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Dimensionality reduction and volume minimization - generalization of the determinant minimization criterion for reduced rank regression problems
Linköpings universitet, Matematiska institutionen, Beräkningsvetenskap. Linköpings universitet, Tekniska högskolan.ORCID-id: 0000-0002-1542-2690
2006 (engelsk)Inngår i: Linear Algebra and its Applications, ISSN 0024-3795, Vol. 418, nr 1, s. 201-214Artikkel i tidsskrift (Fagfellevurdert) Published
Abstract [en]

In this article we propose a generalization of the determinant minimization criterion. The problem of minimizing the determinant of a matrix expression has implicit assumptions that the objective matrix is always nonsingular. In case of singular objective matrix the determinant would be zero and the minimization problem would be meaningless. To be able to handle all possible cases we generalize the determinant criterion to rank reduction and volume minimization of the objective matrix. The generalized minimization criterion is used to solve the following ordinary reduced rank regression problem:

minrank(X)=kdet(B-XA)(B-XA)T,

where A and B are known and X is to be determined. This problem is often encountered in the system identification context.

sted, utgiver, år, opplag, sider
2006. Vol. 418, nr 1, s. 201-214
Emneord [en]
Volume; Minimization criterion; Determinant; Rank deficient matrix
HSV kategori
Identifikatorer
URN: urn:nbn:se:liu:diva-13190DOI: 10.1016/j.laa.2006.01.032OAI: oai:DiVA.org:liu-13190DiVA, id: diva2:18006
Tilgjengelig fra: 2008-04-29 Laget: 2008-04-29 Sist oppdatert: 2013-11-06
Inngår i avhandling
1. Algorithms in data mining using matrix and tensor methods
Åpne denne publikasjonen i ny fane eller vindu >>Algorithms in data mining using matrix and tensor methods
2008 (engelsk)Doktoravhandling, med artikler (Annet vitenskapelig)
Abstract [en]

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 rank approximation of tensors.

The first two papers deal with the reduced rank regression problem, which is encountered in the field of state-space subspace system identification. More specifically the problem is

\[

\min_{\rank(X) = k} \det (B - X A)(B - X A)\tp,

\]

where $A$ and $B$ are given matrices and we want to find $X$ under a certain rank condition that minimizes the determinant. This problem is not properly stated since it involves implicit assumptions on $A$ and $B$ so that $(B - X A)(B - X A)\tp$ is never singular. This deficiency of the determinant criterion is fixed by generalizing the minimization criterion to rank reduction and volume minimization of the objective matrix. The volume of a matrix is defined as the product of its nonzero singular values. We give an algorithm that solves the generalized problem and identify properties of the input and output signals causing a singular objective matrix.

Classification problems occur in many applications. The task is to determine the label or class of an unknown object. The third paper concerns with classification of handwritten digits in the context of tensors or multidimensional data arrays. Tensor and multilinear algebra is an area that attracts more and more attention because of the multidimensional structure of the collected data in various applications. Two classification algorithms are given based on the higher order singular value decomposition (HOSVD). The main algorithm makes a data reduction using HOSVD of 98--99 \% prior the construction of the class models. The models are computed as a set of orthonormal bases spanning the dominant subspaces for the different classes. An unknown digit is expressed as a linear combination of the basis vectors. The resulting algorithm achieves 5\% in classification error with fairly low amount of computations.

The remaining two papers discuss computational methods for the best multilinear

rank approximation problem

\[

\min_{\cB} \| \cA - \cB\|

\]

where $\cA$ is a given tensor and we seek the best low multilinear rank approximation tensor $\cB$. This is a generalization of the best low rank matrix approximation problem. It is well known that for matrices the solution is given by truncating the singular values in the singular value decomposition (SVD) of the matrix. But for tensors in general the truncated HOSVD does not give an optimal approximation. For example, a third order tensor $\cB \in \RR^{I \x J \x K}$ with rank$(\cB) = (r_1,r_2,r_3)$ can be written as the product

\[

\cB = \tml{X,Y,Z}{\cC}, \qquad b_{ijk}=\sum_{\lambda,\mu,\nu}

x_{i\lambda} y_{j\mu} z_{k\nu} c_{\lambda\mu\nu},

\]

