Normalized Cut (Ncut) Algorithm in Spectral Clustering

Spectral clustering is a graph theory-based clustering method, with the Normalized Cut (Ncut) algorithm being one of its classical implementations particularly suitable for scenarios like image segmentation. This algorithm transforms clustering problems into graph partitioning problems by constructing similarity matrices between data points, ultimately employing matrix decomposition techniques to identify optimal data divisions. The core concept of the Ncut algorithm treats datasets as weighted undirected graphs where nodes represent data points and edge weights indicate inter-point similarities. Unlike conventional Min-Cut approaches, Ncut's objective function incorporates normalization terms during partitioning that consider both edge weights and balance between subgraphs, effectively preventing isolated small clusters. Key implementation steps typically involve: - Constructing similarity matrices using proximity measurements between data points, commonly employing Gaussian kernel functions to calculate similarities - Computing degree matrices (diagonal matrices where elements represent sum of edge weights per node) and Laplacian matrices - Performing eigendecomposition on normalized Laplacian matrices and selecting top-k eigenvectors to form new feature spaces - Applying traditional clustering algorithms (e.g., K-means) in the new feature space for final segmentation Implementation Note: In code, the normalized Laplacian is often computed as L_norm = D^(-1/2) * L * D^(-1/2), where D is the degree matrix and L is the unnormalized Laplacian. For image segmentation tasks, the Ncut algorithm effectively captures global similarities between pixels or regions, making it particularly adept at handling images with complex textures or blurred boundaries. Its main advantage lies in not requiring assumptions about data distribution patterns, though high computational and storage demands for similarity matrices may limit large-scale applications. Technical Consideration: Practical implementations often use sparse matrix representations and randomized SVD methods to handle computational complexity for large datasets.