Dimensionality reduction methods are instrumental to visualize high-dimensional data, and t-SNE stands as one of the most widely used methods due to its emphasis on local neighborhood preservation. A central component of t-SNE is the affinity matrix, which expresses pairwise similarities in the form of symmetrized probabilities, over which the optimization problem of t-SNE is defined. We study how the sharpness of this probability distribution affects neighborhood preservation at different scales. We introduce a row-wise power transform controlled by a parameter gamma that can smooth or sharpen each row of the affinity matrix while preserving sparsity and rank order. We show that this transform is equivalent to rescaling the Gaussian bandwidth and thus to changing the perplexity. However, as the sharpness of the probability distribution varies per point, a fixed gamma leads to point-dependent effective perplexities, making it distinct from changing the global perplexity. Empirically, we find that sharpening improves preservation of the very nearest neighbors, while smoothing improves preservation of broader local neighborhoods, outperforming alternative affinity constructions including multiscale methods in the mid-local range.
Figures & tables
Figure 1: (a) Redrawn from Novak et al. (2026) [ 17 ] , showing local versus global neighborhood preservation (here measured as the AUC under the RnX and log-RnX curve respectively, see their paper for details) for a range of state-of-the-art DR methods. One dataset shown here, the paper contains another seven. As can be seen, t-SNE manages to retain local structure substantially better than other DR methods. (b) From Heiter et al. (2024) [ 8 ] , illustrating on a t-SNE projection of mouse cell data [ 19 ] that neighborhood preservation strongly varies across the embeddings. Color corresponds to neighborhood overlap with k=200 .
Figure 2: (a) Affinity row sharpness at ρ=30 for a single point from MNIST (selected as average wrt. summed mass of the top-5 neighbors). Standard t-SNE ( γ=1.0 ) drops steeply after the first few neighbors, while smoothing ( γ=0.5 ) redistributes probability mass more evenly across the neighborhood. (b) Distribution of Effective perplexity across points on MNIST for standard t-SNE ( ρ=30 ), smoothed t-SNE ( γ=0.7 ), and sharpened t-SNE ( γ=1.5 )
Figure 3: NO@k for k=1,…,200 at perplexity ρ=30 with varying γ , on MNIST. Sharpened variants ( γ>1 ) perform better for small k , while smoothed variants ( γ<1 ) perform better for larger k (starting from k≈2/3ρ ).
Figure 4: Standard t-SNE ( ρ=30 ) and smoothed t-SNE ( ρ=30 , γ=0.5 ) embeddings of MNIST. Smoothing produces clearer inter-cluster separation while preserving the overall digit cluster structure.
Figure 5: (a) NO@k comparison of standard t-SNE ( ρ=30 ), smoothed t-SNE ( ρ=30 , γ=0.7 ), and standard t-SNE at ρ=53 , matching the median effective perplexity of smoothed t-SNE. (b) NO@30 for standard and smoothed t-SNE (with constant γ=0.7 ), for varying values of ρ . We observe that both standard t-SNE with same perplexity and matched perplexity have lower broad-local NO. Comparing across perplexities for a specific NO@k , we see that standard t-SNE with higher perplexities also does not achieve the same improved NO.
Figure 6: Sensitivity of t-SNE behavior to γ and ρ on the MNIST dataset. Each cell shows the metric value for a combination of ρ (rows) and γ (columns). (a) Near-local preservation improves consistently with both sharpening and lowering the perplexity. (b) Mid-local preservation peaks at moderate smoothing around γ=0.7 ; extreme smoothing with γ=0 performs worst.
Figure 7: (a) Effect of γ on global structure preservation (Spearman ρ between high- and low-dimensional distances) across four perplexity values on MNIST. In general, lower γ and higher ρ improve global structure preservation. Shaded bands indicate ±1 SD across reruns. (b) NO@k for k=1,…,200 on MNIST, comparing γ variants against alternative affinity constructions. Sharpening ( γ=1.5 ) leads at small k ; smoothing ( γ=0.7 ) outperforms all methods in the mid-local range; γ=0.0 / Uniform and FixedSigmaNN dominate at large k .
Appendix figures & tables9 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 1: Median effective perplexity across ρ and γ settings on MNIST (a), mouse cortex (b), and Adult (c).
Figure 2: Per-point change in effective perplexity after smoothing ( γ=0.7 , ρ=30 ). (left) Versus top-5 conditional mass Mi(5) . (right) Versus t-SNE bandwidth σi across three datasets.
