible reference observations and Vd=πd/2/Γ(d/2+1)V_d=π^d/2/ (d/2+1) the volume of the unit d-ball. The classical estimator is f^k(a)=k/[nℛVdrk(a,ℛ)d] f_k(x_a)=k/[n_RV_dr_k(x_a,R)^d]; in the input space, rk(a,ℛ)=rkX(a,ℛ)r_k(x_a,R)=r_k^X(a,R). Supplementary Section S2 gives the derivation. Our density loss uses the Pearson correlation between input- and embedding-space log-density vectors. Under this correlation, the terms for sample size, neighbor count, and ambient dimension contribute only an additive constant and a positive scale factor. We use the resulting dimension-free quantity logf^k(a)∝−logrkX(a,ℛ). f_k(x_a)\, \,- r_k^X(a,R). (23) In the implementation, the zero self-distance occupies the first position in each query-to-reference distance row, so the radius is the (s+1)(s+1)-th smallest entry. 2.5 Training objective and optimization The dimension-free log-density in Eq. (23) increases from sparse to dense regions. We compare the input- and embedding-space profiles using Pearson correlation. Within a mini-batch of B observations, the correlation is evaluated on a random anchor subset ⊆1,…,BA \1,…,B\ of size A=min(A0,B)A= (A_0,B). We use two neighborhood scales, =⌊k/2⌋,kS=\ k/2 ,\,k\. Let ℛB=1,…,BR_B=\1,…,B\ be the common reference index set for the original observations and their current embeddings. For each anchor a∈a and scale s∈s , define gs,aX g^X_s,a =sg[−log(rsX(a,ℛB)+εlog)], =sg\! [- \! (r_s^X(a,R_B)+ _ ) ], (24) gs,aY g^Y_s,a =−log(rsY(a,ℛB)+εlog). =- \! (r_s^Y(a,R_B)+ _ ). (25) Here, εlog>0 _ >0 stabilizes the logarithm. The stop-gradient operator satisfies sg[u]=usg[u]=u in the forward pass and ∂sg[u]/∂u=0 [u]/∂ u=0. Let sX=(gs,aX)a∈g^X_s=(g^X_s,a)_a and sY=(gs,aY)a∈g^Y_s=(g^Y_s,a)_a be the anchor-indexed log-density vectors at scale s, and let ρ(,)ρ(u,v) denote their Pearson correlation. The density-preservation loss averages one minus this correlation across scales, ℒdens=1||∑s∈[ 1−ρ(sY,sX)].L_dens= 1|S| _s [\,1-ρ (g^Y_s,\,g^X_s ) ]. (26) We compute the correlation by centering each vector and normalizing by its ℓ2 _2 norm: ρ(,)=⟨−u¯,−v¯⟩/(∥−u¯∥∥−v¯∥+ερ)ρ(u,v)= - u,\,v- v / ( - u \, - v + _ρ ). A single-scale version retains only s=ks=k; the ablation study evaluates the contribution of using both scales in S. We jointly optimize the manifold- and density-preservation losses using Eq. (1), with density weight λd≥0 _d≥ 0. 2.6 Model implementation Unless noted otherwise, we train each model for 1000 epochs with AdamW, an initial learning rate of 1×10−31× 10^-3, cosine learning-rate annealing, mixed precision, and a batch size of 4096. The benchmark configuration uses A0=512A_0=512, λd=1.8×10−3 _d=1.8× 10^-3, and k=12k=12. Complete implementation settings and computational measurements are reported in the Supplementary Material. 2.7 Datasets The main benchmark contains nine datasets: a synthetic branching dataset, four scRNA-seq datasets, two image datasets, a text dataset, and a sensor dataset. Supplementary Table S1 summarizes these datasets together with the C. elegans (CELEGAN) developmental case study. The biological data comprise HCL (8), MCA (7), GAST10K (28), EPI, and CELEGAN (17); they cover cell-type atlases, lesion-associated epithelial states, and developmental trajectories. We reserve CELEGAN for the developmental case study because it includes both cell-type and embryonic-time annotations. A separately generated synthetic dyngen trajectory appears only in Supplementary Section S7.1. 2.8 Baselines and evaluation metrics We compare DMT-Dens with six dimensionality-reduction methods: t-SNE (21), UMAP (12; 1), PaCMAP (23), PHATE (13), den-SNE, and densMAP (14). Baseline settings and hyperparameter search grids are reported in the Supplementary Material. The main evaluation considers density preservation and label separability. Following the convention used for den-SNE and densMAP (14), density correlation is the Spearman correlation between local radii in the high-dimensional space and the embedding. Each radius is the mean distance to the k nearest neighbors. The evaluation is computed on the full dataset. The training loss uses k-th-neighbor log-radii and Pearson correlation evaluated on mini-batch anchors at two scales. Label separability is measured by the stratified five-fold cross-validated accuracy of a linear support-vector classifier (SVC) fitted to the two-dimensional embedding. The Supplementary Material reports additional measures of local fidelity, global distance, runtime, and biological trajectory structure. 2.9 Ablation design We conducted single-factor ablations on EPI, HCL, and MNIST. Relative to the full model, four variants use distance affinity, unidirectional matching, all-pair averaging, or only the density scale s=ks=k. A fifth variant removes density regularization. All other settings, including the architecture, augmentation, optimizer, batch size, 1000-epoch schedule, and dataset-specific hyperparameters, are held fixed. Each variant uses the same three seeds. We report density correlation, kNN preservation, and SVC accuracy. 3 Results 3.1 Quantitative results Table 3.1 compares the density correlation and SVC accuracy of DMT-Dens and six baseline methods across nine benchmark datasets. Results are reported over five seeds, and den-SNE is marked OOT on EMNIST because its runtime was 24 h+. Local density correlation and additional embedding-quality metrics are provided in the Supplementary Material. Table 1: Main comparison on density preservation and label separability. Higher is better for both metrics, and the best (highest) value in each row is shown in bold. Entries report mean± deviation over available seeds. The final row of each panel reports the unweighted mean over all nine datasets; this mean is omitted for den-SNE because its EMNIST run did not complete. OOT (out of time) denotes a runtime of 24 h+; this applies to den-SNE on the full EMNIST (byclass, ≈698≈698K points) benchmark. A. Density correlation Dataset t-SNE UMAP den-SNE densMAP PaCMAP PHATE DMT-Dens 1*Synthetic ArtificialTree 0.812±0.0100.812± 0.010 0.626±0.0340.626± 0.034 0.575±0.0310.575± 0.031 0.746±0.0310.746± 0.031 0.653±0.0280.653± 0.028 −0.188±0.098-0.188± 0.098 0.954±0.0040.954± 0.004 4*Biological HCL 0.110±0.0200.110± 0.020 0.036±0.0310.036± 0.031 0.718±0.0170.718± 0.017 0.531±0.0150.531± 0.015 0.083±0.0310.083± 0.031 0.382±0.0090.382± 0.009 0.870±0.0030.870± 0.003 GAST10K 0.061±0.0380.061± 0.038 −0.156±0.017-0.156± 0.017 0.786±0.0110.786± 0.011 0.798±0.0150.798± 0.015 −0.530±0.019-0.530± 0.019 0.169±0.0200.169± 0.020 0.872±0.0080.872± 0.008 EPI 0.222±0.0350.222± 0.035 0.086±0.0290.086± 0.029 0.705±0.0110.705± 0.011 0.712±0.0120.712± 0.012 0.124±0.0250.124± 0.025 0.243±0.0160.243± 0.016 0.792±0.0140.792± 0.014 MCA 0.416±0.0250.416± 0.025 0.112±0.0120.112± 0.012 0.327±0.0150.327± 0.015 0.466±0.0190.466± 0.019 −0.001±0.032-0.001± 0.032 −0.028±0.031-0.028± 0.031 0.644±0.0680.644± 0.068 4*Non-biological MNIST 0.296±0.0480.296± 0.048 −0.081±0.020-0.081± 0.020 0.768±0.0140.768± 0.014 0.756±0.0200.756± 0.020 −0.020±0.040-0.020± 0.040 