subject:(Low rank matrix correction)

Delft University of Technology

1. Swart, Wouter (author). Methods for improving the computational performance of sequentially linear analsysis.

Degree: 2018, Delft University of Technology

URL: http://resolver.tudelft.nl/uuid:dc35a7e3-beb7-4d46-88c6-36e6f980a597

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The numerical simulation of brittle failure with nonlinear finite element analysis (NLFEA) remains a challenge due to robustness issues. These problems are attributed to the… (more)

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The numerical simulation of brittle failure with nonlinear finite element analysis (NLFEA) remains a challenge due to robustness issues. These problems are attributed to the softening material behaviour and the iterative nature of the Newton-Raphson type methods used in NLFEA. However, robust numerical simulations become increasingly important, for example due to recent developments in Groningen. To address these issues, sequentially linear analysis (SLA) was developed which exploits the fact that a linear analysis is inherently stable. By assuming a stepwise material degradation the nonlinear response of a structure can be approximated with a sequence of linear analyses. Although this approach has been proven to be effective for several case studies, the numerical performance is still a problem that has to be solved. After every linear analysis, a single element is damaged resulting in incremental damage. As a result, the system of equations only changes locally between these linear analyses. Traditional solution techniques do not exploit this property and calculate a matrix factorisation every linear analysis, resulting in high computational times per analysis step. Since SLA typically requires many linear analyses to obtain the desired structural response, this leads to unacceptable analysis times. The aim of this thesis is to improve the computational performance of SLA by developing numerical solution techniques which exploit the incremental approach of SLA. To this extend, the following methods have been developed. A direct solution technique has been developed which is based on the Woodbury matrix identity. This identity allows for the numerically cheap computation of the inverse of a low-rank corrected matrix. In this approach, the expensive matrix factorisation does not have to be calculated every linear analysis step. Instead, the old factorisation can be reused along with some additional matrix- and vector multiplications and solving a significantly smaller linear system of equations. An optimal strategy is derived to determine at which point a new factorisation should be calculated. An improved preconditioner for the conjugate gradient (CG) method has been developed. Instead of an incomplete factorisation, the complete factorisation is used as a preconditioner which reduces the number of required CG iterations significantly. The point at which too many CG iterations are required and a new factorisation is necessary, is determined using the same strategy as the first method. From numerical experiments it follows that both methods perform significantly better than the direct solution method, especially for large 3-dimensional problems. The best performance is achieved using the Woodbury matrix identity resulting in the solver no longer being the dominant factor in SLA. Furthermore, significantly larger problems are not solvable in time frames in which previously only smaller problems were solved.

Applied Mathematics

Advisors/Committee Members: van Gijzen, Martin (mentor), Schreppers, G.M.A (graduation committee), Rots, Jan (graduation committee), van Horssen, Wim (graduation committee), Delft University of Technology (degree granting institution).

Subjects/Keywords: Finite Element Analysis; Preconditioning; Structural analysis; Direct method; Iterative method; Low-rank matrix correction

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APA (6^th Edition):

Swart, W. (. (2018). Methods for improving the computational performance of sequentially linear analsysis. (Masters Thesis). Delft University of Technology. Retrieved from http://resolver.tudelft.nl/uuid:dc35a7e3-beb7-4d46-88c6-36e6f980a597

Chicago Manual of Style (16^th Edition):

Swart, Wouter (author). “Methods for improving the computational performance of sequentially linear analsysis.” 2018. Masters Thesis, Delft University of Technology. Accessed January 20, 2021. http://resolver.tudelft.nl/uuid:dc35a7e3-beb7-4d46-88c6-36e6f980a597.

MLA Handbook (7^th Edition):

Swart, Wouter (author). “Methods for improving the computational performance of sequentially linear analsysis.” 2018. Web. 20 Jan 2021.

Vancouver:

Swart W(. Methods for improving the computational performance of sequentially linear analsysis. [Internet] [Masters thesis]. Delft University of Technology; 2018. [cited 2021 Jan 20]. Available from: http://resolver.tudelft.nl/uuid:dc35a7e3-beb7-4d46-88c6-36e6f980a597.

Council of Science Editors:

Swart W(. Methods for improving the computational performance of sequentially linear analsysis. [Masters Thesis]. Delft University of Technology; 2018. Available from: http://resolver.tudelft.nl/uuid:dc35a7e3-beb7-4d46-88c6-36e6f980a597

2. Biradar, Rakesh. Analysis and Prediction of Community Structure Using Unsupervised Learning.

Degree: MS, 2016, Worcester Polytechnic Institute

URL: etd-012616-134431 ; https://digitalcommons.wpi.edu/etd-theses/138

► In this thesis, we perform analysis and prediction for community structures in graphs using unsupervised learning. The methods we use require the data matrices to… (more)

▼ In this thesis, we perform analysis and prediction for community structures in graphs using unsupervised learning. The methods we use require the data matrices to be of low rank, and such matrices appear quite often in real world problems across a broad range of domains. Such a modelling assumption is widely considered by classical algorithms such as principal component analysis (PCA), and the same assumption is often used to achieve dimensionality reduction. Dimension reduction, which is a classic method in unsupervised learning, can be leveraged in a wide array of problems, including prediction of strength of connection between communities from unlabeled or partially labeled data. Accordingly, a low rank assumption addresses many real world problems, and a low rank assumption has been used in this thesis to predict the strength of connection between communities in Amazon product data. In particular, we have analyzed real world data across retail and cyber domains, with the focus being on the retail domain. Herein, our focus is on analyzing the strength of connection between the communities in Amazon product data, where each community represents a group of products, and we are given the strength of connection between the individual products but not between the product communities. We call the strength of connection between individual products first order data and the strength of connection between communities second order data. This usage is inspired by [1] where first order time series are used to compute second order covariance matrices where such covariance matrices encode the strength of connection between the time series. In order to find the strength of connection between the communities, we define various metrics to measure this strength, and one of the goals of this thesis is to choose a good metric, which supports effective predictions. However, the main objective is to predict the strength of connection between most of the communities, given measurements of the strength of connection between only a few communities. To address this challenge, we use modern extensions of PCA such as eRPCA that can provide better predictions and can be computationally efficient for large problems. However, the current theory of eRPCA algorithms is not designed to treat problems where the initial data (such as the second order matrix of communities strength) is both low rank and sparse. Therefore, we analyze the performance of eRPCA algorithm on such data and modify our approaches for the particular structure of Amazon product communities to perform the necessary predictions. Advisors/Committee Members: Randy C. Paffenroth, Advisor, ;.

Subjects/Keywords: eRPCA; Community Prediction; Low Rank; Sparse Matrix

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APA (6^th Edition):

Biradar, R. (2016). Analysis and Prediction of Community Structure Using Unsupervised Learning. (Thesis). Worcester Polytechnic Institute. Retrieved from etd-012616-134431 ; https://digitalcommons.wpi.edu/etd-theses/138

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^th Edition):

Biradar, Rakesh. “Analysis and Prediction of Community Structure Using Unsupervised Learning.” 2016. Thesis, Worcester Polytechnic Institute. Accessed January 20, 2021. etd-012616-134431 ; https://digitalcommons.wpi.edu/etd-theses/138.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^th Edition):

Biradar, Rakesh. “Analysis and Prediction of Community Structure Using Unsupervised Learning.” 2016. Web. 20 Jan 2021.

Vancouver:

Biradar R. Analysis and Prediction of Community Structure Using Unsupervised Learning. [Internet] [Thesis]. Worcester Polytechnic Institute; 2016. [cited 2021 Jan 20]. Available from: etd-012616-134431 ; https://digitalcommons.wpi.edu/etd-theses/138.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Biradar R. Analysis and Prediction of Community Structure Using Unsupervised Learning. [Thesis]. Worcester Polytechnic Institute; 2016. Available from: etd-012616-134431 ; https://digitalcommons.wpi.edu/etd-theses/138

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Temple University

3. Shank, Stephen David. Low-rank solution methods for large-scale linear matrix equations.

Degree: PhD, 2014, Temple University

URL: http://digital.library.temple.edu/u?/p245801coll10,273331

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Mathematics

We consider low-rank solution methods for certain classes of large-scale linear matrix equations. Our aim is to adapt existing low-rank solution methods based on… (more)

Subjects/Keywords: Applied mathematics;

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Shank, S. D. (2014). Low-rank solution methods for large-scale linear matrix equations. (Doctoral Dissertation). Temple University. Retrieved from http://digital.library.temple.edu/u?/p245801coll10,273331

Chicago Manual of Style (16^th Edition):

Shank, Stephen David. “Low-rank solution methods for large-scale linear matrix equations.” 2014. Doctoral Dissertation, Temple University. Accessed January 20, 2021. http://digital.library.temple.edu/u?/p245801coll10,273331.

MLA Handbook (7^th Edition):

Shank, Stephen David. “Low-rank solution methods for large-scale linear matrix equations.” 2014. Web. 20 Jan 2021.

Vancouver:

Shank SD. Low-rank solution methods for large-scale linear matrix equations. [Internet] [Doctoral dissertation]. Temple University; 2014. [cited 2021 Jan 20]. Available from: http://digital.library.temple.edu/u?/p245801coll10,273331.

Council of Science Editors:

Shank SD. Low-rank solution methods for large-scale linear matrix equations. [Doctoral Dissertation]. Temple University; 2014. Available from: http://digital.library.temple.edu/u?/p245801coll10,273331

Colorado School of Mines

4. Yang, Dehui. Structured low-rank matrix recovery via optimization methods.

Degree: PhD, Electrical Engineering, 2018, Colorado School of Mines

URL: http://hdl.handle.net/11124/172154

► From single-molecule microscopy in biology, to collaborative filtering in recommendation systems, to quantum state tomography in physics, many scientific discoveries involve solving ill-posed inverse problems,… (more)

▼ From single-molecule microscopy in biology, to collaborative filtering in recommendation systems, to quantum state tomography in physics, many scientific discoveries involve solving ill-posed inverse problems, where the number of parameters to be estimated far exceeds the number of available measurements. To make these daunting problems solvable, low-dimensional geometric structures are often exploited, and regularizations that promote underlying structures are used for various inference tasks. To date, one of the most effective and plausible low-dimensional models for matrix data is the low-rank structure, which assumes that columns of the data matrix are correlated and lie in a low-dimensional subspace. This helps make certain matrix inverse problems well-posed. However, in some cases, standard low-rank structure is not powerful enough for modeling the underlying data generating process, and additional modeling efforts are desired. This is the main focus of this research. Motivated by applications from different disciplines in engineering and science, in this dissertation, we consider the recovery of three instances of structured matrices from limited measurement data, where additional structures naturally occur in the data matrices beyond simple low-rankness. The structured matrices that we consider include i) low-rank and spectrally sparse matrices in super-resolution imaging; ii) low-rank skew-symmetric matrices in pairwise comparisons; iii) and low-rank positive semidefinite matrices in physical and data sciences. Using optimization as a tool, we develop new regularizers and computationally efficient algorithmic frameworks to account for structured low-rankness in solving these ill-posed inverse problems. For some of the problems considered in this dissertation, theoretical analysis is also carried out for the proposed optimization programs. We show that, under mild conditions, the structured low-rank matrices can be recovered reliably from a minimal number of random measurements. Advisors/Committee Members: Wakin, Michael B. (advisor), Vincent, Tyrone (committee member), Yang, Dejun (committee member), Tang, Gongguo (committee member).

