t-SNE

t-SNE (t-distributed Stochastic Neighbor Embedding) is a nonlinear dimensionality reduction method for visualization: it places high-dimensional points in 2D or 3D so that neighborhoods are preserved. It excels at revealing cluster structure but is a visualization tool, not a general feature transform.

Pairwise Affinities (High Dim)

For each point $i$ define a conditional probability that $i$ would pick $j$ as a neighbor under a Gaussian centered at $x_i$:

$$p_{j\mid i}=\frac{\exp (-\Vert x_i-x_j{\Vert }^2/2{\sigma }_i^2)}{{\sum }_{k\ne i}\exp (-\Vert x_i-x_k{\Vert }^2/2{\sigma }_i^2)}.$$

The bandwidth ${\sigma }_i$ is set per point so the neighborhood has a target size (see perplexity). Symmetrize to $p_{ij}=(p_{j\mid i}+p_{i\mid j})/2N$.

Student-t in Low Dim

In the low-dimensional map, similarities use a Student-t distribution with one degree of freedom (a Cauchy), which has heavy tails:

$$q_{ij}=\frac{(1+\Vert y_i-y_j{\Vert }^2{)}^{-1}}{{\sum }_{k\ne l}(1+\Vert y_k-y_l{\Vert }^2{)}^{-1}}.$$

KL Objective

The map $\{y_i\}$ is found by minimizing the Kullback-Leibler divergence between the high-dim and low-dim affinities via Gradient Descent:

$$C=\mathrm{KL}(P\parallel Q)=\sum _{i\ne j}{p}_{ij}\log \frac{p_{ij}}{q_{ij}}.$$

Because KL is asymmetric, a large $p_{ij}$ modeled by a small $q_{ij}$ (nearby points placed far apart) is heavily penalized, so t-SNE prioritizes preserving local structure over global distances.

Perplexity

Perplexity sets the effective number of neighbors each ${\sigma }_i$ targets:

$$\text{Perp}(P_i)=2^{H(P_i)},H(P_i)=-\sum _j{p}_{j\mid i}{\log }_2{p}_{j\mid i}.$$

Typical values are 5 to 50. Small perplexity emphasizes very local structure; large perplexity blends broader neighborhoods.

The Crowding Problem

In high dimensions there is far more room for a point to have many roughly-equidistant neighbors than in 2D. A Gaussian low-dim kernel would force moderately distant points to crush together. The heavy-tailed Student-t fixes this: it allocates more room at moderate distances, letting clusters spread out and separate cleanly.

Contrast with PCA and UMAP

Aspect	PCA	t-SNE	UMAP
Type	linear	nonlinear	nonlinear
Preserves	global variance	local neighborhoods	local + some global
Deterministic	yes	no (random init)	no
Speed	fast	slow	faster than t-SNE
New-point map	trivial (project)	not natively	supports transform

A common pipeline runs PCA first (to ~50 dims) to denoise and speed things up, then t-SNE for the final 2D plot.

sklearn Usage

from sklearn.decomposition import PCA
from sklearn.manifold import TSNE
 
X50 = PCA(n_components=50).fit_transform(X)     # denoise / speed up
emb = TSNE(
    n_components=2,
    perplexity=30,
    learning_rate="auto",
    init="pca",
    random_state=0,
).fit_transform(X50)

Pitfalls

• Cluster sizes and inter-cluster distances in a t-SNE plot are NOT meaningful; do not read them as real densities or gaps.
• Results depend on perplexity and the random seed; try several settings before trusting a picture.
• t-SNE does not learn a reusable mapping for new data (use UMAP or a parametric model if you need that).
• Do not feed t-SNE coordinates into a downstream classifier; it is for visualization only.