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Topological Data Analysis (TDA) provides mathematically rigorous methods for discovering the "shape" of data. Using tools like persistent homology, Betti numbers, persistence diagrams, and the Mapper algorithm, TDA uncovers multi-scale structural features invisible to classical statistics.
When to Use This Skill
You need shape-aware features that are robust to noise and deformation in point cloud data.
Classical distance-based methods miss higher-order relational structure (loops, voids, clusters).
You are working with time-series, biological sequences, images, or sensor networks where topology is meaningful.
You want to detect connectivity, circular patterns, or cavities in high-dimensional datasets.
You need features that are invariant to continuous deformations of the input space.
You are engineering features for downstream machine learning pipelines and want topologically informed representations.
Typical domains: drug discovery (molecular topology), neuroscience (brain network loops), materials science (pore structures), computer vision (shape matching), and financial data (market regime topology).
Background & Key Concepts
Simplicial Complexes
A simplicial complex is a generalization of a graph that includes higher-dimensional "simplices": vertices (0-simplex), edges (1-simplex), triangles (2-simplex), tetrahedra (3-simplex), and so on. Given a point cloud, we build a nested family of simplicial complexes by varying a distance parameter epsilon.
Vietoris-Rips Complex
For a point cloud X and threshold epsilon, the Vietoris-Rips complex VR(X, epsilon) contains a simplex for every finite subset of X with diameter at most epsilon. As epsilon grows from 0 to infinity, new topological features (connected components, loops, voids) are born and die.
Persistent Homology
Persistent homology tracks topological features across a filtration of simplicial complexes. For each dimension k:
H_0 counts connected components (Betti number beta_0).
H_1 counts independent loops / 1-cycles (Betti number beta_1).
H_2 counts enclosed voids / 2-cycles (Betti number beta_2).
Each feature has a birth time b and a death time d. The persistence (d - b) measures how long the feature survives — long-lived features signal genuine topology; short-lived ones are noise.
Persistence Diagrams and Barcodes
A persistence diagram is a scatter plot of (birth, death) pairs for each topological feature. A barcode is the equivalent interval representation. Features far from the diagonal are topologically significant.
Betti Numbers
The k-th Betti number beta_k counts independent k-dimensional cycles in a topological space:
beta_0 = number of connected components
beta_1 = number of independent loops
beta_2 = number of enclosed voids
Mapper Algorithm
Mapper is a TDA algorithm that compresses high-dimensional point clouds into a graph (the Mapper graph) that captures topological shape. It uses a lens function (e.g., PCA projection, density), a cover of the range, partial clustering within preimages, and nerve construction to yield a navigable network.
Wasserstein and Bottleneck Distance
These metrics measure distances between persistence diagrams. The bottleneck distance is the inf over matchings of the sup of point distances. The p-Wasserstein distance is the inf over matchings of the Lp norm of point distances.
# If gudhi installation fails on some platforms, try conda
conda install -c conda-forge gudhi
No API keys are required for TDA computations. All processing is local.