Subjects/Keywords: matrix completion; models; super-resolution; modal analysis; low-rank; optimization

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Yang, D. (2018). Structured low-rank matrix recovery via optimization methods. (Doctoral Dissertation). Colorado School of Mines. Retrieved from http://hdl.handle.net/11124/172154

Chicago Manual of Style (16^th Edition):

Yang, Dehui. “Structured low-rank matrix recovery via optimization methods.” 2018. Doctoral Dissertation, Colorado School of Mines. Accessed January 20, 2021. http://hdl.handle.net/11124/172154.

MLA Handbook (7^th Edition):

Yang, Dehui. “Structured low-rank matrix recovery via optimization methods.” 2018. Web. 20 Jan 2021.

Vancouver:

Yang D. Structured low-rank matrix recovery via optimization methods. [Internet] [Doctoral dissertation]. Colorado School of Mines; 2018. [cited 2021 Jan 20]. Available from: http://hdl.handle.net/11124/172154.

Council of Science Editors:

Yang D. Structured low-rank matrix recovery via optimization methods. [Doctoral Dissertation]. Colorado School of Mines; 2018. Available from: http://hdl.handle.net/11124/172154

Georgia Tech

5. Rangel Walteros, Pedro Andres. A non-asymptotic study of low-rank estimation of smooth kernels on graphs.

Degree: PhD, Mathematics, 2014, Georgia Tech

URL: http://hdl.handle.net/1853/52988

► This dissertation investigates the problem of estimating a kernel over a large graph based on a sample of noisy observations of linear measurements of the… (more)

Subjects/Keywords: Low-rank matrix completion; Kernels on graphs; High dimensional probability

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APA (6^th Edition):

Rangel Walteros, P. A. (2014). A non-asymptotic study of low-rank estimation of smooth kernels on graphs. (Doctoral Dissertation). Georgia Tech. Retrieved from http://hdl.handle.net/1853/52988

Chicago Manual of Style (16^th Edition):

Rangel Walteros, Pedro Andres. “A non-asymptotic study of low-rank estimation of smooth kernels on graphs.” 2014. Doctoral Dissertation, Georgia Tech. Accessed January 20, 2021. http://hdl.handle.net/1853/52988.

MLA Handbook (7^th Edition):

Rangel Walteros, Pedro Andres. “A non-asymptotic study of low-rank estimation of smooth kernels on graphs.” 2014. Web. 20 Jan 2021.

Vancouver:

Rangel Walteros PA. A non-asymptotic study of low-rank estimation of smooth kernels on graphs. [Internet] [Doctoral dissertation]. Georgia Tech; 2014. [cited 2021 Jan 20]. Available from: http://hdl.handle.net/1853/52988.

Council of Science Editors:

Rangel Walteros PA. A non-asymptotic study of low-rank estimation of smooth kernels on graphs. [Doctoral Dissertation]. Georgia Tech; 2014. Available from: http://hdl.handle.net/1853/52988

Georgia Tech

6. Xia, Dong. Statistical inference for large matrices.

Degree: PhD, Mathematics, 2016, Georgia Tech

URL: http://hdl.handle.net/1853/55632

► This thesis covers two topics on matrix analysis and estimation in machine learning and statistics. The first topic is about density matrix estimation with application… (more)

Subjects/Keywords: Low rank; Matrix estimation; Singular vectors; Random perturbation

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Xia, D. (2016). Statistical inference for large matrices. (Doctoral Dissertation). Georgia Tech. Retrieved from http://hdl.handle.net/1853/55632

Chicago Manual of Style (16^th Edition):

Xia, Dong. “Statistical inference for large matrices.” 2016. Doctoral Dissertation, Georgia Tech. Accessed January 20, 2021. http://hdl.handle.net/1853/55632.

MLA Handbook (7^th Edition):

Xia, Dong. “Statistical inference for large matrices.” 2016. Web. 20 Jan 2021.

Vancouver:

Xia D. Statistical inference for large matrices. [Internet] [Doctoral dissertation]. Georgia Tech; 2016. [cited 2021 Jan 20]. Available from: http://hdl.handle.net/1853/55632.

Council of Science Editors:

Xia D. Statistical inference for large matrices. [Doctoral Dissertation]. Georgia Tech; 2016. Available from: http://hdl.handle.net/1853/55632

Princeton University

7. Zhong, Yiqiao. Spectral methods and MLE: a modern statistical perspective .

Degree: PhD, 2019, Princeton University

URL: http://arks.princeton.edu/ark:/88435/dsp01gb19f8728

► Modern statistical analysis often requires the integration of statistical thinking and algorithmic thinking. There are new challenges posed for classical estimation principles. Indeed, in high-dimensional… (more)

Subjects/Keywords: Eigenvectors; High dimensional statistics; Low rank matrices; Matrix perturbation; Nonconvex optimization; Random matrix theory

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Zhong, Y. (2019). Spectral methods and MLE: a modern statistical perspective . (Doctoral Dissertation). Princeton University. Retrieved from http://arks.princeton.edu/ark:/88435/dsp01gb19f8728

Chicago Manual of Style (16^th Edition):

Zhong, Yiqiao. “Spectral methods and MLE: a modern statistical perspective .” 2019. Doctoral Dissertation, Princeton University. Accessed January 20, 2021. http://arks.princeton.edu/ark:/88435/dsp01gb19f8728.

MLA Handbook (7^th Edition):

Zhong, Yiqiao. “Spectral methods and MLE: a modern statistical perspective .” 2019. Web. 20 Jan 2021.

Vancouver:

Zhong Y. Spectral methods and MLE: a modern statistical perspective . [Internet] [Doctoral dissertation]. Princeton University; 2019. [cited 2021 Jan 20]. Available from: http://arks.princeton.edu/ark:/88435/dsp01gb19f8728.

Council of Science Editors:

Zhong Y. Spectral methods and MLE: a modern statistical perspective . [Doctoral Dissertation]. Princeton University; 2019. Available from: http://arks.princeton.edu/ark:/88435/dsp01gb19f8728

University of Manchester

8. Borsdorf, Ruediger. Structured Matrix Nearness Problems:Theory and Algorithms.

Degree: 2012, University of Manchester

URL: http://www.manchester.ac.uk/escholar/uk-ac-man-scw:162521

► In many areas of science one often has a given matrix, representing forexample a measured data set and is required to find a matrix that… (more)

▼ In many areas of science one often has a given matrix, representing forexample a measured data set and is required to find a matrix that isclosest in a suitable norm to the matrix and possesses additionally a structure,inherited from the model used or coming from the application. We call these problems structured matrix nearness problems. We look at three different groups of these problems that come from real applications, analyze theproperties of the corresponding matrix structure, and propose algorithms to solve them efficiently.The first part of this thesis concerns the nearness problem of finding the nearest k factor correlation matrix C(X) =\diag(I_n -XX^T)+XX^T to a given symmetric matrix, subject to natural nonlinearconstraints on the elements of the n × k matrix X, where distance is measured in the Frobenius norm.Such problems arise, for example, when one is investigating factor models ofcollateralized debt obligations (CDOs) or multivariate time series. We examineseveral algorithms for solving the nearness problem that differ in whether or not they can takeaccount of the nonlinear constraints and in their convergence properties. Our numerical experimentsshow that the performance of the methods depends strongly on the problem, butthat, among our tested methods, the spectral projected gradient method is the clear winner.In the second part we look at two two-sided optimization problems where thematrix of unknowns Y∈\R^{n × p} lies in the Stiefel manifold. These two problems come from an application in atomic chemistry where one is looking for atomic orbitalswith prescribed occupation numbers. We analyze these two problems, propose ananalytic optimal solution of the first and show that an optimal solutionof the second problem can be found by solving a convex quadratic programmingproblem with box constraints and p unknowns. We prove that the latter problem can be solved by the active-set method in atmost 2p iterations. Subsequently, we analyze the set of optimal solutions𝓒={Y∈\R^{n × p}:Y^TY=I_p, Y^TNY=D} of the firstproblem for N symmetric and D diagonal and find that a slight modification of it is a Riemannian manifold. Wederive the geometric objects required to make an optimization over thismanifold possible. We propose an augmented Lagrangian-based algorithm that uses these geometric tools andallows us to optimize an arbitrary smooth function over 𝓒. This algorithm can be used to select a particular solutionout of the latter set 𝓒 by posing a new optimization problem. We compareit numerically with a similar algorithm that,however, does not apply these geometric tools and find that our algorithm yields better performance.The third part is devoted to low rank nearness problems in the Q-norm, where thematrix of interest is additionally of linear structure, meaning it lies inthe set spanned by s predefined matrices U₁,…, U_s∈{0,1}^n × p. These problems are often associated with model reduction, for example in… Advisors/Committee Members: SHARDLOW, TONY T, Higham, Nicholas, Shardlow, Tony.

Subjects/Keywords: correlation matrix; factor structure; matrix embedding; Stiefel manifold; linearly structured matrix; Grassmannian manifold; low rank; optimization over manifolds; matrix nearness problems

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Borsdorf, R. (2012). Structured Matrix Nearness Problems:Theory and Algorithms. (Doctoral Dissertation). University of Manchester. Retrieved from http://www.manchester.ac.uk/escholar/uk-ac-man-scw:162521

Chicago Manual of Style (16^th Edition):

Borsdorf, Ruediger. “Structured Matrix Nearness Problems:Theory and Algorithms.” 2012. Doctoral Dissertation, University of Manchester. Accessed January 20, 2021. http://www.manchester.ac.uk/escholar/uk-ac-man-scw:162521.

