subject:(Stiefel manifolds)

1. Σταθά, Μαρίνα. Μελέτη γεωμετρίας σφαιρών και πολλαπλοτήτων Stiefel.

Degree: 2013, University of Patras

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Σκοπός της εργασίας μας είναι η μελέτη κάποιων αναγωγικών χώρων που παρουσιάζουν ενδιαφέρουσα γεωμετρία. Συγκεκριμένα, μελετάμε τη γεωμετρία της σφαίρας Sⁿ όταν αυτή είναι αμφιδιαφορική… (more)

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Σκοπός της εργασίας μας είναι η μελέτη κάποιων αναγωγικών χώρων που παρουσιάζουν ενδιαφέρουσα γεωμετρία. Συγκεκριμένα, μελετάμε τη γεωμετρία της σφαίρας Sⁿ όταν αυτή είναι αμφιδιαφορική με έναν χώρο πηλίκο G/K και την γεωμετρία των πολλαπλοτήτων Stiefel SO(n)/SO(n-k) (το σύνολο όλων των k-πλαισίων του Rⁿ). Ένας ομογενής χώρος αποτελεί επέκταση των ομάδων Lie, καθώς είναι μια λεία πολλαπλότητα M στην οποία δρα μεταβατικά μια ομάδα Lie G. Κάθε τέτοιος χώρος δίνεται ως M = G/K, όπου K = {g∈ G : gp = p} (p ∈ M). Η βασική γεωμετρική ιδιότητα των ομογενών χώρων είναι ότι αν γνωρίζουμε την τιμή κάποιου γεωμετρικού μεγέθους σε ένα σημείο του χώρου, τότε μπορούμε να υπολογίσουμε την τιμή του μεγέθους αυτού σε οποιοδήποτε άλλο σημείο. Το ιδιαίτερο χαρακτηριστικό των αναγωγικών χώρων G/K είναι ότι υπάρχει ένας Ad(K)-αναλλοίωτος υπόχωρος της άλγεβρας Lie(G). Η περιγραφή όλων των μεταβατικών δράσεων μιας ομάδας Lie σε μια πολλαπλότητα M αποτελεί ένα δύσκολο πρόβλημα. Για την περίπτωση των σφαιρών αυτές έχουν περιγραφτεί το 1953 από τους Montgomery-Samelson-Borel. Στην εργασία μας μελετάμε τη γεωμετρία (καμπυλότητες, μετρικές Einstein) των σφαιρών S³, S⁵ όταν αυτές είναι αμφιδιαφορικές με τα πηλίκα S³ = SO(4)/SO(3) = SU(2) και S⁵ = SO(6)/SO(5) = SU(3)/SU(2). Αντίστοιχα προβλήματα εξετάζονται για τις πολλαπλότητες Stiefel SO(n)/SO(n-k), όπου η περιγραφή όλων των SO(n)-αναλλοίωτων μετρικών παρουσιάζει δυσκολία, λόγω του ότι η ισοτροπική αναπαράστασή τους περιέχει ισοδύναμα υποπρότυπα. Μελετάμε για ποιές από τις συγκεκριμένες πολλαπλότητες η μετρική που επάγεται από τη μορφή Killing είναι μετρική Einstein και περιγράφουμε αναλυτικά τις διαγώνιες SO(n)-αναλλοίωτες μετρικές Einstein στις πολλαπλότητες SO(n)/SO(n-2). Επιπλέον παρουσιάζουμε και ένα καινούργιο αποτέλεσμα, ότι στην πολλαπλότητα SO(5)/SO(2) οι μοναδικές SO(5)-αναλλοίωτες μετρικές Einstein είναι οι μετρικές που είχαν βρεθεί από τον Jensen το 1973.