Core Workflow
Step 1: Generate or Load a Point Cloud
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from sklearn.datasets import make_circles, make_moons
# --- Synthetic example: two concentric circles ---
np.random.seed(42)
X_circles, _ = make_circles(n_samples=300, noise=0.05, factor=0.4)
# --- Synthetic example: noisy torus (parametric) ---defsample_torus(n=500, R=3.0, r=1.0, noise=0.1):
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)
y = (R + r * np.cos(phi)) * np.sin(theta)
z = r * np.sin(phi)
pts = np.column_stack([x, y, z])
pts += noise * np.random.randn(*pts.shape)
return pts
X_torus = sample_torus(n=600)
# --- Loading from a CSV (real data) ---# df = pd.read_csv("my_point_cloud.csv")# X_real = df[["feature_1", "feature_2", "feature_3"]].to_numpy()print(f"Circles shape: {X_circles.shape}")
print(f"Torus shape: {X_torus.shape}")
Step 2: Compute Persistent Homology with ripser
from ripser import ripser
from persim import plot_diagrams
import matplotlib.pyplot as plt
# --- Compute persistent homology up to dimension 2 ---
result_circles = ripser(X_circles, maxdim=1)
result_torus = ripser(X_torus, maxdim=2)
diagrams_circles = result_circles["dgms"] # list: dgms[k] = array of (birth, death) pairs
diagrams_torus = result_torus["dgms"]
# --- Print Betti numbers at a chosen threshold ---defbetti_at_threshold(dgms, threshold):
betti = []
for k, dgm inenumerate(dgms):
iflen(dgm) == 0:
betti.append(0)
continue# Count features alive at threshold (born <= threshold < death)
alive = np.sum((dgm[:, 0] <= threshold) & (dgm[:, 1] > threshold))
betti.append(int(alive))
return betti
eps = 0.5
b_circles = betti_at_threshold(diagrams_circles, eps)
print(f"Betti numbers for circles at eps={eps}: {b_circles}")
# Expected: [1, 2] -> 1 component, 2 loops (inner + outer circle)# --- Plot persistence diagrams ---
fig, axes = plt.subplots(1, 2, figsize=(12, 5))
plt.sca(axes[0])
plot_diagrams(diagrams_circles, show=False)
axes[0].set_title("Persistence Diagram: Two Circles")
plt.sca(axes[1])
plot_diagrams(diagrams_torus, show=False)
axes[1].set_title("Persistence Diagram: Torus")
plt.tight_layout()
plt.savefig("persistence_diagrams.png", dpi=150)
plt.show()
print("Saved persistence_diagrams.png")
Step 3: Compute Persistent Homology with GUDHI
import gudhi
import gudhi.representations
import numpy as np
import matplotlib.pyplot as plt
# --- Build a Rips complex from the circles dataset ---
rips_complex = gudhi.RipsComplex(
points=X_circles,
max_edge_length=2.0# maximum edge length for the filtration
)
# Create a simplex tree up to dimension 2
simplex_tree = rips_complex.create_simplex_tree(max_dimension=2)
print(f"Number of simplices: {simplex_tree.num_simplices()}")
print(f"Number of vertices: {simplex_tree.num_vertices()}")
# --- Compute persistence ---
simplex_tree.compute_persistence()
# Extract persistence pairs for each dimension
diag_gudhi = simplex_tree.persistence() # list of (dim, (birth, death))# Separate by dimensiondefextract_by_dim(diag, dim):
return np.array(
[(b, d) for (k, (b, d)) in diag if k == dim and np.isfinite(d)],
dtype=float
)
h0 = extract_by_dim(diag_gudhi, 0)
h1 = extract_by_dim(diag_gudhi, 1)
print(f"H0 features: {len(h0)}")
print(f"H1 features: {len(h1)}")
# --- Plot barcode ---
gudhi.plot_persistence_barcode(diag_gudhi)
plt.title("Persistence Barcode: Two Circles (GUDHI)")
plt.savefig("barcode_circles.png", dpi=150)
plt.show()
# --- Plot persistence diagram ---
gudhi.plot_persistence_diagram(diag_gudhi)
plt.title("Persistence Diagram: Two Circles (GUDHI)")
