Situations often arise in which the matrix of independent variables is not of full column rank. That is, there are one or more linear dependencies among the independent variables. This paper covers in detail the situation in which the rank is one less than full column rank and extends this coverage to include cases of even greater rank deficiency. The emphasis is on the row geometry of the solu...

Extreme learning machine (ELM) is one of the most popular and important learning algorithms. It comes from single-hidden layer feedforward neural networks. It has been proved that ELM can achieve better performance than support vector machine(SVM) in regression and classification. In this paper, mathematically, with regression problem, the step 3 of ELM is studied. First of all, the equation Hβ...

An n l (l n) space time block code (STBC) C over a complex signal set S consists of a finite number of n l matrices with elements from S. For quasi-static, flat fading channels a primary performance index of C is the minimum of the rank of the difference of any two matrices, called the rank of the code. C is of full rank if its rank is n and is of minimum delay if l = n. The rate R, in bits per...

In 2003 B. Kirchheim-D. Preiss constructed a Lipschitz map in the plane with 5 incompatible gradients, where incompatibility refers to the condition that no two of the five matrices are rank-one connected. The construction is via the method of convex integration and relies on a detailed understanding of the rank-one geometry resulting from a specific set of five matrices. The full computation o...

The performance of reduced-rank linear filtering is studied for the suppression of multiple-access interference. A reduced-rank filter resides in a lower dimensional space, relative to the full-rank filter, which enables faster convergence and tracking. We evaluate the large system output signal-to-interference plus noise ratio (SINR) as a function of filter rank for the multistage Wiener filte...

In this paper we present an efficient algorithm for computing a sparse null space basis for a full row rank matrix. We first apply the ideas of the Markowitz’s pivot selection criterion to a rank reducing algorithm to propose an efficient algorithm for computing sparse null space bases of full row rank matrices. We then describe how we can use the Dulmage-Mendelsohn decomposition to make the re...

Relations between halfand full-lattice CMV operators with scalarand matrix-valued Verblunsky coefficients are investigated. In particular, the decoupling of full-lattice CMV operators into a direct sum of two half-lattice CMV operators by a perturbation of minimal rank is studied. Contrary to the Jacobi case, decoupling a full-lattice CMV matrix by changing one of the Verblunsky coefficients re...