MLA Handbook (7^th Edition):

Borsdorf, Ruediger. “Structured Matrix Nearness Problems:Theory and Algorithms.” 2012. Web. 20 Jan 2021.

Vancouver:

Borsdorf R. Structured Matrix Nearness Problems:Theory and Algorithms. [Internet] [Doctoral dissertation]. University of Manchester; 2012. [cited 2021 Jan 20]. Available from: http://www.manchester.ac.uk/escholar/uk-ac-man-scw:162521.

Council of Science Editors:

Borsdorf R. Structured Matrix Nearness Problems:Theory and Algorithms. [Doctoral Dissertation]. University of Manchester; 2012. Available from: http://www.manchester.ac.uk/escholar/uk-ac-man-scw:162521

9. Haraldson, Joseph. Matrix Polynomials and their Lower Rank Approximations.

Degree: 2019, University of Waterloo

URL: http://hdl.handle.net/10012/14847

► This thesis is a wide ranging work on computing a “lower-rank” approximation of a matrix polynomial using second-order non-linear optimization techniques. Two notions of rank… (more)

▼ This thesis is a wide ranging work on computing a “lower-rank” approximation of a matrix polynomial using second-order non-linear optimization techniques. Two notions of rank are investigated. The first is the rank as the number of linearly independent rows or columns, which is the classical definition. The other notion considered is the lowest rank of a matrix polynomial when evaluated at a complex number, or the McCoy rank. Together, these two notions of rank allow one to compute a nearby matrix polynomial where the structure of both the left and right kernels is prescribed, along with the structure of both the infinite and finite eigenvalues. The computational theory of the calculus of matrix polynomial valued functions is developed and used in optimization algorithms based on second-order approximations. Special functions studied with a detailed error analysis are the determinant and adjoint of matrix polynomials. The unstructured and structured variants of matrix polynomials are studied in a very general setting in the context of an equality constrained optimization problem. The most general instances of these optimization problems are NP hard to approximate solutions to in a global setting. In most instances we are able to prove that solutions to our optimization problems exist (possibly at infinity) and discuss techniques in conjunction with an implementation to compute local minimizers to the problem. Most of the analysis of these problems is local and done through the Karush-Kuhn-Tucker optimality conditions for constrained optimization problems. We show that most formulations of the problems studied satisfy regularity conditions and admit Lagrange multipliers. Furthermore, we show that under some formulations that the second-order sufficient condition holds for instances of interest of the optimization problems in question. When Lagrange multipliers do not exist, we discuss why, and if it is reasonable to do so, how to regularize the problem. In several instances closed form expressions for the derivatives of matrix polynomial valued functions are derived to assist in analysis of the optimality conditions around a solution. From this analysis it is shown that variants of Newton’s method will have a local rate of convergence that is quadratic with a suitable initial guess for many problems. The implementations are demonstrated on some examples from the literature and several examples are cross-validated with different optimization formulations of the same mathematical problem. We conclude with a special application of the theory developed in this thesis is computing a nearby pair of differential polynomials with a non-trivial greatest common divisor, a non-commutative symbolic-numeric computation problem. We formulate this problem as finding a nearby structured matrix polynomial that is rank deficient in the classical sense.

Subjects/Keywords: numerical linear algebra; optimization; matrix polynomial; eigenvalue; gcd; low rank; low rank approximation; polynomial eigenvalue; matrix pencil; smith form; kronecker form; kernel; matrix pencil; approximate gcd

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APA (6^th Edition):

Haraldson, J. (2019). Matrix Polynomials and their Lower Rank Approximations. (Thesis). University of Waterloo. Retrieved from http://hdl.handle.net/10012/14847

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^th Edition):

Haraldson, Joseph. “Matrix Polynomials and their Lower Rank Approximations.” 2019. Thesis, University of Waterloo. Accessed January 20, 2021. http://hdl.handle.net/10012/14847.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^th Edition):

Haraldson, Joseph. “Matrix Polynomials and their Lower Rank Approximations.” 2019. Web. 20 Jan 2021.

Vancouver:

Haraldson J. Matrix Polynomials and their Lower Rank Approximations. [Internet] [Thesis]. University of Waterloo; 2019. [cited 2021 Jan 20]. Available from: http://hdl.handle.net/10012/14847.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Haraldson J. Matrix Polynomials and their Lower Rank Approximations. [Thesis]. University of Waterloo; 2019. Available from: http://hdl.handle.net/10012/14847

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Texas A&M University

10. Sun, Ranye. Thresholding Multivariate Regression and Generalized Principal Components.

Degree: PhD, Statistics, 2014, Texas A&M University

URL: http://hdl.handle.net/1969.1/152564

► As high-dimensional data arises from various fields in science and technology, traditional multivariate methods need to be updated. Principal component analysis and reduced rank regression… (more)

▼ As high-dimensional data arises from various fields in science and technology, traditional multivariate methods need to be updated. Principal component analysis and reduced rank regression are two of the most important multivariate statistical techniques that have seen major changes in recent years. To improving the statistical performance and achieve fast computational efficiency, recent approaches aim at regularizing both the row and column factors of the low-rank matrix approximation by adopting the Lasso-type penalties. Thresholding is another powerful technique for regularizing the row and column factors without solving an optimization problem. This dissertation research covers two novel applications of the idea of thresholding: the thresholding reduced rank multivariate regression and the generalized principal component analysis/singular value decomposition (SVD). The following two para- graphs give brief introductions to each of the two topics, respectively. Uncovering a meaningful relationship between the responses and the predictors is a fundamental goal in multivariate regression problems, which can be very challenging when data are high-dimensional. Dimension reduction and regularization techniques are applied extensively to alleviate the curse of dimensionality. It is desirable to estimate the regression coefficient matrix by low-rank matrices constructed from its SVD. We reduce such regression problems to sparse SVD problems for cor- related data matrices and generalize the fast iterative thresholding for sparse SVDs algorithm to this situation. This generalization inherits the computational and statistical advantages of the original algorithm including its sparse initialization, novel ways of estimating the thresholding levels and the thresholded subspace iterations. It guarantees the orthogonality of the singular vectors and computes them simultaneously and not sequentially as in the existing methods. We also place this algorithm in an optimization framework by introducing a specific bi-convex objective function. An iterative algorithm that minimizes the objective function, via closed form iterates, is proposed and its convergence is established. This enables us to study the large sample properties of the solution of the multivariate regression problem and establishes consistency of the estimators as the sample size tends to infinity. The methodology and the potential adverse impact of dependence on the earlier algorithms are illustrated using simulation and real data. The second part of this dissertation considers transposable data matrices where both their rows and columns are correlated. Such datasets are routinely encountered in fields such as econometrics, bio-informatics, chemometrics, network data and so on. While methods to approximate the high-dimensional data matrices have been extensively researched for uncorrelated and independent situations, they are much less so for the transposable data matrices. A generalization of principal component analysis and the related weighted least… Advisors/Committee Members: Pourahmadi, Mohsen (advisor), Carroll, Raymond J (advisor), Longnecker, Michael T (committee member), Singh, Vijay P (committee member).

Subjects/Keywords: cross-validation; iterative subspace projections; low-rank matrix approximation; regularization; transposable data.

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APA (6^th Edition):

Sun, R. (2014). Thresholding Multivariate Regression and Generalized Principal Components. (Doctoral Dissertation). Texas A&M University. Retrieved from http://hdl.handle.net/1969.1/152564

Chicago Manual of Style (16^th Edition):

Sun, Ranye. “Thresholding Multivariate Regression and Generalized Principal Components.” 2014. Doctoral Dissertation, Texas A&M University. Accessed January 20, 2021. http://hdl.handle.net/1969.1/152564.

MLA Handbook (7^th Edition):

Sun, Ranye. “Thresholding Multivariate Regression and Generalized Principal Components.” 2014. Web. 20 Jan 2021.

Vancouver:

Sun R. Thresholding Multivariate Regression and Generalized Principal Components. [Internet] [Doctoral dissertation]. Texas A&M University; 2014. [cited 2021 Jan 20]. Available from: http://hdl.handle.net/1969.1/152564.

Council of Science Editors:

Sun R. Thresholding Multivariate Regression and Generalized Principal Components. [Doctoral Dissertation]. Texas A&M University; 2014. Available from: http://hdl.handle.net/1969.1/152564

Université Catholique de Louvain

11. Gillis, Nicolas. Nonnegative matrix factorization : complexity, algorithms and applications.

Degree: 2011, Université Catholique de Louvain

URL: http://hdl.handle.net/2078.1/70744

►

Linear dimensionality reduction techniques such as principal component analysis are powerful tools for the analysis of high-dimensional data. In this thesis, we explore a closely… (more)

▼

Linear dimensionality reduction techniques such as principal component analysis are powerful tools for the analysis of high-dimensional data. In this thesis, we explore a closely related problem, namely nonnegative matrix factorization (NMF), a low-rank matrix approximation problem with nonnegativity constraints. More precisely, we seek to approximate a given nonnegative matrix with the product of two low-rank nonnegative matrices. These nonnegative factors can be interpreted in the same way as the data, e.g., as images (described by pixel intensities) or texts (represented by vectors of word counts), and lead to an additive and sparse representation. However, they render the problem much more difficult to solve (i.e., NP-hard). A first goal of this thesis is to study theoretical issues related to NMF. In particular, we make connections with well-known problems in graph theory, combinatorial optimization and computational geometry. We also study computational complexity issues and show, using reductions from the maximum-edge biclique problem, NP-hardness of several low-rank matrix approximation problems, including the rank-one subproblems arising in NMF, a problem involving underapproximation constraints (NMU) and the unconstrained version of the factorization problem where some data is missing or unknown. Our second goal is to improve existing techniques and develop new tools for the analysis of nonnegative data. We propose accelerated variants of several NMF algorithms based on a careful analysis of their computational cost. We also introduce a multilevel approach to speed up their initial convergence. Finally, a new greedy heuristic based on NMU is presented and used for the analysis of hyperspectral images, in which each pixel is measured along hundreds of wavelengths, which allows for example spectroscopy of satellite images.

