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Algorithms in data mining using matrix and tensor methods

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Data mining is the process of finding and extracting valuable knowledge or in-formation from a given and often large set of data.

In many fields of science, engineering, and economics large amounts of data are stored and there is a need to analyze these data in order to extract information for various purposes. Data mining is a general concept involving different tools for performing this kind of analysis. The development of mathematical models and efficient algorithms is of key importance. In this thesis we discuss algorithms for the reduced rank regression problem and algorithms for the computation of the best multilinear…

Contents

1 Introduction and overview
1 Linear systems of equations and linear regression models
2 The determinant minimization criterion
3 Generalization to rank reduction and volume minimization
4 Application to system identication
5 Tensors and numerical multilinear algebra
5.1 Introduction to tensors .
5.2 Basic operations, tensor properties and notation
5.3 Matrix-tensor multiplication
5.4 Canonical tensor matricization
5.5 Contracted products and multilinear algebraic manipulations
6 Tensor rank and low rank tensor approximation
6.1 Application of truncated higher order SVD to handwritten digit classification
6.2 Best low rank tensor approximation
6.3 Optimization on a product of Grassmann manifolds
6.4 The Grassmann gradient and the the Grassmann Hessian
6.5 Newton-Grassmann and quasi-Newton-Grassmann algorithms
7 Future research directions
7.1 Multilinear systems of equations
7.2 Convergence of alternating least squares methods
7.3 Computations with large and sparse tensors
7.4 Attempts for the global minimum
7.5 Other multilinear models
2 Summary of papers
References
Appended manuscripts
I Dimensionality reduction and volume minimization general-ization of the determinant minimization criterion for reduced
rank regression problems
II Rank reduction and volume minimization approach to state-space subspace system identification
III Handwritten digit classification using higher order singular value decomposition
IV A Newton-Grassmann method for computing the best multi-linear rank-(r1, r2, r3) approximation of a tensor
V Best multilinear rank approximation of tensors with quasi-Newton methods on Grassmannians

Author: Savas, Berkant

Source: Linköping University

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