Thesis (Ph.D)--Biomedical Engineering, Georgia Institute of Technology, 2009. / Committee Chair: Hu, Xiaoping; Committee Member: Brummer, Marijn; Committee Member: Duong, Tim; Committee Member: Keilholz, Shella; Committee Member: Schumacher, Eric. Part of the SMARTech Electronic Thesis and Dissertation Collection.

The number of high-dimensional datasets recording multiple aspects of a single phenomenon is ever increasing in many areas of science. This is accompanied by a fundamental need for mathematical frameworks that can compare data tabulated as multiple large-scale matrices of di erent numbers of rows. The only such framework to date, the generalized singular value decomposition (GSVD), is limited to two matrices. This thesis addresses this limitation and de fines a higher-order GSVD (HO GSVD) of N > 2 datasets, that provides a mathematical framework that can compare multiple high-dimensional datasets tabulated as large-scale matrices of different numbers of rows. / text

We describe the use of a tensor mode-1 higher-order singular value decomposition (HOSVD) in the analyses of alignments of 16S and 23S ribosomal RNA (rRNA) sequences, each encoded in a cuboid of frequencies of nucleotides across positions and organisms. This mode-1 HOSVD separates the data cuboids into combinations of patterns of nucleotide frequency variation across the positions and organisms, i.e., "eigenorganisms"' and corresponding nucleotide-specific segments of "eigenpositions," respectively, independent of a-priori knowledge of the taxonomic groups and their relationships, or the rRNA structures. We show that this mode-1 HOSVD provides a mathematical framework for modeling the sequence alignments where the mathematical variables, i.e., the significant eigenpositions and eigenorganisms, are consistent with current biological understanding of the 16S and 23S rRNAs. First, the significant eigenpositions identify multiple relations of similarity and dissimilarity among the taxonomic groups, some known and some previously unknown. Second, the corresponding eigenorganisms identify positions of nucleotides exclusively conserved within the corresponding taxonomic groups, but not among them, that map out entire substructures inserted or deleted within one taxonomic group relative to another. These positions are also enriched in adenosines that are unpaired in the rRNA secondary structure, the majority of which participate in tertiary structure interactions, and some also map to the same substructures. This demonstrates that an organism's evolutionary pathway is correlated and possibly also causally coordinated with insertions or deletions of entire rRNA substructures and unpaired adenosines, i.e., structural motifs which are involved in rRNA folding and function. Third, this mode-1 HOSVD reveals two previously unknown subgenic relationships of convergence and divergence between the Archaea and Microsporidia, that might correspond to two evolutionary pathways, in both the 16S and 23S rRNA alignments. This demonstrates that even on the level of a single rRNA molecule, an organism's evolutionary pathway is composed of different types of changes in structure in reaction to multiple concurrent evolutionary forces. / text

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Face and gait recognition problems are challenging due to largely varying appearances, highly complex pattern distributions, and insufficient training samples. This dissertation focuses on multilinear subspace learning for face and gait recognition, where low-dimensional representations are learned directly from tensorial face or gait objects. This research introduces a unifying multilinear subspace learning framework for systematic treatment of the multilinear subspace learning problem. Three multilinear projections are categorized according to the input-output space mapping as: vector-to-vector projection, tensor-to-tensor projection, and tensor-to-vector projection. Techniques for subspace learning from tensorial data are then proposed and analyzed. Multilinear principal component analysis (MPCA) seeks a tensor-to-tensor projection that maximizes the variation captured in the projected space, and it is further combined with linear discriminant analysis and boosting for better recognition performance. Uncorrelated MPCA (UMPCA) solves for a tensor-to-vector projection that maximizes the captured variation in the projected space while enforcing the zero-correlation constraint. Uncorrelated multilinear discriminant analysis (UMLDA) aims to produce uncorrelated features through a tensor-to-vector projection that maximizes a ratio of the between-class scatter over the within-class scatter defined in the projected space. Regularization and aggregation are incorporated in the UMLDA solution for enhanced performance. Experimental studies and comparative evaluations are presented and analyzed on the PIE and FERET face databases, and the USF gait database. The results indicate that the MPCA-based solution has achieved the best overall performance in various learning scenarios, the UMLDA-based solution has produced the most stable and competitive results with the same parameter setting, and the UMPCA algorithm is effective in unsupervised learning in low-dimensional subspace. Besides advancing the state-of-the-art of multilinear subspace learning for face and gait recognition, this dissertation also has potential impact in both the development of new multilinear subspace learning algorithms and other applications involving tensor objects.

