Glossario IA
Il dizionario completo dell'Intelligenza Artificiale
Spectral Clustering
Unsupervised clustering technique that uses the eigenvalues of a similarity matrix to perform dimensionality reduction before clustering. This method is particularly effective for identifying non-convex structures in complex data.
Similarity Matrix
Symmetric square matrix where each element S[i,j] represents the degree of similarity between data points i and j. It forms the foundation of spectral clustering by encoding the local relationships between observations.
Laplacian Matrix
Discrete differential operator defined as L = D - W, where D is the degree matrix and W is the graph's weight matrix. This matrix captures the geometric structure of the data and its eigenvectors reveal the natural clusters.
Similarity Graph
Graphical representation where nodes correspond to data points and edges weigh their similarity. This structure allows applying theories from graph theory to discover natural groupings in the data.
Spectral K-means
Combination of spectral clustering with the K-means algorithm applied in the eigenvector space. This approach takes advantage of the dimensionality reduction of spectral clustering and the simplicity of K-means implementation.
Eigenspace
Vector subspace spanned by the eigenvectors associated with a given eigenvalue. In spectral clustering, the eigenspace of the smallest non-zero eigenvalues forms a new representation of the data that facilitates clustering.
Adjacency Matrix
Square matrix where A[i,j] = 1 if an edge connects vertices i and j in the graph, otherwise 0. It represents the connectivity structure of the similarity graph used in spectral clustering.
Degree Matrix
Diagonal matrix where each diagonal element D[i,i] represents the sum of the weights of the edges incident to vertex i. This matrix is essential for constructing the Laplacian matrix and normalizing the similarity graph.
Spectral Normalization
Technique for normalizing the Laplacian matrix to stabilize numerical calculations and improve clustering quality. Symmetric normalization L_sym = I - D^(-1/2)WD^(-1/2) is commonly used.
Spectral Segmentation
Application of spectral clustering to partition images or volumes into coherent regions. This method exploits spectral similarities between pixels to detect natural boundaries in visual data.
Clustering Coefficient
Measure quantifying the local density of connections around a node in the graph. This coefficient influences the construction of the similarity matrix and can affect the quality of clusters identified by spectral clustering.
K-Nearest Neighbors Graph
Type of similarity graph where each node is connected to its k nearest neighbors according to a given metric. This approach reduces graph density while preserving the local structure essential for clustering.
Gaussian Kernel
Similarity function K(x,y) = exp(-||x-y||²/2σ²) used to calculate weights in the similarity matrix. The σ parameter controls the similarity scale and influences graph connectivity.
Normalized Laplacian Matrix
Variant of the normalized Laplacian matrix to obtain eigenvalues in the range [0,2]. The symmetric form L_rw = I - D^(-1)W is particularly suitable for clustering data of variable sizes.
Eigenvalue Decomposition
Mathematical process of finding eigenvalues and eigenvectors of a matrix. In spectral clustering, this crucial step transforms the clustering problem into a partitioning problem in a reduced-dimension space.
Graph Partitioning
Fundamental problem of dividing graph vertices into disjoint sets while minimizing connections between sets. Spectral clustering solves this problem using eigenvectors of the Laplacian matrix.