t-SNE and UMAP: Visualising High-Dimensional Data

You have trained a model that embeds 50,000 documents into 768-dimensional vectors. Are there natural groups? Are two classes confused? PCA to two dimensions often shows a shapeless cloud, because the structure lies on a curved, non-linear manifold. t-SNE and UMAP are non-linear methods designed to reveal such structure in 2-D plots. They are indispensable tools — and among the most frequently misinterpreted.

t-SNE: matching neighbour probabilities#

t-distributed Stochastic Neighbour Embedding (van der Maaten & Hinton, 2008) works in three steps.

1. High-dimensional similarities. For each point $i$, define a Gaussian-based probability that $i$ would pick $j$ as its neighbour:

The bandwidth $\sigma_i$ is chosen per point so that the distribution's perplexity matches a user-set value — roughly, the effective number of neighbours. Symmetrise: $p_{ij} = (p_{j \mid i} + p_{i \mid j})/2n$.

2. Low-dimensional similarities use a heavy-tailed Student-t distribution with one degree of freedom:

3. Optimise the 2-D positions $\mathbf{y}_i$ by gradient descent to minimise

Why the heavy tail? In high dimensions, many points can be moderately far from a given point, but in 2-D there is not enough room — the crowding problem. The Student-t lets moderately distant points be placed much farther apart in 2-D, producing well-separated clusters.

UMAP: fuzzy topology#

Uniform Manifold Approximation and Projection (McInnes, Healy & Melville, 2018) builds a weighted k-nearest-neighbour graph in high dimensions (grounded in fuzzy simplicial sets from topology), then optimises a low-dimensional layout whose graph matches it using a cross-entropy objective with attractive forces along edges and repulsive forces via negative sampling.

Compared with t-SNE, UMAP typically:

• runs much faster and scales to millions of points;
• preserves more global structure (relative positions of clusters are somewhat more meaningful, though still not reliable);
• can transform new points into an existing embedding;
• works as a general-purpose reduction for clustering (e.g. UMAP → HDBSCAN on text embeddings).

Hyperparameters#

import numpy as np
from sklearn.datasets import load_digits
from sklearn.manifold import TSNE
from sklearn.decomposition import PCA
import matplotlib.pyplot as plt
# pip install umap-learn
import umap

X, y = load_digits(return_X_y=True)
X50 = PCA(n_components=30, random_state=0).fit_transform(X)   # common pre-step for speed/denoising

emb = {
    "PCA": PCA(n_components=2).fit_transform(X),
    "t-SNE (perplexity 30)": TSNE(perplexity=30, init="pca", random_state=0).fit_transform(X50),
    "UMAP (n_neighbors 15)": umap.UMAP(n_neighbors=15, min_dist=0.1, random_state=0).fit_transform(X50),
}
fig, axes = plt.subplots(1, 3, figsize=(16, 5))
for ax, (name, Z) in zip(axes, emb.items()):
    ax.scatter(Z[:, 0], Z[:, 1], c=y, cmap="tab10", s=5); ax.set_title(name)
plt.tight_layout(); plt.show()

PCA shows overlapping digit groups; t-SNE and UMAP show ten crisp clusters.

How NOT to read these plots#

Wattenberg, Viégas and Johnson's interactive article "How to Use t-SNE Effectively" demonstrates these pitfalls vividly; every student should explore it.

Good practice#

• Standardise features; for very high-dimensional data, reduce to 30–50 dimensions with PCA first.
• Try multiple perplexities / neighbour counts and seeds; trust only structure that persists.
• Colour points by known labels, metadata and model predictions to generate hypotheses — then test those hypotheses with proper methods.
• Report hyperparameters with every plot.

Uses in ML practice#

• Inspecting learned embeddings (words, sentences, images).
• Diagnosing classifier confusion — misclassified points often lie between clusters.
• Detecting label noise and data-quality problems (points of one class inside another's cluster).
• Exploring single-cell biology data, where UMAP plots are standard.