The purpose of our work is to study homogeneous spaces that present interesting geometry. These include the geometry of the sphere Sⁿ diffeomorphic to a quotient space G/K and the geometry of Stiefel manifolds SO(n)/SO(n-k) (the set of all k-planes in Rⁿ). A homogeneous space is a smooth manifold M in which a Lie group acts transitively. Any such space is given as M = G/K where K = {g∈ G : gp = p} (p∈ M). The basic geometric property of homogeneous space is that if we know the value of a geometrical object at a point of the space, then we can estimate the value of thiw quantity at any other point. The special feature of reductive homogeneous space G/K is that there exists an Ad(K)-invariant subspace of the Lie algebra Lie(G). The description of all transitive actions of a Lie group into a manifold M is a difficult problem. In the case of spheres such actions have been described in 1953 by the Montgomery, Samelson and Borel. In our work we study the geometry (curvature, Einstein metrics) of the sphere S³ = SO(4)/SO(3) = SU(2), S⁵ = SO(6)/SO(5) = SU(3)/SU(2). Similar problems are examined for the Stiefel…

Advisors/Committee Members: Αρβανιτογεώργος, Ανδρέας, Statha, Marina, Παπαντωνίου, Βασίλειος, Κοτσιώλης, Αθανάσιος.

Subjects/Keywords: Ομάδες Lie; Ομογενείς χώροι; Αναλλοίωτες μετρικές; Μετρικές Einstein; Πολλαπλότητες Stiefel; 516.35; Lie groups; Homogeneous spaces; Invariant metrics; Einstein metrics; Stiefel manifolds

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

APA (6^th Edition):

Σταθά, �. (2013). Μελέτη γεωμετρίας σφαιρών και πολλαπλοτήτων Stiefel. (Masters Thesis). University of Patras. Retrieved from http://hdl.handle.net/10889/7985

Chicago Manual of Style (16^th Edition):

Σταθά, Μαρίνα. “Μελέτη γεωμετρίας σφαιρών και πολλαπλοτήτων Stiefel.” 2013. Masters Thesis, University of Patras. Accessed February 26, 2021. http://hdl.handle.net/10889/7985.

MLA Handbook (7^th Edition):

Σταθά, Μαρίνα. “Μελέτη γεωμετρίας σφαιρών και πολλαπλοτήτων Stiefel.” 2013. Web. 26 Feb 2021.

Vancouver:

Σταθά �. Μελέτη γεωμετρίας σφαιρών και πολλαπλοτήτων Stiefel. [Internet] [Masters thesis]. University of Patras; 2013. [cited 2021 Feb 26]. Available from: http://hdl.handle.net/10889/7985.

Council of Science Editors:

Σταθά �. Μελέτη γεωμετρίας σφαιρών και πολλαπλοτήτων Stiefel. [Masters Thesis]. University of Patras; 2013. Available from: http://hdl.handle.net/10889/7985

2. Barbosa, Alex Melges [UNESP]. Classes de Stiefel-Whitney e de Euler.

Degree: 2017, Universidade Estadual Paulista

URL: http://hdl.handle.net/11449/148923

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Neste trabalho, apresentaremos uma descrição axiomática das classes de Stiefel-Whitney e, assumindo válidos estes axiomas, mostraremos algumas de suas aplicações. Posteriormente, definiremos as classes de… (more)

Subjects/Keywords: Variedades C∞; Fibrados Vetoriais; Classes de Stiefel-Whitney; Fibrados Orientados; Classe de Euler; C∞ Manifolds; Vector Bundles; Stiefel-Whitney Classes; Oriented Bundles; Euler Class

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

APA (6^th Edition):

Barbosa, A. M. [. (2017). Classes de Stiefel-Whitney e de Euler. (Thesis). Universidade Estadual Paulista. Retrieved from http://hdl.handle.net/11449/148923

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):

Barbosa, Alex Melges [UNESP]. “Classes de Stiefel-Whitney e de Euler.” 2017. Thesis, Universidade Estadual Paulista. Accessed February 26, 2021. http://hdl.handle.net/11449/148923.

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

MLA Handbook (7^th Edition):

Barbosa, Alex Melges [UNESP]. “Classes de Stiefel-Whitney e de Euler.” 2017. Web. 26 Feb 2021.

Vancouver:

Barbosa AM[. Classes de Stiefel-Whitney e de Euler. [Internet] [Thesis]. Universidade Estadual Paulista; 2017. [cited 2021 Feb 26]. Available from: http://hdl.handle.net/11449/148923.

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

Council of Science Editors:

Barbosa AM[. Classes de Stiefel-Whitney e de Euler. [Thesis]. Universidade Estadual Paulista; 2017. Available from: http://hdl.handle.net/11449/148923

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

University of Manchester

3. 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 February 26, 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. 26 Feb 2021.

