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Topic: Matrix factorization

	PlanetMath: matrix inverse
		Some caveats: computing the matrix inverse for ill-conditioned matrices is error-prone; special care must be taken and there are sometimes special algorithms to calculate the inverse of certain classes of matrices (for example, Hilbert matrices).
		The need to find the matrix inverse depends on the situation- whether done by hand or by computer, and whether the matrix is simply a part of some equation or expression or not.
		However, the matrix inverse may exist in the case of the elements being members of a commutative ring, provided that the determinant of the matrix is a unit in the ring.
	planetmath.org /encyclopedia/MatrixInverse.html (465 words)

	Fast and effective algorithms for graph partitioning and sparse-matrix ordering
		MD-based heuristics are local greedy heuristics that reorder the columns of a symmetric sparse matrix such that the column with the fewest nonzeros at a given stage of factorization is the next one to be eliminated at that stage.
		Therefore, during numerical factorization, the columns corresponding to the separator nodes are eliminated after all columns corresponding to the nodes of the two subgraphs have been eliminated.
		Many implementations of sparse factorization switch to dense-matrix factorization at this stage because the cost of additional arithmetic operations is more than compensated for by the advantages of dense factorization, such as the absence of indirect addressing and better utilization of memory hierarchy.
	www.research.ibm.com /journal/rd/411/gupta.html (5952 words)

	PlanetMath: matrix factorization
		See the entries for these and other matrix factorizations for details on the contents of the factor matrices, where to apply them, and how to best calculate them.
		A related problem is to diagonalize or tridiagonalize many matrices using the same matrix.
		This is version 6 of matrix factorization, born on 2004-03-12, modified 2005-08-05.
	www.planetmath.org /encyclopedia/MatrixFactorization.html (206 words)

	PlanetMath: matrix factorization
		See the entries for these and other matrix factorizations for details on the contents of the factor matrices, where to apply them, and how to best calculate them.
		A related problem is to diagonalize or tridiagonalize many matrices using the same matrix.
		This is version 6 of matrix factorization, born on 2004-03-12, modified 2005-08-05.
	planetmath.org /encyclopedia/FactorMatrix.html (210 words)

	Matrix Decompositions
		The LU decomposition of a matrix is frequently used as part of a Gaussian elimination process for solving a matrix equation.
		This factorization has a number of uses, one of which is that, because it is a triangular factorization, it can be used to solve systems of equations involving symmetric positive definite matrices.
		This matrix cannot be diagonalized because the matrix of eigenvectors is singular.
	documents.wolfram.com /v5/Built-inFunctions/AdvancedDocumentation/LinearAlgebra/4.5.html (2005 words)

	DSSKYF(3DXML)
		The inertia of the symmetric matrix A is the triplet of integers (ipeigen, ineigen, izeigen), consisting of the number of positive, negative and zero eigenvalues, respectively.
		In addition to the transp(U)DU factorization, you can also obtain the determinant of A, the number of positive and negative eigenvalues of the matrix A and an indication of the existence of zero eigenvalues.
		If the factorization process is stopped at row i due to a small pivot, then the inertia, determinant and statistics on the matrix are evaluated for rows ibeg + 1 through (i-1).
	www.uwm.edu /cgi-bin/IMT/wwwman?topic=dsskyf(3dxml)&msection=3 (1485 words)

	Chapter 3: Matrix Factorization
		Factorization is used to solve large sparse matrix A.
		Use only one matrix A. After the factorization, matrix A is destroyed and replaced with L and U. The diagonal of matrix L is not stored.
		This factorization is applicable for sparse matrices with half-storage.
	www.geocities.com /sparsematrix.geo/pflow/bch03.html (1030 words)

