About TopoMetry

TopoMetry is a geometry-aware Python toolkit for exploring high-dimensional data via diffusion/Laplacian operators. It learns neighborhood graphs → Laplace–Beltrami–type operators → spectral scaffolds → refined graphs and then finds clusters and builds low-dimensional layouts for analysis and visualization.

• AnnData/Scanpy wrappers for single-cell workflows
• scikit-learn–style transformers with a high-level orchestrator
• Fixed-time & multiscale spectral scaffolds (no .X mutation; namespaced outputs)
• Operator-native metrics to quantify geometry preservation and Riemannian diagnostics to evaluate distortion in visualizations
• Designed for large, diverse datasets (e.g., single-cell omics)

For background, see our preprint: https://doi.org/10.1101/2022.03.14.484134

Geometry-first rationale (short)

We approximate the Laplace–Beltrami operator (LBO) by learning well-weighted similarity graphs and their Laplacian/diffusion operators. The eigenfunctions of these operators form an orthonormal basis—the spectral scaffold—that captures the dataset’s intrinsic geometry across scales. This view connects to Diffusion Maps, Laplacian Eigenmaps, and related kernel eigenmaps, and enables downstream tasks such as clustering and graph-layout optimization with geometry preserved.

When to use TopoMetry

Use TopoMetry when you want:

• Geometry-faithful representations beyond variance maximization (e.g., PCA)
• Robust low-dimensional views and clustering from operator-grounded features
• Quantitative operator-native metrics to compare methods and parameter choices
• Reproducible, non-destructive pipelines (no mutation of adata.X)

Empirically, TopoMetry often outperforms PCA-based pipelines and stand-alone layouts. Still, let the data decide—TopoMetry includes metrics and reports to support evidence-based choices.

When not to use TopoMetry

• Very small sample sizes where the manifold hypothesis is weak
• Workflows needing streaming/online updates or inverse transforms (embedding new points without recomputing operators is not currently supported). If that’s critical, consider UMAP or parametric/autoencoder approaches—and you can still use TopoMetry to audit geometry or estimate intrinsic dimensionality to guide model design.

Installation

Prior to installing TopoMetry, make sure you have cmake, scikit-build and setuptools available in your system. If using Linux:

sudo apt-get install cmake
pip install scikit-build setuptools

Then you can install TopoMetry from PyPI:

pip install topometry

Neighbor search uses hnswlib when it is installed (pip install hnswlib), which is recommended for all but small datasets. Without it, TopoMetry falls back to nmslib if present, and otherwise to scikit-learn's exact search. PaCMAP layouts need pip install pacmap.

Tutorials and documentation

Check TopoMetry's documentation for tutorials, guided analyses and other documentation.

Minimal example (current API)

import scanpy as sc
import topo as tp

adata = sc.datasets.pbmc3k_processed()

# Fit TopoMetry end-to-end (non-destructive; outputs are namespaced)
tg = tp.sc.fit_adata(adata, n_jobs=1, verbosity=0, random_state=7)

# Plot some results
sc.pl.embedding(adata, basis='spectral_scaffold', color='topo_clusters')
sc.pl.embedding(adata, basis='TopoMAP', color='topo_clusters')
sc.pl.embedding(adata, basis='TopoPaCMAP', color='topo_clusters')

# Save cleanly (I/O-safe)
adata.write_h5ad("pbmc3k_topometry.h5ad")

Changelog

v1.1.1 — Fixes to standing issues

⚠️ Results computed with a cosine metric change, and base_metric='cosine' is the default. Analyses run with any release from 0.2.0.0 to 1.1.0 on a cosine metric should be re-run. Euclidean graphs and the default kernels are unchanged on the hnswlib and nmslib backends.

• Cosine neighbor graphs held similarities instead of distances. Within each neighborhood the closest points got the lowest kernel weights. Neighbor sets were right; the weights were not. Cosine kernels, geodesics and intrinsic-dimension estimates are now computed on angles, and kernels have no self-loops.
• Neighbor search. A missing backend falls back to the next available of hnswlib, nmslib and scikit-learn with a warning; every backend returns the same number of neighbors; backend='nmslib' is honoured; nmslib no longer returns squared distances for dense input.
• Kernel options that did not do what they say now do: CkNN (delta is passed on and usually needs tuning), neighborhood expansion, use_angular=False, pairwise=True.
• Projections. Isomap and the MDE recipes are run on distances rather than on the diffusion operator; UMAP and landmarks work; a standalone Projector builds proper affinities.
• Reproducibility. With n_jobs=1 and a random_state, two fits agree bit for bit. Eigenvector signs no longer flip between runs.
• TopOGraph. global_id, local_ids(), global_id_mle() and global_id_fsa() return the estimates (the scaffold size is n_scaffold_components); pseudotime and spectral_selectivity pair each eigenvector with its own eigenvalue; saving no longer strips the fitted object.
• Riemannian diagnostics keep the Laplacian sparse, and the plotting wrapper honours diffusion_t and groupby.
• Single-cell wrappers. tp.sc.preprocess no longer modifies its input; BBKNN integration builds kernels from distances; PaCMAP is optional; fit_adata works without hnswlib.
• Compatibility with current scikit-learn, SciPy and matplotlib.

v1.1.0 — Batch integration and data mapping

• CCA-anchor batch correction (Seurat v3-style) via tp.sc.run_cca_integration
• Reference atlas persistence (save_cca_reference / load_cca_reference) and sequential query mapping (map_to_cca_reference)
• High-level preparation utilities (prepare_for_integration, prepare_for_mapping, find_mapping_order)
• Neighbourhood-based integration quality metrics (compute_all_integration_metrics: kNN purity, kNN mixing, iLISI, cLISI, ARI, NMI)

v1.0.x — Complete overhaul

• Redesigned user API with tp.sc.fit_adata and tp.sc.run_and_report one-liner workflows
• New utilities for single-cell analysis: intrinsic dimensionality, spectral selectivity, feature modes, graph-signal filtering, imputation
• Overhauled geometry-preservation metrics (PF1, PJS, SP) and Riemannian diagnostics (pullback metric, deformation maps)
• Full compatibility with the scverse ecosystem (scanpy, scVelo, AnnData)

Citation

---

@article {Oliveira2022.03.14.484134,
	author = {Oliveira, David S and Domingos, Ana I. and Velloso, Licio A},
	title = {TopoMetry systematically learns and evaluates the latent geometry of single-cell data},
	elocation-id = {2022.03.14.484134},
	year = {2025},
	doi = {10.1101/2022.03.14.484134},
	publisher = {Cold Spring Harbor Laboratory},
	URL = {https://www.biorxiv.org/content/early/2025/10/15/2022.03.14.484134},
	eprint = {https://www.biorxiv.org/content/early/2025/10/15/2022.03.14.484134.full.pdf},
	journal = {bioRxiv}
}