Vancouver:

Borsdorf R. Structured Matrix Nearness Problems:Theory and Algorithms. [Internet] [Doctoral dissertation]. University of Manchester; 2012. [cited 2021 Feb 26]. 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

University of North Texas

4. Green, Michael Douglas, 1965-. Manifolds, Vector Bundles, and Stiefel-Whitney Classes.

Degree: 1990, University of North Texas

URL: https://digital.library.unt.edu/ark:/67531/metadc504181/

► The problem of embedding a manifold in Euclidean space is considered. Manifolds are introduced in Chapter I along with other basic definitions and examples. Chapter… (more)

Subjects/Keywords: manifolds; vector bundles; Stiefel-Whitney classes; Euclidean space; Manifolds (Mathematics); Vector bundles.

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

APA (6^th Edition):

Green, Michael Douglas, 1. (1990). Manifolds, Vector Bundles, and Stiefel-Whitney Classes. (Thesis). University of North Texas. Retrieved from https://digital.library.unt.edu/ark:/67531/metadc504181/

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):

Green, Michael Douglas, 1965-. “Manifolds, Vector Bundles, and Stiefel-Whitney Classes.” 1990. Thesis, University of North Texas. Accessed February 26, 2021. https://digital.library.unt.edu/ark:/67531/metadc504181/.

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

MLA Handbook (7^th Edition):

Green, Michael Douglas, 1965-. “Manifolds, Vector Bundles, and Stiefel-Whitney Classes.” 1990. Web. 26 Feb 2021.

Vancouver:

Green, Michael Douglas 1. Manifolds, Vector Bundles, and Stiefel-Whitney Classes. [Internet] [Thesis]. University of North Texas; 1990. [cited 2021 Feb 26]. Available from: https://digital.library.unt.edu/ark:/67531/metadc504181/.

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

Council of Science Editors:

Green, Michael Douglas 1. Manifolds, Vector Bundles, and Stiefel-Whitney Classes. [Thesis]. University of North Texas; 1990. Available from: https://digital.library.unt.edu/ark:/67531/metadc504181/

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

Colorado State University

5. Lui, Yui Man. Geometric methods on special manifolds for visual recognition.

Degree: PhD, Computer Science, 2010, Colorado State University

URL: http://hdl.handle.net/10217/39042

► Many computer vision methods assume that the underlying geometry of images is Euclidean. This assumption is generally not valid. Therefore, this dissertation introduces new nonlinear… (more)

▼ Many computer vision methods assume that the underlying geometry of images is Euclidean. This assumption is generally not valid. Therefore, this dissertation introduces new nonlinear geometric frameworks based upon special manifolds, namely Graβmann and Stiefel manifolds, for visual recognition. The motivation for this thesis is driven by the intrinsic geometry of visual data in which the visual data can be either a still image or video. Visual data are represented as points in appropriately chosen parameter spaces. The idiosyncratic aspects of the data in these spaces are then exploited for pattern classification. Three major research results are presented in this dissertation: face recognition for illumination spaces on Stiefel manifolds, face recognition on Graβmann registration manifolds, and action classification on product manifolds. Previous work has shown that illumination cones are idiosyncratic for face recognition in illumination spaces. However, it has not been addressed how a single image relates to an illumination cone. In this dissertation, a Bayesian model is employed to relight a single image to a set of illuminated variants. The subspace formed by these illuminated variants is characterized on a Stiefel manifold. A new distance measure called Canonical Stiefel Quotient (CSQ) is introduced. CSQ performs two projections on a tangent space of a Stiefel manifold and uses the quotient for classification. The proposed method demonstrates that illumination cones can be synthesized by relighting a single image to a set of images, and the synthesized illumination cones are discriminative for face recognition. Experiments on the CMU-PIE and YaleB data sets reveal that CSQ not only achieves high recognition accuracies for generic faces but also is robust to the choice of training sets. Subspaces can be realized as points on Graβmann manifolds. Motivated by image perturbation and the geometry of Graβmann manifolds, we present a method called Graβmann Registration Manifolds (GRM) for face recognition. First, a tangent space is formed by a set of affine perturbed images where the tangent space admits a vector space structure. Second, the tangent spaces are embedded on a Graβmann manifold and chordal distance is used to compare subspaces. Experiments on the FERET database suggest that the proposed method yields excellent results using both holistic and local features. Specifically, on the FERET Dup2 data set, which is generally considered the most difficult data set on FERET, the proposed method achieves the highest rank one identification rate among all non-trained methods currently in the literature. Human actions compose a series of movements and can be described by a sequence of video frames. Since videos are multidimensional data, data tensors are the natural choice for data representation. In this dissertation, a data tensor is expressed as a point on a product manifold and classification is performed on this product space. First, we factorize a data tensor using a modified High Order Singular Value… Advisors/Committee Members: Beveridge, J. Ross (advisor), Kirby, Michael, 1961- (committee member), Draper, Bruce A. (Bruce Austin), 1962- (committee member), Whitley, L. Darrell (committee member).

