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