<p>Capturing high dimensional complex ensembles of data is becoming commonplace in a variety of application areas. Some examples include</p><p>biological studies exploring relationships between genetic mutations and diseases, atmospheric and spatial data, and internet usage and online behavioral data. These large complex data present many challenges in their modeling and statistical analysis. Motivated by high dimensional data applications, in this thesis, we focus on building scalable Bayesian nonparametric regression algorithms and on developing models for joint distributions of complex object ensembles.</p><p>We begin with a scalable method for Gaussian process regression, a commonly used tool for nonparametric regression, prediction and spatial modeling. A very common bottleneck for large data sets is the need for repeated inversions of a big covariance matrix, which is required for likelihood evaluation and inference. Such inversion can be practically infeasible and even if implemented, highly numerically unstable. We propose an algorithm utilizing random projection ideas to construct flexible, computationally efficient and easy to implement approaches for generic scenarios. We then further improve the algorithm incorporating some structure and blocking ideas in our random projections and demonstrate their applicability in other contexts requiring inversion of large covariance matrices. We show theoretical guarantees for performance as well as substantial improvements over existing methods with simulated and real data. A by product of the work is that we discover hitherto unknown equivalences between approaches in machine learning, random linear algebra and Bayesian statistics. We finally connect random projection methods for large dimensional predictors and large sample size under a unifying theoretical framework.</p><p>The other focus of this thesis is joint modeling of complex ensembles of data from different domains. This goes beyond traditional relational modeling of ensembles of one type of data and relies on probability mixing measures over tensors. These models have added flexibility over some existing product mixture model approaches in letting each component of the ensemble have its own dependent cluster structure. We further investigate the question of measuring dependence between variables of different types and propose a very general novel scaled measure based on divergences between the joint and marginal distributions of the objects. Once again, we show excellent performance in both simulated and real data scenarios.</p> / Dissertation

Face and gait recognition problems are challenging due to largely varying appearances, highly complex pattern distributions, and insufficient training samples. This dissertation focuses on multilinear subspace learning for face and gait recognition, where low-dimensional representations are learned directly from tensorial face or gait objects. This research introduces a unifying multilinear subspace learning framework for systematic treatment of the multilinear subspace learning problem. Three multilinear projections are categorized according to the input-output space mapping as: vector-to-vector projection, tensor-to-tensor projection, and tensor-to-vector projection. Techniques for subspace learning from tensorial data are then proposed and analyzed. Multilinear principal component analysis (MPCA) seeks a tensor-to-tensor projection that maximizes the variation captured in the projected space, and it is further combined with linear discriminant analysis and boosting for better recognition performance. Uncorrelated MPCA (UMPCA) solves for a tensor-to-vector projection that maximizes the captured variation in the projected space while enforcing the zero-correlation constraint. Uncorrelated multilinear discriminant analysis (UMLDA) aims to produce uncorrelated features through a tensor-to-vector projection that maximizes a ratio of the between-class scatter over the within-class scatter defined in the projected space. Regularization and aggregation are incorporated in the UMLDA solution for enhanced performance. Experimental studies and comparative evaluations are presented and analyzed on the PIE and FERET face databases, and the USF gait database. The results indicate that the MPCA-based solution has achieved the best overall performance in various learning scenarios, the UMLDA-based solution has produced the most stable and competitive results with the same parameter setting, and the UMPCA algorithm is effective in unsupervised learning in low-dimensional subspace. Besides advancing the state-of-the-art of multilinear subspace learning for face and gait recognition, this dissertation also has potential impact in both the development of new multilinear subspace learning algorithms and other applications involving tensor objects.

