Finding a low-rank basis in a matrix subspace

For a given matrix subspace, how can we find a basis that consists of low-rank matrices? This is a generalization of the sparse vector problem. It turns out that when the subspace is spanned by rank-1 matrices, the matrices can be obtained by the tensor CP decomposition. For the higher rank case, the situation is not as straightforward. In this work we present an algorithm based on a greedy process applicable to higher rank problems. Our algorithm first estimates the minimum rank by applying soft singular value thresholding to a nuclear norm relaxation, and then computes a matrix with that rank using the method of alternating projections. We provide local convergence results, and compare our algorithm with several alternative approaches. Applications include data compression beyond the classical truncated SVD, computing accurate eigenvectors of a near-multiple eigenvalue, image separation and graph Laplacian eigenproblems

Numerical Methods for Parabolic Partial Differential Equations on Metric Graphs

The major motivation for this work arose from the problem of simulating diffusion type processes in the human brain network. This thesis addresses numerical methods for parabolic partial differential equations (PDEs) on network structures interpreted as metric spaces (metric graphs). Such domains frequently occur in the context of quantum graphs, where they are studied together with a differential operator and coupling conditions at the vertices of the metric graph. Quantum graphs are popular models for thin, branched structures, and there is a great interest in their studies also from the theoretical point of view. The present work aims to bridge the gap between the theoretical work and the practical usage of quantum graph models by studying arising numerical problems. The main focus is on initial boundary value problems governed by (semilinear) parabolic partial differential equations that involve a second order spatial derivative posed on the edges of the graph. The particularity of these problems are the coupling conditions of the PDEs on their common vertices. The two central methods studied in this thesis are a Galerkin discretization with linear finite elements and a spectral Galerkin discretization with basis functions obtained from an eigenvalue problem on the metric graph. Both approaches follow the method of lines, i.e., Galerkin’s method is applied for the spatial discretization resulting in a system of ordinary differential equations. Spectral accuracy can be obtained with the spectral discretization in space for sufficiently smooth functions that fulfill certain coupling conditions at the vertices. In the finite element approach, the semidiscretization is solved with classical implicit-explicit time stepping methods combined with a graph specific multigrid solver for the arising systems of linear equations in each time step. In the spectral method, the stiffness matrix is diagonal such that exponential integrators can be applied efficiently to solve the semidiscretized system. The difficulty of the spectral method, by contrast, is the computation of an eigenfunction basis. The computation of quantum graph spectra thus is the last important aspect of this work. The problem of computing eigenfunctions can be reduced to a nonlinear eigenvalue problem (NEP). In the particular case of equilateral graphs, the NEP even simplifies to a linear eigenvalue problem in the size of the number of vertices of the underlying graph. The proposed NEP solver applies equilateral approximations combined with a nested iteration approach to obtain initial guesses for a Newton-trace iteration. Human connectomes interpreted as metric graphs are consulted to test the applicability of the methods to real world, large scale problems. Experiments on simulating distribution of tau proteins in the brain of Alzheimer’s disease patients complete this work

A Unified Analysis of Multi-task Functional Linear Regression Models with Manifold Constraint and Composite Quadratic Penalty

This work studies the multi-task functional linear regression models where both the covariates and the unknown regression coefficients (called slope functions) are curves. For slope function estimation, we employ penalized splines to balance bias, variance, and computational complexity. The power of multi-task learning is brought in by imposing additional structures over the slope functions. We propose a general model with double regularization over the spline coefficient matrix: i) a matrix manifold constraint, and ii) a composite penalty as a summation of quadratic terms. Many multi-task learning approaches can be treated as special cases of this proposed model, such as a reduced-rank model and a graph Laplacian regularized model. We show the composite penalty induces a specific norm, which helps to quantify the manifold curvature and determine the corresponding proper subset in the manifold tangent space. The complexity of tangent space subset is then bridged to the complexity of geodesic neighbor via generic chaining. A unified convergence upper bound is obtained and specifically applied to the reduced-rank model and the graph Laplacian regularized model. The phase transition behaviors for the estimators are examined as we vary the configurations of model parameters

Degeneracies in the Eigenvalue Spectrum of Quantum Graphs

In this dissertation, we analyze the spectrum of the Laplace operator on graphs. In particular, we are interested in generic eigenpairs. We consider a wide range of vertex conditions on vertices of a quantum graph. Furthermore, we also investigate the eigenfunctions, showing that generically they do not vanish on vertices, unless this is unavoidable due to presence of looping edges. In the proof, the simplicity of eigenvalues and non-vanishing of eigenvalues are tightly interconnected; each property is assisting in the proof of the other (the proof is done by induction). The proof is geometric in nature and uses local modifications of the graph to reduce it to previously considered cases. We also consider an application of the result to the study of the secular manifold of a graph, showing that for large classes of graphs, the set of smooth points of the manifold has exactly two connected components. The spectrum of a symmetric quantum graph is also considered. We aim to give explicit and computation-oriented formulas for extracting the part of a Schrödinger operator on a graph which corresponds to a particular irreducible representation of the graph's symmetry. Starting with a representation of the symmetry by its action on the space of directed bonds of the graph, we find a basis which block-diagonalizes both the representation and the bond scattering matrix of the graph. The latter leads to a factorization of the secular determinant into factors that correspond to irreducible representation of the symmetry group

