AFCM (Full Graph Embedding) Documentation

🔍 Overview

This version of Adaptive Fuzzy C-Means (AFCM) includes a graph embedding step using a Laplacian matrix derived from a k-nearest neighbors graph. The embedding regularizes fuzzy clustering using both manifold structure and adaptive memberships.

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⚙️ Class Definition

Class Name: AFCM

class AFCM:
    def __init__(self, c: int, lambda_: float = 1.0, m: float = 2.0, max_iter: int = 100, tol: float = 1e-5, n_neighbors: int = 5):
        self.c = c
        self.lambda_ = lambda_
        self.m = m
        self.max_iter = max_iter
        self.tol = tol
        self.n_neighbors = n_neighbors

This class implements the full AFCM algorithm with manifold embedding and adaptive fuzzy clustering.

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📋 Parameters

Parameter	Type	Default	Description
c	int	—	Number of clusters
lambda_	float	1.0	Regularization strength for embedding
m	float	2.0	Fuzziness degree
max_iter	int	100	Maximum number of iterations
tol	float	1e-5	Tolerance for convergence
n_neighbors	int	5	Number of neighbors used to build Laplacian graph

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🚀 Usage Examples

from soft_clustering._afcm._afcm import AFCM
import numpy as np

X = np.vstack([
    np.random.normal(loc=[0, 0], scale=0.5, size=(50, 2)),
    np.random.normal(loc=[4, 4], scale=0.5, size=(50, 2)),
])

model = AFCM(c=2, lambda_=1.0)
labels, U = model.fit_predict(X)

print("Cluster Labels:", labels)
print("Membership Matrix:", U)

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📥 Input / 📤 Output

• Input to fit_predict(X):

  • X (np.ndarray): Data matrix of shape (N x D)

• Returns:

  • labels (np.ndarray): Final cluster assignments

  • U (np.ndarray): Final fuzzy membership matrix

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🛠️ Methods

• __init__(...): Initializes the full AFCM model with graph embedding.

• fit_predict(X): Applies fuzzy clustering in embedded space based on graph Laplacian and membership matrix.

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📝 Implementation Notes

• The algorithm uses Laplacian + regularization term B to find a manifold-preserving embedding X̃.

• Embedding is updated at every iteration based on current membership matrix U.

• Suitable for datasets with complex or nonlinear structure.

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📚 Reference

This implementation is based on: “Adaptive Fuzzy C-Means with Graph Embedding”,
Zhang, Liu, Liu, and Tao.