0.152±0.0220.152± 0.022 0.700±0.0280.700± 0.028 EMNIST 0.241±0.0250.241± 0.025 0.025±0.0250.025± 0.025 OOT 0.600±0.0220.600± 0.022 0.094±0.0180.094± 0.018 0.166±0.0430.166± 0.043 0.678±0.0150.678± 0.015 NG20 0.386±0.0290.386± 0.029 0.281±0.0210.281± 0.021 0.758±0.0070.758± 0.007 0.797±0.0100.797± 0.010 0.190±0.0370.190± 0.037 0.609±0.0150.609± 0.015 0.674±0.0170.674± 0.017 ACT 0.355±0.0510.355± 0.051 0.147±0.0370.147± 0.037 0.902±0.0070.902± 0.007 0.840±0.0110.840± 0.011 0.135±0.0430.135± 0.043 0.394±0.0930.394± 0.093 0.832±0.0070.832± 0.007 Mean 0.3220.322 0.1200.120 – 0.6940.694 0.0810.081 0.2110.211 0.7800.780 B. SVC accuracy Dataset t-SNE UMAP den-SNE densMAP PaCMAP PHATE DMT-Dens 1*Synthetic ArtificialTree 0.848±0.0190.848± 0.019 0.548±0.0220.548± 0.022 0.589±0.0350.589± 0.035 0.610±0.0200.610± 0.020 0.672±0.0260.672± 0.026 0.227±0.0640.227± 0.064 0.796±0.0220.796± 0.022 4*Biological HCL 0.601±0.0200.601± 0.020 0.304±0.0190.304± 0.019 0.752±0.0320.752± 0.032 0.282±0.0040.282± 0.004 0.755±0.0140.755± 0.014 0.337±0.0290.337± 0.029 0.781±0.0150.781± 0.015 GAST10K 0.654±0.0140.654± 0.014 0.597±0.0260.597± 0.026 0.770±0.0510.770± 0.051 0.593±0.0130.593± 0.013 0.772±0.0240.772± 0.024 0.736±0.0160.736± 0.016 0.792±0.0310.792± 0.031 EPI 0.839±0.0120.839± 0.012 0.848±0.0100.848± 0.010 0.831±0.0260.831± 0.026 0.843±0.0100.843± 0.010 0.858±0.0120.858± 0.012 0.788±0.0040.788± 0.004 0.884±0.0220.884± 0.022 MCA 0.435±0.0180.435± 0.018 0.326±0.0250.326± 0.025 0.807±0.0230.807± 0.023 0.290±0.0140.290± 0.014 0.603±0.0310.603± 0.031 0.713±0.0160.713± 0.016 0.774±0.0590.774± 0.059 4*Non-biological MNIST 0.931±0.0060.931± 0.006 0.962±0.0030.962± 0.003 0.922±0.0070.922± 0.007 0.945±0.0040.945± 0.004 0.959±0.0020.959± 0.002 0.706±0.0320.706± 0.032 0.950±0.0070.950± 0.007 EMNIST 0.601±0.0200.601± 0.020 0.655±0.0080.655± 0.008 OOT 0.646±0.0120.646± 0.012 0.626±0.0120.626± 0.012 0.468±0.0050.468± 0.005 0.639±0.0080.639± 0.008 NG20 0.198±0.0330.198± 0.033 0.219±0.0080.219± 0.008 0.294±0.0210.294± 0.021 0.264±0.0170.264± 0.017 0.245±0.0210.245± 0.021 0.239±0.0190.239± 0.019 0.297±0.0340.297± 0.034 ACT 0.859±0.0070.859± 0.007 0.823±0.0190.823± 0.019 0.831±0.0160.831± 0.016 0.811±0.0320.811± 0.032 0.842±0.0150.842± 0.015 0.783±0.0210.783± 0.021 0.876±0.0030.876± 0.003 Mean 0.6630.663 0.5870.587 – 0.5870.587 0.7040.704 0.5550.555 0.7540.754 Based on the reported means, DMT-Dens had the highest density correlation on ArtificialTree, HCL, GAST10K, EPI, MCA, and EMNIST. den-SNE had the highest value on MNIST and ACT, and densMAP had the highest value on NG20. DMT-Dens ranked first or second in mean SVC accuracy on seven datasets and third on MNIST and EMNIST. Figure 1c reports the mean density-correlation and SVC-accuracy ranks over the eight datasets with complete results for all methods. EMNIST was excluded because the den-SNE runtime was 24 h+. DMT-Dens had the lowest mean rank for both metrics and lay on the Pareto frontier when all seven methods were considered. Among the baselines, densMAP and den-SNE had the two lowest mean density-correlation ranks, while PaCMAP and den-SNE had the two lowest mean SVC-accuracy ranks. 3.2 Biological embedding visualizations Figure 4 displays the two-dimensional embeddings produced by DMT-Dens, t-SNE, UMAP, den-SNE, and densMAP for GAST10K, HCL, and EPI. Points are colored by the cell-type annotations used for evaluation. Within the DMT-Dens panels, local point density differs across annotated populations, with both compact and diffuse regions visible in GAST10K, HCL, and EPI. The full seven-method comparison for all datasets is provided in Supplementary Figure S2. Figure 3: Two-dimensional embeddings of three single-cell datasets. Rows correspond to GAST10K, HCL, and EPI, and columns correspond to t-SNE, UMAP, den-SNE, densMAP, and DMT-Dens. Points are colored by cell type, and axis limits are matched across methods within each dataset. DMT-Dens is shown in the rightmost column. Supplementary Figure S2 includes MCA, PHATE, and PaCMAP. Figure 4: Two-dimensional embeddings of the C. elegans embryonic time-course data produced by DMT-Dens, t-SNE, PHATE, and densMAP. (A) Embeddings colored by annotated cell type. (B) The same embeddings colored by observed developmental time. Quantitative developmental-time metrics are reported in Supplementary Table S11. 3.3 Ablation results Table 3.3 reports the ablations described in Section 2.9. Across the three datasets, all-pair averaging produced the highest mean density correlation, together with the lowest mean kNN preservation and SVC accuracy. Removing density regularization produced the opposite pattern: density correlation was the lowest among the six settings, whereas kNN preservation and SVC accuracy were higher than those of the full model. This variant attained the highest kNN preservation on MNIST and the highest SVC accuracy on HCL and MNIST. Replacing rank affinity with distance affinity lowered all three metrics relative to the full model. The unidirectional variant increased kNN preservation on every dataset and attained the highest values on EPI and HCL, while its density correlation remained below that of the full model. The single-scale-density variant also had lower density correlation than the full model and its SVC accuracy increased on EPI and decreased slightly on HCL and MNIST. Using the mean of density correlation and SVC accuracy, the full model obtained the highest value among the six settings on EPI, HCL, and MNIST. No ablation variant exceeded the full model on all three reported metrics for any dataset. Table 2: Ablation results on EPI, HCL, and MNIST. Each row changes one component of the full model at a time, as defined in Section 2.9. Dens., kNN, and SVC denote density correlation, kNN preservation, and SVC accuracy, respectively. Entries report mean± standard deviation over three matched seeds (42–44). Higher is better, and the highest value in each dataset–metric column is shown in bold. EPI HCL MNIST -4-7-10 Dens. kNN SVC Dens. kNN SVC Dens. kNN SVC Remove or replace one component of Full Full 0.817±0.0010.817± 0.001 0.155±0.0020.155± 0.002 0.885±0.0190.885± 0.019 0.863±0.0030.863± 0.003 0.092±0.0010.092± 0.001 0.790±0.0300.790± 0.030 0.697±0.0030.697± 0.003 0.310±0.0020.310± 0.002 0.950±0.0010.950± 0.001 Distance affinity 0.674±0.0020.674± 0.002 0.108±0.0000.108± 0.000 0.736±0.0040.736± 0.004 0.818±0.0040.818± 0.004 0.062±0.0030.062± 0.003 0.485±0.0100.485± 0.010 0.607±0.0430.607± 0.043 0.250±0.0030.250± 0.003 0.745±0.0120.745± 0.012 Unidirectional 0.765±0.0170.765± 0.017 0.166±0.0020.166± 0.002 0.870±0.0480.870± 0.048 0.844±0.0050.844± 0.005 0.101±0.0010.101± 0.001 0.801±0.0020.801± 0.002 0.673±0.0060.673± 0.006 0.313±0.0010.313± 0.001 0.952±0.0030.952± 0.003 All-pair 0.905±0.0020.905± 0.002 0.035±0.0050.035± 0.005 0.447±0.0110.447± 0.011 0.960±0.0020.960± 0.002 0.028±0.0010.028± 0.001 0.380±0.0260.380± 0.026 0.896±0.0110.896± 0.011 0.117±0.0200.117± 0.020 0.691±0.0670.691± 0.067 Single-scale density 0.791±0.0110.791± 0.011 0.153±0.0030.153± 0.003 0.906±0.0080.906± 0.008 