where $\cC \in \RR^{r_1 \x r_2 \x r_3}$ and $X \in \RR^{I \times r_1}$, $Y \in \RR^{J \times r_2}$, and $Z \in \RR^{K \times r_3}$ are matrices of full column rank. Since it is no restriction to assume that $X$, $Y$, and $Z$ have orthonormal columns and due to these constraints, the approximation problem can be considered as a nonlinear optimization problem defined on a product of Grassmann manifolds. We introduce novel techniques for multilinear algebraic manipulations enabling means for theoretical analysis and algorithmic implementation. These techniques are used to solve the approximation problem using Newton and Quasi-Newton methods specifically adapted to operate on products of Grassmann manifolds. The presented algorithms are suited for small, large and sparse problems and, when applied on difficult problems, they clearly outperform alternating least squares methods, which are standard in the field.

sted, utgiver, år, opplag, sider
Matematiska institutionen, 2008. s. 29
Serie
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1178
Emneord
Volume, Minimization criterion, Determinant, Rank deficient matrix, Reduced rank regression, System identification, Rank reduction, Volume minimization, General algorithm, Handwritten digit classification, Tensors, Higher order singular value decomposition, Tensor approximation, Least squares, Tucker model, Multilinear algebra, Notation, Contraction, Tensor matricization, Newton's method, Grassmann manifolds, Product manifolds, Quasi-Newton algorithms, BFGS and L-BFGS, Symmetric tensor approximation, Local/intrinsic coordinates, Global/embedded coordinates;
HSV kategori
Identifikatorer
urn:nbn:se:liu:diva-11597 (URN)978-91-7393-907-2 (ISBN)
Disputas
2008-05-27, Glashuset, B-huset, ing. 25, Campus Valla, Linköpings universitet, Linköping, 10:15 (engelsk)
Opponent
Veileder
Tilgjengelig fra: 2008-04-29 Laget: 2008-04-29 Sist oppdatert: 2013-10-11
2. Algorithms in data mining: reduced rank regression and classification by tensor methods
Åpne denne publikasjonen i ny fane eller vindu >>Algorithms in data mining: reduced rank regression and classification by tensor methods
2005 (engelsk)Licentiatavhandling, med artikler (Annet vitenskapelig)
Abstract [en]

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, which consists of three appended manuscripts, we discuss algorithms for reduced rank regression and for classification in the context of tensor theory.

The first two manuscripts deal with the reduced rank regression problem, which is encountered in the field of state-space subspace system identification. More specifically the problem is

where A and B are given matrices and we want to find X under a certain rank condition that minimizes the determinant. This problem is not properly stated since it involves implicit assumptions on A and B so that (B - XA)(B - XA)T is never singular. This deficiency of the determinant criterion is fixed by generalizing the minimization criterion to rank reduction and volume minimization of the objective matrix. The volume of a matrix is defined as the product of its nonzero singular values. We give an algorithm that solves the generalized problem and identify properties of the input and output signals causing singularity on the objective matrix.

Classification problems occur in many applications. The task is to determine the label or class of an unknown object. The third appended manuscript concerns with classification of hand written digits in the context of tensors or multidimensional data arrays. Tensor theory is also an area that attracts more and more attention because of the multidimensional structure of the collected data in a various applications. Two classification algorithms are given based on the higher order singular value decomposition (HOSVD). The main algorithm makes a data reduction using HOSVD of 98%- 99% prior the construction of the class models. The models are computed as a set of orthonormal bases spanning the dominant subspaces for the different classes. An unknown digit is expressed as a linear combination of the basis vectors. The amount of computations is fairly low and the performance reasonably good, 5% in error rate.

sted, utgiver, år, opplag, sider
Linköping: Linköpings universitet, 2005. s. 65
Serie
Linköping Studies in Science and Technology. Thesis, ISSN 0280-7971 ; 1214
HSV kategori
Identifikatorer
urn:nbn:se:liu:diva-30272 (URN)15789 (Lokal ID)91-85457-81-7 (ISBN)15789 (Arkivnummer)15789 (OAI)
Tilgjengelig fra: 2009-10-09 Laget: 2009-10-09 Sist oppdatert: 2013-11-06

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