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Topological Data Analysis

Persistent homology beyond H0 and H1
A Python and Rust implementation built for performance

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TDA computes Vietoris–Rips persistent homology from point clouds or precomputed distance matrices. The performance-critical implementation is written in Rust and is available through both Python and Rust APIs.

[!CAUTION] TDA is in an early stage of development. APIs may change between releases.

Installation

TDA requires Python 3.11 or newer. Pre-built wheels for common macOS, Windows, and Linux platforms are published on PyPI:

python -m pip install tda

Compiling from source

Building from source requires Rust/Cargo and maturin:

git clone https://github.com/antonio-leitao/topological-data-analysis.git
cd topological-data-analysis

python3 -m venv .venv
source .venv/bin/activate  # Windows: .venv\Scripts\activate
python -m pip install maturin numpy

cd crates/python
maturin develop --release

Python usage

import numpy as np
import tda

# Point cloud: an (n, d) float32 array.
points = np.random.rand(200, 3).astype(np.float32)
barcode = tda.persistent_homology(points, max_dim=2)

# barcode[d] is a (k_d, 2) array of [birth, death] intervals.
print(barcode[0])  # H0
print(barcode[1])  # H1
print(barcode[2])  # H2

Precomputed distance matrices are also supported:

distances = np.asarray(my_distance_matrix, dtype=np.float32)
barcode = tda.persistent_homology(
    distances,
    max_dim=1,
    distance_matrix=True,
)

Use filtration_size to inspect the size of the truncated filtration built with the same options:

size = tda.filtration_size(points, max_dim=2, peel=True)
print(size)

Parameters

Both persistent_homology and filtration_size accept the following parameters:

Parameter Type Default Description
data np.ndarray Two-dimensional float32 array. Shape (n, d) for a point cloud or (n, n) for a distance matrix. Inputs are copied into row-major internal storage.
max_dim int 1 Highest homology dimension to compute. Capped at 4.
threshold float | None None Maximum filtration value. None uses the enclosing radius; an explicit value is capped by that radius.
distance_matrix bool False Interpret data as a square distance matrix. Symmetry, zero diagonal, and non-negativity are assumed rather than validated.
quotient bool False Use the smaller quotient-cover filtration. This is an approximation with a log(3) interleaving guarantee, not the exact Vietoris–Rips barcode.
peel bool False Apply an exact strong-collapse reduction before computing the result.
parallel bool True Enable parallel preprocessing, sorting, and candidate assembly for sufficiently large inputs.

persistent_homology returns a list of length max_dim + 1. Entry barcode[d] is a NumPy array of shape (k_d, 2) whose rows are [birth, death] intervals. A death value of inf marks an essential feature.

Rust usage

The Rust crate is published as tda_core:

[dependencies]
tda_core = "0.3"

For a point cloud, pass a flat row-major (n, d) slice. The ambient dimension is inferred from the slice length and n:

use tda_core::{persistent_homology, Error};

fn main() -> Result<(), Error> {
    let points: Vec<f32> = vec![
        0.0, 0.0,
        1.0, 0.0,
        0.0, 1.0,
        1.0, 1.0,
    ];

    let barcode = persistent_homology(
        &points,
        4,     // number of points
        1,     // max_dim
        None,  // threshold
        false, // distance_matrix
        false, // quotient
        false, // peel
        true,  // parallel
    )?;

    for (dim, intervals) in barcode.intervals.iter().enumerate() {
        for interval in intervals {
            println!("H{dim}: [{}, {})", interval.birth, interval.death);
        }
    }

    Ok(())
}

Distance matrices use the same function with distance_matrix = true:

use tda_core::{persistent_homology, Error};

fn main() -> Result<(), Error> {
    let distances: Vec<f32> = vec![
        0.0, 1.0, 2.0,
        1.0, 0.0, 1.5,
        2.0, 1.5, 0.0,
    ];

    let barcode = persistent_homology(
        &distances,
        3,
        1,
        None,
        true,  // distance_matrix
        false, // quotient
        false, // peel
        true,  // parallel
    )?;

    println!("{:?}", barcode.intervals);
    Ok(())
}

Invalid inputs return tda_core::Error, including shape mismatches, invalid thresholds, too few or too many points, and unsupported dimensions.

License

TDA is distributed under the MIT License.

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0.3.0 This release

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0.1.0

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