0.860±0.0040.860± 0.004 0.092±0.0010.092± 0.001 0.788±0.0060.788± 0.006 0.683±0.0100.683± 0.010 0.309±0.0010.309± 0.001 0.943±0.0040.943± 0.004 No density 0.211±0.0070.211± 0.007 0.162±0.0010.162± 0.001 0.904±0.0080.904± 0.008 0.068±0.0340.068± 0.034 0.097±0.0020.097± 0.002 0.842±0.0030.842± 0.003 −0.054±0.041-0.054± 0.041 0.340±0.0010.340± 0.001 0.955±0.0010.955± 0.001 3.4 Biological case study We evaluated DMT-Dens on the C. elegans embryonic time course (CELEGAN) (17). Each cell has a cell-type annotation and an embryonic-time annotation recorded as a discrete interval in minutes after fertilization. Figure 4 displays the embeddings colored separately by these two annotations. Figure 5 reports density correlation and SVC accuracy. Supplementary Section S7.2 and Supplementary Table S11 report pseudotime correlation, ordering accuracy, time continuity, DEMaP, and the reachable fraction of cells. Pseudotime correlation and ordering accuracy are computed on the reachable subset when an embedding graph is fragmented, and the corresponding reachable fractions are reported in the same table. Figure 5: Density correlation versus SVC accuracy on the C. elegans case study (mean ± s.d. over five seeds). The dashed line marks the baseline Pareto frontier. DMT-Dens had a mean density correlation of 0.9090.909 and a mean SVC accuracy of 0.7210.721; the highest baseline values for these metrics were 0.8410.841 for densMAP and 0.8280.828 for t-SNE, respectively. 3.5 Supplementary results Supplementary Table S9 reports additional embedding metrics, and Supplementary Figures S2 and S3 show the full seven-method embedding comparisons. Supplementary Tables S3–S5 report the analyses of the density weight λd _d, neighborhood scale k, and number of density anchors. Runtime and peak GPU memory for DMT-Dens are reported in Supplementary Tables S6 and S7. Supplementary Section S7.1 reports results for all methods on synthetic single-cell data generated with dyngen (5). The simulation provides the trajectory topology, branch assignments, and simulation time. 4 Discussion DMT-Dens integrates rank-affinity manifold alignment, hard-pair optimization, and an explicit density-consistency regularizer. Across the benchmark datasets, the resulting embeddings showed high density correlation, particularly on the biological datasets, while remaining competitive in SVC accuracy. The ablation results distinguish the roles of the objective components. Removing density regularization markedly reduced density correlation, whereas all-pair averaging increased density correlation but lowered kNN preservation and SVC accuracy. Unidirectional matching and single-scale density estimation produced dataset-dependent changes. No ablation improved all three metrics within any dataset, indicating that density fidelity, neighborhood preservation, and label separability are related but distinct objectives. Nonlinear projections can alter local point density and the apparent extent of cell populations (14; 6). The density-consistency term penalizes disagreement between density estimates in the processed input and the embedding, encouraging compact and diffuse regions in two dimensions to follow the corresponding input-space variation. For rare transition cells (29) and continuous differentiation processes (19), this information complements neighborhood structure and annotation-based views of the embedding. The meaning of the retained density requires care. DMT-Dens operates on the processed input data, whose density reflects experimental sampling and preprocessing. Tissue dissociation can also induce transcriptional changes (20). The area occupied by a population should therefore be interpreted as observed sample density, not as a calibrated estimate of biological abundance. This distinction is especially important for rare and transitional states, which are sensitive to sampling variation. Continuous cell-state density estimators such as Mellon (16) offer a basis for evaluating alternative density definitions. The current formulation retains two user-specified choices: the neighborhood scale and the density-loss weight. Their effects varied across datasets in the sensitivity analyses, motivating further study of adaptive neighborhood selection and data-dependent loss calibration. The dyngen evaluation also shows that density fidelity and trajectory recovery need not align: DMT-Dens led the density metrics, while t-SNE and den-SNE attained the highest Branch SVC and topology fidelity, respectively. Density-aware projections should therefore be evaluated alongside topology-specific criteria when trajectory structure is central to the analysis. The parametric mapping supports projection of new samples, and the density regularizer could be evaluated in other parametric projection models (26). Further work should also test its interaction with trajectory-inference (18; 24) and RNA-velocity-based fate-mapping workflows (2; 11). DMT-Dens provides a parametric approach for retaining relative density patterns while balancing neighborhood structure and label separability. Its embeddings remain representations of sampled and preprocessed data, and their biological interpretation should explicitly account for technical and sampling effects. Evaluation on larger datasets and in downstream tasks is needed to establish when improved density fidelity changes biological analyses. Conflicts of interest The authors declare no competing interests. Funding Funding information will be added in the final version. Data availability All datasets used in this work are publicly available. Source code and reproducible scripts are available at https://github.com/Ruizhe-wang/DMT-Dens. Author contributions statement Author contributions will be completed after the author list is finalized. 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Nature Communications 12 (1), p. 5609. External Links: Document Cited by: §4. Supplementary Material DMT-Dens: Density-preserving manifold visualization for biological data S1 Datasets Table S1 summarizes the nine benchmark datasets and the C. elegans (CELEGAN) case-study dataset described in the main-text Methods. The separately generated dyngen simulation is described in Section S7.1. The table reports the modality, sample size n, feature dimension D, number of class labels, and evaluation annotation for each listed dataset. Labels are excluded from representation training but are used in the combined hyperparameter-selection criterion and in downstream evaluation and visualization. The KaugK_aug-nearest-neighbor graph used for data augmentation is computed once in a 64-dimensional principal-component space. We use Kaug=200K_aug=200 for the main nine-dataset benchmark, Kaug=50K_aug=50 for the CELEGAN case study, and Kaug=500K_aug=500 for the dyngen case study. ArtificialTree is generated with the synthetic tree simulator in the PHATE package, which samples points from a branching hierarchy with Gaussian noise. The implementation is documented under phate.tree.gen_dla. Table S1: Datasets used in the