Topological Data Analysis

A skill for applying topological data analysis (TDA) methods to research data. Covers persistent homology, Vietoris-Rips complexes, persistence diagrams, the Mapper algorithm, and vectorization methods for integrating topological features into machine learning pipelines.

Core Concepts

Simplicial Complexes from Data

TDA extracts topological features (connected components, loops, voids) from data by building simplicial complexes at multiple scales:

| Complex | Construction | Computational Cost | |---------|-------------|-------------------| | Vietoris-Rips | Edge if distance < epsilon | O(n^d) for d-simplices | | Cech | Ball intersection (exact) | Computationally expensive | | Alpha | Delaunay-based (exact in low dim) | Efficient in R^2, R^3 | | Cubical | Grid-based (for images) | Linear in pixels |

Filtration and Persistence

Scale epsilon:  0.1    0.3    0.5    0.7    1.0
                |------|------|------|------|------|
Components:      10      6      3      2      1
  (H0 features born at 0, die at merging scale)

Loops:           0      0      1      2      0
  (H1 features born when loop forms, die when filled)

A feature that persists across many scales is a genuine topological signal; short-lived features are noise.

Persistent Homology with Ripser

Computing Persistence Diagrams

python
import numpy as np
from ripser import ripser
from persim import plot_diagrams

def compute_persistence(point_cloud: np.ndarray,
                         max_dim: int = 2,
                         max_edge: float = 2.0) -> dict:
    """
    Compute persistent homology of a point cloud.
    point_cloud: (n_points, n_dimensions) array
    max_dim: maximum homology dimension to compute
    max_edge: maximum edge length in Rips complex
    Returns persistence diagrams for each dimension.
    """
    result = ripser(
        point_cloud,
        maxdim=max_dim,
        thresh=max_edge,
    )

    diagrams = result["dgms"]
    summary = {}

    for dim, dgm in enumerate(diagrams):
        # Filter out infinite death times for H0
        finite = dgm[dgm[:, 1] < np.inf] if len(dgm) > 0 else dgm
        lifetimes = finite[:, 1] - finite[:, 0] if len(finite) > 0 else np.array([])

        summary[f"H{dim}"] = {
            "n_features": len(finite),
            "max_persistence": float(lifetimes.max()) if len(lifetimes) > 0 else 0,
            "mean_persistence": float(lifetimes.mean()) if len(lifetimes) > 0 else 0,
            "birth_death_pairs": finite.tolist(),
        }

    return summary

# Example: torus point cloud
def sample_torus(n=1000, R=3.0, r=1.0, noise=0.1):
    """Sample points from a torus in R^3."""
    theta = np.random.uniform(0, 2 * np.pi, n)
    phi = np.random.uniform(0, 2 * np.pi, n)
    x = (R + r * np.cos(phi)) * np.cos(theta) + np.random.normal(0, noise, n)
    y = (R + r * np.cos(phi)) * np.sin(theta) + np.random.normal(0, noise, n)
    z = r * np.sin(phi) + np.random.normal(0, noise, n)
    return np.column_stack([x, y, z])

torus = sample_torus(500)
persistence = compute_persistence(torus, max_dim=2)
# Expected: H0 has 1 long-lived component,
#           H1 has 2 prominent loops (the two fundamental cycles),
#           H2 has 1 prominent void (the cavity)

Persistence Vectorization

Converting Persistence to Feature Vectors

To use topological features in machine learning, persistence diagrams must be vectorized:

python
from sklearn.base import BaseEstimator, TransformerMixin

class PersistenceStatistics(BaseEstimator, TransformerMixin):
    """
    Extract statistical features from persistence diagrams.
    Produces a fixed-length feature vector from variable-length diagrams.
    """

    def __init__(self, max_dim: int = 1):
        self.max_dim = max_dim

    def fit(self, X, y=None):
        return self

    def transform(self, diagrams_list: list) -> np.ndarray:
        features = []
        for diagrams in diagrams_list:
            row = []
            for dim in range(self.max_dim + 1):
                dgm = diagrams[dim]
                lifetimes = dgm[:, 1] - dgm[:, 0]
                lifetimes = lifetimes[np.isfinite(lifetimes)]

                if len(lifetimes) == 0:
                    row.extend([0, 0, 0, 0, 0, 0])
                else:
                    row.extend([
                        len(lifetimes),              # count
                        np.sum(lifetimes),            # total persistence
                        np.max(lifetimes),            # max persistence
                        np.mean(lifetimes),           # mean persistence
                        np.std(lifetimes),            # std persistence
                        np.sum(lifetimes ** 2),       # persistence entropy proxy
                    ])
            features.append(row)
        return np.array(features)

Persistence Images

python
def persistence_image(diagram: np.ndarray, resolution: int = 20,
                       sigma: float = 0.1,
                       weight_fn=None) -> np.ndarray:
    """
    Compute a persistence image from a persistence diagram.
    Transforms birth-death pairs into a stable, fixed-size representation.
    """
    if weight_fn is None:
        weight_fn = lambda birth, persistence: persistence

