Concepts

A short tour of the core objects. See Math details for the theory and API Reference for signatures.

Kernel graph

The package builds a sparse affinity graph from nearest-neighbor distances. The Kernel class wraps kNN construction (topo.base.ann.kNN) and several bandwidth/density-correction schemes (adaptive bandwidth, continuous k-NN, fuzzy simplicial sets, Gaussian).

Diffusion operator

A Markov-type graph operator derived from the kernel, used for diffusion maps and related spectral decompositions. Powers of the operator describe diffusion of a signal across the graph over time t.

Laplacian Eigenmaps

Eigenvectors of a graph Laplacian represent the geometry of the graph in a low-dimensional, orthonormal basis. The EigenDecomposition class computes these spectral scaffolds (diffusion-map DM, multiscale msDM, and Laplacian-eigenmap variants).

Spectral scaffold

The collection of eigenfunctions forms the spectral scaffold — an orthonormal basis capturing intrinsic geometry across scales. Refined kNN graphs and kernels are then built in scaffold space before computing 2-D layouts.

Topology-preservation metrics

topo.eval provides metrics comparing local and global geometric structure between the original data representation and learned embeddings, plus Riemannian distortion diagnostics. These scores evaluate geometry/topology preservation, not supervised target predictiveness; for target-aware checks see the Practical FAQ.

High-level orchestrator

TopOGraph ties the pipeline together: kNN → kernel → eigenbasis → scaffold → refined graph → 2-D layouts, with a scikit-learn-style fit / transform interface.