nine-dataset benchmark and the C. elegans case study. The separately generated dyngen simulation is described in Section S7.1. The table reports modality, sample size n, feature dimension D, number of class labels, and the annotation used for evaluation. Modality n D Classes Evaluation annotation synthetic 80,000 1,000 80 Synthetic branch labels MNIST image 60,000 784 10 Digit class (00–99) EMNIST image 697,932 784 62 Character class NG20 text 18,846 100 20 Newsgroup topic ACT sensor 10,299 561 6 Activity type HCL scRNA-seq 49,551 3,038 57 Human cell types across tissues MCA scRNA-seq 23,341 9,120 19 Mouse cell types GAST10K scRNA-seq 10,638 1,458 12 Human gastric-mucosa cell types CELEGAN scRNA-seq 54,649 2,766 36 C. elegans embryonic cell types; developmental time EPI scRNA-seq 100,000 500 10 Epithelial cell/state S2 Derivation of the dimension-free kNN log-density Let rk()r_k(x) denote the distance from a query observation x to its k-th nearest neighbor among n reference observations in d dimensions, excluding the query itself when it belongs to the reference sample. Assuming that the density f is approximately constant within this neighborhood gives kn≈f()Vdrk()d,Vd=πd/2Γ(d2+1), kn≈ f(x)\,V_d\,r_k(x)^d, 16.38895ptV_d= π^d/2 \! ( d2+1 ), (S1) where VdV_d is the volume of the unit d-ball. The corresponding kNN density estimator and its logarithm are f^k()=knVdrk()d,logf^k()=C−dlogrk(),C=logk−logn−logVd. f_k(x)= knV_dr_k(x)^d, 16.38895pt f_k(x)=C-d r_k(x), 16.38895ptC= k- n- V_d. (S2) For fixed n, k, and d, the term C is constant across points and d is a positive scale factor. Because Pearson correlation is invariant to positive affine transformations of either density vector, neither term affects the density-preservation objective. We therefore use the dimension-free quantity logf^k()∝−logrk() f_k(x) - r_k(x). S3 Model setup The architecture and training settings reported here are the effective configurations used in the final five-seed benchmark. Input features are normalized with BatchNorm1d and compressed by a learned low-rank map into M=32M=32 latent tokens with rank r=16r=16. Each latent is independently expanded to token width dt=224d_t=224 and augmented with a learned latent-identity embedding. Two pre-LayerNorm Transformer blocks with four attention heads and feed-forward width 4dt4d_t operate within the latent-token set of each observation. Both residual and attention dropout are zero. A final LayerNorm and mean pooling aggregate the tokens, after which a linear layer followed by batch normalization produces the 40-dimensional representation i=eϕ(i)z_i=e_φ(x_i). S3.1 Training and hyperparameter settings For the main benchmark, we set the density-loss weight to λd=1.8×10−3 _d=1.8× 10^-3, the neighborhood scale to k=12k=12, the maximum number of density anchors to A0=512A_0=512, the rank-kernel base to η=0.4η=0.4, the degrees of freedom of the embedding-space t kernel to ν=0.01ν=0.01, and the hard-pair threshold rank to m0=100m_0=100. We use εdist=10−12 _dist=10^-12, εaff=10−6 _aff=10^-6, εBCE=10−7 _BCE=10^-7, εlog=10−8 _ =10^-8, and ερ=10−8 _ρ=10^-8 for numerical stability. Neighborhood augmentation uses Kaug=200K_aug=200 and Dpca=64D_pca=64, with independent interpolation weights αi∼Uniform(0.05,1) _i (0.05,1). We train the model for 10001000 epochs using AdamW with a learning rate of 1×10−31× 10^-3, cosine annealing, a five-epoch warmup, mixed precision on one GPU, and a batch size of 40964096. Sensitivity to λd _d, k, and A0A_0 is reported in Section S4. S3.2 Baseline hyperparameter search We perform hyperparameter selection in two stages. First, each baseline is evaluated on the two-axis structural grid in Table S2, yielding 5×5=255× 5=25 configurations per method. Second, the selected structural configuration for den-SNE and densMAP is held fixed while the density-strength parameter is varied over λ∈0.1,0.25,0.5,1,2,4λ∈\0.1,0.25,0.5,1,2,4\. At both stages, configurations are ranked by the arithmetic mean of density correlation and SVC accuracy. Each selected configuration is then evaluated over five seeds for final reporting. Thus, class labels contribute to hyperparameter selection and evaluation but not to representation training. DMT-Dens settings are selected using the same combined criterion; sensitivity to its density-estimation parameters is reported in Section S4. Table S2: Hyperparameter search grids for the baseline methods. Hyperparameter Search values -SNE perplexity 15, 30, 50, 80, 12015,\ 30,\ 50,\ 80,\ 120 early exaggeration 4, 8, 12, 18, 244,\ 8,\ 12,\ 18,\ 24 UMAP n_neighbors 10, 15, 20, 40, 8010,\ 15,\ 20,\ 40,\ 80 min_dist 0.001, 0.01, 0.05, 0.08, 0.150.001,\ 0.01,\ 0.05,\ 0.08,\ 0.15 den-SNE perplexity 15, 30, 50, 80, 12015,\ 30,\ 50,\ 80,\ 120 early exaggeration 4, 8, 12, 18, 244,\ 8,\ 12,\ 18,\ 24 density strength λ (stage 2) 0.1, 0.25, 0.5, 1, 2, 40.1,\ 0.25,\ 0.5,\ 1,\ 2,\ 4 densMAP n_neighbors 10, 15, 20, 40, 8010,\ 15,\ 20,\ 40,\ 80 min_dist 0.001, 0.01, 0.05, 0.08, 0.150.001,\ 0.01,\ 0.05,\ 0.08,\ 0.15 density strength λ (stage 2) 0.1, 0.25, 0.5, 1, 2, 40.1,\ 0.25,\ 0.5,\ 1,\ 2,\ 4 PaCMAP n_neighbors 10, 15, 20, 40, 8010,\ 15,\ 20,\ 40,\ 80 (MN_ratio, FP_ratio) (0.3,1),(0.5,1),(0.5,2),(1,2),(2,5)(0.3,1),\ (0.5,1),\ (0.5,2),\ (1,2),\ (2,5) PHATE knn 5, 10, 15, 20, 405,\ 10,\ 15,\ 20,\ 40 decay 10, 20, 40, 80, 12010,\ 20,\ 40,\ 80,\ 120 S4 Parameter sensitivity We assess one-at-a-time sensitivity to three density-estimation hyperparameters: the maximum number of density anchors A0A_0, the neighborhood scale k, and the density-loss weight λd _d. In each table, the other two parameters are held at the main-benchmark setting (A0=512A_0=512, k=12k=12, and λd=1.8×10−3 _d=1.8× 10^-3). The central setting is represented by the same five baseline runs with seeds 42–46 in all three tables. Table S3: Sensitivity to the maximum number of density anchors A0A_0 on GAST10K, MCA, HCL, and NG20. Entries are mean± deviation over five seeds; higher is better for all three reported metrics. Setting Density Local dens. SVC 10K A0=128A_0=128 0.821±0.0220.821± 0.022 0.824±0.0210.824± 0.021 0.794±0.0450.794± 0.045 A0=256A_0=256 0.853±0.0130.853± 0.013 0.848±0.0130.848± 0.013 0.768±0.0540.768± 0.054 A0=512A_0=512 0.873±0.0090.873± 0.009 0.856±0.0110.856± 0.011 0.787±0.0510.787± 0.051 A0=768A_0=768 0.871±0.0130.871± 0.013 0.848±0.0140.848± 0.014 0.765±0.0370.765± 0.037 A0=1024A_0=1024 0.878±0.0120.878± 0.012 0.850±0.0150.850± 0.015 0.727±0.0670.727± 0.067 A0=1536A_0=1536 0.884±0.0100.884± 0.010 0.852±0.0150.852± 0.015 0.789±0.0430.789± 0.043 A0=128A_0=128 0.661±0.0400.661± 0.040 0.612±0.0430.612± 0.043 0.739±0.0830.739± 0.083 A0=256A_0=256 0.648±0.0440.648± 0.044 0.581±0.0400.581± 0.040 0.804±0.0290.804± 0.029 A0=512A_0=512 0.642±0.0630.642± 0.063 0.551±0.0860.551± 0.086 0.788±0.0260.788± 0.026 A0=768A_0=768 0.612±0.0710.612± 0.071 0.529±0.0690.529± 0.069 0.752±0.0450.752± 0.045 A0=1024A_0=1024 0.608±0.0300.608± 0.030 0.516±0.0450.516± 0.045 0.750±0.0420.750± 0.042 A0=1536A_0=1536 0.597±0.0140.597± 0.014 0.504±0.0310.504± 0.031 0.783±0.0840.783± 