Metric learning has been shown to be highly effective to improve the performance of nearest neighbor classification. In this paper, we address the problem of metric learning for symmetric positive definite (SPD) matrices such as covariance matrices, which arise in many real-world applications. Naively using standard Mahalanobis metric learning methods under the Euclidean geometry for SPD matric...

Motivated by applications in microwave engineering and diffusion tensor imaging, we study the problem of deconvolution density estimation on the space of positive definite symmetric matrices. We develop a nonparametric estimator for the density function of a random sample of positive definite matrices. Our estimator is based on the Helgason-Fourier transform and its inversion, the natural tools...

Usage of the Sherman-Morrison-Woodbury formula to update linear systems after low rank modifications of the system matrix is widespread in machine learning. However, it is well known that this formula can lead to serious instabilities in the presence of roundoff error. If the system matrix is symmetric positive definite, it is almost always possible to use a representation based on the Cholesky...

in this paper a linear fuzzy fredholm integral equation(ffie) with arbitrary fuzzy function input and symmetric triangular (fuzzy interval) output is considered. for each variable, output is the nearest triangular fuzzy number (fuzzy interval) to the exact fuzzy solution of (ffie).

We give a constructive proof of a theorem of Marshall and Olkin that any real symmetric positive definite matrix can be symmetrically scaled by a positive diagonal matrix to have arbitrary positive row sums. We give a slight extension of the result, showing that given a sign pattern, there is a unique diagonal scaling with that sign pattern, and we give upper and lower bounds on the entries of ...

Let A be a connected integer symmetric matrix, i.e., A = (aij) ∈ Mn(Z) for some n, A = AT , and the underlying graph (vertices corresponding to rows, with vertex i joined to vertex j if aij 6= 0) is connected. We show that if all the eigenvalues of A are strictly positive, then tr(A) ≥ 2n− 1. There are two striking corollaries. First, the analogue of the Schur-SiegelSmyth trace problem is solve...

STRONG RANK REVEALING CHOLESKY FACTORIZATION M. GU AND L. MIRANIAN y Abstract. For any symmetric positive definite n nmatrixAwe introduce a definition of strong rank revealing Cholesky (RRCh) factorization similar to the notion of strong rank revealing QR factorization developed in the joint work of Gu and Eisenstat. There are certain key properties attached to strong RRCh factorization, the im...

In this paper, amodulus-based generalized skew-Hermitian triangular splitting (MGSTS) iteration method is present for solving a class of linear complementarity problems with the system matrix either being an H+-matrix with non-positive off-diagonal entries or a symmetric positive definite matrix. The convergence of the MGSTS iterationmethod is studied in detail. By choosing different parameters...

In this paper a linear Fuzzy Fredholm Integral Equation(FFIE) with arbitrary Fuzzy Function input and symmetric triangular (Fuzzy Interval) output is considered. For each variable, output is the nearest triangular fuzzy number (fuzzy interval) to the exact fuzzy solution of (FFIE).