What is t-SNE, and when should you use it instead of PCA?
Answer
t-SNE (t-distributed stochastic neighbor embedding) is a non-linear visualization method that turns pairwise distances in high-dimensional space into neighborhood probabilities, then arranges low-dimensional points so their own neighborhood probabilities, computed with a heavy-tailed Student-t kernel, match them by minimizing the KL divergence between the two distributions. Use it instead of PCA when the goal is a 2D map that reveals cluster structure: PCA preserves global variance along linear axes, while t-SNE sacrifices global geometry to keep close neighbors close. Do not use it as a preprocessing step for a downstream model: it is transductive (no transform for new points), stochastic across runs, and its cluster sizes and separations are not quantitative.
(1) What It Optimizes: match high-dimensional affinities 
(Gaussian, calibrated per point) with low-dimensional affinities 
(Student-t) via KL divergence; the heavy t-tails let dissimilar points sit far apart without penalty, solving the crowding problem.
(2) Perplexity Is the Knob: it acts as the effective neighbor count per point. Too small fragments clusters, too large merges them; 5 to 50 is the usual range, and results vary with the random seed.
(3) Production Pattern: PCA first, neighbor embedding second. 10x Genomics’ Cell Ranger pipeline runs PCA on gene expression, then computes t-SNE or UMAP projections for its Loupe Browser; UMAP scales better than t-SNE on large cell counts, which is why it has become the default projection at that scale.
Figure 1: The same clustered data two ways: PCA (left) preserves global spread but overlaps clusters that are not linearly separable; t-SNE (right) separates them cleanly. The t-SNE panel’s axes and inter-cluster gaps carry no units and no meaning.
Mathematical Formulation:
Where:
is the probability that 
picks 
as a neighbor; the per-point is set so the perplexity matches the chosen value, and the conditionals are symmetrized into 
.
is the same affinity computed on the map points 
with a Student-t kernel (1 degree of freedom), whose heavy tails prevent crowding.
is the KL divergence between the two affinity matrices, minimized by gradient descent directly on the map coordinates.
is the probability that 
picks 
as a neighbor; the per-point 
is set so the perplexity matches the chosen value, and the conditionals are symmetrized into 
with a Student-t kernel (1 degree of freedom), whose heavy tails prevent crowding.
is the KL divergence between the two affinity matrices, minimized by gradient descent directly on the map coordinates.| Feature | t-SNE | PCA |
|---|---|---|
| Type | Non-linear, neighbor-graph based | Linear orthogonal projection |
| Preserves | Local neighborhoods | Global variance and large-scale structure (projection can only shrink distances) |
| Output | 2-3D coordinates only, no transform | Reusable projection, any  |
| Determinism | Stochastic; reruns differ | Deterministic up to sign flips |
| Right Use | Final-step cluster visualization | Preprocessing, compression, de-correlation |
Figure 2: Perplexity sweeps the effective neighborhood size: at 5 the clusters fragment into fake sub-clusters, at 30 the structure is clean, at 100 distinct clusters merge. Always check a second perplexity before trusting a t-SNE picture.
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