Spectral clustering on arvau

Wikipedia / Spectral clustering

In multivariate statistics, spectral clustering techniques make use of the spectrum (eigenvalues) of the similarity matrix of the data to perform dimensionality reduction before clustering in fewer dimensions. The similarity matrix is provided as an input and consists of a quantitative assessment of the relative similarity of each pair of points in the dataset.

In application to image segmentation, spectral clustering is known as segmentation-based object categorization.

Definitions

Given an enumerated set of data points, the similarity matrix may be defined as a symmetric matrix. The general approach to spectral clustering is to use a standard clustering method (there are many such methods, k-means is discussed below) on relevant eigenvectors of a Laplacian matrix. There are many different ways to define a Laplacian which have different mathematical interpretations, and so the clustering will also have different interpretations. The eigenvectors that are relevant are the ones that correspond to several smallest eigenvalues of the Laplacian except for the smallest eigenvalue which will have a value of 0. For computational efficiency, these eigenvectors are often computed as the eigenvectors corresponding to the largest several eigenvalues of a function of the Laplacian.

Laplacian matrixLaplacian matrix

Spectral clustering is well known to relate to partitioning of a mass-spring system, where each mass is associated with a data point and each spring stiffness corresponds to a weight of an edge describing a similarity of the two related data points, as in the spring system. Specifically, the classical reference explains that the eigenvalue problem describing transversal vibration modes of a mass-spring system is exactly the same as the eigenvalue problem for the graph Laplacian matrix defined as

is the diagonal matrix

and A is the adjacency matrix.

The masses that are tightly connected by the springs in the mass-spring system evidently move together from the equilibrium position in low-frequency vibration modes, so that the components of the eigenvectors corresponding to the smallest eigenvalues of the graph Laplacian can be used for meaningful clustering of the masses. For example, assuming that all the springs and the masses are identical in the 2-dimensional spring system pictured, one would intuitively expect that the loosest connected masses on the right-hand side of the system would move with the largest amplitude and in the opposite direction to the rest of the masses when the system is shaken — and this expectation will be confirmed by analyzing components of the eigenvectors of the graph Laplacian corresponding to the smallest eigenvalues, i.e., the smallest vibration frequencies.

Laplacian_matrix#Laplacian_matrix_normalization_2Laplacian matrix normalization

The goal of normalization is making the diagonal entries of the Laplacian matrix to be all unit, also scaling off-diagonal entries correspondingly. In a weighted graph, a vertex may have a large degree because of a small number of connected edges but with large weights just as well as due to a large number of connected edges with unit weights.

A popular normalized spectral clustering technique is the normalized cuts algorithm or Shi–Malik algorithm introduced by Jianbo Shi and Jitendra Malik, commonly used for image segmentation based on the eigenvector corresponding to the second-smallest eigenvalue of the symmetric normalized Laplacian defined as

is also the eigenvector corresponding to the second-largest eigenvalue of the symmetrically normalized adjacency matrix

The random walk (or left) normalized Laplacian is defined as

and can also be used for spectral clustering. A mathematically equivalent algorithm takes the eigenvector corresponding to the largest eigenvalue of the random walk normalized adjacency.

Cluster analysisCluster analysis via Spectral Embedding

-dimensional vector space components, which may be done in various ways.

, called the Fiedler vector, thus bi-partitioning the graph and labeling the data points with two labels. This sign-based approach follows the intuitive explanation of spectral clustering via the mass-spring model — in the low frequency vibration mode that the Fiedler vector represents, one cluster data points identified with mutually strongly connected masses would move together in one direction, while in the complement cluster data points identified with remaining masses would move together in the opposite direction. The algorithm can be used for hierarchical clustering by repeatedly partitioning the subsets in the same fashion.

, any vector clustering technique can be used, e.g., DBSCAN.

Algorithms

Basic Algorithm
(or the normalized Laplacian)
)
Cluster the graph nodes based on these features (e.g., using k-means clustering)

has not already been explicitly constructed, the efficiency of spectral clustering may be improved if the solution to the corresponding eigenvalue problem is performed in a matrix-free fashion (without explicitly manipulating or even computing the similarity matrix), as in the Lanczos algorithm.

For large-sized graphs, the second eigenvalue of the (normalized) graph Laplacian matrix is often ill-conditioned, leading to slow convergence of iterative eigenvalue solvers. Preconditioning is a key technology accelerating the convergence, e.g., in the matrix-free LOBPCG method. Spectral clustering has been successfully applied on large graphs by first identifying their community structure, and then clustering communities.

Spectral clustering is closely related to nonlinear dimensionality reduction, and dimension reduction techniques such as locally-linear embedding can be used to reduce errors from noise or outliers.

Costs

, it is important to estimate the memory footprint vector of the labels in memory.

The need to construct the graph Laplacian is common for all distance- or correlation-based clustering methods. Computing the eigenvectors is specific to spectral clustering only.

Constructing graph Laplacian

graph adjacency matrix.

Moreover, a normalized Laplacian has exactly the same eigenvectors as the normalized adjacency matrix, but with the order of the eigenvalues reversed. Thus, instead of computing the eigenvectors corresponding to the smallest eigenvalues of the normalized Laplacian, one can equivalently compute the eigenvectors corresponding to the largest eigenvalues of the normalized adjacency matrix, without even talking about the Laplacian matrix.