	BioMed Central \| Full text \| Biclustering of gene expression data by non-smooth non-negative matrix factorization
		Matrix factorization at k = 3 was able to correctly partition the set of genes and conditions belonging to each one of the embedded patterns.
		When the matrix was sorted by the values of the first basis experiment and the corresponding basis gene the set of genes and conditions belonging to Pla were grouped together in upper-left corner of the array.
		The interpretation of the new factorization is then two fold: data can be faithful reconstructed using additive combinations of a reduced set of factors and, at the same time, interpretation of the factors is easier due to the intuitive sparse, non-overlapped part-based representation of the data.
	www.biomedcentral.com /1471-2105/7/78 (8763 words)

	The use of Positive Matrix Factorization in the analysis of molecular line spectra from the Thumbprint Nebula
		Positive matrix factorization (PMF) analyzes the matrix containing the measured spectra by calculating a small number of basic spectral profiles and the weights with which each of these basic components or factors is present in each individual spectrum.
		Factor analysis (FA) and principal component analysis (PCA) are older methods, which are in principle also capable of doing this, when applied to a matrix of spectra.
		A clear change in velocity is seen from the maps of the weights of the three first factors and in the western part of the cloud the strongest component has a well-defined intensity peak.
	www.mrao.cam.ac.uk /yerac/juvela/juvela.html (1548 words)

	BioMed Central \| Full text \| LS-NMF: A modified non-negative matrix factorization algorithm utilizing uncertainty ...
		For microarray data, the matrix D provides the estimates of transcriptional levels, such that each column corresponds to the estimate for a single condition, with each matrix element in a column corresponding to the estimate for a single gene (or probe set) in that condition.
		Matrix P is then of size K × J, with each row representing the relative mRNA levels of a metagene across the conditions.
		For figures 3a and 3b, the cell cycle coregulation groups identified previously from known transcriptional response and transcription factor binding analyses in the yeast cell cycle are used [28].
	www.biomedcentral.com /1471-2105/7/175 (5706 words)

	BioMed Central \| Full text \| bioNMF: a versatile tool for non-negative matrix factorization in biology
		Each factor (column) in the matrix W corresponds to a semantic feature (described as weighted sum of terms) while each column in H corresponds to the new representation for a gene as a linear combination of semantic factors (gene semantic profile).
		Matrix S, on the other hand, is denoted as smoothing matrix and its task is to demand sparseness in both W and H.
		Once the factorization has been completed, results can be explored using a graphical user interface (see Figure 3), allowing the selection of interesting patterns by removing those genes and samples with very low values in their factor and encoding vector.
	www.biomedcentral.com /1471-2105/7/366 (5085 words)

	Linear Equations
		The factorization for a general tridiagonal matrix is like that for a general band matrix with kl = 1 and ku = 1.
		The factorization for a symmetric positive definite band matrix with k superdiagonals (or subdiagonals) has the same form as for a symmetric positive definite matrix, but the factor U (or L) is a band matrix with k superdiagonals (subdiagonals).
		While the primary use of a matrix factorization is to solve a system of equations, other related tasks are provided as well.
	www.netlib.org /lapack/lug/node38.html (756 words)

	Matrix Factorization using Distributed Panels on the Fujitsu AP1000 (Site not responding. Last check: 2007-10-17)
		Most studies on matrix factorization and similar algorithms have assumed that the block-partition size or panel width for the algorithm, w, to be the same as the matrix distribution block size, r, where a rectangular block-cyclic matrix distribution is being employed.
		Here the choice of w=r is essentially determined by the off-processor memory level of the memory hierarchy, with the value of w being a tradeoff between communication startup overhead and load balance considerations.
		In this paper, we re-examine this assumption in the context of LU and Cholesky factorization of block-cyclic distributed matrices on scalar-based distributed memory multiprocessors, such as the Fujitsu AP1000.
	cs.anu.edu.au /~Peter.Strazdins/papers/MatFactDistBlock.html (349 words)

	Guide and Reference
		The array specified for AP should not be altered between calls to the factorization and solve subroutines; otherwise unpredictable results may occur.
		On output, all elements of this matrix A are 1.0.
		Example 3: This example shows a factorization of the same positive definite symmetric matrix A of order 9 used in Example 1, but stored in lower storage mode.
	www.cs.unb.ca /docs/spdocs/essl/essl330.html (1482 words)