0.084 A0=128A_0=128 0.837±0.0110.837± 0.011 0.818±0.0180.818± 0.018 0.774±0.0130.774± 0.013 A0=256A_0=256 0.847±0.0100.847± 0.010 0.819±0.0190.819± 0.019 0.778±0.0080.778± 0.008 A0=512A_0=512 0.864±0.0050.864± 0.005 0.829±0.0110.829± 0.011 0.763±0.0080.763± 0.008 A0=768A_0=768 0.867±0.0110.867± 0.011 0.826±0.0180.826± 0.018 0.774±0.0160.774± 0.016 A0=1024A_0=1024 0.871±0.0110.871± 0.011 0.831±0.0180.831± 0.018 0.764±0.0200.764± 0.020 A0=1536A_0=1536 0.871±0.0150.871± 0.015 0.821±0.0190.821± 0.019 0.769±0.0310.769± 0.031 20 A0=128A_0=128 0.578±0.0110.578± 0.011 0.589±0.0050.589± 0.005 0.335±0.0100.335± 0.010 A0=256A_0=256 0.649±0.0200.649± 0.020 0.650±0.0180.650± 0.018 0.345±0.0260.345± 0.026 A0=512A_0=512 0.678±0.0230.678± 0.023 0.676±0.0230.676± 0.023 0.334±0.0270.334± 0.027 A0=768A_0=768 0.714±0.0350.714± 0.035 0.700±0.0260.700± 0.026 0.331±0.0370.331± 0.037 A0=1024A_0=1024 0.723±0.0310.723± 0.031 0.702±0.0240.702± 0.024 0.334±0.0440.334± 0.044 A0=1536A_0=1536 0.759±0.0260.759± 0.026 0.728±0.0210.728± 0.021 0.307±0.0190.307± 0.019 Table S4: Sensitivity to the density neighborhood scale k on GAST10K, MCA, HCL, and NG20. Entries are mean± deviation over five seeds; higher is better for all three reported metrics. Setting Density Local dens. SVC 10K k=5k=5 0.809±0.0210.809± 0.021 0.796±0.0240.796± 0.024 0.792±0.0450.792± 0.045 k=10k=10 0.859±0.0120.859± 0.012 0.838±0.0100.838± 0.010 0.749±0.0330.749± 0.033 k=12k=12 0.873±0.0090.873± 0.009 0.856±0.0110.856± 0.011 0.787±0.0510.787± 0.051 k=15k=15 0.868±0.0090.868± 0.009 0.860±0.0130.860± 0.013 0.728±0.0540.728± 0.054 k=20k=20 0.871±0.0110.871± 0.011 0.876±0.0100.876± 0.010 0.782±0.0580.782± 0.058 k=25k=25 0.863±0.0050.863± 0.005 0.881±0.0050.881± 0.005 0.760±0.0720.760± 0.072 k=30k=30 0.852±0.0080.852± 0.008 0.885±0.0100.885± 0.010 0.781±0.0280.781± 0.028 k=40k=40 0.800±0.0130.800± 0.013 0.874±0.0100.874± 0.010 0.765±0.0500.765± 0.050 k=5k=5 0.566±0.0650.566± 0.065 0.496±0.0760.496± 0.076 0.747±0.0790.747± 0.079 k=10k=10 0.639±0.0620.639± 0.062 0.535±0.0690.535± 0.069 0.793±0.0260.793± 0.026 k=12k=12 0.642±0.0630.642± 0.063 0.551±0.0860.551± 0.086 0.788±0.0260.788± 0.026 k=15k=15 0.612±0.0370.612± 0.037 0.559±0.0330.559± 0.033 0.764±0.0450.764± 0.045 k=20k=20 0.647±0.0840.647± 0.084 0.621±0.0820.621± 0.082 0.792±0.0100.792± 0.010 k=25k=25 0.666±0.0440.666± 0.044 0.647±0.0420.647± 0.042 0.786±0.0440.786± 0.044 k=30k=30 0.596±0.0560.596± 0.056 0.592±0.0590.592± 0.059 0.749±0.0700.749± 0.070 k=40k=40 0.514±0.0600.514± 0.060 0.516±0.0620.516± 0.062 0.770±0.0300.770± 0.030 k=5k=5 0.797±0.0610.797± 0.061 0.767±0.0680.767± 0.068 0.755±0.0190.755± 0.019 k=10k=10 0.853±0.0110.853± 0.011 0.817±0.0190.817± 0.019 0.768±0.0210.768± 0.021 k=12k=12 0.864±0.0050.864± 0.005 0.829±0.0110.829± 0.011 0.763±0.0080.763± 0.008 k=15k=15 0.865±0.0090.865± 0.009 0.838±0.0150.838± 0.015 0.761±0.0140.761± 0.014 k=20k=20 0.869±0.0050.869± 0.005 0.860±0.0090.860± 0.009 0.781±0.0210.781± 0.021 k=25k=25 0.859±0.0090.859± 0.009 0.859±0.0140.859± 0.014 0.791±0.0160.791± 0.016 k=30k=30 0.841±0.0120.841± 0.012 0.854±0.0100.854± 0.010 0.779±0.0260.779± 0.026 k=40k=40 0.792±0.0110.792± 0.011 0.833±0.0120.833± 0.012 0.791±0.0170.791± 0.017 20 k=5k=5 0.526±0.0190.526± 0.019 0.526±0.0170.526± 0.017 0.327±0.0340.327± 0.034 k=10k=10 0.671±0.0360.671± 0.036 0.666±0.0350.666± 0.035 0.297±0.0350.297± 0.035 k=12k=12 0.678±0.0230.678± 0.023 0.676±0.0230.676± 0.023 0.334±0.0270.334± 0.027 k=15k=15 0.680±0.0200.680± 0.020 0.683±0.0180.683± 0.018 0.318±0.0410.318± 0.041 k=20k=20 0.709±0.0170.709± 0.017 0.718±0.0180.718± 0.018 0.320±0.0240.320± 0.024 k=25k=25 0.694±0.0110.694± 0.011 0.713±0.0140.713± 0.014 0.308±0.0360.308± 0.036 k=30k=30 0.689±0.0110.689± 0.011 0.727±0.0100.727± 0.010 0.315±0.0400.315± 0.040 k=40k=40 0.660±0.0090.660± 0.009 0.726±0.0130.726± 0.013 0.329±0.0100.329± 0.010 Table S5: Sensitivity to the density loss weight λd _d on GAST10K, MCA, HCL, and NG20. Entries are mean± deviation over five seeds; higher is better for all three reported metrics. Setting Density Local dens. SVC 10K λd=0.0001 _d=0.0001 0.223±0.0630.223± 0.063 0.186±0.0680.186± 0.068 0.726±0.0620.726± 0.062 λd=0.0005 _d=0.0005 0.742±0.0200.742± 0.020 0.665±0.0300.665± 0.030 0.752±0.0490.752± 0.049 λd=0.001 _d=0.001 0.837±0.0070.837± 0.007 0.803±0.0140.803± 0.014 0.753±0.0690.753± 0.069 λd=0.0018 _d=0.0018 0.873±0.0090.873± 0.009 0.856±0.0110.856± 0.011 0.787±0.0510.787± 0.051 λd=0.003 _d=0.003 0.883±0.0020.883± 0.002 0.876±0.0060.876± 0.006 0.760±0.0420.760± 0.042 λd=0.006 _d=0.006 0.907±0.0040.907± 0.004 0.904±0.0070.904± 0.007 0.767±0.0450.767± 0.045 λd=0.01 _d=0.01 0.920±0.0070.920± 0.007 0.920±0.0080.920± 0.008 0.746±0.0440.746± 0.044 λd=0.02 _d=0.02 0.936±0.0060.936± 0.006 0.937±0.0050.937± 0.005 0.724±0.0500.724± 0.050 λd=0.0001 _d=0.0001 0.323±0.0510.323± 0.051 0.239±0.0610.239± 0.061 0.790±0.0650.790± 0.065 λd=0.0005 _d=0.0005 0.434±0.0400.434± 0.040 0.359±0.0400.359± 0.040 0.821±0.0300.821± 0.030 λd=0.001 _d=0.001 0.564±0.0600.564± 0.060 0.489±0.0570.489± 0.057 0.813±0.0170.813± 0.017 λd=0.0018 _d=0.0018 0.642±0.0630.642± 0.063 0.551±0.0860.551± 0.086 0.788±0.0260.788± 0.026 λd=0.003 _d=0.003 0.719±0.0780.719± 0.078 0.646±0.0830.646± 0.083 0.786±0.0440.786± 0.044 λd=0.006 _d=0.006 0.778±0.0190.778± 0.019 0.728±0.0310.728± 0.031 0.758±0.0520.758± 0.052 λd=0.01 _d=0.01 0.834±0.0190.834± 0.019 0.802±0.0260.802± 0.026 0.689±0.0270.689± 0.027 λd=0.02 _d=0.02 0.831±0.0240.831± 0.024 0.809±0.0290.809± 0.029 0.657±0.0380.657± 0.038 λd=0.0001 _d=0.0001 0.441±0.0250.441± 0.025 0.420±0.0240.420± 0.024 0.825±0.0160.825± 0.016 λd=0.0005 _d=0.0005 0.751±0.0070.751± 0.007 0.703±0.0140.703± 0.014 0.804±0.0180.804± 0.018 λd=0.001 _d=0.001 0.819±0.0100.819± 0.010 0.769±0.0230.769± 0.023 0.796±0.0140.796± 0.014 λd=0.0018 _d=0.0018 0.864±0.0050.864± 0.005 0.829±0.0110.829± 0.011 0.763±0.0080.763± 0.008 λd=0.003 _d=0.003 0.886±0.0080.886± 0.008 0.861±0.0140.861± 0.014 0.750±0.0200.750± 0.020 λd=0.006 _d=0.006 0.914±0.0040.914± 0.004 0.898±0.0040.898± 0.004 0.711±0.0280.711± 0.028 λd=0.01 _d=0.01 0.924±0.0030.924± 0.003 0.915±0.0050.915± 0.005 0.695±0.0150.695± 0.015 λd=0.02 _d=0.02 0.934±0.0030.934± 0.003 0.930±0.0020.930± 0.002 0.625±0.0150.625± 0.015 20 λd=0.0001 _d=0.0001 0.196±0.0260.196± 0.026 0.229±0.0210.229± 0.021 0.331±0.0220.331± 0.022 λd=0.0005 _d=0.0005 0.565±0.0380.565± 0.038 0.539±0.0300.539± 0.030 0.322±0.0080.322± 0.008 λd=0.001 _d=0.001 0.646±0.0220.646± 0.022 0.625±0.0210.625± 0.021 0.319±0.0360.319± 0.036 λd=0.0018 _d=0.0018 0.678±0.0230.678± 0.023 0.676±0.0230.676± 0.023 0.334±0.0270.334± 0.027 λd=0.003 _d=0.003 0.712±0.0190.712± 0.019 0.710±0.0200.710± 0.020 0.301±0.0540.301± 0.054 λd=0.006 _d=0.006 0.762±0.0160.762± 0.016 0.766±0.0120.766± 0.012 0.286±0.0350.286± 0.035 λd=0.01 _d=0.01 0.812±0.0080.812± 0.008 0.819±0.0090.819± 0.009 0.259±0.0130.259± 0.013 λd=0.02 _d=0.02 0.895±0.0150.895± 0.015 0.899±0.0170.899± 0.017 0.183±0.0220.183± 0.022 S5 Runtime, peak GPU memory, and hardware Matched runtime and peak-memory measurements were collected on a workstation with two