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Article
ImplicitQR factorization of a product of three matrices
We present a numerical algorithm for computing the implicit QR factorization of a product of three matrices, and we illustrate the technique by applying it to the generalized total least squares and the restri...
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Article
The product-product singular value decomposition of matrix triplets
A new decomposition of a matrix triplet (A, B, C) corresponding to the singular value decomposition of the matrix productABC is developed in this paper, which will be termed theProduct-Product Singular Value Deco...
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Article
An efficient Total Least Squares algorithm based on a rank-revealing two-sided orthogonal decomposition
Solving Total Least Squares (TLS) problemsAX≈B requires the computation of the noise subspace of the data matrix [A;B]. The widely used tool for doing this is the Singular Value Decomposition (SVD). However, the ...
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Article
A look-ahead algorithm for the solution of general Hankel systems
The solution of systems of linear equations with Hankel coefficient matrices can be computed with onlyO(n 2) arithmetic operations, as compared toO(n 3) operations for the general cases. However, the classical Ha...
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Chapter and Conference Paper
The Canonical Correlations of Matrix Pairs and their Numerical Computation
This paper is concerned with the analysis of canonical correlations of matrix pairs and their numerical computation. We first develop a decomposition theorem for matrix pairs having the same number of rows whi...
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Article
A note on constructing a symmetric matrix with specified diagonal entries and eigenvalues
A finite step algorithm is given such that for any two vectorsa, λ ∈R n witha majorized by λ, it computes a symmetric matrixH ∈R n x n with ...
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Article
Computing the generalized singular values/vectors of large sparse or structured matrix pairs
We present a numerical algorithm for computing a few extreme generalized singular values and corresponding vectors of a sparse or structured matrix pair \(\{A,B\}\) . The algori...
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Article
An algorithm and a stability theory for downdating the ULV decomposition
An alternative to performing the singular value decomposition is to factor a matrixA into $$A = U\left( {\begin{array}{*{20}c} C \\...
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Article
Computing the optimal commuting matrix pairs
A matrix can be modified by an additive perturbation so that it commutes with any given matrix. In this paper, we discuss several algorithms for computing the smallest perturbation in the Frobenius norm for a ...
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Chapter and Conference Paper
Large-scale SVD and subspace-based methods for information retrieval
A theoretical foundation for latent semantic indexing (LSI) is proposed by adapting a model first used in array signal processing to the context of information retrieval using the concept of subspaces. It is show...
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Article
Modifying the generalized singular value decomposition with application with application in direction-of-arrival finding
We consider updating and downdating problems for the generalized singular value decomposition (GSVD) of matrix pairs when new rows are added to one of the matrices or old rows are deleted. Two classes of algor...
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Chapter and Conference Paper
Unsupervised Learning: Self-aggregation in Scaled Principal Component Space*
We demonstrate that data clustering amounts to a dynamic process of self-aggregation in which data objects move towards each other to form clusters, revealing the inherent pattern of similarity. Selfaggregation i...
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Chapter and Conference Paper
Extracting Shared Topics of Multiple Documents
In this paper, we present a weighted graph based method to simultaneously compare the textual content of two or more documents and extract the shared (sub)topics of them, if available. A set of documents are m...
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Chapter and Conference Paper
Nonlinear Dimension Reduction via Local Tangent Space Alignment
In this paper we present a new algorithm for manifold learning and nonlinear dimension reduction. Based on a set of unorganized data points sampled with noise from the manifold, we represent the local geometry...
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Chapter and Conference Paper
Spectral Clustering for Robust Motion Segmentation
In this paper, we propose a robust motion segmentation method using the techniques of matrix factorization and subspace separation. We first show that the shape interaction matrix can be derived using QR decompos...
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Article
Linear low-rank approximation and nonlinear dimensionality reduction
We present our recent work on both linear and nonlinear data reduction methods and algorithms: for the linear case we discuss results on structure analysis of SVD of columnpartitioned matrices and sparse low-r...
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Article
Open AccessTowards Inferring Protein Interactions: Challenges and Solutions
Discovering interacting proteins has been an essential part of functional genomics. However, existing experimental techniques only uncover a small portion of any interactome. Furthermore, these data often have...
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Chapter and Conference Paper
Optimizing Surface Triangulation Via Near Isometry with Reference Meshes
Optimization of the mesh quality of surface triangulation is critical for advanced numerical simulations and is challenging under the constraints of error minimization and density control. We derive a new meth...
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Chapter and Conference Paper
IKNN: Informative K-Nearest Neighbor Pattern Classification
The K-nearest neighbor (KNN) decision rule has been a ubiquitous classification tool with good scalability. Past experience has shown that the optimal choice of K depends upon the data, making it laborious to tun...
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Article
Analysis of an alignment algorithm for nonlinear dimensionality reduction
The goal of dimensionality reduction or manifold learning for a given set of high-dimensional data points, is to find a low-dimensional parametrization for them. Usually it is easy to carry out this parametriz...