(FSA 3) – UCL, 2011

Advisors/Committee Members: UCL - SST/ICTM/INMA - Pôle en ingénierie mathématique, Glineur, François, Lefèvre, Philippe, Van Dooren, Paul, Absil, Pierre-Antoine, Henrion, Didier, Van Huffel, Sabine, Vavasis, Stephen, Fiorini, Samuel.

Subjects/Keywords: Low-rank matrix approximation; Nonnegative matrices; Computational complexity; Optimization; Underapproximation; Data mining; Hyperspectral image analysis

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APA (6^th Edition):

Gillis, N. (2011). Nonnegative matrix factorization : complexity, algorithms and applications. (Thesis). Université Catholique de Louvain. Retrieved from http://hdl.handle.net/2078.1/70744

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^th Edition):

Gillis, Nicolas. “Nonnegative matrix factorization : complexity, algorithms and applications.” 2011. Thesis, Université Catholique de Louvain. Accessed January 20, 2021. http://hdl.handle.net/2078.1/70744.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^th Edition):

Gillis, Nicolas. “Nonnegative matrix factorization : complexity, algorithms and applications.” 2011. Web. 20 Jan 2021.

Vancouver:

Gillis N. Nonnegative matrix factorization : complexity, algorithms and applications. [Internet] [Thesis]. Université Catholique de Louvain; 2011. [cited 2021 Jan 20]. Available from: http://hdl.handle.net/2078.1/70744.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Gillis N. Nonnegative matrix factorization : complexity, algorithms and applications. [Thesis]. Université Catholique de Louvain; 2011. Available from: http://hdl.handle.net/2078.1/70744

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

12. Zhu, Ziwei. Distributed and Robust Statistical Learning .

Degree: PhD, 2018, Princeton University

URL: http://arks.princeton.edu/ark:/88435/dsp01d217qs22x

► Decentralized and corrupted data are nowadays ubiquitous, which impose fundamental challenges for modern statistical analysis. Illustrative examples are massive and decentralized data produced by distributed… (more)

Subjects/Keywords: distributed learning; high-dimensional statistics; low-rank matrix recovery; principal component analysis; regression; robust statistics

1 more image…

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APA (6^th Edition):

Zhu, Z. (2018). Distributed and Robust Statistical Learning . (Doctoral Dissertation). Princeton University. Retrieved from http://arks.princeton.edu/ark:/88435/dsp01d217qs22x

Chicago Manual of Style (16^th Edition):

Zhu, Ziwei. “Distributed and Robust Statistical Learning .” 2018. Doctoral Dissertation, Princeton University. Accessed January 20, 2021. http://arks.princeton.edu/ark:/88435/dsp01d217qs22x.

MLA Handbook (7^th Edition):

Zhu, Ziwei. “Distributed and Robust Statistical Learning .” 2018. Web. 20 Jan 2021.

Vancouver:

Zhu Z. Distributed and Robust Statistical Learning . [Internet] [Doctoral dissertation]. Princeton University; 2018. [cited 2021 Jan 20]. Available from: http://arks.princeton.edu/ark:/88435/dsp01d217qs22x.

Council of Science Editors:

Zhu Z. Distributed and Robust Statistical Learning . [Doctoral Dissertation]. Princeton University; 2018. Available from: http://arks.princeton.edu/ark:/88435/dsp01d217qs22x

University of Illinois – Urbana-Champaign

13. Balasubramanian, Arvind. Applications of low-rank matrix recovery methods in computer vision.

Degree: PhD, 1200, 2012, University of Illinois – Urbana-Champaign

URL: http://hdl.handle.net/2142/31929

► The ubiquitous availability of high-dimensional data such as images and videos has generated a lot of interest in high-dimensional data analysis. One of the key… (more)

▼ The ubiquitous availability of high-dimensional data such as images and videos has generated a lot of interest in high-dimensional data analysis. One of the key issues that needs to be addressed in real applications is the presence of large-magnitude non-Gaussian errors. For image data, the problem of deformations or domain transformations also poses interesting challenges. In this thesis, we harness recent advances in low-rank matrix recovery via convex optimization techniques to solve real problems in computer vision. This thesis also provides some theoretical analysis that extends existing results to new observation models. Low-rank matrix approximations are a popular tool in data analysis. The well-known Principal Component Analysis (PCA) algorithm is a good example. Recently, it was shown that low-rank matrices can be recovered exactly from grossly corrupted measurements via convex optimization. This framework, called Principal Component Pursuit (PCP), constitutes a powerful tool that allows us to handle corrupted measurements and even missing entries in a principled way. In this thesis, we extend existing theoretical results to the case when a large majority of the entries of the matrix are badly corrupted. On the application side, we first briefly look at the image formation model that naturally gives rise to a low-rank matrix structure, and see how PCP can be used effectively in the photometric stereo problem. We then extend the existing PCP framework in a non-trivial fashion to effectively handle domain transformations in images. The proposed ideas are used to align multiple images simultaneously with one another, as well as to represent structured and symmetric textures in a novel way that is invariant to deformations. In addition to achieving excellent performance on real data, these methods are potentially very useful for other vision tasks like 3D structure recovery for urban images, automatic camera calibration, etc. Finally, we provide some theoretical guarantees for the new observation model encountered in the aforementioned applications. In particular, we show that under some conditions it is possible to recover most low-rank matrices even when a small linear fraction of their entries has been badly corrupted and, furthermore, when only linear measurements of the corrupted matrix are available. Besides being one of the first theoretical results for this case, this dissertation opens up many exciting avenues for future research in this direction. Advisors/Committee Members: Ma, Yi (advisor), Ma, Yi (Committee Chair), Huang, Thomas S. (committee member), Milenkovic, Olgica (committee member), Meyn, Sean P. (committee member).

Subjects/Keywords: Image Alignment; Texture Rectification; Low-Rank Matrix Recovery; Convex Optimization; Photometric Stereo; Principal Component Pursuit

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APA (6^th Edition):

Balasubramanian, A. (2012). Applications of low-rank matrix recovery methods in computer vision. (Doctoral Dissertation). University of Illinois – Urbana-Champaign. Retrieved from http://hdl.handle.net/2142/31929

Chicago Manual of Style (16^th Edition):

Balasubramanian, Arvind. “Applications of low-rank matrix recovery methods in computer vision.” 2012. Doctoral Dissertation, University of Illinois – Urbana-Champaign. Accessed January 20, 2021. http://hdl.handle.net/2142/31929.

MLA Handbook (7^th Edition):

Balasubramanian, Arvind. “Applications of low-rank matrix recovery methods in computer vision.” 2012. Web. 20 Jan 2021.

Vancouver:

Balasubramanian A. Applications of low-rank matrix recovery methods in computer vision. [Internet] [Doctoral dissertation]. University of Illinois – Urbana-Champaign; 2012. [cited 2021 Jan 20]. Available from: http://hdl.handle.net/2142/31929.

Council of Science Editors:

Balasubramanian A. Applications of low-rank matrix recovery methods in computer vision. [Doctoral Dissertation]. University of Illinois – Urbana-Champaign; 2012. Available from: http://hdl.handle.net/2142/31929

Georgia Tech

14. Zhou, Fan. Statistical inference for high dimensional data with low rank structure.

Degree: PhD, Mathematics, 2018, Georgia Tech

URL: http://hdl.handle.net/1853/60750

► We study two major topics on statistical inference for high dimensional data with low rank structure occurred in many machine learning and statistics applications. The… (more)

Subjects/Keywords: Nonparametric statistics; Matrix completion; Low rank; Nuclear norm; Tensor; Singular vector perturbation

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Zhou, F. (2018). Statistical inference for high dimensional data with low rank structure. (Doctoral Dissertation). Georgia Tech. Retrieved from http://hdl.handle.net/1853/60750

Chicago Manual of Style (16^th Edition):

Zhou, Fan. “Statistical inference for high dimensional data with low rank structure.” 2018. Doctoral Dissertation, Georgia Tech. Accessed January 20, 2021. http://hdl.handle.net/1853/60750.

MLA Handbook (7^th Edition):

Zhou, Fan. “Statistical inference for high dimensional data with low rank structure.” 2018. Web. 20 Jan 2021.

Vancouver:

Zhou F. Statistical inference for high dimensional data with low rank structure. [Internet] [Doctoral dissertation]. Georgia Tech; 2018. [cited 2021 Jan 20]. Available from: http://hdl.handle.net/1853/60750.

Council of Science Editors:

Zhou F. Statistical inference for high dimensional data with low rank structure. [Doctoral Dissertation]. Georgia Tech; 2018. Available from: http://hdl.handle.net/1853/60750

University of Texas – Austin

15. Bhojanapalli, Venkata Sesha Pavana Srinadh. Large scale matrix factorization with guarantees: sampling and bi-linearity.

Degree: PhD, Electrical and Computer Engineering, 2015, University of Texas – Austin

URL: http://hdl.handle.net/2152/32832

► Low rank matrix factorization is an important step in many high dimensional machine learning algorithms. Traditional algorithms for factorization do not scale well with the… (more)

Subjects/Keywords: Matrix completion; Non-convex optimization; Low rank approximation; Semi-definite optimization; Tensor factorization; Scalable algorithms

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APA (6^th Edition):

Bhojanapalli, V. S. P. S. (2015). Large scale matrix factorization with guarantees: sampling and bi-linearity. (Doctoral Dissertation). University of Texas – Austin. Retrieved from http://hdl.handle.net/2152/32832

Chicago Manual of Style (16^th Edition):

Bhojanapalli, Venkata Sesha Pavana Srinadh. “Large scale matrix factorization with guarantees: sampling and bi-linearity.” 2015. Doctoral Dissertation, University of Texas – Austin. Accessed January 20, 2021. http://hdl.handle.net/2152/32832.

MLA Handbook (7^th Edition):

Bhojanapalli, Venkata Sesha Pavana Srinadh. “Large scale matrix factorization with guarantees: sampling and bi-linearity.” 2015. Web. 20 Jan 2021.

Vancouver:

Bhojanapalli VSPS. Large scale matrix factorization with guarantees: sampling and bi-linearity. [Internet] [Doctoral dissertation]. University of Texas – Austin; 2015. [cited 2021 Jan 20]. Available from: http://hdl.handle.net/2152/32832.

