We give an explicit comparison of eigenvalues and eigenvectors of XY Hamiltonians of an open linear spin-1/2 chain and a closed spin-1/2 ring with periodic in space coefficients. It is shown that the Hamiltonian of a k-periodic chain with nk − 1 sites has (n − 1)k multiplicity one eigenvalues which are eigenvalues of multiplicity two for a Hamiltonian of a k-periodic ring with 2kn sites. For th...

We show that in a deep neural network trained with ReLU, the low-lying layers should be replaceable with truncated linearly activated layers. We derive the gradient descent equations in this truncated linear model and demonstrate that –if the distribution of the training data is stationary during training– the optimal choice for weights in these low-lying layers is the eigenvectors of the covar...

We propose a robust approach to discriminant kernel-based feature extraction for face recognition and verification. We show, for the first time, how to perform the eigen analysis of the within-class scatter matrix directly in the feature space. This eigen analysis provides the eigenspectrum of its range space and the corresponding eigenvectors as well as the eigenvectors spanning its null space...

We consider N × N Hermitian random matrices with i.i.d. entries. The matrix is normalized so that the average spacing between consecutive eigenvalues is of order 1/N . We study the connection between eigenvalue statistics on microscopic energy scales η ≪ 1 and (de)localization properties of the eigenvectors. Under suitable assumptions on the distribution of the single matrix elements, we first ...

In this class, we introduced the random walk on graphs. The last lecture shows Perron-Frobenius theory to the analysis of primary eigenvectors which is the stationary distribution. In this lecture we will study the second eigenvector. To analyze the properties of the graph, we construct two matrices: one is (unnormalized) graph Laplacian and the other is normalized graph Laplacian. In the first...

In this paper, we consider the problem of approximating a set of arbitrary, discrete-time, Gaussian random processes by a single, representative wavelet-based, Gaussian process. We measure the similarity between the original processes and the wavelet-based process with the Bhattacharyya coefficient. By manipulating the Bhattacharyya coefficient, we reduce the task of defining the representative...

This paper presents a method for the extraction of articulatory parameters from direct processing of raw images of the lips. The system architecture is made of three independent parts. First, a new greyscale mouth image is centred and downsampled. Second, the image is aligned and projected onto a basis of artificial images. These images are the eigenvectors computed from a PCA applied on a set ...

We present new O(n3) algorithms that compute eigenvalues and eigenvectors to high relative accuracy in floating point arithmetic for the following types of matrices: symmetric Cauchy, symmetric diagonally scaled Cauchy, symmetric Vandermonde, and symmetric totally nonnegative matrices when they are given as products of nonnegative bidiagonal factors. The algorithms are divided into two stages: ...

This paper presents a new nonlinear approximate indexing method for highdimensional data such as multimedia data. The new indexing method is designed for approximate similarity searches and all the work is performed in the transformed Gaussian space. In this indexing method, we first map the input space into a feature space via the Gaussian mapping, and then compute the top eigenvectors in the ...

A new fast algorithm for calculating a few maximum (or minimum) eigenvalues-&nd the corresponding eigenvectors of large N x N Hermitian m&rices is presented. The method is based on a molecular dynamics algorithm for N coupled harmonic-oscillators. The time step for iteration is chosen so that only the normal mode with _ _. _ the maximum eigenvalue grows exponentially. Other eigenvalues and eige...