Recover a dataset's latent geometry from the spectrum of an operator built on its neighbourhood graph, rather than assuming a shape in advance.

Spectral Geometry

The idea

Build a neighbourhood graph on your data, form an operator on it — a Laplacian, a diffusion operator — and read the geometry off its eigenvectors. The low eigenvectors give coordinates that respect local structure, and the eigenvalues say how much structure each coordinate carries.

The mathematical basis is that the Laplace–Beltrami operator on a manifold determines a great deal of its geometry, and that a graph Laplacian on samples drawn from a manifold converges to it. So the spectrum of a matrix you can compute approximates the geometry of a manifold you cannot observe. Laplacian eigenmaps and diffusion maps are the classical instances; diffusion maps add a time parameter that controls how far structure is propagated before it is read.

This is a different lineage from persistent homology, and the difference is the point. persistent-homology is topological — it counts holes and is deliberately blind to how far apart things are. Spectral geometry is metric: it is about distances, densities, and the diffusion between points, and it produces coordinates rather than invariants.

Why single-cell work runs on it

Single-cell expression data is the case that made these methods routine. Cells sit in thousands of dimensions, the biologically meaningful variation is low-dimensional, and the structure is frequently continuous — a differentiation trajectory is a path, not a cluster.

Spectral methods suit that shape. They do not assume clusters, they represent continuity natively, and their coordinates are ordered by how much structure each explains. Nearly every step of a standard workflow is spectral underneath: the neighbourhood graph, the diffusion-based denoising, the embedding.

The problem nobody solved by picking a default

Every spectral pipeline embeds a chain of choices: which distance metric, how many neighbours, which kernel, which normalization of the Laplacian, how many eigenvectors, and how to lay the result out in two dimensions for viewing. Each choice changes the answer, and the field’s usual practice is to adopt one chain as a default and stop asking.

That is the practice topometry was built to attack, and it is why the fixture exists here. Its argument is that the right response to an unresolvable choice is to evaluate it rather than fix it — construct many latent geometries from the same data, and score how well each preserves the original structure, rather than defending one pipeline. The evaluation is the contribution; the individual operators mostly predate it.

The corollary is worth stating plainly for anyone reading a single-cell embedding: a two-dimensional layout is the end of a long chain of decisions, and the visual impression it gives of “how many populations there are” is not a finding. Distances between clusters in a UMAP plot are not distances.

What to watch

Neighbourhood size is the dominant knob. Too small fragments a continuous trajectory into spurious clusters; too large merges genuinely distinct populations. It deserves a sensitivity check, not a default.
Density confounds geometry. A densely sampled region and a genuinely tight region look alike to a plain nearest-neighbour graph. Diffusion-based normalizations exist specifically to separate the two, and the choice of whether to apply them is a modelling decision about your data.
Layout is not embedding. The spectral coordinates and the 2-D picture drawn from them are different objects with different reliability, and conclusions should be drawn from the former.

What implements it here

topometry — the systematic approach: many latent geometries built and evaluated against each other rather than one pipeline asserted.
Fixtures: topometry-environment, topometry-1.1-environment (a pinned earlier line), and phate-environment for the diffusion-potential embedding used widely in trajectory work.

persistent-laplacian shares the spectral machinery but asks a topological question with it. The two are easy to confuse from the names and are not the same practice: one recovers coordinates for a dataset, the other summarizes shape across a filtration.