Council of Science Editors:

Bhojanapalli VSPS. Large scale matrix factorization with guarantees: sampling and bi-linearity. [Doctoral Dissertation]. University of Texas – Austin; 2015. Available from: http://hdl.handle.net/2152/32832

University of Pennsylvania

16. Yang, Dan. Singular Value Decomposition for High Dimensional Data.

Degree: 2012, University of Pennsylvania

URL: https://repository.upenn.edu/edissertations/595

► Singular value decomposition is a widely used tool for dimension reduction in multivariate analysis. However, when used for statistical estimation in high-dimensional low rank matrix… (more)

Subjects/Keywords: Cross validation; Denoise; Low rank matrix approximation; PCA; Penalization; Thresholding; Statistics and Probability

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APA (6^th Edition):

Yang, D. (2012). Singular Value Decomposition for High Dimensional Data. (Thesis). University of Pennsylvania. Retrieved from https://repository.upenn.edu/edissertations/595

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^th Edition):

Yang, Dan. “Singular Value Decomposition for High Dimensional Data.” 2012. Thesis, University of Pennsylvania. Accessed January 20, 2021. https://repository.upenn.edu/edissertations/595.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^th Edition):

Yang, Dan. “Singular Value Decomposition for High Dimensional Data.” 2012. Web. 20 Jan 2021.

Vancouver:

Yang D. Singular Value Decomposition for High Dimensional Data. [Internet] [Thesis]. University of Pennsylvania; 2012. [cited 2021 Jan 20]. Available from: https://repository.upenn.edu/edissertations/595.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Yang D. Singular Value Decomposition for High Dimensional Data. [Thesis]. University of Pennsylvania; 2012. Available from: https://repository.upenn.edu/edissertations/595

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Michigan Technological University

17. Azzam, Joy. Sub-Sampled Matrix Approximations.

Degree: PhD, Department of Mathematical Sciences, 2020, Michigan Technological University

URL: https://digitalcommons.mtu.edu/etdr/1002

► Matrix approximations are widely used to accelerate many numerical algorithms. Current methods sample row (or column) spaces to reduce their computational footprint and approximate… (more)

Subjects/Keywords: Matrix Approximation; Inverse Matrix Approximation; Low Rank Approximation; Quasi-Newton; Randomized Numerical Linear Algebra; Preconditioner; Sub-Sampled; Other Applied Mathematics

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APA (6^th Edition):

Azzam, J. (2020). Sub-Sampled Matrix Approximations. (Doctoral Dissertation). Michigan Technological University. Retrieved from https://digitalcommons.mtu.edu/etdr/1002

Chicago Manual of Style (16^th Edition):

Azzam, Joy. “Sub-Sampled Matrix Approximations.” 2020. Doctoral Dissertation, Michigan Technological University. Accessed January 20, 2021. https://digitalcommons.mtu.edu/etdr/1002.

MLA Handbook (7^th Edition):

Azzam, Joy. “Sub-Sampled Matrix Approximations.” 2020. Web. 20 Jan 2021.

Vancouver:

Azzam J. Sub-Sampled Matrix Approximations. [Internet] [Doctoral dissertation]. Michigan Technological University; 2020. [cited 2021 Jan 20]. Available from: https://digitalcommons.mtu.edu/etdr/1002.

Council of Science Editors:

Azzam J. Sub-Sampled Matrix Approximations. [Doctoral Dissertation]. Michigan Technological University; 2020. Available from: https://digitalcommons.mtu.edu/etdr/1002

18. Amadeo, Lily. Large Scale Matrix Completion and Recommender Systems.

Degree: MS, 2015, Worcester Polytechnic Institute

URL: etd-090415-162439 ; https://digitalcommons.wpi.edu/etd-theses/1021

► "The goal of this thesis is to extend the theory and practice of matrix completion algorithms, and how they can be utilized, improved, and scaled… (more)

Subjects/Keywords: low rank matrix; robust principal component analysis; convex relaxation; principal component analysis; recommender systems; matrix completion

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APA (6^th Edition):

Amadeo, L. (2015). Large Scale Matrix Completion and Recommender Systems. (Thesis). Worcester Polytechnic Institute. Retrieved from etd-090415-162439 ; https://digitalcommons.wpi.edu/etd-theses/1021

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^th Edition):

Amadeo, Lily. “Large Scale Matrix Completion and Recommender Systems.” 2015. Thesis, Worcester Polytechnic Institute. Accessed January 20, 2021. etd-090415-162439 ; https://digitalcommons.wpi.edu/etd-theses/1021.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^th Edition):

Amadeo, Lily. “Large Scale Matrix Completion and Recommender Systems.” 2015. Web. 20 Jan 2021.

Vancouver:

Amadeo L. Large Scale Matrix Completion and Recommender Systems. [Internet] [Thesis]. Worcester Polytechnic Institute; 2015. [cited 2021 Jan 20]. Available from: etd-090415-162439 ; https://digitalcommons.wpi.edu/etd-theses/1021.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Amadeo L. Large Scale Matrix Completion and Recommender Systems. [Thesis]. Worcester Polytechnic Institute; 2015. Available from: etd-090415-162439 ; https://digitalcommons.wpi.edu/etd-theses/1021

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Georgia Tech

19. Lee, Joonseok. Local approaches for collaborative filtering.

Degree: PhD, Computer Science, 2015, Georgia Tech

URL: http://hdl.handle.net/1853/53846

► Recommendation systems are emerging as an important business application as the demand for personalized services in E-commerce increases. Collaborative filtering techniques are widely used for… (more)

▼ Recommendation systems are emerging as an important business application as the demand for personalized services in E-commerce increases. Collaborative filtering techniques are widely used for predicting a user's preference or generating a list of items to be recommended. In this thesis, we develop several new approaches for collaborative filtering based on model combination and kernel smoothing. Specifically, we start with an experimental study that compares a wide variety of CF methods under different conditions. Based on this study, we formulate a combination model similar to boosting but where the combination coefficients are functions rather than constant. In another contribution we formulate and analyze a local variation of matrix factorization. This formulation constructs multiple local matrix factorization models and then combines them into a global model. This formulation is based on the local low-rank assumption, a slightly different but more plausible assumption about the rating matrix. We apply this assumption to both rating prediction and ranking problems, with both empirical validations and theoretical analysis. We contribute with this thesis in four aspects. First, the local approaches we present significantly improve the accuracy of recommendations both in rating prediction and ranking problems. Second, with the more realistic local low-rank assumption, we fundamentally change the underlying assumption for matrix factorization-based recommendation systems. Third, we present highly efficient and scalable algorithms which take advantage of parallelism, suited for recent large scale datasets. Lastly, we provide an open source software implementing the local approaches in this thesis as well as many other recent recommendation algorithms, which can be used both in research and production. Advisors/Committee Members: Chau, Duen Horng (Polo) (advisor), Lebanon, Guy (committee member), Zha, Hongyuan (committee member), Song, Le (committee member), Singer, Yoram (committee member).

Subjects/Keywords: Recommendation systems; Collaborative filtering; Machine learning; Local low-rank assumption; Matrix factorization; Matrix approximation; Ensemble collaborative ranking

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APA (6^th Edition):

Lee, J. (2015). Local approaches for collaborative filtering. (Doctoral Dissertation). Georgia Tech. Retrieved from http://hdl.handle.net/1853/53846

Chicago Manual of Style (16^th Edition):

Lee, Joonseok. “Local approaches for collaborative filtering.” 2015. Doctoral Dissertation, Georgia Tech. Accessed January 20, 2021. http://hdl.handle.net/1853/53846.

MLA Handbook (7^th Edition):

Lee, Joonseok. “Local approaches for collaborative filtering.” 2015. Web. 20 Jan 2021.

Vancouver:

Lee J. Local approaches for collaborative filtering. [Internet] [Doctoral dissertation]. Georgia Tech; 2015. [cited 2021 Jan 20]. Available from: http://hdl.handle.net/1853/53846.

Council of Science Editors:

Lee J. Local approaches for collaborative filtering. [Doctoral Dissertation]. Georgia Tech; 2015. Available from: http://hdl.handle.net/1853/53846

20. MIAO WEIMIN. Matrix Completion Models with Fixed Basis Coefficients and Rank Regularized Problems with Hard Constraints.

Degree: 2013, National University of Singapore

URL: http://scholarbank.nus.edu.sg/handle/10635/37889

Subjects/Keywords: matrix completion; rank minimization; low rank; error bound; rank consistency; semi-nuclear norm

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APA (6^th Edition):

WEIMIN, M. (2013). Matrix Completion Models with Fixed Basis Coefficients and Rank Regularized Problems with Hard Constraints. (Thesis). National University of Singapore. Retrieved from http://scholarbank.nus.edu.sg/handle/10635/37889

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^th Edition):

WEIMIN, MIAO. “Matrix Completion Models with Fixed Basis Coefficients and Rank Regularized Problems with Hard Constraints.” 2013. Thesis, National University of Singapore. Accessed January 20, 2021. http://scholarbank.nus.edu.sg/handle/10635/37889.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^th Edition):

WEIMIN, MIAO. “Matrix Completion Models with Fixed Basis Coefficients and Rank Regularized Problems with Hard Constraints.” 2013. Web. 20 Jan 2021.

