The Classic Ncut Algorithm in Spectral Clustering

Spectral clustering is a graph theory-based clustering method whose core idea is to treat data points as vertices in a graph and achieve clustering through graph partitioning. Among these methods, the Normalized Cut (Ncut) algorithm stands out as one of the most representative spectral clustering algorithms. The Ncut algorithm primarily involves the following key steps: First, constructing a similarity matrix between data points, which essentially serves as the graph's adjacency matrix. Next, computing the degree matrix of the graph - a diagonal matrix where diagonal elements represent the degree of each vertex. Then, through the construction of the Laplacian matrix, the clustering problem is transformed into a graph partitioning problem. During the solution process, the algorithm requires eigenvalue decomposition. Specifically, it performs eigenvalue decomposition on the normalized Laplacian matrix and selects the eigenvectors corresponding to the k smallest eigenvalues. These eigenvectors form a new feature space where traditional clustering methods (such as k-means) are applied for final clustering. MATLAB, as a widely-used tool in scientific computing, is particularly suitable for implementing such matrix operation-intensive algorithms. The Ncut algorithm can be efficiently implemented in MATLAB by leveraging its powerful matrix computation capabilities and built-in eigenvalue decomposition functions like `eig()` or `eigs()`. This implementation demonstrates excellent performance with medium-scale datasets, balancing computational accuracy with runtime efficiency. A distinctive feature of the Ncut algorithm is its consideration of the graph's global structure information. By minimizing the ratio between inter-cluster connection weights and intra-cluster connection weights, it produces more balanced clustering results. This gives it significant advantages over traditional clustering algorithms when handling non-convex distributed data, particularly through its normalized cut criterion that prevents biased partitioning.