Subjects/Keywords: action classification; visual recognition; special manifolds; geometric methods; face recognition; Human face recognition (Computer science); Grassmann manifolds; Stiefel manifolds

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

APA (6^th Edition):

Lui, Y. M. (2010). Geometric methods on special manifolds for visual recognition. (Doctoral Dissertation). Colorado State University. Retrieved from http://hdl.handle.net/10217/39042

Chicago Manual of Style (16^th Edition):

Lui, Yui Man. “Geometric methods on special manifolds for visual recognition.” 2010. Doctoral Dissertation, Colorado State University. Accessed February 26, 2021. http://hdl.handle.net/10217/39042.

MLA Handbook (7^th Edition):

Lui, Yui Man. “Geometric methods on special manifolds for visual recognition.” 2010. Web. 26 Feb 2021.

Vancouver:

Lui YM. Geometric methods on special manifolds for visual recognition. [Internet] [Doctoral dissertation]. Colorado State University; 2010. [cited 2021 Feb 26]. Available from: http://hdl.handle.net/10217/39042.

Council of Science Editors:

Lui YM. Geometric methods on special manifolds for visual recognition. [Doctoral Dissertation]. Colorado State University; 2010. Available from: http://hdl.handle.net/10217/39042

Univerzitet u Beogradu

6. Radovanović, Marko S. Гребнерове базе за многострукости застава и примене.

Degree: Matematički fakultet, 2016, Univerzitet u Beogradu

URL: https://fedorabg.bg.ac.rs/fedora/get/o:11333/bdef:Content/get

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Математика - Алгебра / Mathematics - Algebra

о Бореловом опису, целобројна и мод 2 кохомологија многостру- кости застава дата је као полиномијална алгебра посечена по… (more)

Subjects/Keywords: Grobner bases; cohomology of ag manifolds; quantum cohomology; symmetric functions; Kostka numbers; cup-length; Schubert calculus; Chern classes; Stiefel-Whitney classes; immersions

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

APA (6^th Edition):

Radovanović, M. S. (2016). Гребнерове базе за многострукости застава и примене. (Thesis). Univerzitet u Beogradu. Retrieved from https://fedorabg.bg.ac.rs/fedora/get/o:11333/bdef:Content/get

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):

Radovanović, Marko S. “Гребнерове базе за многострукости застава и примене.” 2016. Thesis, Univerzitet u Beogradu. Accessed February 26, 2021. https://fedorabg.bg.ac.rs/fedora/get/o:11333/bdef:Content/get.

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

MLA Handbook (7^th Edition):

Radovanović, Marko S. “Гребнерове базе за многострукости застава и примене.” 2016. Web. 26 Feb 2021.

Vancouver:

Radovanović MS. Гребнерове базе за многострукости застава и примене. [Internet] [Thesis]. Univerzitet u Beogradu; 2016. [cited 2021 Feb 26]. Available from: https://fedorabg.bg.ac.rs/fedora/get/o:11333/bdef:Content/get.

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

Council of Science Editors:

Radovanović MS. Гребнерове базе за многострукости застава и примене. [Thesis]. Univerzitet u Beogradu; 2016. Available from: https://fedorabg.bg.ac.rs/fedora/get/o:11333/bdef:Content/get

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

University of Manchester

7. 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 · 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 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 February 26, 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. 26 Feb 2021.

Vancouver:

Borsdorf R. Structured matrix nearness problems : theory and algorithms. [Internet] [Doctoral dissertation]. University of Manchester; 2012. [cited 2021 Feb 26]. 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

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