This paper presents a diffusion based probabilistic interpretation of spectral clustering and dimensionality reduction algorithms that use the eigenvectors of the normalized graph Laplacian. Given the pairwise adjacency matrix of all points, we define a diffusion distance between any two data points and show that the low dimensional representation of the data by the first few eigenvectors of th...

 = αI +βT, where T is defined by the preceding formula. This matrix arises in many applications, such as n coupled harmonic oscillators and solving the Laplace equation numerically. Clearly M and T have the same eigenvectors and their respective eigenvalues are related by μ = α+βλ . Thus, to understand M it is sufficient to work with the simpler matrix T . Eigenvalues and Eigenvectors of T Usu...

in this paper, the rayleigh's quotient and the inverse vector iteration method are presented. the latter approach helps to obtain the natural frequencies and mode shapes of a structure. inverse vector iteration method with shifting enables to determine the higher modes. some basic theorems of linear algebra are presented and extended to study the free vibration of structures. the variation...

The principle of dimensionality reduction with PCA is the representation of the dataset ‘X’in terms of eigenvectors ei ∈ RN  of its covariance matrix. The eigenvectors oriented in the direction with the maximum variance of X in RN carry the most      relevant information of X. These eigenvectors are called principal components [8]. Ass...

In spectral clustering, one defines a similarity matrix for a collection of data points, transforms the matrix to get the Laplacian matrix, finds the eigenvectors of the Laplacian matrix, and obtains a partition of the data using the leading eigenvectors. The last step is sometimes referred to as rounding, where one needs to decide how many leading eigenvectors to use, to determine the number o...

The principle of dimensionality reduction with PCA is the representation of the dataset ‘X’in terms of eigenvectors ei ∈ RN  of its covariance matrix. The eigenvectors oriented in the direction with the maximum variance of X in RN carry the most      relevant information of X. These eigenvectors are called principal components [8]. Ass...

We will see that the number of eigenvalues is n for an n× n matrix. Regarding eigenvectors, if x is an eigenvector then so is ax for any scalar a. However, if we consider only one eigenvector for each ax family, then there is a 1-1 correspondence of such eigenvectors to eigenvalues. Typically, we consider eigenvectors of unit length. Diagonal matrices are simple, the eigenvalues are the entries...

Given a set of N such shape samples {s, . . . , s}, a parametric statistical subspace of the object’s shape variance can be retrieved by first applying Generalised Procrustes Analysis on the shapes to normalise them with respect to the global similarity transform (i.e., scale, in-plane rotation and translation) and then using Principal Component Analysis (PCA). The returned shape subspace is fu...

The large systems of complex linear equations that are generated in QCD problems often have multiple right-hand sides (for multiple sources) and multiple shifts (for multiple masses). Deflated GMRES methods have previously been developed for solving multiple right-hand sides. Eigenvectors are generated during solution of the first right-hand side and used to speed up convergence for the other r...