Intel Xeon Gold 5118 CPUs at 2.30 GHz (24 physical cores and 48 threads in total), 376 GiB of system memory, and eight NVIDIA GeForce RTX 2080 Ti GPUs with 11 GB of GDDR6 memory each. The system ran Ubuntu 22.04.5 LTS with NVIDIA driver 545.23.08, for which nvidia-smi reported CUDA 12.3 compatibility. DMT-Dens was implemented in Python 3.10.19 using PyTorch 2.5.1+cu121, Lightning 2.5.4, and cuDNN 9.1.0; the PyTorch build used CUDA 12.1. Baselines were implemented with openTSNE 1.0.4 for t-SNE, umap-learn 0.5.9.post2 for UMAP and densMAP, PaCMAP 0.8.0, PHATE 2.0.0, and the repository-bundled 2020 implementation of den-SNE. Evaluation used scikit-learn 1.7.2. Each DMT-Dens run used one RTX 2080 Ti GPU, whereas the non-parametric baselines (t-SNE, UMAP, den-SNE, densMAP, PHATE, and PaCMAP) were run on CPU. The runtime analysis reports wall-clock fitting time from the matched runtime sweep. Values are the mean± deviation over three seeds, measured in seconds. Peak GPU memory under the full training configuration is reported in Table S7. Table S6: Runtime scaling with dataset size. Entries are wall-clock fitting times in seconds, reported as mean± deviation over three seeds. n t-SNE UMAP den-SNE densMAP PHATE DMT-Dens 2,000 12.9±0.212.9± 0.2 35.6±0.335.6± 0.3 21.8±0.421.8± 0.4 29.8±0.229.8± 0.2 16.5±0.516.5± 0.5 119.2±12.4119.2± 12.4 5,000 25.9±0.525.9± 0.5 56.4±0.856.4± 0.8 61.8±0.961.8± 0.9 47.7±8.547.7± 8.5 30.5±0.630.5± 0.6 152.7±6.4152.7± 6.4 10,000 82.7±1.682.7± 1.6 71.7±1.271.7± 1.2 141.8±9.8141.8± 9.8 72.2±8.572.2± 8.5 35.1±1.035.1± 1.0 232.9±64.7232.9± 64.7 20,000 108.2±1.2108.2± 1.2 71.9±1.371.9± 1.3 376.9±14.2376.9± 14.2 78.7±1.378.7± 1.3 48.1±0.748.1± 0.7 321.0±3.9321.0± 3.9 40,000 146.3±5.8146.3± 5.8 105.6±1.4105.6± 1.4 891.7±42.2891.7± 42.2 152.4±5.3152.4± 5.3 70.4±0.270.4± 0.2 427.2±39.6427.2± 39.6 60,000 176.9±3.4176.9± 3.4 138.2±1.9138.2± 1.9 1635.9±27.81635.9± 27.8 213.4±7.4213.4± 7.4 102.7±4.1102.7± 4.1 508.5±12.7508.5± 12.7 2,000 12.7±0.212.7± 0.2 35.0±0.235.0± 0.2 26.7±0.026.7± 0.0 29.9±0.229.9± 0.2 15.8±0.015.8± 0.0 40.4±1.140.4± 1.1 5,000 27.3±1.127.3± 1.1 56.8±0.956.8± 0.9 75.7±1.475.7± 1.4 56.0±0.356.0± 0.3 29.6±0.629.6± 0.6 73.0±3.073.0± 3.0 10,000 82.2±0.782.2± 0.7 72.0±1.572.0± 1.5 134.0±0.7134.0± 0.7 78.9±2.078.9± 2.0 34.8±1.434.8± 1.4 98.5±1.998.5± 1.9 20,000 112.6±3.8112.6± 3.8 73.2±0.573.2± 0.5 319.1±1.4319.1± 1.4 96.2±0.896.2± 0.8 45.1±0.745.1± 0.7 155.3±4.8155.3± 4.8 40,000 148.3±1.5148.3± 1.5 104.2±0.9104.2± 0.9 976.5±7.2976.5± 7.2 159.9±1.8159.9± 1.8 70.1±0.670.1± 0.6 307.8±2.5307.8± 2.5 60,000 179.1±3.4179.1± 3.4 139.1±0.5139.1± 0.5 1746.3±1.51746.3± 1.5 233.7±2.1233.7± 2.1 104.0±1.0104.0± 1.0 460.4±2.1460.4± 2.1 10K 2,000 19.2±0.519.2± 0.5 42.9±1.542.9± 1.5 29.4±0.429.4± 0.4 37.6±1.037.6± 1.0 20.8±1.020.8± 1.0 89.1±7.689.1± 7.6 5,000 38.1±2.238.1± 2.2 72.3±1.772.3± 1.7 73.9±1.773.9± 1.7 71.8±0.371.8± 0.3 39.5±1.039.5± 1.0 127.4±5.0127.4± 5.0 10,000 52.3±1.652.3± 1.6 97.9±1.297.9± 1.2 138.9±12.5138.9± 12.5 97.8±0.597.8± 0.5 46.2±1.446.2± 1.4 169.3±11.1169.3± 11.1 10,638 53.7±1.753.7± 1.7 87.7±0.287.7± 0.2 157.6±8.0157.6± 8.0 85.8±0.885.8± 0.8 45.9±0.745.9± 0.7 185.0±9.7185.0± 9.7 2,000 31.7±1.231.7± 1.2 60.5±2.560.5± 2.5 50.6±1.750.6± 1.7 51.5±1.551.5± 1.5 33.2±1.433.2± 1.4 107.6±6.6107.6± 6.6 5,000 65.1±3.365.1± 3.3 96.5±1.896.5± 1.8 105.1±3.7105.1± 3.7 93.1±2.493.1± 2.4 60.1±1.760.1± 1.7 173.9±12.0173.9± 12.0 10,000 87.9±3.187.9± 3.1 123.9±3.2123.9± 3.2 191.9±7.4191.9± 7.4 124.1±1.9124.1± 1.9 65.1±1.665.1± 1.6 217.2±49.8217.2± 49.8 20,000 128.6±0.2128.6± 0.2 170.1±5.3170.1± 5.3 403.7±24.5403.7± 24.5 156.5±1.3156.5± 1.3 79.9±0.779.9± 0.7 399.4±78.5399.4± 78.5 40,000 184.5±6.9184.5± 6.9 328.4±6.5328.4± 6.5 1004.4±43.61004.4± 43.6 261.1±3.4261.1± 3.4 119.9±1.0119.9± 1.0 636.0±8.3636.0± 8.3 49,551 215.3±8.6215.3± 8.6 430.8±15.7430.8± 15.7 1371.0±84.61371.0± 84.6 312.5±8.5312.5± 8.5 139.0±1.9139.0± 1.9 852.7±149.7852.7± 149.7 2,000 79.2±1.379.2± 1.3 123.7±2.9123.7± 2.9 358.2±10.8358.2± 10.8 101.2±2.2101.2± 2.2 76.8±2.576.8± 2.5 210.3±103.6210.3± 103.6 5,000 163.8±0.5163.8± 0.5 189.4±4.1189.4± 4.1 451.4±13.0451.4± 13.0 177.1±2.9177.1± 2.9 132.1±3.1132.1± 3.1 293.0±48.9293.0± 48.9 10,000 199.8±4.0199.8± 4.0 251.7±5.1251.7± 5.1 579.2±25.1579.2± 25.1 214.8±0.9214.8± 0.9 141.7±3.1141.7± 3.1 392.6±21.1392.6± 21.1 20,000 362.5±2.9362.5± 2.9 369.3±7.8369.3± 7.8 804.6±37.1804.6± 37.1 274.3±3.1274.3± 3.1 158.2±2.6158.2± 2.6 459.4±31.2459.4± 31.2 23,341 320.9±5.1320.9± 5.1 424.6±10.9424.6± 10.9 915.6±50.1915.6± 50.1 301.8±6.9301.8± 6.9 163.3±2.3163.3± 2.3 511.4±13.2511.4± 13.2 Table S7 reports peak GPU memory for the nine benchmark datasets, defined as the maximum CUDA memory allocated during training. Peak usage does not exceed approximately 6.76.7 GB. Table S7: Peak GPU memory of DMT-Dens under the full training configuration for the nine benchmark datasets. Values are the maximum CUDA memory allocated during training. Peak GPU mem. (MB) 6032.2 EMNIST 5923.4 EPI 5898.7 GAST10K 5982.4 HCL 6120.4 MCA 6649.8 MNIST 5923.4 NG20 5864.8 ArtificialTree 5943.2 S6 Full quantitative metrics This section reports local-structure metrics that complement the main-text summaries of density correlation and SVC accuracy (Table 3.1). Table S8 reports local density correlation and scattered point intrusion rate (SPIR), and Table S9 reports continuity and kNN preservation. Density correlation compares pointwise kNN-radius estimates over the full dataset and thus summarizes the overall ordering of dense and sparse regions. Local density correlation evaluates density agreement within local neighborhoods, making it more sensitive to short-range density variation. SPIR quantifies spurious neighborhood intrusions in the embedding, in which points that are distant in the input space become locally close after projection; lower values indicate fewer such intrusions. Table S8: Density-preservation metrics for DMT-Dens and six baselines across the benchmark datasets. Higher local density correlation and lower scattered point intrusion rate (SPIR) are better. The best value in each row is shown in bold. Entries are mean± deviation over the available seeds. The final row of each panel reports the unweighted mean over all nine datasets; this mean is omitted for den-SNE because its EMNIST run did not complete. OOT (out of time) denotes a runtime of 24 h+ for den-SNE on the full EMNIST dataset. Density-preservation metrics Local density correlation t-SNE UMAP den-SNE densMAP PHATE PaCMAP DMT-Dens 0.791±0.0150.791± 0.015 0.566±0.0470.566± 0.047 0.535±0.0340.535± 0.034 0.638±0.0580.638± 0.058 −0.421±0.082-0.421± 0.082 0.685±0.0310.685± 0.031 0.904±0.0110.904± 0.011 ACT 0.408±0.0610.408± 0.061 0.220±0.0520.220± 0.052 0.908±0.0090.908± 0.009 0.866±0.0110.866± 0.011 0.449±0.0980.449± 0.098 0.159±0.0420.159± 0.042 0.839±0.0060.839± 0.006 EMNIST 0.228±0.0350.228± 0.035 0.045±0.0240.045± 0.024 OOT 0.605±0.0250.605± 0.025 0.167±0.0550.167± 0.055 0.097±0.0200.097± 0.020 0.673±0.0200.673± 0.020 MNIST 0.262±0.0460.262± 0.046 −0.110±0.028-0.110± 0.028 0.796±0.0150.796± 0.015 0.781±0.0210.781± 