Naive constructions of the graph adjacency matrix entries of the matrix. Nystrom method can be used to approximate the similarity matrix, but the approximate matrix is not elementwise positive, i.e. cannot be interpreted as a distance-based similarity.

Algorithms to construct the graph adjacency matrix as a sparse matrix are typically based on a nearest neighbor search non-zero entries, and the calculations can be trivially run in parallel.

Computing eigenvectors

as determined by the Fiedler vector.

data points. Moreover, matrix-free eigenvalue solvers such as LOBPCG can efficiently run in parallel, e.g., on multiple GPUs with distributed memory, resulting not only in high quality clusters, which spectral clustering is famous for, but also top performance.

Software

Free software implementing spectral clustering is available in large open source projects like scikit-learn using LOBPCG with multigrid preconditioning or ARPACK, MLlib for pseudo-eigenvector clustering using the power iteration method, and R.

Relationship with other clustering methods

The ideas behind spectral clustering may not be immediately obvious. It may be useful to highlight relationships with other methods. In particular, it can be described in the context of kernel clustering methods, which reveals several similarities with other approaches.

Relationship with k-means

Spectral clustering is closely related to the k-means algorithm, especially in how cluster assignments are ultimately made. Although the two methods differ fundamentally in their initial formulations—spectral clustering being graph-based and k-means being centroid-based—the connection becomes clear when spectral clustering is viewed through the lens of kernel methods.

In particular, weighted kernel k-means provides a key theoretical bridge between the two. Kernel k-means is a generalization of the standard k-means algorithm, where data is implicitly mapped into a high-dimensional feature space through a kernel function, and clustering is performed in that space. Spectral clustering, especially the normalized versions, performs a similar operation by mapping the input data (or graph nodes) to a lower-dimensional space defined by the eigenvectors of the graph Laplacian. These eigenvectors correspond to the solution of a relaxation of the normalized cut or other graph partitioning objectives.

Mathematically, the objective function minimized by spectral clustering can be shown to be equivalent to the objective function of weighted kernel k-means in this transformed space. This was formally established in works such as where they demonstrated that normalized cuts are equivalent to a weighted version of kernel k-means applied to the rows of the normalized Laplacian’s eigenvector matrix.

Because of this equivalence, spectral clustering can be viewed as performing kernel k-means in the eigenspace defined by the graph Laplacian. This theoretical insight has practical implications: the final clustering step in spectral clustering typically involves running the standard k-means algorithm on the rows of the matrix formed by the first k eigenvectors of the Laplacian. These rows can be thought of as embedding each data point or node in a low-dimensional space where the clusters are more well-separated and hence, easier for k-means to detect.

Additionally, multi-level methods have been developed to directly optimize this shared objective function. These methods work by iteratively coarsening the graph to reduce problem size, solving the problem on a coarse graph, and then refining the solution on successively finer graphs. This leads to more efficient optimization for large-scale problems, while still capturing the global structure preserved by the spectral embedding.

Relationship to DBSCAN

Spectral clustering is also conceptually related to DBSCAN (Density-Based Spatial Clustering of Applications with Noise), particularly in the special case where the spectral method is used to identify connected graph components of a graph. In this trivial case—where the goal is to identify subsets of nodes with no interconnecting edges between them—the spectral method effectively reduces to a connectivity-based clustering approach, much like DBSCAN.

DBSCAN operates by identifying density-connected regions in the input space: points that are reachable from one another via a sequence of neighboring points within a specified radius (ε), and containing a minimum number of points (minPts). The algorithm excels at discovering clusters of arbitrary shape and separating out noise without needing to specify the number of clusters in advance.

In spectral clustering, when the similarity graph is constructed using a hard connectivity criterion (i.e., binary adjacency based on whether two nodes are within a threshold distance), and no normalization is applied to the Laplacian, the resulting eigenstructure of the graph Laplacian directly reveals disconnected components of the graph. This mirrors DBSCAN's ability to isolate density-connected components. The zeroth eigenvectors of the unnormalized Laplacian correspond to these components, with one eigenvector per connected region.

This connection is most apparent when spectral clustering is used not to optimize a soft partition (like minimizing the normalized cut), but to identify exact connected components—which corresponds to the most extreme form of “density-based” clustering, where only directly or transitively connected nodes are grouped together. Therefore, spectral clustering in this regime behaves like a spectral version of DBSCAN, especially in sparse graphs or when constructing ε-neighborhood graphs.

While DBSCAN operates directly in the data space using density estimates, spectral clustering transforms the data into an eigenspace where global structure and connectivity are emphasized. Both methods are non-parametric in spirit, and neither assumes convex cluster shapes, which further supports their conceptual alignment.

Measures to compare clusterings

Ravi Kannan, Santosh Vempala and Adrian Vetta proposed a bicriteria measure to define the quality of a given clustering. They said that a clustering was an (α, ε)-clustering if the conductance of each cluster (in the clustering) was at least α and the weight of the inter-cluster edges was at most ε fraction of the total weight of all the edges in the graph. They also look at two approximation algorithms in the same paper.

History and related literatures

Spectral clustering has a long history. Spectral clustering as a machine learning method was popularized by Shi & Malik and Ng, Jordan, & Weiss.

Ideas and network measures related to spectral clustering also play an important role in a number of applications apparently different from clustering problems. For instance, networks with stronger spectral partitions take longer to converge in opinion-updating models used in sociology and economics.