    # Transform to birth-persistence coordinates
    births = diagram[:, 0]
    persistences = diagram[:, 1] - diagram[:, 0]

    # Create grid
    x_range = np.linspace(births.min() - sigma, births.max() + sigma, resolution)
    y_range = np.linspace(0, persistences.max() + sigma, resolution)
    xx, yy = np.meshgrid(x_range, y_range)

    image = np.zeros((resolution, resolution))

    for b, p in zip(births, persistences):
        if not np.isfinite(p):
            continue
        w = weight_fn(b, p)
        gaussian = w * np.exp(-((xx - b)**2 + (yy - p)**2) / (2 * sigma**2))
        image += gaussian

    return image

The Mapper Algorithm

Constructing Mapper Graphs

Mapper provides a compressed topological summary of high-dimensional data:

python
import kmapper as km
from sklearn.cluster import DBSCAN

def run_mapper(data: np.ndarray, lens_fn=None, n_cubes: int = 10,
               overlap: float = 0.3) -> dict:
    """
    Run the Mapper algorithm to produce a simplicial complex
    summarizing the shape of the data.
    data: (n_samples, n_features) array
    lens_fn: filter function (default: first two PCA components)
    """
    mapper = km.KeplerMapper(verbose=0)

    # Compute lens (filter function)
    if lens_fn is None:
        from sklearn.decomposition import PCA
        lens = mapper.fit_transform(data, projection=PCA(n_components=2))
    else:
        lens = lens_fn(data)

    # Build the Mapper graph
    graph = mapper.map(
        lens, data,
        cover=km.Cover(n_cubes=n_cubes, perc_overlap=overlap),
        clusterer=DBSCAN(eps=0.5, min_samples=3),
    )

    # Summary statistics
    n_nodes = len(graph["nodes"])
    n_edges = sum(len(v) for v in graph["links"].values()) // 2

    return {
        "n_nodes": n_nodes,
        "n_edges": n_edges,
        "node_sizes": [len(v) for v in graph["nodes"].values()],
        "graph": graph,
    }

Mapper Parameters

| Parameter | Effect | Guidance | |-----------|--------|---------| | Filter function | Projects data to low dimensions | PCA, eccentricity, density | | Number of intervals | Controls resolution of cover | 10-30 typical | | Overlap percentage | Controls connectivity | 20-50%, higher = more edges | | Clustering algorithm | Groups points within intervals | DBSCAN, single-linkage |

Stability and Statistical Significance

Bottleneck and Wasserstein Distances

python
from persim import bottleneck, wasserstein

def compare_persistence_diagrams(dgm1: np.ndarray,
                                   dgm2: np.ndarray) -> dict:
    """
    Compare two persistence diagrams using standard TDA distances.
    """
    bn_dist = bottleneck(dgm1, dgm2)
    ws_dist = wasserstein(dgm1, dgm2, order=2)

    return {
        "bottleneck_distance": round(bn_dist, 6),
        "wasserstein_2_distance": round(ws_dist, 6),
    }

Permutation Test for Topological Features

python
def permutation_test_persistence(data1: np.ndarray, data2: np.ndarray,
                                   n_permutations: int = 1000,
                                   dim: int = 1) -> dict:
    """
    Test whether two point clouds have significantly different
    topological features using a permutation test on Wasserstein distance.
    """
    from persim import wasserstein

    # Observed distance
    dgm1 = ripser(data1, maxdim=dim)["dgms"][dim]
    dgm2 = ripser(data2, maxdim=dim)["dgms"][dim]
    observed = wasserstein(dgm1, dgm2)

    # Permutation distribution
    combined = np.vstack([data1, data2])
    n1 = len(data1)
    perm_distances = []

    for _ in range(n_permutations):
        perm = np.random.permutation(len(combined))
        perm_d1 = combined[perm[:n1]]
        perm_d2 = combined[perm[n1:]]
        perm_dgm1 = ripser(perm_d1, maxdim=dim)["dgms"][dim]
        perm_dgm2 = ripser(perm_d2, maxdim=dim)["dgms"][dim]
        perm_distances.append(wasserstein(perm_dgm1, perm_dgm2))

    p_value = np.mean(np.array(perm_distances) >= observed)

    return {
        "observed_distance": round(observed, 6),
        "p_value": round(p_value, 4),
        "significant_at_005": p_value < 0.05,
    }

Tools and Libraries

• Ripser / ripser.py: Fast Vietoris-Rips persistence computation
• GUDHI: Comprehensive TDA library (C++ with Python bindings)
• persim: Persistence diagram distances and visualization
• KeplerMapper: Python Mapper algorithm implementation
• giotto-tda: TDA integrated with scikit-learn API
• Dionysus 2: Persistent homology and cohomology
• scikit-tda: Meta-package bundling ripser, persim, kepler-mapper, tadasets