t-SNE and UMAP: Nonlinear Embeddings

Question

PCA gives you a linear projection, but sometimes the structure you care about lives on a curved manifold that no linear projection can flatten.

Explain why PCA can miss curved structure, how t-SNE works (converting pairwise distances into probabilities and minimising a KL divergence between high- and low-dimensional neighbour distributions), what the perplexity parameter controls, and the crowding problem. Explain why cluster sizes and inter-cluster distances in a t-SNE plot are not meaningful, how UMAP differs, and why these are visualisation tools rather than feature generators. Give trading context.

Accepted Answer

PCA finds the directions of maximum variance and projects onto them. Because it is a linear map (a rotation followed by dropping axes), it can only preserve structure that lies along straight directions. If the data lives on a curved manifold — the classic "Swiss roll", where points far apart along the rolled-up sheet are close in 3-D space — PCA flattens the roll and superimposes regions that are actually far apart on the manifold. It cannot unroll curved structure. Nonlinear methods like t-SNE and UMAP are designed exactly for this: they try to preserve local neighbourhood relationships rather than global linear geometry.

How t-SNE workst-SNE (t-distributed Stochastic Neighbour Embedding) turns the geometry into probability distributions over neighbours. In the high-dimensional space, it converts the distance between points $i$ and $j$ into a conditional probability that $i$ would pick $j$ as a neighbour, using a Gaussian kernel centred on $i$: \begin{equation} p_{j|i} = \frac{\exp(-\lVert x_i - x_j Vert^2 / 2\sigma_i^2)}{\sum_{k 
eq i} \exp(-\lVert x_i - x_k Vert^2 / 2\sigma_i^2)} \end{equation} These are symmetrised into joint probabilities $p_{ij}$. In the low-dimensional map, it defines analogous probabilities $q_{ij}$ but using a Student-t (Cauchy) kernel with one degree of freedom rather than a Gaussian: \begin{equation} q_{ij} = \frac{(1 + \lVert y_i - y_j Vert^2)^{-1}}{\sum_{k 
eq l}(1 + \lVert y_k - y_l Vert^2)^{-1}} \end{equation} It then places the low-dimensional points $y_i$ so that $q$ matches $p$ as closely as possible, by minimising the Kullback-Leibler divergence \begin{equation} KL(P \| Q) = \sum_{i 
eq j} p_{ij} \ln \frac{p_{ij}}{q_{ij}} \end{equation} via gradient descent. Because KL divergence is asymmetric, it heavily penalises placing true neighbours (large $p_{ij}$) far apart, so t-SNE faithfully preserves local structure — who is near whom — while being relatively indifferent to where distant clusters end up.

PerplexityThe per-point bandwidth $\sigma_i$ is set so that each point's neighbour distribution has a target perplexity, which behaves like "the effective number of neighbours each point considers" (typical values 5 to 50). Low perplexity focuses on very local structure and can fragment real clusters; high perplexity smooths over fine detail and blends clusters together. The plot can change substantially with perplexity, so it should always be varied.

The crowding problemIn high dimensions there is far more room than in 2-D: a point can have many roughly-equidistant neighbours, but there simply isn't enough area in the plane to place them all at moderate distance. With a Gaussian in the low-D space, moderately distant points would be crushed together. The heavy-tailed Student-t kernel solves this — it allows dissimilar points to sit far apart in the map without incurring a large penalty, giving the well-separated blobs t-SNE is known for. This fix is precisely why distances between clusters are distorted.

Why cluster sizes and distances are not meaningfulTwo consequences follow directly. First, t-SNE rescales density locally (each $\sigma_i$ adapts), so a tight cluster and a diffuse cluster can render at the same on-screen size — the area of a blob tells you nothing about its true spread or how many points it contains relative to others. Second, the algorithm optimises local neighbourhoods and uses a heavy-tailed kernel, so the gaps between clusters carry no quantitative meaning: a cluster drawn far from another is not necessarily more different than one drawn nearby. The reliable information is only the within-cluster grouping (which points are neighbours). Reading global geometry off a t-SNE plot is the classic mistake.

How UMAP differsUMAP (Uniform Manifold Approximation and Projection) builds a weighted nearest-neighbour graph grounded in manifold theory and topology, then optimises a low-dimensional layout of that graph using a cross-entropy-style objective (with both attractive and repulsive forces) rather than a KL divergence over t-kernels. In practice UMAP is much faster and scales to large datasets, tends to preserve global structure somewhat better (so inter-cluster arrangement is a little more trustworthy, though still not to be over-interpreted), and exposes intuitive knobs: n_neighbors (local vs global emphasis, analogous to perplexity) and min_dist (how tightly points may pack). It is largely a drop-in replacement for t-SNE and usually the default choice today.

These are visualisation tools, not feature generatorsA crucial warning: do not feed t-SNE or UMAP coordinates into a predictive model as features. Both are non-parametric and stochastic — different random seeds give different embeddings, there is no stable, well-defined transform to apply to new out-of-sample points (t-SNE has none at all; UMAP's transform exists but is approximate), and the embedding is fit to make a picture, not to be a faithful linear-algebraic projection. They are for exploring and explaining data to humans. When you need dimensionality reduction inside a pipeline, use a method with a stable, invertible-in-spirit mapping such as PCA or an autoencoder.

Trading contextThese embeddings are excellent for exploratory analysis. You might embed daily market states (described by volatility, breadth, factor returns, term-structure features) and visually inspect whether distinct regimes — calm, trending, stressed — separate into clusters, or embed instruments by their return profiles to spot natural groupings beyond sector labels. But treat the picture as a hypothesis generator: confirm any apparent regime with a proper out-of-sample test, never trade on the coordinates themselves, and remember that the distance between two blobs on a t-SNE plot of market regimes does not tell you how different those regimes really are.

Title	Category	Subcategory	Difficulty	Status
How Many Principal Components?	Machine Learning	Dimensionality Reduction	Medium	Unlock
Principal Component Analysis	Machine Learning	Dimensionality Reduction	Hard	Unlock

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t-SNE and UMAP: Nonlinear Embeddings

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