	Optimizing Matrix Factorizations for Real-Time Applications (Site not responding. Last check: 2007-10-17)
		In a typical case, the QR factorization is performed on M samples of the clutter and jamming taken across the N channels.
		The samples could be used to explicitly form a covariance matrix, but moving to the power domain worsens the condition of the matrix, and hence many prefer to perform the QR factorization on the measured (voltage domain) signals.
		All but one of the factorized columns are kicked out of cache, then blocks of the remaining columns are brought into cache and orthgonalized against the single factorized column until all remaining columns are processed.
	www.dtc.army.mil /hpcw/1999/lindermn (3055 words)

	Tim Davis: technical reports
		Timothy A. Davis A new method for sparse LU factorization is presented that combines a left-looking column pre-ordering strategy with a right-looking unsymmetric-pattern multifrontal numerical factorization.
		The sparse inverse subset problem is the computation of the entries of the inverse of a sparse matrix for which the corresponding entry is nonzero in the factors of the matrix.
		Sparse matrix factorization algorithms for general problems are typically characterized by irregular memory access patterns that limit their performance on parallel-vector supercomputers.
	www.cise.ufl.edu /~davis/techreports.html (5674 words)

	Engineering and Scientific Subroutine Library for AIX Version 3 Release 3: Guide and Reference - SPPF, DPPF, SPOF, ... (Site not responding. Last check: 2007-10-17)
		Matrix A is factored using Cholesky factorization as
		The array specified for AP should not be altered between calls to the factorization and solve subroutines; otherwise unpredictable results may occur.
		On output, all elements of this matrix A are 1.0.
	webdocs.caspur.it /ibm/essl/essl146.html (1734 words)

	Machine Learning at the University of Toronto (Site not responding. Last check: 2007-10-17)
		Matrices that can be factored into a product of two simpler matrices can serve as a useful and often natural model in the analysis of tabulated or high-dimensional data.
		The most common use of matrix factorization is Principal Component Analysis, in which one seeks a rank-constrained factorization minimizing the sum-squared difference to fully observed data.
		In this talk, I will discuss several issues related to learning with matrix factorization: the optimization problem of finding factorizations (eg for weighted sum-squared loss, or other loss measures); the asymptotic biases in estimating a low-rank subspace with a general noise model; formulations appropriate for collaborative filtering and generalization error bounds for those formulations.
	learning.cs.toronto.edu /abstracts/nati04.html (140 words)

	Video summarization via similarity matrix factorization
		The singular values are used to determine the effective rank of the matrix, which corresponds to the number of significant segment clusters in the source video.
		We use the singular vectors to decompose the similarity matrix into a matrix sum of rank-one terms, each of which correspond to one of the segment clusters.
		This linear matrix decomposition is designed to build a purely additive, or parts-based matrix sum, approximating the original similarity matrix.
	www.fxpal.com /?p=vidSumNMF (406 words)

	Abstracts (Site not responding. Last check: 2007-10-17)
		In earlier work, the kernel matrix was learned by maximizing the variance in feature space while preserving the distances and angles between nearest neighbors.
		Noting that the kernel matrix implicitly maps the data into a nonlinear feature space, we show how to discover a mapping that unfolds the underlying manifold from which the data was sampled.
		The kernel matrix is constructed by maximizing the variance in feature space subject to local constraints that preserve the angles and distances between nearest neighbors.
	www.cis.upenn.edu /~lsaul/abstracts.html (3164 words)

	Guide and Reference (Site not responding. Last check: 2007-10-17)
		The output from these factorization subroutines should be used only as input to the solve subroutines SGTS and DGTS, respectively.
		It is used to record column interchanges in the tridiagonal matrix due to partial pivoting.
		This factorization can then be used by SGTS or DGTS, respectively, to solve tridiagonal systems of linear equations.
	www.pdc.kth.se /doc/SP/manuals/essl/html/essl347.html (632 words)

