We propose a method based on Cholesky decomposition for Non-negative Matrix Factorization (NMF). NMF enables to learn local representation due to its non-negative constraint. However, when utilizing NMF as a representation leaning method, the issues due to the non-orthogonality of the learned representation has not been dealt with. Since NMF learns both feature vectors and data vectors in the f...

Recent approaches to independent component analysis have used kernel independence measures to obtain very good performance in ICA, particularly in areas where classical methods experience difficulty (for instance, sources with near-zero kurtosis). In this chapter, we compare two efficient extensions of these methods for large-scale problems: random subsampling of entries in the Gram matrices us...

It is well-known that analysis of incomplete Cholesky and LU decompositions with a general dropping is very difficult and of limited applicability, see, for example, the results on modified decompositions [1], [2], [3] and later results based on similar concepts. This is true not only for the dropping based on magnitude of entries but it also applies to algorithms that use a prescribed sparsity...

In image processing, textures are generally represented as homogeneous random fields, homogeneous meaning stationary or second-order stationary. This paper presents a generalization of the second-order stationarity to the second-order invariance under a group of transforms. Some examples of interesting groups are given. The Cholesky factorization is applied for the synthesis of random fields sh...

We present the first order error bound for the Lyapunov equation AX +XA∗ = −GG∗, where A is perturbed to A+ δA. We use the structure of the solution of the Lyapunov equation X = m ∑ k=1 WkW ∗ k , where Wk is the k-th matrix obtained by the Low Rank Cholesky Factor ADI (LRCF-ADI) algorithm using the set of ADI parameters equal to exact eigenvalues of A, that is with ADI parameters {p1, . . . , p...

Efficient execution of numerical algorithms requires adapting the code to the underlying execution platform. In this paper we show the process of fine tuning our sparse Hypermatrix Cholesky factorization in order to exploit efficiently two important machine resources: processor and memory. Using the techniques we presented in previous papers we tune our code on a different platform. Then, we ex...

The sparse Cholesky factorization of some large matrices can require a two dimensional partitioning of the matrix. The sparse hypermatrix storage scheme produces a recursive 2D partitioning of a sparse matrix. The subblocks are stored as dense matrices so BLAS3 routines can be used. However, since we are dealing with sparse matrices some zeros may be stored in those dense blocks. The overhead i...

We have derived exact Langevin equations for a model of quasispecies dynamics. The inherent multiplicative reaction noise is complex and its statistical properties are specified completely. The numerical simulation of the complex Langevin equations is carried out using the Cholesky decomposition for the noise covariance matrix. This internal noise, which is due to diffusion-limited reactions, p...

The transmission matrix, introduced by Friedland in 1957, can be used to characterize a linear, time invariant system having an emprically-determined impulse response. The Wiener-Kalman filter can be determined by Cholesky factorization of a covariance matrix formed from the transmission matrix. An analogous result is given for linear, quadratic control. The method is illustrated by several exa...

A simple method to compute the minimum distance Dmin in multiuser code division multiple access (CDMA) systems based on the Cholesky decomposition of the positive-definite symmetric correlation matrix is proposed. Although finding Dmin is known to be NP-hard, this decomposition allows the computation of the minimum distance and the asymptotic efficiency of optimum multiuser detection to be perf...