The text of Laplace, Sur l’application du calcul des probabilités à la philosophie naturelle, (Théorie Analytique des Probabilités. Troisième Édition. Premier Supplément), 1820, is quoted in the context of the Gram-Schmidt algorithm. We provide an English translation of Laplace’s manuscript (originally in French) and interpret the algorithms of Laplace in a contemporary context. The two algorit...

This paper focuses on the performance analysis of a linear system solving based on Cholesky decomposition and QR factorization, implemented on 16bits fixed-point DSP-chip (TMS320C6474). The classical method of Cholesky decomposition has the advantage of low execution time. However, the modified Gram-Schmidt QR factorization performs better in term of robustness against the round-off error propa...

We derive coupled on-line learning rules for the singular value decomposition (SVD) of a cross-covariance matrix. In coupled SVD rules, the singular value is estimated alongside the singular vectors, and the effective learning rates for the singular vector rules are influenced by the singular value estimates. In addition, we use a first-order approximation of Gram-Schmidt orthonormalization as ...

In this paper we show how to compute recursively an approximation of the left and right dominant singular subspaces of a given matrix. In order to perform as few as possible operations on each column of the matrix, we use a variant of the classical Gram–Schmidt algorithm to estimate this subspace. The method is shown to be particularly suited for matrices with many more rows than columns. Bound...

In this paper we consider the design of spectrally efficient time-limited pulses for ultra-wideband (UWB) systems using an overlapping pulse position modulation scheme. For this we investigate an orthogonalization method, which was developed in 1950 by Löwdin [1, 2]. Our objective is to obtain a set of N orthogonal (Löwdin) pulses, which remain time-limited and spectrally efficient for UWB syst...

A novel significant vector (SV) regression algorithm is proposed in this paper based on an analysis of Chen's orthogonal least squares (OLS) regression algorithm. The proposed regularized SV algorithm finds the significant vectors in a successive greedy process in which, compared to the classical OLS algorithm, the orthogonalization has been removed from the algorithm. The performance of the pr...

Abstract This paper proposes a cost-effective and variable-channel floating-point fast independent component analysis (FastICA) hardware architecture and implementation for EEG signal processing. The Gram-Schmidt orthonormalization based whitening process is utilized to eliminate the use of the dedicated hardware for eigenvalue decomposition (EVD) in the FastICA algorithm. The proposed two proc...