40 research outputs found
Quaternion Singular Value Decomposition based on Bidiagonalization to a Real Matrix using Quaternion Householder Transformations
We present a practical and efficient means to compute the singular value
decomposition (svd) of a quaternion matrix A based on bidiagonalization of A to
a real bidiagonal matrix B using quaternionic Householder transformations.
Computation of the svd of B using an existing subroutine library such as lapack
provides the singular values of A. The singular vectors of A are obtained
trivially from the product of the Householder transformations and the real
singular vectors of B. We show in the paper that left and right quaternionic
Householder transformations are different because of the noncommutative
multiplication of quaternions and we present formulae for computing the
Householder vector and matrix in each case
A New Double Color Image Watermarking Algorithm Based on the SVD and Arnold Scrambling
We propose a new image watermarking scheme based on the real SVD and Arnold scrambling to embed a color watermarking image into a color host image. Before embedding watermark, the color watermark image W with size of M×M is scrambled by Arnold transformation to obtain a meaningless image W~. Then, the color host image A with size of N×N is divided into nonoverlapping N/M×N/M pixel blocks. In each (i,j) pixel block Ai,j, we form a real matrix Ci,j with the red, green, and blue components of Ai,j and perform the SVD of Ci,j. We then replace the three smallest singular values of Ci,j by the red, green, and blue values of W~ij with scaling factor, to form a new watermarked host image A~ij. With the reserve procedure, we can extract the watermark from the watermarked host image. In the process of the algorithm, we only need to perform real number algebra operations, which have very low computational complexity and are more effective than the one using the quaternion SVD of color image
A structure-preserving one-sided Jacobi method for computing the SVD of a quaternion matrix
Abstract(#br)In this paper, we propose a structure-preserving one-sided cyclic Jacobi method for computing the singular value decomposition of a quaternion matrix. In our method, the columns of the quaternion matrix are orthogonalized in pairs by using a sequence of orthogonal JRS-symplectic Jacobi matrices to its real counterpart. We establish the quadratic convergence of our method specially. We also give some numerical examples to illustrate the effectiveness of the proposed method
Non-normal Recurrent Neural Network (nnRNN): learning long time dependencies while improving expressivity with transient dynamics
A recent strategy to circumvent the exploding and vanishing gradient problem
in RNNs, and to allow the stable propagation of signals over long time scales,
is to constrain recurrent connectivity matrices to be orthogonal or unitary.
This ensures eigenvalues with unit norm and thus stable dynamics and training.
However this comes at the cost of reduced expressivity due to the limited
variety of orthogonal transformations. We propose a novel connectivity
structure based on the Schur decomposition and a splitting of the Schur form
into normal and non-normal parts. This allows to parametrize matrices with
unit-norm eigenspectra without orthogonality constraints on eigenbases. The
resulting architecture ensures access to a larger space of spectrally
constrained matrices, of which orthogonal matrices are a subset. This crucial
difference retains the stability advantages and training speed of orthogonal
RNNs while enhancing expressivity, especially on tasks that require
computations over ongoing input sequences