Symmetric positive semi-definite (SPSD) matrix approximation methods have been extensively used to speed up large-scale eigenvalue computation and kernel learning methods. The standard sketch based method, which we call the prototype model, produces relatively accurate approximations, but is inefficient on large square matrices. The Nyström method is highly efficient, but can only achieve low a...

Fixed-point simulation results are used for the performance measure of inverting matrices by Cholesky decomposition. The fixed-point Cholesky decomposition algorithm is implemented using a fixed-point reconfigurable processing element. The reconfigurable processing element provides all mathematical operations required by Cholesky decomposition. The fixed-point word length analysis is based on s...

In this paper we discuss matrix decompositions in the symmetrized max-plus algebra. The max-plus algebra has maximization and addition as basic operations. In contrast to linear algebra many fundamental problems in the max-plus algebra still have to be solved. In this paper we discuss max-algebraic analogues of some basic matrix decompositions from linear algebra. We show that we can use algori...

Let complex matrices $A$ and $B$ have the same sizes. Using the singular value decomposition, we characterize the $g$-inverse $B^{(1)}$ of $B$ such that the distance between a given $g$-inverse of $A$ and the set of all $g$-inverses of the matrix $B$ reaches minimum under the unitarily invariant norm. With this result, we derive additive and multiplicative perturbation bounds of the nearest per...

We introduce a provably correct learning algorithm for latent-variable PCFGs. The algorithm relies on two steps: first, the use of a matrix-decomposition algorithm applied to a co-occurrence matrix estimated from the parse trees in a training sample; second, the use of EM applied to a convex objective derived from the training samples in combination with the output from the matrix decomposition...

We consider a certain decomposition of the matrix algebra Mn(F ), where F is a field. The commutation relations of that decomposition yield an n × n matrix Mn , which determines the multilinear polynomial identities of Mn(F ). Thus if char(F ) = 0, the matrix Mn ) determines the polynomial identities of Mn(F ). We show that M Mn(F ) is very close to the tensor product of two n × n Vandermonde m...

Given an undamped gyroscopic system GðλÞ 1⁄4 Mλ þ CλþK with M , K symmetric and C skew-symmetric, this paper presents a real-valued spectral decomposition of GðλÞ by a real standard pair ðX;TÞ and a skew-symmetric parameter matrix S . When T is assumed to be a block diagonal matrix, the parameter matrix S has a special structure. This spectral decomposition is applied to solve the quadratic inv...

In recent works, we have studied linear predictive models constrained by time-domain filters. In the present study, we will study the one-dimensional case in more detail. Firstly, we obtain root-exchange properties between the roots of an all-pole model and corresponding constraints. Secondly, using the root-exchange property we can construct a novel matrix decomposition A RA = I, where R is a ...

Recently, the low rank and sparse (LRS) matrix decomposition has been introduced as an effective mean to solve the multi-view registration. It views each available relative motion as a block element to reconstruct one matrix so as to approximate the low rank matrix, where global motions can be recovered for multi-view registration. However, this approach is sensitive to the sparsity of the reco...

We show how to cheaply estimate the Fr echet derivative and the condition number for a general class of matrix functions (the class includes the matrix sign function and functions that can be expressed as power series) via the Schur decomposition. In the case of the matrix sign function we also give a method to compute the Fr echet derivative exactly. We also show that often this general method...