Skip to main content

PyTorch Implementation of Spectral Clustering

Spectral Clustering is a powerful technique for detecting clusters in data with complex structures. PyTorch, typically associated with deep learning, can also be used to implement Spectral Clustering. This article will show you how to perform Spectral Clustering using PyTorch, with detailed explanations of each step.

1. Introduction​

In the previous articles, we explored the theory of Spectral Clustering and implemented it in both scikit-learn and TensorFlow. Now, we'll implement Spectral Clustering in PyTorch. This implementation allows for flexibility and speed when working with larger datasets or integrating clustering into PyTorch-based projects.

2. Importing Required Libraries​

We start by importing the necessary libraries:

import torch
import numpy as np
import matplotlib.pyplot as plt
from sklearn.datasets import make_moons
from sklearn.metrics import pairwise_distances
  • torch: For performing tensor operations and eigenvalue decomposition.
  • numpy: For numerical operations.
  • matplotlib: For visualizing the data and clustering results.
  • scikit-learn: To generate a synthetic dataset using make_moons.

3. Generating the Dataset​

We'll generate a two-dimensional dataset with two interlocking half-moon shapes using make_moons:

# Generate the dataset
X, y = make_moons(n_samples=300, noise=0.05, random_state=42)

# Visualize the dataset
plt.scatter(X[:, 0], X[:, 1], c=y, cmap='viridis')
plt.title('Dataset with Two Interlocking Half-Moons')
plt.xlabel('Feature 1')
plt.ylabel('Feature 2')
plt.show()

4. Implementing Spectral Clustering in PyTorch​

4.1 Step 1: Construct the Similarity Graph​

First, we need to construct the similarity graph by calculating the pairwise distances between the data points and applying the RBF (Gaussian) kernel to compute the similarities.

def rbf_kernel(X, gamma=1.0):
    # Compute the pairwise distances
    pairwise_dists = pairwise_distances(X, metric='euclidean')
    
    # Apply the RBF kernel
    K = np.exp(-gamma * pairwise_dists ** 2)
    
    return K

# Construct the similarity graph
gamma = 1.0
similarity_graph = rbf_kernel(X, gamma=gamma)

4.2 Step 2: Compute the Laplacian Matrix​

We now compute the Laplacian matrix, which will be used to perform the eigenvalue decomposition.

def compute_laplacian(similarity_graph):
    # Compute the degree matrix
    degree_matrix = np.diag(np.sum(similarity_graph, axis=1))
    
    # Compute the Laplacian matrix
    laplacian_matrix = degree_matrix - similarity_graph
    
    return laplacian_matrix

# Compute the Laplacian matrix
laplacian_matrix = compute_laplacian(similarity_graph)

4.3 Step 3: Eigenvalue Decomposition in PyTorch​

Next, we perform eigenvalue decomposition on the Laplacian matrix using PyTorch. PyTorch’s eig function allows us to efficiently compute the eigenvalues and eigenvectors.

def eigen_decomposition(laplacian_matrix, n_clusters):
    # Convert Laplacian matrix to a PyTorch tensor
    laplacian_tensor = torch.tensor(laplacian_matrix, dtype=torch.float32)
    
    # Perform eigenvalue decomposition
    eigenvalues, eigenvectors = torch.eig(laplacian_tensor, eigenvectors=True)
    
    # Select the eigenvectors corresponding to the smallest eigenvalues
    eigenvectors = eigenvectors[:, :n_clusters]
    
    return eigenvectors

n_clusters = 2
embedding = eigen_decomposition(laplacian_matrix, n_clusters)

4.4 Step 4: K-Means Clustering in the Embedded Space​

After reducing the data using eigenvectors, we apply K-Means clustering in the embedded space:

from sklearn.cluster import KMeans

# Perform K-Means clustering in the embedded space
kmeans = KMeans(n_clusters=n_clusters, random_state=42)
labels = kmeans.fit_predict(embedding.detach().numpy())

4.5 Visualizing the Clustering Results​

Let's visualize the results of the clustering:

# Visualize the clusters
plt.scatter(X[:, 0], X[:, 1], c=labels, cmap='viridis')
plt.title('Spectral Clustering Results (PyTorch Implementation)')
plt.xlabel('Feature 1')
plt.ylabel('Feature 2')
plt.show()

You should see that Spectral Clustering correctly identifies the two interlocking half-moon shapes as separate clusters.

5. Analyzing the Results​

This PyTorch-based implementation of Spectral Clustering demonstrates that PyTorch can be effectively used for clustering tasks, especially when integrated into existing PyTorch workflows. The steps, though slightly different from scikit-learn or TensorFlow, follow the same principles.

Key Points:​

  • Flexibility in PyTorch: This implementation highlights PyTorch’s flexibility for operations beyond deep learning, such as eigenvalue decomposition and clustering.
  • Effective for Complex Clusters: Spectral Clustering can detect non-convex clusters and performs well on datasets like the interlocking half-moons.

6. Conclusion​

Spectral Clustering is a versatile algorithm capable of detecting clusters with complex structures. Implementing it in PyTorch demonstrates the utility of the framework beyond traditional deep learning tasks, enabling efficient clustering on challenging datasets. By following this guide, you should now be able to implement Spectral Clustering using PyTorch in your own machine learning workflows.