0.021 0.169±0.0210.169± 0.021 −0.042±0.044-0.042± 0.044 0.712±0.0350.712± 0.035 NG20 0.446±0.0320.446± 0.032 0.350±0.0220.350± 0.022 0.737±0.0080.737± 0.008 0.830±0.0080.830± 0.008 0.665±0.0130.665± 0.013 0.259±0.0440.259± 0.044 0.666±0.0190.666± 0.019 EPI 0.226±0.0330.226± 0.033 0.103±0.0320.103± 0.032 0.697±0.0120.697± 0.012 0.716±0.0100.716± 0.010 0.251±0.0150.251± 0.015 0.143±0.0240.143± 0.024 0.774±0.0180.774± 0.018 GAST10K 0.059±0.0400.059± 0.040 −0.164±0.021-0.164± 0.021 0.793±0.0160.793± 0.016 0.810±0.0150.810± 0.015 0.187±0.0170.187± 0.017 −0.526±0.028-0.526± 0.028 0.858±0.0100.858± 0.010 HCL 0.120±0.0250.120± 0.025 0.041±0.0390.041± 0.039 0.716±0.0160.716± 0.016 0.542±0.0200.542± 0.020 0.407±0.0100.407± 0.010 0.142±0.0380.142± 0.038 0.838±0.0040.838± 0.004 MCA 0.389±0.0220.389± 0.022 0.096±0.0160.096± 0.016 0.301±0.0150.301± 0.015 0.493±0.0220.493± 0.022 −0.029±0.032-0.029± 0.032 0.004±0.0330.004± 0.033 0.557±0.0680.557± 0.068 0.3250.325 0.1270.127 – 0.6980.698 0.2050.205 0.1020.102 0.7580.758 SPIR (lower is better) t-SNE UMAP den-SNE densMAP PHATE PaCMAP DMT-Dens 0.615±0.0310.615± 0.031 0.637±0.0320.637± 0.032 0.634±0.0300.634± 0.030 0.639±0.0210.639± 0.021 0.604±0.0360.604± 0.036 0.626±0.0160.626± 0.016 0.619±0.0260.619± 0.026 ACT 0.448±0.0160.448± 0.016 0.472±0.0240.472± 0.024 0.435±0.0050.435± 0.005 0.494±0.0190.494± 0.019 0.463±0.0520.463± 0.052 0.378±0.0130.378± 0.013 0.362±0.0110.362± 0.011 EMNIST 0.733±0.0190.733± 0.019 0.744±0.0240.744± 0.024 OOT 0.768±0.0200.768± 0.020 0.802±0.0000.802± 0.000 0.703±0.0200.703± 0.020 0.674±0.0190.674± 0.019 MNIST 0.670±0.0190.670± 0.019 0.702±0.0070.702± 0.007 0.649±0.0280.649± 0.028 0.705±0.0150.705± 0.015 0.748±0.0110.748± 0.011 0.647±0.0180.647± 0.018 0.624±0.0320.624± 0.032 NG20 0.608±0.0280.608± 0.028 0.564±0.0160.564± 0.016 0.621±0.0260.621± 0.026 0.505±0.0130.505± 0.013 0.560±0.0200.560± 0.020 0.584±0.0120.584± 0.012 0.660±0.0320.660± 0.032 EPI 0.778±0.0150.778± 0.015 0.829±0.0170.829± 0.017 0.688±0.0170.688± 0.017 0.668±0.0200.668± 0.020 0.794±0.0140.794± 0.014 0.632±0.0270.632± 0.027 0.612±0.0330.612± 0.033 GAST10K 0.341±0.0200.341± 0.020 0.393±0.0230.393± 0.023 0.207±0.0260.207± 0.026 0.162±0.0250.162± 0.025 0.214±0.0220.214± 0.022 0.180±0.0130.180± 0.013 0.215±0.0220.215± 0.022 HCL 0.691±0.0060.691± 0.006 0.720±0.0260.720± 0.026 0.590±0.0220.590± 0.022 0.681±0.0270.681± 0.027 0.650±0.0170.650± 0.017 0.590±0.0160.590± 0.016 0.573±0.0160.573± 0.016 MCA 0.740±0.0240.740± 0.024 0.789±0.0120.789± 0.012 0.656±0.0190.656± 0.019 0.764±0.0190.764± 0.019 0.695±0.0240.695± 0.024 0.654±0.0130.654± 0.013 0.658±0.0120.658± 0.012 0.6250.625 0.6500.650 – 0.5980.598 0.6140.614 0.5550.555 0.5550.555 Table S9: Neighbor-ranking metrics for DMT-Dens and six baselines across the benchmark datasets. Higher continuity and kNN preservation are better, and the best value in each row is shown in bold. Seed averaging, unweighted means, and the OOT marker follow Table S8. Neighbor-ranking metrics Continuity t-SNE UMAP den-SNE densMAP PHATE PaCMAP DMT-Dens 0.997±0.0000.997± 0.000 0.997±0.0000.997± 0.000 0.986±0.0010.986± 0.001 0.998±0.0000.998± 0.000 0.958±0.0350.958± 0.035 0.992±0.0010.992± 0.001 0.996±0.0010.996± 0.001 ACT 0.972±0.0000.972± 0.000 0.969±0.0020.969± 0.002 0.971±0.0010.971± 0.001 0.970±0.0010.970± 0.001 0.967±0.0020.967± 0.002 0.975±0.0010.975± 0.001 0.973±0.0010.973± 0.001 EMNIST 0.898±0.0030.898± 0.003 0.887±0.0040.887± 0.004 OOT 0.892±0.0030.892± 0.003 0.910±0.0010.910± 0.001 0.889±0.0020.889± 0.002 0.882±0.0090.882± 0.009 MNIST 0.948±0.0020.948± 0.002 0.947±0.0020.947± 0.002 0.949±0.0020.949± 0.002 0.949±0.0020.949± 0.002 0.949±0.0030.949± 0.003 0.939±0.0030.939± 0.003 0.941±0.0030.941± 0.003 NG20 0.879±0.0090.879± 0.009 0.884±0.0060.884± 0.006 0.891±0.0020.891± 0.002 0.912±0.0020.912± 0.002 0.892±0.0070.892± 0.007 0.888±0.0030.888± 0.003 0.864±0.0040.864± 0.004 EPI 0.929±0.0060.929± 0.006 0.929±0.0060.929± 0.006 0.917±0.0080.917± 0.008 0.932±0.0060.932± 0.006 0.931±0.0050.931± 0.005 0.920±0.0040.920± 0.004 0.921±0.0070.921± 0.007 GAST10K 0.831±0.0110.831± 0.011 0.826±0.0060.826± 0.006 0.762±0.0060.762± 0.006 0.827±0.0060.827± 0.006 0.804±0.0060.804± 0.006 0.739±0.0030.739± 0.003 0.761±0.0090.761± 0.009 HCL 0.797±0.0110.797± 0.011 0.697±0.0210.697± 0.021 0.737±0.0160.737± 0.016 0.710±0.0080.710± 0.008 0.792±0.0020.792± 0.002 0.753±0.0040.753± 0.004 0.759±0.0080.759± 0.008 MCA 0.740±0.0230.740± 0.023 0.799±0.0150.799± 0.015 0.607±0.0250.607± 0.025 0.825±0.0060.825± 0.006 0.617±0.0270.617± 0.027 0.666±0.0270.666± 0.027 0.751±0.0560.751± 0.056 0.8880.888 0.8820.882 – 0.8910.891 0.8690.869 0.8620.862 0.8720.872 kNN preservation t-SNE UMAP den-SNE densMAP PHATE PaCMAP DMT-Dens 0.913±0.0030.913± 0.003 0.708±0.0080.708± 0.008 0.783±0.0080.783± 0.008 0.752±0.0050.752± 0.005 0.289±0.0660.289± 0.066 0.833±0.0060.833± 0.006 0.900±0.0050.900± 0.005 ACT 0.423±0.0030.423± 0.003 0.370±0.0040.370± 0.004 0.392±0.0040.392± 0.004 0.298±0.0040.298± 0.004 0.250±0.0140.250± 0.014 0.300±0.0050.300± 0.005 0.297±0.0070.297± 0.007 EMNIST 0.347±0.0050.347± 0.005 0.320±0.0050.320± 0.005 OOT 0.302±0.0080.302± 0.008 0.168±0.0030.168± 0.003 0.328±0.0050.328± 0.005 0.312±0.0040.312± 0.004 MNIST 0.371±0.0020.371± 0.002 0.318±0.0040.318± 0.004 0.322±0.0020.322± 0.002 0.246±0.0060.246± 0.006 0.178±0.0070.178± 0.007 0.325±0.0050.325± 0.005 0.315±0.0030.315± 0.003 NG20 0.337±0.0100.337± 0.010 0.297±0.0050.297± 0.005 0.274±0.0050.274± 0.005 0.210±0.0070.210± 0.007 0.142±0.0080.142± 0.008 0.303±0.0090.303± 0.009 0.257±0.0040.257± 0.004 EPI 0.181±0.0040.181± 0.004 0.140±0.0030.140± 0.003 0.157±0.0040.157± 0.004 0.128±0.0030.128± 0.003 0.115±0.0040.115± 0.004 0.146±0.0040.146± 0.004 0.154±0.0070.154± 0.007 GAST10K 0.083±0.0030.083± 0.003 0.068±0.0040.068± 0.004 0.076±0.0030.076± 0.003 0.042±0.0010.042± 0.001 0.042±0.0020.042± 0.002 0.049±0.0020.049± 0.002 0.056±0.0010.056± 0.001 HCL 0.117±0.0050.117± 0.005 0.080±0.0030.080± 0.003 0.098±0.0050.098± 0.005 0.077±0.0030.077± 0.003 0.062±0.0030.062± 0.003 0.090±0.0050.090± 0.005 0.088±0.0070.088± 0.007 MCA 0.097±0.0050.097± 0.005 0.055±0.0020.055± 0.002 0.092±0.0070.092± 0.007 0.067±0.0040.067± 0.004 0.064±0.0040.064± 0.004 0.080±0.0060.080± 0.006 0.084±0.0040.084± 0.004 0.3190.319 0.2620.262 – 0.2360.236 0.1460.146 0.2730.273 0.2740.274 S7 Additional evidence We complement the main benchmark with two evaluations based on external reference information: synthetic dyngen trajectories with a known backbone and a C. elegans embryonic time course with observed developmental-time annotations. S7.1 Synthetic trajectories (dyngen) We evaluate DMT-Dens on synthetic single-cell data generated with dyngen (5), which simulates cells along a prescribed trajectory backbone governed by a controlled gene-regulatory network. The resulting topology, branch assignments, and simulation times provide reference information for evaluating the recovered branching structure. Simulation setup. We generate single-cell data along a tree-structured, multifurcating trajectory backbone. Each simulated cell has a reference