plt.savefig("diagram_circles_gudhi.png", dpi=150)
plt.show()
Step 4: Persistence Landscape and Vectorization
import gudhi.representations as gr
import numpy as np
# GUDHI representations for machine learning pipelines# Diagrams as list-of-arrays format expected by gudhi.representations
diag_list_h1 = [h1] # wrap in list for transformer API# --- Persistence Landscape ---
landscape = gr.Landscape(num_landscapes=5, resolution=100)
L = landscape.fit_transform(diag_list_h1)
print(f"Landscape shape: {L.shape}") # (1, 5*100)# --- Betti Curve ---
betti_curve = gr.BettiCurve(resolution=100)
B = betti_curve.fit_transform(diag_list_h1)
print(f"Betti curve shape: {B.shape}") # (1, 100)# --- Persistence Image ---
pers_image = gr.PersistenceImage(bandwidth=0.05, weight=lambda x: x[1], resolution=[20, 20])
PI = pers_image.fit_transform(diag_list_h1)
print(f"Persistence image shape: {PI.shape}") # (1, 400)# --- Silhouette ---
silhouette = gr.Silhouette(weight=lambda x: x[1]**2, resolution=100)
S = silhouette.fit_transform(diag_list_h1)
print(f"Silhouette shape: {S.shape}") # (1, 100)# Plot persistence imageimport matplotlib.pyplot as plt
plt.figure(figsize=(5, 5))
plt.imshow(PI[0].reshape(20, 20), origin="lower", cmap="hot_r")
plt.colorbar(label="Persistence density")
plt.title("Persistence Image (H1)")
plt.savefig("persistence_image.png", dpi=150)
plt.show()
Advanced Usage
Mapper Graph Construction
import numpy as np
import matplotlib.pyplot as plt
from sklearn.datasets import make_circles
from sklearn.decomposition import PCA
from sklearn.cluster import DBSCAN
import networkx as nx
# Attempt to use kmapper (part of scikit-tda / KeplerMapper)try:
import kmapper as km
from kmapper import Cover
np.random.seed(0)
X, labels = make_circles(n_samples=500, noise=0.03, factor=0.5)
# Initialize Mapper
mapper = km.KeplerMapper(verbose=1)
# Lens: 1D PCA projection
lens = mapper.fit_transform(X, projection=PCA(n_components=1))
# Build the graph
graph = mapper.map(
lens,
X,
cover=Cover(n_cubes=10, perc_overlap=0.5),
clusterer=DBSCAN(eps=0.1, min_samples=5)
)
# Visualize
mapper.visualize(
graph,
path_html="mapper_circles.html",
title="Mapper: Two Circles",
color_values=labels,
color_function_name="Ground truth label"
)
print("Mapper graph saved to mapper_circles.html")
# Convert to NetworkX for analysis
G_mapper = nx.Graph()
for node in graph["nodes"]:
G_mapper.add_node(node, size=len(graph["nodes"][node]))
for node, neighbors in graph["links"].items():
for neighbor in neighbors:
G_mapper.add_edge(node, neighbor)
print(f"Mapper graph: {G_mapper.number_of_nodes()} nodes, {G_mapper.number_of_edges()} edges")
print(f"Connected components: {nx.number_connected_components(G_mapper)}")
except ImportError:
print("kmapper not installed. Install with: pip install kmapper")
TDA-Based Feature Engineering for Classification
import numpy as np
import pandas as pd
from ripser import ripser
from persim import PersStats
from sklearn.pipeline import Pipeline
from sklearn.preprocessing import StandardScaler
from sklearn.ensemble import RandomForestClassifier
from sklearn.model_selection import cross_val_score
from sklearn.datasets import make_classification
# Generate multi-class dataset
np.random.seed(42)
X_raw, y = make_classification(n_samples=200, n_features=10, n_informative=5, n_classes=3, random_state=42)
defcompute_tda_features(X_point_cloud, maxdim=1):
"""
Compute a fixed-length TDA feature vector from a point cloud.
Features: persistence statistics for each homology dimension.
