Inverse Problems of Matrix Data Reconstruction
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2010-03-26
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Abstract
Mathematical modeling is an indispensable task in almost every discipline of sciences. If a
model for a specific phenomenon can be correctly established, then it empowers the practitioners
to analyze, predict, and delegate an onward decision which may have important applications and consequences. However, since most of the information gathering devices or methods, including our best intellectual endeavor for understanding, have only ï¬ nite bandwidth, we cannot avoid the fact that the models employed often are not exact. A constant update or modification of an existing model based on the newest information therefore is in demand. Such a task is generally referred to as an inverse problem. While in a forward problem the concern usually is to express the behavior of a certain physical system in terms of its system parameters, in the inverse problem the concern is to express the parameters in term of the behavior. This thesis addresses a small portion of the mass domain of inverse problems. The specific focus has been on matrix data reconstruction subject to some intrinsic or prescribed constraints. The purpose of this investigation is to develop theoretic understanding and numerical algorithms for model reconstruction so that the inexactness and uncertainty are reduced while certain specific conditions are satisfied. Explained and illustrated in this thesis are some most
frequently used methodologies of matrix data reconstruction so that for a given dataset, these
techniques can be employed to construct or update various (known) structural properties, or to
classify or purify certain (unknown) embedded characteristics. Areas of applications include,
for example, the applied mechanics where systems of bodies move in response to the values of
their known endogenous parameters and the medical or social sciences where the causes (variables) of the observed incidences neither are known a priori nor can be precisely quantified.
All of these could be considered as an inverse problem of matrix data reconstruction. This
research revolves around two specific topics – quadratic inverse eigenvalue problems and low
rank approximations – and some other related problems, both in theory and in computation.
An immediate and the most straightforward application of the quadratic inverse eigenvalue
problem would be the construction of a vibration system from its observed or desirable dynamical behavior. Its mathematical model is associated to the quadratic matrix polynomial
Q(λ) = M λ^2 + C λ + K whose eigenvalues and eigenvectors govern the vibrational behavoir.
Tremendous complexities and difficulties in recovering cofficient matrices M , C , K arise when the predetermined inner-connectivity among its elements and the mandatory nonnegativity of
its parameters must be taken into account. Considerable efforts have been taken to derive theory and numerical methods for solving inverse eigenvalue problems, but techniques developed thus
far can handle the inverse problems only on a case by case basis. The ï¬ rst contribution in this
investigation is an efficient, reliable semi-definite programming technique for inverse eigenvalues problems subject to specified spectral and structural constraints. Of particular concern is the issue of inexactness of the prescribed or measured eigeninformation, which is almost inevitable in practice, since inaccurate data will affect the solvability of this inverse eigenvalue problem. To address this issue, a second contribution in this investigation is a methodical approach by
using the notion of truncated QR decomposition to ï¬ rst determine whether a nearby inverse
problem is solvable and, if it is so, the method computes the approximate coefficient matrices while providing an estimate of the residual error. Both methods enjoy the advantages of
preserving inter-connectivity structures and other important properties embedded in the original problems. More importantly, both approaches allow more flexibilities and robustness in handling highly structured problems than other special-purpose algorithms.
Low rank approximations have become increasingly important and ubiquitous in this era of
information. Generally, there is no uniï¬ ed approach because the technique often is data type
dependent. This research studies and proposes new factorization techniques for three different
type of data. The ï¬ rst algorithm aims to perform a nonnegative matrix factorization of a
nonnegative data matrix by recasting the problem geometrically as the approximation of a given
polytope on the probability simplex by a simpler polytope with fewer facets. This view leads to
a convenient way of decomposing the data by computing the proximity map which, in contrast
to most existing algorithm where only an approximate map is used, ï¬ nds the unique and global minimum per iteration. The second algorithm investigates the factorization of integer matrices which is more realistic and important in informatics. Searching through the literature, it appears that there does not exist a suitable algorithm which can handle this type of problem well owing to its discrete nature. Two effective approaches for computing integer matrix factorization are proposed in this investigation — one is based on hamming distance and the other on Euclidean distance. A lower rank approximation of a matrix A ∈ Z^{m×n} ≈ U V with factors U ∈ Z_2^{m×k}, V ∈ Z^{k×n} , where columns of U are mutually exclusive and integer k < min{m, n}
is given. The third algorithm concerns expressing a nonnegative matrix as the shortest sum of
nonnegative rank one matrices, the so called nonnegative rank factorization. Till now, only a
few abstract results which are somewhat too conceptual for numerical implementation have been
developed in the literature. Employing the Wedderburn rank reduction formula, a numerical
procedure detecting whether a nonnegative rank factorization exists is presented. In the event
that such a factorization does not exist, it is able to compute the maximum nonnegative rank
splitting.
This thesis includes a detailed analysis of inverse eigenvalue problems and low rank
factorizations. Some of the theories are classical, but new insights are obtained and their implementation for numerical computation are developed. On the other hand, this investigation leads to quite a few innovative algorithms which are effective and robust in tackling the otherwise very difficult inverse problems. The research is ongoing and several interesting research problems are identified in this thesis.
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Keywords
nonnegative rank, eigenstructure completion, quadratic model, nonnegative rank factorization, Wedderburn rank reduction formula, inverse eigenvalue problem, quadratic matrix polynomial, model updating, spill-over, connectivity, linear inequality system, nonnegativity, low rank approximation, quadratic programming, maximin problem, semi-deï¬ nite programming, structural constraint, nonnegative matrix factorization, polytope approximation, Hahn–Banach theorem, probability simplex, Euclidean distance matrix, pattern discovery, supporting hyperplane, matrix factorization, classiï¬ cation, clustering, nonnegative matrix, completely positive matrix, cp-rank
Citation
Degree
PhD
Discipline
Applied Mathematics