branch assignment and simulation time. Evaluation. We report the high-dimensional-to-embedding density correlation and local density correlation used in the main benchmark. Separately, the dyngen branch assignments and trajectory backbone are used to assess branch-label separability with a support-vector classifier (Branch SVC) and topology fidelity, defined as the F1F_1 agreement between the embedding neighborhood graph and the reference backbone. We compare DMT-Dens with the same six baselines used in the main text. Each method is subsequently evaluated over five seeds using its selected configuration. Table S10: Quantitative comparison on synthetic dyngen trajectories with a known reference topology. Higher values are better, and the best value in each column is shown in bold. Each reported configuration maximizes (densitycorrelation+BranchSVC)/2(density\ correlation+Branch\ SVC)/2 over the corresponding hyperparameter sweep. Entries are mean± deviation over five seeds. Density corr. Local density corr. Branch SVC Topology fidelity -SNE 0.064±0.0220.064± 0.022 0.057±0.0190.057± 0.019 0.922±0.0030.922± 0.003 0.885±0.0000.885± 0.000 UMAP 0.267±0.0180.267± 0.018 0.442±0.0150.442± 0.015 0.888±0.0330.888± 0.033 0.854±0.0420.854± 0.042 den-SNE 0.222±0.0270.222± 0.027 0.259±0.0210.259± 0.021 0.915±0.0180.915± 0.018 0.923±0.0000.923± 0.000 densMAP 0.754±0.0090.754± 0.009 0.801±0.0030.801± 0.003 0.881±0.0090.881± 0.009 0.877±0.0320.877± 0.032 PaCMAP 0.044±0.0280.044± 0.028 0.160±0.0200.160± 0.020 0.874±0.0060.874± 0.006 0.900±0.0210.900± 0.021 PHATE 0.269±0.0330.269± 0.033 0.338±0.0350.338± 0.035 0.896±0.0350.896± 0.035 0.754±0.0340.754± 0.034 DMT-Dens 0.811±0.0110.811± 0.011 0.880±0.0080.880± 0.008 0.853±0.0430.853± 0.043 0.846±0.0380.846± 0.038 Across five seeds, DMT-Dens records the highest density correlation (0.811±0.0110.811± 0.011) and local density correlation (0.880±0.0080.880± 0.008), compared with 0.754±0.0090.754± 0.009 and 0.801±0.0030.801± 0.003, respectively, for densMAP. Its Branch SVC is 0.853±0.0430.853± 0.043, while t-SNE records the highest value of 0.922±0.0030.922± 0.003. Its topology fidelity is 0.846±0.0380.846± 0.038, while den-SNE records the highest value of 0.923±0.0000.923± 0.000. Figure S1 shows the embeddings colored by reference branch assignment. Figure S1: Embeddings of synthetic dyngen data with a known branching topology. Colors denote the reference branch assignment. S7.2 Developmental-time preservation (C. elegans) We next assess how each embedding represents developmental time in the C. elegans embryonic time course (17). Each cell has a binned embryonic-time annotation, which we map to a numeric value. Because the dataset does not provide a lineage backbone, the evaluation is based on the observed time annotation and the two-dimensional embedding. Pseudotime correlation is the Spearman correlation between observed embryonic time and embedding pseudotime. Embedding pseudotime is defined as geodesic distance from the earliest-time cells on the embedding k-nearest-neighbor graph (k=15k=15). Ordering accuracy is the tie-aware fraction of cell pairs with different observed times whose embedding-pseudotime order agrees with their observed-time order. Time continuity is one minus the mean absolute observed-time difference across embedding kNN edges, normalized by the corresponding mean for random pairs; higher values indicate greater temporal similarity between neighboring cells. DEMaP is the Spearman correlation between ambient-space geodesic distances and embedding-space Euclidean distances. We also report reach, the fraction of cells reachable from the earliest-time root on the embedding graph. For fragmented graphs, pseudotime correlation and ordering accuracy are computed only on the reachable subset and must therefore be interpreted together with reach. Entries are mean± standard deviation over five seeds. Unmarked den-SNE and densMAP rows use their default density weights (0.1 and 2.0, respectively); rows marked “(λ)(λ)” use the density weight selected in the second-stage search while holding the structural hyperparameters fixed. Table S11: Developmental-time preservation relative to observed embryonic-time annotations in the C. elegans time course. Entries are mean± s.d. over five seeds; higher values are better. For fragmented embeddings, pseudotime correlation and ordering accuracy are computed on the reachable subset and must be interpreted together with reach. No rank-based highlighting is applied. Rows marked “(λ)(λ)” use the density weight selected in the second-stage search; unmarked rows use the default weight. Reach Pseudotime corr. Ordering acc. Time continuity DEMaP -SNE 0.921±0.0110.921± 0.011 0.285±0.0890.285± 0.089 0.615±0.0360.615± 0.036 0.776±0.0010.776± 0.001 0.428±0.0180.428± 0.018 UMAP 0.630±0.0430.630± 0.043 −0.133±0.113-0.133± 0.113 0.446±0.0460.446± 0.046 0.716±0.0020.716± 0.002 0.354±0.0200.354± 0.020 den-SNE 0.624±0.0000.624± 0.000 0.184±0.0000.184± 0.000 0.568±0.0000.568± 0.000 0.771±0.0000.771± 0.000 0.484±0.0000.484± 0.000 densMAP 1.000±0.0001.000± 0.000 0.212±0.0480.212± 0.048 0.585±0.0190.585± 0.019 0.448±0.0050.448± 0.005 0.565±0.0130.565± 0.013 PaCMAP 0.430±0.0850.430± 0.085 0.518±0.0890.518± 0.089 0.713±0.0420.713± 0.042 0.783±0.0030.783± 0.003 0.502±0.0070.502± 0.007 PHATE 1.000±0.0001.000± 0.000 0.392±0.0500.392± 0.050 0.657±0.0200.657± 0.020 0.557±0.0120.557± 0.012 0.817±0.0140.817± 0.014 den-SNE (λ) 1.000±0.0001.000± 0.000 0.234±0.0000.234± 0.000 0.593±0.0000.593± 0.000 0.649±0.0000.649± 0.000 0.629±0.0000.629± 0.000 densMAP (λ) 1.000±0.0001.000± 0.000 0.263±0.0490.263± 0.049 0.604±0.0200.604± 0.020 0.401±0.0060.401± 0.006 0.615±0.0050.615± 0.005 DMT-Dens 0.997±0.0010.997± 0.001 0.556±0.0520.556± 0.052 0.732±0.0270.732± 0.027 0.739±0.0050.739± 0.005 0.643±0.0110.643± 0.011 DMT-Dens obtains a reach of 0.997±0.0010.997± 0.001, a pseudotime correlation of 0.556±0.0520.556± 0.052, and an ordering accuracy of 0.732±0.0270.732± 0.027. The corresponding PaCMAP values are 0.430±0.0850.430± 0.085, 0.518±0.0890.518± 0.089, and 0.713±0.0420.713± 0.042, respectively. Time continuity is 0.739±0.0050.739± 0.005 for DMT-Dens, 0.776±0.0010.776± 0.001 for t-SNE, and 0.783±0.0030.783± 0.003 for PaCMAP. DEMaP is 0.643±0.0110.643± 0.011 for DMT-Dens and 0.817±0.0140.817± 0.014 for PHATE. S8 Full embedding comparisons Figures S2 and S3 extend the representative embedding comparison in main-text Figure 4 to all seven methods. Dataset columns, color scales, and point styling match those in the main text. Figure S2: Full embedding comparison across seven methods on the four single-cell datasets (GAST10K, HCL, EPI, and MCA). Points are colored by cell-type label. This figure extends main-text Figure 4. Figure S3: Full embedding comparison across seven methods on MNIST, EMNIST, 20 Newsgroups, and ArtificialTree. Points are colored by class label. The panels illustrate qualitative differences in density variation, overlap, and fragmentation; quantitative density and label-separability results are reported separately.