Vancouver:

WEIMIN M. Matrix Completion Models with Fixed Basis Coefficients and Rank Regularized Problems with Hard Constraints. [Internet] [Thesis]. National University of Singapore; 2013. [cited 2021 Jan 20]. Available from: http://scholarbank.nus.edu.sg/handle/10635/37889.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

WEIMIN M. Matrix Completion Models with Fixed Basis Coefficients and Rank Regularized Problems with Hard Constraints. [Thesis]. National University of Singapore; 2013. Available from: http://scholarbank.nus.edu.sg/handle/10635/37889

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

University of Manchester

21. Borsdorf, Ruediger. Structured matrix nearness problems : theory and algorithms.

Degree: PhD, 2012, University of Manchester

URL: https://www.research.manchester.ac.uk/portal/en/theses/structured-matrix-nearness-problemstheory-and-algorithms(554f944d-9a78-4b54-90c2-1ef06866c402).html ; http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.554165

► In many areas of science one often has a given matrix, representing for example a measured data set and is required to find a matrix… (more)

▼ In many areas of science one often has a given matrix, representing for example a measured data set and is required to find a matrix that is closest in a suitable norm to the matrix and possesses additionally a structure, inherited from the model used or coming from the application. We call these problems structured matrix nearness problems. We look at three different groups of these problems that come from real applications, analyze the properties of the corresponding matrix structure, and propose algorithms to solve them efficiently. The first part of this thesis concerns the nearness problem of finding the nearest k factor correlation matrix C(X) = diag(I_n -XX T)+XX T to a given symmetric matrix, subject to natural nonlinear constraints on the elements of the n x k matrix X, where distance is measured in the Frobenius norm. Such problems arise, for example, when one is investigating factor models of collateralized debt obligations (CDOs) or multivariate time series. We examine several algorithms for solving the nearness problem that differ in whether or not they can take account of the nonlinear constraints and in their convergence properties. Our numerical experiments show that the performance of the methods depends strongly on the problem, but that, among our tested methods, the spectral projected gradient method is the clear winner. In the second part we look at two two-sided optimization problems where the matrix of unknowns Y ε R {n x p} lies in the Stiefel manifold. These two problems come from an application in atomic chemistry where one is looking for atomic orbitals with prescribed occupation numbers. We analyze these two problems, propose an analytic optimal solution of the first and show that an optimal solution of the second problem can be found by solving a convex quadratic programming problem with box constraints and p unknowns. We prove that the latter problem can be solved by the active-set method in at most 2p iterations. Subsequently, we analyze the set of optimal solutions C}= {Y ε R n x p:Y TY=I_p,Y TNY=D} of the first problem for N symmetric and D diagonal and find that a slight modification of it is a Riemannian manifold. We derive the geometric objects required to make an optimization over this manifold possible. We propose an augmented Lagrangian-based algorithm that uses these geometric tools and allows us to optimize an arbitrary smooth function over C. This algorithm can be used to select a particular solution out of the latter set C by posing a new optimization problem. We compare it numerically with a similar algorithm that ,however, does not apply these geometric tools and find that our algorithm yields better performance. The third part is devoted to low rank nearness problems in the Q-norm, where the matrix of interest is additionally of linear structure, meaning it lies in the set spanned by s predefined matrices U₁,..., U_s ε {0,1} n x p. These problems are often associated with model reduction, for example in speech encoding, filter design, or latent semantic indexing. We…

Subjects/Keywords: 025.04; correlation matrix; factor structure; matrix embedding; Stiefel manifold; linearly structured matrix; Grassmannian manifold; low rank; optimization over manifolds; matrix nearness problems

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APA (6^th Edition):

Borsdorf, R. (2012). Structured matrix nearness problems : theory and algorithms. (Doctoral Dissertation). University of Manchester. Retrieved from https://www.research.manchester.ac.uk/portal/en/theses/structured-matrix-nearness-problemstheory-and-algorithms(554f944d-9a78-4b54-90c2-1ef06866c402).html ; http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.554165

Chicago Manual of Style (16^th Edition):

Borsdorf, Ruediger. “Structured matrix nearness problems : theory and algorithms.” 2012. Doctoral Dissertation, University of Manchester. Accessed January 20, 2021. https://www.research.manchester.ac.uk/portal/en/theses/structured-matrix-nearness-problemstheory-and-algorithms(554f944d-9a78-4b54-90c2-1ef06866c402).html ; http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.554165.

MLA Handbook (7^th Edition):

Borsdorf, Ruediger. “Structured matrix nearness problems : theory and algorithms.” 2012. Web. 20 Jan 2021.

Vancouver:

Borsdorf R. Structured matrix nearness problems : theory and algorithms. [Internet] [Doctoral dissertation]. University of Manchester; 2012. [cited 2021 Jan 20]. Available from: https://www.research.manchester.ac.uk/portal/en/theses/structured-matrix-nearness-problemstheory-and-algorithms(554f944d-9a78-4b54-90c2-1ef06866c402).html ; http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.554165.

Council of Science Editors:

Borsdorf R. Structured matrix nearness problems : theory and algorithms. [Doctoral Dissertation]. University of Manchester; 2012. Available from: https://www.research.manchester.ac.uk/portal/en/theses/structured-matrix-nearness-problemstheory-and-algorithms(554f944d-9a78-4b54-90c2-1ef06866c402).html ; http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.554165

University of Michigan

22. Nayar, Himanshu. Application of Random Matrix Theory to Multimodal Fusion.

Degree: PhD, Electrical Engineering: Systems, 2017, University of Michigan

URL: http://hdl.handle.net/2027.42/140962

► Multimodal data fusion is an interesting problem and its applications can be seen in image processing, signal processing and machine learning. In applications where we… (more)

Subjects/Keywords: Data driven fusion; Random Matrix Theory; Factor analysis; Low rank decomposition; Clique recovery; Electrical Engineering; Engineering

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APA (6^th Edition):

Nayar, H. (2017). Application of Random Matrix Theory to Multimodal Fusion. (Doctoral Dissertation). University of Michigan. Retrieved from http://hdl.handle.net/2027.42/140962

Chicago Manual of Style (16^th Edition):

Nayar, Himanshu. “Application of Random Matrix Theory to Multimodal Fusion.” 2017. Doctoral Dissertation, University of Michigan. Accessed January 20, 2021. http://hdl.handle.net/2027.42/140962.

MLA Handbook (7^th Edition):

Nayar, Himanshu. “Application of Random Matrix Theory to Multimodal Fusion.” 2017. Web. 20 Jan 2021.

Vancouver:

Nayar H. Application of Random Matrix Theory to Multimodal Fusion. [Internet] [Doctoral dissertation]. University of Michigan; 2017. [cited 2021 Jan 20]. Available from: http://hdl.handle.net/2027.42/140962.

Council of Science Editors:

Nayar H. Application of Random Matrix Theory to Multimodal Fusion. [Doctoral Dissertation]. University of Michigan; 2017. Available from: http://hdl.handle.net/2027.42/140962

Wayne State University

23. Wang, Lijun. Complex data analytics via sparse, low-rank matrix approximation.

Degree: PhD, Computer Science, 2012, Wayne State University

URL: https://digitalcommons.wayne.edu/oa_dissertations/583

► Today, digital data is accumulated at a faster than ever speed in science, engineering, biomedicine, and real-world sensing. Data mining provides us an effective… (more)

▼ Today, digital data is accumulated at a faster than ever speed in science, engineering, biomedicine, and real-world sensing. Data mining provides us an effective way for the exploration and analysis of hidden patterns from these data for a broad spectrum of applications. Usually, these datasets share one prominent characteristic: tremendous in size with tens of thousands of objects and features. In addition, data is not only collected over a period of time, but the relationship between data points can change over that period too. Besides, knowledge is very sparsely encoded because the patterns are usually active only in a local area. The ubiquitous phenomenon of massive, dynamic, and sparse data imposes considerable challenges in data mining research. Recently, techniques that can expand the human ability to comprehend large-scale data have attracted significant attention in the research community. In this dissertation, we present approaches to solve the problems of complex data analysis in various applications. Specifically, we have achieved the following: 1) we develop Exemplar-based low-rank sparse Matrix Decomposition (EMD), a novel method for fast clustering large-scale data by incorporating low-rank approximations into matrix decomposition-based clustering; 2) we propose ECKF, a general model for large-scale Evolutionary Clustering based on low-rank Kernel matrix Factorization; by monitoring the low-rank approximation errors at every time step, ECKF can analyze if the underlying structure of the data or the nature of the relationship between the data points has changed over different time steps; based on this, a decision to either succeed the previous clustering or perform a new clustering is made; 3) we propose a Multi-level Low-rank Approximation (MLA) framework for fast spectral clustering, which is empirically shown to cluster large-scale data very efficiently; 4) we extend the MLA framework with a non-linear kernel and apply it to HD image segmentation; with sufficient data samples selected by fast sampling strategy, our method shows superior performance compared with other leading approximate spectral clusterings; 5) we develop a fast algorithm to detect abnormal crowd behavior in surveillance videos by employing low-rank matrix approximations to model crowd behavior patterns; through experiments performed on simulation crowd videos, we demonstrate the effectiveness of our method. Advisors/Committee Members: Ming Dong.

Subjects/Keywords: abnormal event detection, Big data analytics, clustering, evolutionary clustering, large-scale data analysis, low-rank matrix approximation; Computer Sciences

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APA (6^th Edition):

Wang, L. (2012). Complex data analytics via sparse, low-rank matrix approximation. (Doctoral Dissertation). Wayne State University. Retrieved from https://digitalcommons.wayne.edu/oa_dissertations/583

Chicago Manual of Style (16^th Edition):

Wang, Lijun. “Complex data analytics via sparse, low-rank matrix approximation.” 2012. Doctoral Dissertation, Wayne State University. Accessed January 20, 2021. https://digitalcommons.wayne.edu/oa_dissertations/583.

MLA Handbook (7^th Edition):

Wang, Lijun. “Complex data analytics via sparse, low-rank matrix approximation.” 2012. Web. 20 Jan 2021.

Vancouver:

Wang L. Complex data analytics via sparse, low-rank matrix approximation. [Internet] [Doctoral dissertation]. Wayne State University; 2012. [cited 2021 Jan 20]. Available from: https://digitalcommons.wayne.edu/oa_dissertations/583.

Council of Science Editors:

Wang L. Complex data analytics via sparse, low-rank matrix approximation. [Doctoral Dissertation]. Wayne State University; 2012. Available from: https://digitalcommons.wayne.edu/oa_dissertations/583

Northeastern University

24. Shao, Ming. Efficient transfer feature learning and its applications on social media.

Degree: PhD, Department of Electrical and Computer Engineering, 2016, Northeastern University

URL: http://hdl.handle.net/2047/D20213068

► In the era of social media, more and more social characteristics are conveyed by multimedia, i.e., images, videos, audios, and webpages with rich media information.… (more)

▼ In the era of social media, more and more social characteristics are conveyed by multimedia, i.e., images, videos, audios, and webpages with rich media information. With the emergence and popularity of the Internet, collecting multimedia data cannot be much easier than today. However, tons of data available on-line are lack of organization and their tags or labels are sparse or loosely organized for certain problems we are interested in. The basic question is how to learn discriminant features or representations given very limited or poor training data for social media analytics.; In this thesis, we focus on the popular social media data such as, face, object, digital number images, and study the problems of social media analytics in two lines: (1) developing efficient and effective machine learning tools given limited or poor training data by considering the structure of the data from different domains, (2) applying existing or developed machine learning tools to novel social media problems, e.g., kinship verification, family photo understanding. These two lines are detailed as followings:; For knowledge-based machine learning algorithms, label or tag is critical in training the discriminative model. However, labeling data is not an easy task because these data are either too costly to obtain or too expensive to hand-label. For that reason, researchers use labeled, yet relevant, data from different databases to facilitate learning process. This is exactly transfer learning that studies how to transfer the knowledge gained from an existing and well-established data (source) to a new problem (target). To this end, we propose two novel methods to align the structure of the source and target data in the learned subspace to mitigate the domain divergence, which provide robust, scalable, and discriminant features for learning tasks. The first method utilizes a low-rank regularizer to guide the discriminant subspace learning. The second method manages to take advantage of the unlabeled source data for efficient large-scale transfer learning.; Transfer learning approaches above are appropriate for social media analytics problems such as kinship verification. A critical observation is that faces of parents captured while they were young are more like their children's compared with images captured when they are old. Therefore, we can readily apply the proposed transfer learning methods to kinship verification defined above, where kin relation between young parent and child is the source problem, while that between old parent and child is the target. Promising research outcome can be extended to real-world applications: family album management, image retrieval and annotation, missing children search, etc.

