Ng–Jordan–Weiss (NJW) method is one of the most widely used spectral clustering algorithms. For a K clustering problem, this method partitions data using the largest K eigenvectors of the normalized affinity matrix derived from the dataset. It has been demonstrated that the spectral relaxation solution of K-way grouping is located on the subspace of the largest K eigenvectors. However, we find ...

We first show that the eigenvector of a tensor is well-defined. The differences between the eigenvectors of a tensor and its E-eigenvectors are the eigenvectors on the nonsingular projective variety S = {x ∈ P | n

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]. assume that n images in a set are originally represented in mat...

be an orthonormal basis of H consisting of eigenvectors of A where {ek : k ∈ N} is a set of eigenvectors of A corresponding to nonzero eigenvalues of A and {fm : m ∈ I} is a set of eigenvectors of A corresponding to the zero eigenvalue (we know that we may assume that I is a countable set by exercise 6.10). We make the assumption that the elements of {ek : k ∈ N} are ordered so that the first e...

The search for a canonical set of eigenvectors of the discrete Fourier transform has been ongoing for more than three decades. The goal is to find an orthogonal basis of eigenvectors which would approximate Hermite functions – the eigenfunctions of the continuous Fourier transform. This eigenbasis should also have some degree of analytical tractability and should allow for efficient numerical c...

In coupled learning rules for principal component analysis, eigenvectors and eigenvalues are simultaneously estimated in a coupled system of equations. Coupled single-neuron rules have favorable convergence properties. For the estimation of multiple eigenvectors, orthonormalization methods have to be applied, either full Gram-Schmidt orthonormalization, its first-order approximation as used in ...

Usually, principal components analysis is carried out by calculating the eigenvalues and eigenvectors of the correlation matrix. With N cases and P variables, if we write X for the N × P matrix which has been standardised so that columns have zero mean and unit standard deviation, we find the eigenvalues and eigenvectors of the P × P matrix X.X (which is N or (N − 1) times the correlation matri...