Conceptual and empirical comparison of dimensionality reduction algorithms (PCA, KPCA, LDA, MDS, SVD, LLE, ISOMAP, LE, ICA, t-SNE)

The present communication relates to the theory of the solution, in positive and negative integral numbers, of systems of linear indeterminate equations, having integral coefficients. In connexion with this theory, a solution is also given of certain problems relating to rectangular matrices, composed of integral numbers, which are of frequent use in the higher arithmetic. Of this kind are the two following:—1. 1. \"Given (in integral numbers) the values of the determinants of any rectangular matrix of given dimensions, to find all the matrices, the constituents of which are integers, and the determinants of which have those given values.

http://rspl.royalsocietypublishing.org/content/11/86.full.pdf

I. On systems of linear indeterminate equations and congruences

https://sci2s.ugr.es/keel/pdf/specific/articulo/chen_information_2008.pdf

An information granulation based data mining approach for classifying imbalanced data

Blended or simultaneous sources have been the focus of a good deal of interest recently. In conventional acquisition the time intervals between successive sources are large enough to avoid interference in time. In blended acquisition, temporal overlap between source responses is allowed. The procedure of retrieving data as if they were acquired in the conventional way is called deblending. This is an essential step if standard processing flows are to be applied. Several inversion techniques have been proposed for solving this ill‐posed problem. We study the properties of an iterative estimation and subtraction algorithm that integrates a coherency‐pass filter in a dedicated iteration. We begin by stating the problem we wish to solve and develop a new, more general, interpretation of the method. We then apply an algebraic analysis of the iteration to establish the convergence characteristics of the algorithm. In order to facilitate this analysis, the notion of leakage subspace is introduced, i.e., a subspace where energy that cannot be uniquely assigned to one of the sources resides. We find that a unique solution exists, if, and only if, there is no leakage subspace. If a unique solution does not exist, then the iteration converges to a least‐norm solution contaminated by the projection of the initial guess onto the leakage subspace. The insights gained by this analysis lead us to the development of a simple tool that can provide valuable information during the design of a blended survey. Finally we present results from the application of this method to real blended marine data and then draw our conclusions.

Convergence analysis of a coherency‐constrained inversion for the separation of blended data

https://hal.archives-ouvertes.fr/hal-00476223/document

On the bit-complexity of sparse polynomial and series multiplication

https://msvlab.hre.ntou.edu.tw/grades/now/inte/App%20SVD.pdf

Applications of singular-value decomposition (SVD)

The deterministic recursive pivot-free algorithms for computing the generalized Bruhat decomposition of the matrix in the field and for the computation of the inverse matrix are presented. This method has the same complexity as algorithm of matrix multiplication, and it is suitable for the parallel computer systems.

Fast generalized bruhat decomposition

Given an m×n matrix A, with m ≥ n, the four subspaces associated with it are shown in Fig. 1 (see [1]). Fig. 1. The row spaces and the nullspaces of A and A T ; a 1 through a n and h 1 through h m are abbreviations of the alignerframe and hangerframe vectors respectively (see [2]). The Fundamental Theorem of Linear Algebra tells us that N (A) is the orthogonal complement of R(A T). These four subspaces tell the whole story of the Linear System Ax = y. So, for example, the absence of N (A T) indicates that a solution always exists, whereas the absence of N (A) indicates that this solution 123 is unique. Given the importance of these subspaces, computing bases for them is the gist of Linear Algebra. In \" Classical \" Linear Algebra, bases for these subspaces are computed using Gaussian Elimination; they are orthonormalized with the help of the Gram-Schmidt method. Continuing our previous work [3] and following Uhl's excellent approach [2] we use SVD analysis to compute orthonormal bases for the four subspaces associated with A, and give a 3D explanation. We then state and prove what we call the \" SVD-Fundamental Theorem \" of Linear Algebra, and apply it in solving systems of linear equations.

/pdf/the-svd-fundamental-theorem-of-linear-algebra-4aahjinwyl.pdf

The SVD-Fundamental Theorem of Linear Algebra

Effective matrix methods for solving standard linear algebra problems in a commutative domains are discussed. Two of them are new. There are a methods for computing adjoined matrices and solving system of linear equations in a commutative domains.

Effective Matrix Methods in Commutative Domains

In 1917 Pell (1) and Gordon used sylvester2, Sylvester’s little known and hardly ever used matrix of 1853, to compute(2) the coefficients of a Sturmian remainder — obtained in applying in Q[x], Sturm’s algorithm on two polynomials f, g ∈ Z[x] of degree n — in terms of the determinants (3) of the corresponding submatrices of sylvester2. Thus, they solved a problem that had eluded both J. J. Sylvester, in 1853, and E. B. Van Vleck, in 1900. (4) In this paper we extend the work by Pell and Gordon and show how to compute (2) the coefficients of an Euclidean remainder — obtained in finding in Q[x], the greatest common divisor of f, g ∈ Z[x] of degree n — in terms of the determinants (5) of the corresponding submatrices of sylvester1, Sylvester’s widely known and used matrix of 1840. (1) See the link http://en.wikipedia.org/wiki/Anna_Johnson_Pell_Wheeler for her biography (2) Both for complete and incomplete sequences, as defined in the sequel. (3) Also known as modified subresultants. (4) Using determinants Sylvester and Van Vleck were able to compute the coefficients of Sturmian remainders only for the case of complete sequences. (5) Also known as (proper) subresultants.

/pdf/on-the-remainders-obtained-in-finding-the-greatest-common-jlddsjg65y.pdf

Gennadi Malaschonok

Papers

Applications of singular-value decomposition (SVD)

Fast generalized bruhat decomposition

The SVD-Fundamental Theorem of Linear Algebra

Effective Matrix Methods in Commutative Domains

On the Remainders Obtained in Finding the Greatest Common Divisor of Two Polynomials