Skein theory and topological quantum field theory

Skein modules arise naturally when mathematicians try to generalize the Jones polynomial of knots. In the first part of this work, we study properties of skein modules. The Temperley-Lieb algebra and some of its generalizations are skein modules. We construct a bases for these skein modules. With this basis, we are able to compute some gram determinants of bilinear forms on these skein modules. Also we use this basis to prove that the Mahler measures of colored Jones polynomial of a sequence of knots converges to the Mahler measure of some two variable polynomial. The topological quantum field theory constructed by Blanchet, Habegger, Mas- baum and Vogel can be considered as a generalization of quantum invariants. It assigns modules to surfaces and linear maps to cobordisms. In particular, it assigns the ground ring to empty surface and constants to cobordisms of empty surface to itself, which are closed 3-manifolds. In this way, we get quantum invariants of 3-manifolds back. In the second part of the work, knot invariants are constructed using topological quantum field theory from quantum invariants of tangles. We prove that this is another way to compute the Turaev-Viro polynomial of knots and related invariants

Generalized genetical genomics : advanced methods and applications

Generalized genetical genomics (GGG) is a systems genetics approach that combines the analysis of genetic variation with population-wide assessment of variation in molecular traits in multiple environments to identify genotype-by-environment interactions. This thesis starts by introducing the generalized genetical genomics strategy (Chapter 1). Then, we present a newly developed software, designGG for designing optimal GGG experiments (Chapter 2). Next, two important statistical issues relevant to GGG studies were addressed. We discussed the critical concerns on causal inference with genetic data. In addition, we examined the permutation method used for determining the significance of quantitative trait loci (QTL) hotspots in linkage and association studies (Chapter 3−4). Furthermore, we applied the GGG strategy to three pilot studies: In the first of these, we showed that heritable differences in the plastic responses of gene expression are largely regulated in “trans''. In the second pilot study, we demonstrated that heritable differences in transcript abundance are highly sensitive to cellular differentiation stage. In the third study, we found that the alternative splicing machinery exhibits a general genetic robustness in C. elegans and that only a minor fraction of genes shows heritable variation in splicing forms and relative abundance. (Chapter 5−7). Finally, we conclude by discussing various fundamental issues involved in data preprocessing, QTL mapping, result interpretation and network reconstruction and suggesting future directions yet to be explored in order to expand the reach of systems genetics (Chapter 8).

機械学習と通信のための劣モジュラ・スパース最適化手法

学位の種別: 課程博士審査委員会委員 : （主査）東京大学教授 岩田 覚, 東京大学教授 定兼 邦彦, 東京大学教授 山本 博資, 東京大学准教授 武田 朗子, 東京大学准教授 平井 広志University of Tokyo(東京大学

Anion Photoelectron Spectroscopic Studies: Antioxidants, Actinide Clusters, and Molecular Activation

Gas phase anion photoelectron spectroscopy is uniquely suited to study chemistry at the molecular level, as atoms, molecules, and clusters are isolated and thus unperturbed by confounding environmental effects which often complicate analyses carried out in the liquid or solid state. Photoelectron spectroscopy provides information about the electronic structure of anions, as well as the geometry of the anions and corresponding neutral species when combined with theoretical calculations. A variety of ion sources were employed to generate the anions in these studies: electrospray ionization (ESI), laser vaporization (LVS), and pulsed arc cluster ion source (PACIS). Using these techniques, two antioxidants, a range of actinide containing clusters, and multiple activation reactions were studied. Additionally, a new double rod laser vaporization source was designed and constructed to generate single atom catalyst (SAC) mimics. Chapter III presents the studies of this thesis and is divided into three major sections based on ion source: ESI, LVS, and PACIS. ESI brought the water-soluble antioxidants (ascorbate, deprotonated ascorbate, propyl gallate, and gallate) into the gas phase. LVS ablated uranium and thorium rods to generate gas phase atoms and actinide containing clusters, as well as highlighted the reaction between iridium and hydroxylamine and the phenomenon of intramolecular electron-induced proton transfer. Finally, using PACIS, two thorium clusters were generated, and CO2 activation with two metal hydrides were studied