New Holder - Type Inequalities for the Tracy-Singh and Khatri-Rao Products of Positive Matrices

Recently, the authors establised a number of inequalities involving Khatri-Rao product of two positives matrices. Here, in this paper, the result are establised in three ways. First, we find new Holder-type inequalities for Tracy-Singh and Khatri-Rao products products of positive semi-devinite matrices. Second, the result are extended to provide estimates of sums of the Khatri-Rao and Tracy-Singh products of any finite number of positive semi-definite matrices. Three, the result lead to inequalities involving the Hadamard and Kronecker, as a special case

Wideband DOA Estimation via Sparse Bayesian Learning over a Khatri-Rao Dictionary

This paper deals with the wideband direction-of-arrival (DOA) estimation by exploiting the multiple measurement vectors (MMV) based sparse Bayesian learning (SBL) framework. First, the array covariance matrices at different frequency bins are focused to the reference frequency by the conventional focusing technique and then transformed into the vector form. Then a matrix called the Khatri-Rao dictionary is constructed by using the Khatri-Rao product and the multiple focused array covariance vectors are set as the new observations. DOA estimation is to find the sparsest representations of the new observations over the Khatri-Rao dictionary via SBL. The performance of the proposed method is compared with other well-known focusing based wideband algorithms and the Cramer-Rao lower bound (CRLB). The results show that it achieves higher resolution and accuracy and can reach the CRLB under relative demanding conditions. Moreover, the method imposes no restriction on the pattern of signal power spectral density and due to the increased number of rows of the dictionary, it can resolve more sources than sensors

Matrix inequalities involving the Khatri-Rao product

summary:We extend three inequalities involving the Hadamard product in three ways. First, the results are extended to any partitioned blocks Hermitian matrices. Second, the Hadamard product is replaced by the Khatri-Rao product. Third, the necessary and sufficient conditions under which equalities occur are presented. Thereby, we generalize two inequalities involving the Khatri–Rao product

Khatri-Rao sums for Hilbert space operators

We generalize the notions of Khatri-Rao sums for matrices and tensor sums for Hilbert space operators to KhatriRao sums for Hilbert space operators. This kind of operator sum is compatible with algebraic operations and order relations. We investigate its analytic properties, including continuity, convergence, and norm bounds. We also discuss the role of selection operator that relates Khatri-Rao sums to Tracy-Singh sums and Khatri-Rao products. Binomial theorem involving Khatri-Rao sums and its consequences are then established

Sparsity Order Estimation from a Single Compressed Observation Vector

We investigate the problem of estimating the unknown degree of sparsity from compressive measurements without the need to carry out a sparse recovery step. While the sparsity order can be directly inferred from the effective rank of the observation matrix in the multiple snapshot case, this appears to be impossible in the more challenging single snapshot case. We show that specially designed measurement matrices allow to rearrange the measurement vector into a matrix such that its effective rank coincides with the effective sparsity order. In fact, we prove that matrices which are composed of a Khatri-Rao product of smaller matrices generate measurements that allow to infer the sparsity order. Moreover, if some samples are used more than once, one of the matrices needs to be Vandermonde. These structural constraints reduce the degrees of freedom in choosing the measurement matrix which may incur in a degradation in the achievable coherence. We thus also address suitable choices of the measurement matrices. In particular, we analyze Khatri-Rao and Vandermonde matrices in terms of their coherence and provide a new design for Vandermonde matrices that achieves a low coherence

Fast Exact Leverage Score Sampling from Khatri-Rao Products with Applications to Tensor Decomposition

We present a data structure to randomly sample rows from the Khatri-Rao product of several matrices according to the exact distribution of its leverage scores. Our proposed sampler draws each row in time logarithmic in the height of the Khatri-Rao product and quadratic in its column count, with persistent space overhead at most the size of the input matrices. As a result, it tractably draws samples even when the matrices forming the Khatri-Rao product have tens of millions of rows each. When used to sketch the linear least squares problems arising in CANDECOMP / PARAFAC tensor decomposition, our method achieves lower asymptotic complexity per solve than recent state-of-the-art methods. Experiments on billion-scale sparse tensors validate our claims, with our algorithm achieving higher accuracy than competing methods as the decomposition rank grows.Comment: To appear at the 37th Conference on Neural Information Processing Systems (Neurips'23). 28 pages, 10 figures, 6 table

Generalizations and Some Applications of Kronecker and Hadamard Products of Matrices