	Engineering and Scientific Subroutine Library for AIX Version 3 Release 3: Guide and Reference - SPBF, DPBF, SPBCHF, ...
		For a description of how a positive definite symmetric band matrix is stored in lower-band-packed storage mode in an array, see Positive Definite Symmetric Band Matrix.
		Matrix A is not positive definite (for SPBF and DPBF).
		Matrix A is not positive definite (for SPBCHF and DPBCHF).
	www.nersc.gov /vendor_docs/ibm/essl/essl161.html (778 words)

	CSAIL Event Calendar
		Matrices that can be factored into a product of two simpler matrices
		formulation, and two novel analysis approaches to matrix factorization
		entries in a partially observed matrix of user preferences).
	www.csail.mit.edu /events/eventcalendar/calendar.php?show=event&id=150 (221 words)

	Linear Equations (Site not responding. Last check: 2007-10-17)
		The factorization for a general tridiagonal matrix is like that for a general band matrix with
		While the primary use of a matrix factorization is to solve a system of equations, other related tasks are provided as well.
		factorize (obviously not needed for triangular matrices);
	www.ma.utexas.edu /documentation/lapack/node40.html (654 words)

	[No title] (Site not responding. Last check: 2007-10-17)
		The theory of such highly structured matrix equations as the Lyapunov and Riccati equations, which are widely used in applications, represents an advanced part of matrix analysis.
		Mostly attention is paid to the definitions and properties of invariant subspaces of regular linear matrix pencils, which are a straightforward generalization of invariant subspaces of matrices.
		A brief survey of the theory of linear matrix equations is followed by a summary of the main results on the Lyapunov equations.
	www.elsevier.com /homepage/saj/523281/h03.htm (396 words)

	Distributed BLAS and Matrix factorization Research
		Further research by Dr Strazdins on advanced methods for matrix factorizations has continued, with new techniques to reduce communication costs for algorithmic blocking, and to optimize memory access patterns, having been developed.
		For this platform, a storage block size of 4 and an algorithmic blocking factor of 32 was found to be optimal for double precision.
		It should be noted that the number of nodes on the ANU machine is too small to enable algorithmic blocking to achieve its load balance advantage that it has on say a 64 node machine.
	cap.anu.edu.au /cap/reports/report99/node36.html (342 words)

	Engineering and Scientific Subroutine Library for AIX Version 3 Release 3: Guide and Reference - SPTF and ...
		The output from these factorization subroutines should be used only as input to the solve subroutines SPTS and DPTS, respectively.
		The factorization of positive definite symmetric tridiagonal matrix A is computed using Gaussian elimination.
		This factorization can then be used by SPTS or DPTS, respectively, to solve the tridiagonal systems of linear equations.
	www.nersc.gov /vendor_docs/ibm/essl/essl168.html (465 words)

	A Sequential Factorization Method for Recovering Shape and Motion from Image Streams (Site not responding. Last check: 2007-10-17)
		We present a sequential factorization method for recovering the three-dimensional shape of an object and the motion of the camera from a sequence of images, using tracked features.
		A covariance-like matrix is stored instead of feature positions, and its size remains constant as the number of frames increases.
		First, the measurement matrix is factorized into two matrices of rank 3 using the singular value decomposition.
	www.cs.cmu.edu /afs/cs/usr/tk/www/Projects_www/IUW94morita.fm.html (4030 words)

	4.4 Calculating the Determinant, Inverse, and Condition Number (Site not responding. Last check: 2007-10-17)
		Factorizations are useful for many calculations besides solutions to systems of equations.
		Matrix inverses are often used in describing matrix algorithms.
		The condition number of a matrix describes how sensitive it is to errors in the right-side vector when solving systems of equations.
	www.roguewave.com /support/docs/hppdocs/lpkug/4-4.html (286 words)

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