t-Distributed Stochastic Neighbor Embedding (t-SNE)

t-SNE converts high-dimensional Euclidean distances into conditional probabilities that represent similarities. It computes the probability that a point x_i would pick x_j as its neighbor if neighbors were picked in proportion to their probability density under a Gaussian kernel. This focuses the algorithm on preserving local neighborhoods rather than global distances.

In the low-dimensional embedding space, t-SNE uses a Student t-distribution with one degree of freedom (a Cauchy distribution) to model similarities. This heavy-tailed distribution alleviates the crowding problem by allowing moderately distant points in the high-dimensional space to be modeled by larger distances in the low-dimensional map, preventing all points from collapsing to the center.

The algorithm minimizes the Kullback-Leibler (KL) divergence between the high-dimensional and low-dimensional probability distributions using gradient descent. This cost function is asymmetric, meaning t-SNE heavily penalizes mapping points that are close in the original space as far apart in the embedding, but tolerates mapping distant points as close. This results in a strong preservation of local clusters.

Perplexity is a crucial hyperparameter that balances attention between local and global aspects of the data. It can be interpreted as a smooth measure of the effective number of neighbors. Typical values range between 5 and 50. In patient stratification, lower perplexity reveals fine-grained cellular subtypes, while higher perplexity captures broader disease trajectories. The algorithm is robust to changes in perplexity within a reasonable range.

The t-SNE cost function is non-convex, meaning different initializations can lead to different visualizations. It is standard practice to run the algorithm multiple times and select the embedding with the lowest KL divergence. Researchers should never rely on a single run for biological conclusions; consistent cluster separation across multiple runs validates the robustness of identified patient subgroups.

t-SNE does not preserve global data structure reliably. The distance between clusters and the relative size of clusters in the embedding plot are meaningless. A large, spread-out cluster in a t-SNE plot does not necessarily represent a larger patient population or greater variance. For preserving global structure, complementary techniques like UMAP or PCA should be used alongside t-SNE.

A linear dimensionality reduction technique that transforms data into a set of orthogonal principal components ordered by variance. Unlike t-SNE, PCA preserves global structure and is deterministic, making it ideal for feature engineering and initial data exploration. However, PCA fails to capture non-linear relationships in complex biological data.

• Key difference: PCA preserves large pairwise distances; t-SNE preserves small ones
• Use case: Preprocessing step before applying t-SNE to reduce noise
• Limitation: Cannot separate entangled manifolds like concentric circles

A manifold learning technique that competes directly with t-SNE for visualizing high-dimensional data. UMAP better preserves global data structure and runs significantly faster on large datasets. It is built on rigorous Riemannian geometry and algebraic topology, making it the preferred choice for single-cell genomics workflows.

• Advantage over t-SNE: Better preservation of inter-cluster distances
• Speed: Often 10-100x faster than t-SNE on datasets with >100k points
• Mathematical foundation: Assumes data lies on a locally connected manifold

The most critical tuning parameter in t-SNE, controlling the balance between local and global structure. Perplexity can be interpreted as a smooth measure of the effective number of neighbors. Typical values range from 5 to 50, but the optimal setting depends on dataset size.

• Low perplexity (5-10): Emphasizes very fine local structure, may create fragmented clusters
• High perplexity (50-100): Captures broader patterns but may merge distinct groups
• Rule of thumb: Set perplexity between 5 and the number of samples divided by 3

The fundamental challenge that t-SNE was designed to solve. In high-dimensional spaces, the volume of a sphere grows exponentially with radius, causing moderate-distance points to overwhelm the available low-dimensional space. t-SNE uses a heavy-tailed Student-t distribution in the low-dimensional map to alleviate this crowding, allowing dissimilar points to be modeled by larger pairwise distances.

• Why it matters: Without addressing crowding, all points collapse to the center
• Solution: The t-distribution's long tails create repulsive forces between dissimilar points
• Result: Distinct clusters separate naturally in the visualization

The asymmetric cost function minimized by t-SNE during optimization. It measures how well the low-dimensional probability distribution Q approximates the high-dimensional distribution P. The asymmetry penalizes failing to preserve local structure more heavily than failing to preserve distant relationships.

• Interpretation: KL(P||Q) = sum over points of P * log(P/Q)
• Asymmetry consequence: t-SNE is better at preserving clusters than inter-cluster distances
• Optimization: Minimized via gradient descent with momentum to avoid poor local minima

A trick used during the initial phase of t-SNE optimization where all high-dimensional probabilities P are multiplied by a factor (typically 4-12). This amplifies attractive forces between similar points, encouraging the formation of tightly clustered groups before repulsive forces spread them apart.

• Purpose: Creates well-separated clusters early in optimization
• Duration: Applied for the first 100-250 iterations, then removed
• Effect: Without it, clusters may remain diffuse and overlapping in the final map