In this thesis, generalizations of Kronecker, Hadamard and usual products (sums) that depend on the partitioned of matrices are studied and defined. Namely: Tracy- Singh, Khatri-Rao, box, strong Kronecker, block Kronecker, block Hadamard, restricted Khatri-Rao products (sums) which are extended the meaning of Kronecker, Hadamard and usual products (sums). The matrix convolution products, namely: matrix convolution, Kronecker convolution and Hadamard convolution products of matrices with entries in set of functions are also considered. The connections among them are derived and most useful properties are studied in order to find new applications of Tracy-Singh and Khatri-Rao products (sums). These applications are: a family of generalized inverses, a family of coupled singular matrix problems, a family of matrix inequalities and a family of geometric means. In the theory of generalized inverses of matrices and their applications, the five generalized inverses, namely Moore-Penrose, weighted Moore-Penrose, Drazin, weighted Drazin and group inverses and their expressions and properties are studied. Moreover, some new numerous matrix expressions involving these generalized inverses and weighted matrix norms of the Tracy-Singh products matrices are also derived. In addition, we establish some necessary and sufficient conditions for the reverse order law of Drazin and weighted Drazin inverses. These results play a central role in our applications and many other applications. In the field of system identification and matrix products work, we propose several algorithms for computing the solutions of the coupled matrix differential equations, coupled matrix convolution differential, coupled matrix equations, restricted coupled singular matrix equations, coupled matrix least-squares problems and weighted Least -squares problems based on idea of Kronecker (Hadamard) and Tracy-Singh(Khatri-Rao) products (sums) of matrices. The way exists which transform the coupled matrix problems and coupled matrix differential equations into forms for which solutions may be readily computed. The common vector exact solutions of these coupled are presented and, subsequently, construct a computationally - efficient solution of coupled matrix linear least-squares problems and nonhomogeneous coupled matrix differential equations. We give new applications for the representations of weighted Drazin, Drazin and Moore-Penrose inverses of Kronecker products to the solutions of restricted singular matrix and coupled matrix equations. The analysis indicates that the Kronecker (Hadamard) structure method can achieve good efficient while the Hadamard structure method achieve more efficient when the unknown matrices are diagonal. Several special cases of these systems are also considered and solved, and then we prove the existence and uniqueness of the solution of each case, which includes the well-known coupled Sylvester matrix equations. We show also that the solutions of non-homogeneous matrix differential equations can be written in convolution forms. The analysis indicates also that the algorithms can be easily to find the common exact solutions to the coupled matrix and matrix differential equations for partitioned matrices by using the connections between Tracy-Singh, Block Kronecker and Khatri -Rao products and partitioned vector row (column) and our definition which is the so-called partitioned diagonal extraction operators. Unlike Matrix algebra, which is based on matrices, analysis must deal with estimates. In other words, Inequalities lie at the core of analysis. For this reason, it’s of great importance to give bounds and inequalities involving matrices. In this situation, the results are organized in the following five ways: First, we find some extensions and generalizations of the inequalities involving Khatri-Rao products of positive (semi) definite matrices. We turn to results relating Khatri-Rao and Tracy- Singh powers and usual powers, extending and generalizing work of previous authors. Second, we derive some new attractive inequalities involving Khatri-Rao products of positive (semi) definite matrices. We remark that some known inequalities and many other new interesting inequalities can easily be found by using our approaches. Third, we study some sufficient and necessary conditions under which inequalities below become equalities. Fourth, some counter examples are considered to show that some inequalities do not hold in general case. Fifth, we find Hölder-type inequalities for Tracy-Singh and Khatri-Rao products of positive (semi) definite matrices. The results lead to inequalities involving Hadamard and Kronecker products, as a special case, which includes the well-known inequalities involving Hadamard product of matrices, for instance, Kantorovich-type inequalities and generalization of Styan's inequality. We utilize the commutativity of the Hadamard product (sum) for possible to develop and improve some interesting inequalities which do not follow simply from the work of researchers, for example, Visick's inequality. Finally, a family of geometric means for positive two definite matrices is studied; we discuss possible definitions of the geometric means of positive definite matrices. We study the geometric means of two positive definite matrices to arrive the definitions of the weighted operator means of positive definite matrices. By means of several examples, we show that there is no known definition which is completely satisfactory. We have succeeded to find many new desirable properties and connections for geometric means related to Tracy-Singh products in order to obtain new unusual estimates for the Khatri-Rao (Tracy-Singh) products of several positive definite matrices