"""
result = ripser(X_point_cloud, maxdim=maxdim)
dgms = result["dgms"]
features = []
for k, dgm inenumerate(dgms):
# Filter out infinite deaths
finite_mask = np.isfinite(dgm[:, 1])
dgm_finite = dgm[finite_mask]
iflen(dgm_finite) == 0:
# Pad with zeros if no features in this dimension
features.extend([0.0] * 6)
continue
persistence = dgm_finite[:, 1] - dgm_finite[:, 0]
features.extend([
float(np.sum(persistence)), # total persistencefloat(np.max(persistence)), # max persistencefloat(np.mean(persistence)), # mean persistencefloat(np.std(persistence)), # std of persistencefloat(len(persistence)), # number of featuresfloat(np.median(persistence)), # median persistence
])
return np.array(features)
# For each sample, treat its 10 features as a 1D time-series point cloud# In real applications, each sample might be a separate point cloud
tda_features = np.array([
compute_tda_features(X_raw[i].reshape(-1, 1))
for i inrange(len(X_raw))
])
print(f"TDA feature matrix shape: {tda_features.shape}")
# Build a classification pipeline
clf_pipeline = Pipeline([
("scaler", StandardScaler()),
("clf", RandomForestClassifier(n_estimators=100, random_state=42))
])
scores = cross_val_score(clf_pipeline, tda_features, y, cv=5, scoring="accuracy")
print(f"TDA-feature classification accuracy: {scores.mean():.3f} +/- {scores.std():.3f}")
import gudhi
import numpy as np
import matplotlib.pyplot as plt
from skimage import data, filters
# Load a grayscale image and compute its cubical persistent homology
image = data.coins() # 303x384 grayscale# Downsample to speed up computationfrom skimage.transform import resize
image_small = resize(image, (60, 80), anti_aliasing=True)
# GUDHI cubical complex
cc = gudhi.CubicalComplex(
dimensions=list(image_small.shape),
top_dimensional_cells=image_small.flatten().tolist()
)
cc.compute_persistence()
diag_img = cc.persistence()
print(f"Number of persistence pairs: {len(diag_img)}")
# Plot
gudhi.plot_persistence_diagram(diag_img)
plt.title("Cubical Persistence: Coins Image")
plt.savefig("cubical_coins.png", dpi=150)
plt.show()
Troubleshooting
Problem: ripser is slow on large point clouds (N > 5000)
# Use the fast C++ backend via scikit-tda / ripser++
pip install ripser
# For even larger clouds, consider subsampling with farthest point sampling
deffarthest_point_sample(X, n_samples):
"""Greedy farthest-point subsampling of a point cloud."""
n = len(X)
selected = [np.random.randint(n)]
dists = np.full(n, np.inf)
for _ inrange(n_samples - 1):
last = selected[-1]
d = np.sum((X - X[last]) ** 2, axis=1)
dists = np.minimum(dists, d)
selected.append(int(np.argmax(dists)))
return X[selected]
X_sub = farthest_point_sample(X_torus, 300)
Problem: GUDHI installation fails on Windows
# Use conda-forge channel
conda install -c conda-forge gudhi
# Or use WSL/Linux for native builds
Problem: Persistence diagram has too many noise points near the diagonal
# Filter low-persistence features before vectorizationdeffilter_diagram(dgm, min_persistence=0.05):
persistence = dgm[:, 1] - dgm[:, 0]
return dgm[persistence >= min_persistence]
dgm_filtered = filter_diagram(diagrams_circles[1], min_persistence=0.1)
print(f"Features before filter: {len(diagrams_circles[1])}")
print(f"Features after filter: {len(dgm_filtered)}")
Problem: kmapper graph is empty or disconnected
# Increase overlap percentage or reduce number of cubes
graph = mapper.map(
lens, X,
cover=Cover(n_cubes=8, perc_overlap=0.7), # more overlap
clusterer=DBSCAN(eps=0.2, min_samples=3) # looser clustering
)
Problem: Memory error when computing high-dimensional Rips complex
# Limit max dimension and max edge length
result = ripser(X, maxdim=1, thresh=1.5) # thresh caps the filtration
Problem: Inconsistent results between ripser and GUDHI
Both libraries implement the standard persistence algorithm but may differ in tie-breaking for equal filtration values. This is expected. For reproducibility, fix np.random.seed and use consistent preprocessing.