Figure 3: NO@k for k=1,…,200 at ρ=30 with varying γ , on mouse cortex (a) and Adult (b). The MNIST result is shown in the main paper.
Figure 4: Standard t-SNE versus smoothed t-SNE ( γ=0.5 ) on mouse cortex (a) and Adult (b). The MNIST embedding is shown in the main paper.
Figure 5: NO@k comparison between standard t-SNE, smoothed t-SNE ( γ=0.7 ), and matched-perplexity standard t-SNE on mouse cortex (a) and Adult (b). The MNIST result is shown in the main paper.
Figure 6: NO@30 as a function of perplexity for standard t-SNE and smoothed t-SNE ( γ=0.7 , not tuned) on mouse cortex (a) and Adult (b). Smoothed t-SNE peaks at a lower perplexity than standard t-SNE and attains a higher peak NO@30 . The MNIST result is shown in the main paper.
Figure 7: Sensitivity of neighborhood preservation to γ and ρ on mouse cortex (a, b) and Adult (c, d). The MNIST result is shown in the main paper.
Figure 8: Effect of γ on global structure preservation on mouse cortex (a) and Adult (b). The MNIST result is shown in the main paper.
Figure 9: NO@k for k=1,…,200 , comparing γ variants against alternative affinity constructions on mouse cortex (a) and Adult (b). The overall pattern from MNIST holds, with γ=0.7 outperforming other methods in the mid-local range. On Adult, γ=0.7 maintains its advantage more broadly, while on mouse cortex the multiscale methods are more competitive at larger k .
Dimensionality reduction methods such as t-SNE are designed to preserve local neighborhood structure but do not explicitly account for how probability mass is distributed, often leading to distortions of data density. We reformulate dimensionality reduction as the joint alignment of two components: (i) conditional structure, capturing local relationships, and (ii) relative density structure, captured via local density statistics. Based on this perspective, we introduce Density-Regularized SNE (DR-SNE), which augments the stochastic neighbor embedding objective with a density regularization term derived from normalized log-density estimates. Unlike prior approaches such as DensMAP and DenSNE, which rely on local scale consistency, DR-SNE directly aligns normalized density estimates, providing a simple and scale-invariant mechanism for preserving relative density variations. Empirically, DR-SNE improves density preservation while maintaining competitive neighborhood fidelity, and yields gains on density-sensitive tasks such as anomaly detection across multiple datasets. These results suggest that incorporating density information complements geometry-focused objectives in dimensionality reduction.
Dimensionality-reduction (DR) methods are routinely judged by how well each point's k nearest neighbors survive the 2-D embedding (recall@k, trustworthiness, continuity). We argue this family is a biased measure of distance fidelity: its per-point variable radius and hard inclusion threshold favor neighbor-graph methods (t-SNE, UMAP) and penalize methods that preserve absolute distances. We instead score DR fidelity with a fixed-radius distance-band Shepard rho: the Spearman correlation between high-D and 2-D pairwise distances, restricted to cumulative distance bands so that near and global structure are reported separately, with every point judged on the same absolute radius. On synthetic datasets with known ground-truth geometry (non-uniform density, dense clusters, a closed-loop transition, off-subspace outliers, imbalanced two-population data) at realistic noise (SNR=1, D=768, N=1000), we benchmark eight methods -- PCA, Isomap, t-SNE, UMAP, PyMDE, PCC, DREAMS, and the closed-source toorPIA -- and show that (i) high global Shepard rho can coexist with a ~93x collapse of within-cluster scale, invisible to rank-based scores but obvious in a value-based over-compression metric; (ii) recall@k and the fixed-radius band disagree systematically, in the direction the bias predicts; (iii) a membership-restricted Shepard rho resolves single-point and minority-population questions that many-pair statistics cannot -- questions on which even DREAMS, a recent local-plus-global hybrid, fails silently. A supplementary out-of-sample (addplot) test asks whether a never-seen anomaly lands outside the normal region and whether its direction identifies its source. All metrics are computed exactly on all pairwise distances, independently of any method's internals, and every number is reproducible offline: the closed-source method's output coordinates (not its algorithm) are committed to the artifact.
t-SNE has gained popularity as a dimension reduction technique, especially for visualizing data. It is well-known that all dimension reduction techniques may lose important features of the data. We provide a mathematical framework for understanding this loss for t-SNE by establishing a number of results in different scenarios showing how important features of data are lost by using t-SNE.