2.1. Delve Into the Mathematical Framework

Riemannian geometry is the mathematics of curved surfaces. It’s how we describe spaces where “straight lines” don’t exist — like the surface of a sphere or the folds of a crumpled sheet.

UMAP assumes your high-dimensional data doesn’t fill the entire space — it lies along a curved manifold. Each small neighborhood of points is approximately flat (locally Euclidean), but when you move farther, the curvature reveals itself.

So UMAP’s task is:

• Respect local geometry (short distances).
• Understand global structure (how local patches connect).
• Flatten it carefully without tearing (manifold unfolding).

Think of it like unfolding a paper globe — you can flatten small parts without distortion, but large parts need careful cutting and joining to preserve relationships.

In a high-dimensional world, not every pair of points is meaningfully connected. UMAP approximates connectivity by building small local neighborhoods — the fuzzy simplicial sets we learned earlier.

Within each neighborhood, distances are small enough that Euclidean geometry works fine. Across neighborhoods, UMAP relies on topology — how these neighborhoods connect — rather than the exact distances.

That’s what makes UMAP resilient to noise and scaling: it doesn’t care about absolute positions, only how points relate locally.

So, local neighborhoods act as “flat tiles” on a curved surface, and UMAP learns how to stitch those tiles together consistently.

In curved spaces, the shortest distance between two points isn’t a straight line — it’s a geodesic (the “as-straight-as-possible” curve on the surface).

UMAP approximates these geodesic distances through its k-nearest-neighbor graph.

• Each edge in the graph represents a short, trustworthy local connection.
• Longer paths are built by chaining local edges together — tracing how the manifold folds through space.

This graph-based approximation allows UMAP to estimate true manifold distances without ever computing them directly — an elegant shortcut that makes it both powerful and fast.

If a straight line cuts through the mountain, UMAP prefers to take the hiking trail that actually winds along its surface — more steps, but more faithful to the terrain.

Before optimization begins, UMAP gives itself a head start by using spectral embedding, a mathematically guided initialization method.

Here’s how it works:

• Compute the graph Laplacian — a matrix that captures how connected each point is to its neighbors.
• Perform eigen-decomposition on that matrix to extract its key structure.
• Use the first few eigenvectors (think of them like “vibration modes” of the data graph) to position points in low dimensions.

This gives UMAP a near-optimal starting layout that already reflects the manifold’s structure — making optimization smoother and faster.

Think of it like sketching the rough outline of your painting before filling in the details — it saves time, prevents mistakes, and ensures a balanced composition.

In Riemannian geometry, every manifold has a metric tensor, a function that defines how distances are measured locally.

For two nearby points, the infinitesimal distance $ds$ is defined as:

Where:

• $g_{ij}$ → elements of the metric tensor (a local scaling matrix)
• $dx^i, dx^j$ → small coordinate differences

In simple terms, $g_{ij}$ encodes how “stretched” or “compressed” space is in different directions.

The graph Laplacian is defined as:

Where:

• $W$ → adjacency matrix (connection strengths between points)
• $D$ → diagonal matrix of node degrees (sum of connections per node)

UMAP uses eigenvectors of $L$ to get a spectral embedding — a low-dimensional map that respects the smooth connectivity of the graph.

• “Riemannian geometry means UMAP computes actual curvature.” → No, it approximates curvature implicitly through local neighborhoods.
• “Spectral embedding is just PCA.” → Spectral methods use graph connectivity, not raw variance — they preserve topology, not linear directions.
• “Geodesic distances are measured directly.” → They’re approximated through shortest paths on the kNN graph.