Subjects/Keywords: domain adaptation; kinship verification; low-rank matrix analysis; transfer learning; Computer vision; Machine learning; Social media; Biometric identification; Mathematical models; Matrices

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APA (6^th Edition):

Shao, M. (2016). Efficient transfer feature learning and its applications on social media. (Doctoral Dissertation). Northeastern University. Retrieved from http://hdl.handle.net/2047/D20213068

Chicago Manual of Style (16^th Edition):

Shao, Ming. “Efficient transfer feature learning and its applications on social media.” 2016. Doctoral Dissertation, Northeastern University. Accessed January 20, 2021. http://hdl.handle.net/2047/D20213068.

MLA Handbook (7^th Edition):

Shao, Ming. “Efficient transfer feature learning and its applications on social media.” 2016. Web. 20 Jan 2021.

Vancouver:

Shao M. Efficient transfer feature learning and its applications on social media. [Internet] [Doctoral dissertation]. Northeastern University; 2016. [cited 2021 Jan 20]. Available from: http://hdl.handle.net/2047/D20213068.

Council of Science Editors:

Shao M. Efficient transfer feature learning and its applications on social media. [Doctoral Dissertation]. Northeastern University; 2016. Available from: http://hdl.handle.net/2047/D20213068

University of Illinois – Chicago

25. Bhoi, Amlaan. Invariant Kernels for Few-shot Learning.

Degree: 2019, University of Illinois – Chicago

URL: http://hdl.handle.net/10027/23714

► Recent advances in few-shot learning algorithms focus on the development of meta-learning or improvements in distance-based algorithms. However, the majority of these approaches do not… (more)

Subjects/Keywords: few-shot learning; invariance learning; adversarial learning; low-rank matrix approximation; image classification; deep learning; kernel learning

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APA (6^th Edition):

Bhoi, A. (2019). Invariant Kernels for Few-shot Learning. (Thesis). University of Illinois – Chicago. Retrieved from http://hdl.handle.net/10027/23714

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^th Edition):

Bhoi, Amlaan. “Invariant Kernels for Few-shot Learning.” 2019. Thesis, University of Illinois – Chicago. Accessed January 20, 2021. http://hdl.handle.net/10027/23714.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^th Edition):

Bhoi, Amlaan. “Invariant Kernels for Few-shot Learning.” 2019. Web. 20 Jan 2021.

Vancouver:

Bhoi A. Invariant Kernels for Few-shot Learning. [Internet] [Thesis]. University of Illinois – Chicago; 2019. [cited 2021 Jan 20]. Available from: http://hdl.handle.net/10027/23714.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Bhoi A. Invariant Kernels for Few-shot Learning. [Thesis]. University of Illinois – Chicago; 2019. Available from: http://hdl.handle.net/10027/23714

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

University of Iowa

26. Wang, Tianming. Non-convex methods for spectrally sparse signal reconstruction via low-rank Hankel matrix completion.

Degree: PhD, Applied Mathematical and Computational Sciences, 2018, University of Iowa

URL: https://ir.uiowa.edu/etd/6331

► Spectrally sparse signals arise in many applications of signal processing. A spectrally sparse signal is a mixture of a few undamped or damped complex… (more)

Subjects/Keywords: low-rank Hankel matrix completion; NMR spectroscopy; projected gradient descent; Riemannian optimization; spectrally sparse signals; Applied Mathematics

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APA (6^th Edition):

Wang, T. (2018). Non-convex methods for spectrally sparse signal reconstruction via low-rank Hankel matrix completion. (Doctoral Dissertation). University of Iowa. Retrieved from https://ir.uiowa.edu/etd/6331

Chicago Manual of Style (16^th Edition):

Wang, Tianming. “Non-convex methods for spectrally sparse signal reconstruction via low-rank Hankel matrix completion.” 2018. Doctoral Dissertation, University of Iowa. Accessed January 20, 2021. https://ir.uiowa.edu/etd/6331.

MLA Handbook (7^th Edition):

Wang, Tianming. “Non-convex methods for spectrally sparse signal reconstruction via low-rank Hankel matrix completion.” 2018. Web. 20 Jan 2021.

Vancouver:

Wang T. Non-convex methods for spectrally sparse signal reconstruction via low-rank Hankel matrix completion. [Internet] [Doctoral dissertation]. University of Iowa; 2018. [cited 2021 Jan 20]. Available from: https://ir.uiowa.edu/etd/6331.

Council of Science Editors:

Wang T. Non-convex methods for spectrally sparse signal reconstruction via low-rank Hankel matrix completion. [Doctoral Dissertation]. University of Iowa; 2018. Available from: https://ir.uiowa.edu/etd/6331

Rice University

27. Darvish Rouhani, Bita. A Resource-Aware Streaming-based Framework for Big Data Analysis.

Degree: MS, Engineering, 2015, Rice University

URL: http://hdl.handle.net/1911/87764

► The ever growing body of digital data is challenging conventional analytical techniques in machine learning, computer vision, and signal processing. Traditional analytical methods have been… (more)

▼ The ever growing body of digital data is challenging conventional analytical techniques in machine learning, computer vision, and signal processing. Traditional analytical methods have been mainly developed based on the assumption that designers can work with data within the confines of their own computing environment. The growth of big data, however, is changing that paradigm especially in scenarios where severe memory and computational resource constraints exist. This thesis aims at addressing major challenges in big data learning problem by devising a new customizable computing framework that holistically takes into account the data structure and underlying platform constraints. It targets a widely used class of analytical algorithms that model the data dependencies by iteratively updating a set of matrix parameters, including but not limited to most regression methods, expectation maximization, and stochastic optimizations, as well as the emerging deep learning techniques. The key to our approach is a customizable, streaming-based data projection methodology that adaptively transforms data into a new lower-dimensional embedding by simultaneously considering both data and hardware characteristics. It enables scalable data analysis and rapid prototyping of an arbitrary matrix-based learning task using a sparse-approximation of the collection that is constantly updated inline with the data arrival. Our work is supported by a set of user-friendly Application Programming Interfaces (APIs) that ensure automated adaptation of the proposed framework to various datasets and System on Chip (SoC) platforms including CPUs, GPUs, and FPGAs. Proof of concept evaluations using a variety of large contemporary datasets corroborate the practicability and scalability of our approach in resource-limited settings. For instance, our results demonstrate 50-fold improvement over the best known prior-art in terms of memory, energy, power, and runtime for training and execution of deep learning models in deployment of different sensing applications including indoor localization and speech recognition on constrained embedded platforms used in today's IoT enabled devices such as autonomous vehicles, robots, and smartphone. Advisors/Committee Members: Koushanfar, Farinaz (advisor), Aazhang, Behnaam (committee member), Baraniuk, Richard (committee member).

Subjects/Keywords: Streaming model; Big data; Dense matrix; Low-rank approximation; HW/SW co-design; Deep Learning; Scalable machine learning

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Darvish Rouhani, B. (2015). A Resource-Aware Streaming-based Framework for Big Data Analysis. (Masters Thesis). Rice University. Retrieved from http://hdl.handle.net/1911/87764

Chicago Manual of Style (16^th Edition):

Darvish Rouhani, Bita. “A Resource-Aware Streaming-based Framework for Big Data Analysis.” 2015. Masters Thesis, Rice University. Accessed January 20, 2021. http://hdl.handle.net/1911/87764.

MLA Handbook (7^th Edition):

Darvish Rouhani, Bita. “A Resource-Aware Streaming-based Framework for Big Data Analysis.” 2015. Web. 20 Jan 2021.

Vancouver:

Darvish Rouhani B. A Resource-Aware Streaming-based Framework for Big Data Analysis. [Internet] [Masters thesis]. Rice University; 2015. [cited 2021 Jan 20]. Available from: http://hdl.handle.net/1911/87764.

Council of Science Editors:

Darvish Rouhani B. A Resource-Aware Streaming-based Framework for Big Data Analysis. [Masters Thesis]. Rice University; 2015. Available from: http://hdl.handle.net/1911/87764

28. Vinyes, Marina. Convex matrix sparsity for demixing with an application to graphical model structure estimation : Parcimonie matricielle convexe pour les problèmes de démixage avec une application à l'apprentissage de structure de modèles graphiques.