Diplominiame darbe nagrinėjamas specialusis atraminių elementų erdvės atvejis, kai atraminis elementas yra II – os eilės viršvektorius ( p = 2 ). Pirmieji du darbo paragrafai yra referatyvinio pobūdžio. Pirmajame paragrafe gautos diferencijuojamos daugdaros struktūrinės lygtys, pateikiamos kai kurių diferencialinių geometrinių objektų ( skaliarinės funkcijos, tenzorinio lauko, viršvektorinio lauko ir afiniosios sieties objekto ) diferencialinės lygtis. Antrajame paragrafe afiniųjų ir tenzorinių siečių pagalba apibrėžta tenzoriaus invariantinio diferencijavimo operacija ir surastos tenzoriaus kovariantinės išvestinės. Surastos sąlygos kada tenzoriaus kovariantinės išvestinės, apibrėžto afiniosios sieties pagalba sutampa su šio tenzoriaus kovariantinėmis išvestinėmis, apibrėžtomis tenzorinės sieties pagalba. Trečiajame paragrafe apibrėžiama antros eilės atraminių viršvektorių erdvė, surastos šios erdvės struktūrinės lygtys ir ištirta liečiamosios ir koliečiamosios erdvių struktūra. Ketvirtajame paragrafe nagrinėjamos koliečiamosios erdvės normalizacijos. Tiesinės sieties pagalba kolie���iamojoje erdvėje išskiriami du invariantiniai poerdviai. Iš tiesinės sieties komponenčių ir jų pirmojo tęsinio sukonstruotos šios erdvės nepilna afinioji ir nepilna tenzorinė sietys, surasti šių siečių tarpusavio ryšiai, surastos pakankamos sąlygos, kad nepilnoji afinioji sietis sutaptų su nepilnosios tenzorinės sieties indukuota afiniąja sietimi. Čia tai pat surastos erdvės su nepilnąja... [toliau žr. visą tekstą] / In work it is examined special case of spaces support elements, when support elements are the super – vectors of the second order ( p = 2 ). The space of the second order support super – vectors is certain, the structural equations of this space are found and the structure tangential and cotangential spaces is researched. Further it is examined to space tangential spaces normalization. By means of linear connectivity in tangential space are detach two invariant subspaces. By means of components of linear connectivity and their first continuation are designed incomplete affine and incomplete tensors connectivity of this space, also connection between them is found. Further by means of linear connectivity natural normalization of space is examined, it is proved that the object of linear connectivity and its first continuation induces incomplete super – vector connectivity of this space. The structural equations of space with incomplete super – vector and complete super – vector connectivity are found. As operation of invariant differentiation of a super - vector field is certain, analogues of Richi and Bianca identities are received.

Due to the exponential growth of the dimension of the space of tensors V_(1)⊗• • •⊗V_(n), any naive method of representing these tensors is intractable on a computer. In practice, we consider feasible subspaces (subvarieties) which are defined to reduce the storage cost and the computational complexity. In this thesis, we study two such types of subvarieties: the third secant variety of the product of n projective spaces, and tensor network states. For the third secant variety of the product of n projective spaces, we determine set-theoretic defining equations, and give an upper bound of the degrees of these equations. For tensor network states, we answer a question of L. Grasedyck that arose in quantum information theory, showing that the limit of tensors in a space of tensor network states need not be a tensor network state. We also give geometric descriptions of spaces of tensor networks states corresponding to trees and loops.

Diffusion Tensor Imaging (DTI) is a medical imaging method that measures tissue structure. This is valuable when applied to the central nervous system (CNS) because it can provide structural information about white matter tracts. DTI of the spinal cord has been suggested as the next great leap in clinical diagnostics for spinal cord injury and disease because it may provide a measurable correlate of the physical structure of the cord with the associated functional deficit. Collecting precise structural information from the site of injury could be used to improve diagnostics and guide treatments. While these are the long term goals of DTI research, there are currently fundamental questions with regards to image resolution and motion-related artifacts in spinal cord which have not been thoroughly addressed. DTI is a sensitive imaging method which requires multiple mathematical calculations and approximations to complete. The limitations of the method compound with the limitations of imaging the spinal cord leading to the query: How reliable is DTI in the spinal cord? It is the goal of this study to begin to address these concerns. First, the effect of spinal cord motion on tissue discrimination was examined by comparing DTI results obtained in the presence and absence of a correctional measure for cardiac-induced motion called 'cardiac gating'. Tissue discriminability was found to be greatest in the cervical cord. Second, DTI results were subjected to two classification algorithms and compared with known anatomy to assess tissue discrimination accuracy as well as the types of associated errors. The proportion of errors in tissue classification was very high, presenting itself in all subjects. This result indicated that the DTI values associated with particular tissues were not unique to only those tissues. Finally, a theoretical model was implemented to assess the degree to which image resolution specifically affected the tissue classification accuracy obtained in the above experiments, as opposed to other errors such as MRI ghosting, blurring or distortions. It was found that DTI provides a systematically biased representation of spinal cord tissues. To overcome this limitation, future studies should concentrate efforts on increasing image resolution. / Thesis (Master, Neuroscience Studies) -- Queen's University, 2010-09-24 02:29:32.619