Degree: Docteur es, Signal, Image, Automatique, 2018, Université Paris-Est

URL: http://www.theses.fr/2018PESC1130

►

En apprentissage automatique on a pour but d'apprendre un modèle, à partir de données, qui soit capable de faire des prédictions sur des nouvelles données… (more)

▼

En apprentissage automatique on a pour but d'apprendre un modèle, à partir de données, qui soit capable de faire des prédictions sur des nouvelles données (pas explorées auparavant). Pour obtenir un modèle qui puisse se généraliser sur les nouvelles données, et éviter le sur-apprentissage, nous devons restreindre le modèle. Ces restrictions sont généralement une connaissance a priori de la structure du modèle. Les premières approches considérées dans la littérature sont la régularisation de Tikhonov et plus tard le Lasso pour induire de la parcimonie dans la solution. La parcimonie fait partie d'un concept fondamental en apprentissage automatique. Les modèles parcimonieux sont attrayants car ils offrent plus d'interprétabilité et une meilleure généralisation (en évitant le sur-apprentissage) en induisant un nombre réduit de paramètres dans le modèle. Au-delà de la parcimonie générale et dans de nombreux cas, les modèles sont structurellement contraints et ont une représentation simple de certains éléments fondamentaux, comme par exemple une collection de vecteurs, matrices ou tenseurs spécifiques. Ces éléments fondamentaux sont appelés atomes. Dans ce contexte, les normes atomiques fournissent un cadre général pour estimer ce type de modèles. périodes de modèles. Le but de cette thèse est d'utiliser le cadre de parcimonie convexe fourni par les normes atomiques pour étudier une forme de parcimonie matricielle. Tout d'abord, nous développons un algorithme efficace basé sur les méthodes de Frank-Wolfe et qui est particulièrement adapté pour résoudre des problèmes convexes régularisés par une norme atomique. Nous nous concentrons ensuite sur l'estimation de la structure des modèles graphiques gaussiens, où la structure du modèle est encodée dans la matrice de précision et nous étudions le cas avec des variables manquantes. Nous proposons une formulation convexe avec une approche algorithmique et fournissons un résultat théorique qui énonce les conditions nécessaires pour récupérer la structure souhaitée. Enfin, nous considérons le problème de démixage d'un signal en deux composantes ou plus via la minimisation d’une somme de normes ou de jauges, encodant chacune la structure a priori des composants à récupérer. En particulier, nous fournissons une garantie de récupération exacte dans le cadre sans bruit, basée sur des mesures d'incohérence

The goal of machine learning is to learn a model from some data that will make accurate predictions on data that it has not seen before. In order to obtain a model that will generalize on new data, and avoid overfitting, we need to restrain the model. These restrictions are usually some a priori knowledge of the structure of the model. First considered approaches included a regularization, first ridge regression and later Lasso regularization for inducing sparsity in the solution. Sparsity, also known as parsimony, has emerged as a fundamental concept in machine learning. Parsimonious models are appealing since they provide more interpretability and better generalization (avoid…

Advisors/Committee Members: Komodakis, Nikos (thesis director).

Subjects/Keywords: Normes atomiques; Optimisation convexe; Parcimonie matricielle; Rang faible; Parcimonie; Atomic norms; Convex optimisation; Matrix sparsity; Low rank; Sparsity

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APA (6^th Edition):

Vinyes, M. (2018). Convex matrix sparsity for demixing with an application to graphical model structure estimation : Parcimonie matricielle convexe pour les problèmes de démixage avec une application à l'apprentissage de structure de modèles graphiques. (Doctoral Dissertation). Université Paris-Est. Retrieved from http://www.theses.fr/2018PESC1130

Chicago Manual of Style (16^th Edition):

Vinyes, Marina. “Convex matrix sparsity for demixing with an application to graphical model structure estimation : Parcimonie matricielle convexe pour les problèmes de démixage avec une application à l'apprentissage de structure de modèles graphiques.” 2018. Doctoral Dissertation, Université Paris-Est. Accessed January 20, 2021. http://www.theses.fr/2018PESC1130.

MLA Handbook (7^th Edition):

Vinyes, Marina. “Convex matrix sparsity for demixing with an application to graphical model structure estimation : Parcimonie matricielle convexe pour les problèmes de démixage avec une application à l'apprentissage de structure de modèles graphiques.” 2018. Web. 20 Jan 2021.

Vancouver:

Vinyes M. Convex matrix sparsity for demixing with an application to graphical model structure estimation : Parcimonie matricielle convexe pour les problèmes de démixage avec une application à l'apprentissage de structure de modèles graphiques. [Internet] [Doctoral dissertation]. Université Paris-Est; 2018. [cited 2021 Jan 20]. Available from: http://www.theses.fr/2018PESC1130.

Council of Science Editors:

Vinyes M. Convex matrix sparsity for demixing with an application to graphical model structure estimation : Parcimonie matricielle convexe pour les problèmes de démixage avec une application à l'apprentissage de structure de modèles graphiques. [Doctoral Dissertation]. Université Paris-Est; 2018. Available from: http://www.theses.fr/2018PESC1130

University of Lund

29. Grussler, Christian. Rank Reduction with Convex Constraints.

Degree: 2017, University of Lund

URL: https://lup.lub.lu.se/record/54cb814f-59fe-4bc9-a7ef-773cbcf06889 ; https://portal.research.lu.se/ws/files/19595129/Thesis.pdf

► This thesis addresses problems which require low-rank solutions under convex constraints. In particular, the focus lies on model reduction of positive systems, as well as… (more)

▼ This thesis addresses problems which require low-rank solutions under convex constraints. In particular, the focus lies on model reduction of positive systems, as well as finite dimensional optimization problems that are convex, apart from a low-rank constraint. Traditional model reduction techniques try to minimize the error between the original and the reduced system. Typically, the resulting reduced models, however, no longer fulfill physically meaningful constraints. This thesis considers the problem of model reduction with internal and external positivity constraints. Both problems are solved by means of balanced truncation. While internal positivity is shown to be preserved by a symmetry property; external positivity preservation is accomplished by deriving a modified balancing approach based on ellipsoidal cone invariance.In essence, positivity preserving model reduction attempts to find an infinite dimensional low-rank approximation that preserves nonnegativity, as well as Hankel structure. Due to the non-convexity of the low-rank constraint, this problem is even challenging in a finite dimensional setting. In addition to model reduction, the present work also considers such finite dimensional low-rank optimization problems with convex constraints. These problems frequently appear in applications such as image compression, multivariate linear regression, matrix completion and many more. The main idea of this thesis is to derive the largest convex minorizers of rank-constrained unitarily invariant norms. These minorizers can be used to construct optimal convex relaxations for the original non-convex problem. Unlike other methods such as nuclear norm regularization, this approach benefits from having verifiable a posterior conditions for which a solution to the convex relaxation and the corresponding non-convex problem coincide. It is shown that this applies to various numerical examples of well-known low-rank optimization problems. In particular, the proposed convex relaxation performs significantly better than nuclear norm regularization. Moreover, it can be observed that a careful choice among the proposed convex relaxations may have a tremendous positive impact on matrix completion. Computational tractability of the proposed approach is accomplished in two ways. First, the considered relaxations are shown to be representable by semi-definite programs. Second, it is shown how to compute the proximal mappings, for both, the convex relaxations, as well as the non-convex problem. This makes it possible to apply first order method such as so-called Douglas-Rachford splitting. In addition to the convex case, where global convergence of this algorithm is guaranteed, conditions for local convergence in the non-convex setting are presented. Finally, it is shown that the findings of this thesis also extend to the general class of so-called atomic norms that allow us to cover other non-convex constraints.

Subjects/Keywords: Control Engineering; low-rank approximation; model reduction; non-convex optimization; Douglas-Rachford; matrix completion; overlapping norm; k-support norm; atomic norm

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Grussler, C. (2017). Rank Reduction with Convex Constraints. (Doctoral Dissertation). University of Lund. Retrieved from https://lup.lub.lu.se/record/54cb814f-59fe-4bc9-a7ef-773cbcf06889 ; https://portal.research.lu.se/ws/files/19595129/Thesis.pdf

Chicago Manual of Style (16^th Edition):

Grussler, Christian. “Rank Reduction with Convex Constraints.” 2017. Doctoral Dissertation, University of Lund. Accessed January 20, 2021. https://lup.lub.lu.se/record/54cb814f-59fe-4bc9-a7ef-773cbcf06889 ; https://portal.research.lu.se/ws/files/19595129/Thesis.pdf.

MLA Handbook (7^th Edition):

Grussler, Christian. “Rank Reduction with Convex Constraints.” 2017. Web. 20 Jan 2021.

Vancouver:

Grussler C. Rank Reduction with Convex Constraints. [Internet] [Doctoral dissertation]. University of Lund; 2017. [cited 2021 Jan 20]. Available from: https://lup.lub.lu.se/record/54cb814f-59fe-4bc9-a7ef-773cbcf06889 ; https://portal.research.lu.se/ws/files/19595129/Thesis.pdf.

Council of Science Editors:

Grussler C. Rank Reduction with Convex Constraints. [Doctoral Dissertation]. University of Lund; 2017. Available from: https://lup.lub.lu.se/record/54cb814f-59fe-4bc9-a7ef-773cbcf06889 ; https://portal.research.lu.se/ws/files/19595129/Thesis.pdf

University of Lund

30. Jiang, Fangyuan. Low Rank Matrix Factorization and Relative Pose Problems in Computer Vision.

Degree: 2015, University of Lund

URL: https://lup.lub.lu.se/record/5368358 ; https://portal.research.lu.se/ws/files/5379996/5368400.pdf

► This thesis is focused on geometric computer vision problems. The first part of the thesis aims at solving one fundamental problem, namely low-rank matrix factorization.… (more)

Subjects/Keywords: Mathematics; Computer Vision and Robotics (Autonomous Systems); Geometric Computer Vision; Low-rank Matrix Factorization; Relative Pose

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Jiang, F. (2015). Low Rank Matrix Factorization and Relative Pose Problems in Computer Vision. (Doctoral Dissertation). University of Lund. Retrieved from https://lup.lub.lu.se/record/5368358 ; https://portal.research.lu.se/ws/files/5379996/5368400.pdf

Chicago Manual of Style (16^th Edition):

Jiang, Fangyuan. “Low Rank Matrix Factorization and Relative Pose Problems in Computer Vision.” 2015. Doctoral Dissertation, University of Lund. Accessed January 20, 2021. https://lup.lub.lu.se/record/5368358 ; https://portal.research.lu.se/ws/files/5379996/5368400.pdf.

MLA Handbook (7^th Edition):

Jiang, Fangyuan. “Low Rank Matrix Factorization and Relative Pose Problems in Computer Vision.” 2015. Web. 20 Jan 2021.

Vancouver:

Jiang F. Low Rank Matrix Factorization and Relative Pose Problems in Computer Vision. [Internet] [Doctoral dissertation]. University of Lund; 2015. [cited 2021 Jan 20]. Available from: https://lup.lub.lu.se/record/5368358 ; https://portal.research.lu.se/ws/files/5379996/5368400.pdf.

Council of Science Editors:

Jiang F. Low Rank Matrix Factorization and Relative Pose Problems in Computer Vision. [Doctoral Dissertation]. University of Lund; 2015. Available from: https://lup.lub.lu.se/record/5368358 ; https://portal.research.lu.se/ws/files/5379996/5368400.pdf

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Open Access Theses and Dissertations