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You searched for subject:(Regular convergence). Showing records 1 – 3 of 3 total matches.

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Texas A&M University

1. Goldsmith, Aaron Seth. LASSO Asymptotics For Heavy Tailed Errors.

Degree: 2015, Texas A&M University

We consider the asymptotic behavior of the l1 regularized least squares estimator (LASSO) for the linear regression model Y=X(beta)+xi with training data (X,Y) in RnxpxRn, true parameter beta in Rp, and observation noise xi in Rn. The LASSO estimator, defined by betahat in argminu in Rp||Xu-Y||2+lambda||u||1 introduces a bias toward 0 to encourage sparse estimates. LASSO has become a staple in the statistician?s breadbasket; it behaves very well and is quickly computed. In the case that xii are i.i.d. with E|xii|alpha<alphat}=t-alpha for some 1<alpha<2, Chatterjee and Lahiri found the exact rate, almost surely, for which the LASSO betahat tends to beta. We consider instead xi I that are i.i.d., possess all moments less than alpha, and eventually nearly follow a Pareto tail P{|xii|>t}=t-alpha Specifically, we only require that the tails of xii to be regularly varying. We center and scale both the quantity inside the arg min and betahat itself to prepare for a CLT. We find conditions that promise both convergence (uniformly over a class of designs X) of the quantity inside the arg min and uniform tightness of the centered, scaled bethahat. Then, we use a standard theorem to pass to uniform convergence of the centered, scaled betahat. Finally, we use a basic inequality to prove rate consistency for betahat when p is allowed to increase with n. Advisors/Committee Members: Zinn, Joel (advisor), Schlumprecht, Thomas (committee member), Rojas, Maurice (committee member), Mueller-Harknett, Ursula (committee member).

Subjects/Keywords: LASSO; Weak Convergence; Regular Variation; Heavy Tails

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APA (6th Edition):

Goldsmith, A. S. (2015). LASSO Asymptotics For Heavy Tailed Errors. (Thesis). Texas A&M University. Retrieved from http://hdl.handle.net/1969.1/156229

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

Chicago Manual of Style (16th Edition):

Goldsmith, Aaron Seth. “LASSO Asymptotics For Heavy Tailed Errors.” 2015. Thesis, Texas A&M University. Accessed July 14, 2020. http://hdl.handle.net/1969.1/156229.

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

MLA Handbook (7th Edition):

Goldsmith, Aaron Seth. “LASSO Asymptotics For Heavy Tailed Errors.” 2015. Web. 14 Jul 2020.

Vancouver:

Goldsmith AS. LASSO Asymptotics For Heavy Tailed Errors. [Internet] [Thesis]. Texas A&M University; 2015. [cited 2020 Jul 14]. Available from: http://hdl.handle.net/1969.1/156229.

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

Council of Science Editors:

Goldsmith AS. LASSO Asymptotics For Heavy Tailed Errors. [Thesis]. Texas A&M University; 2015. Available from: http://hdl.handle.net/1969.1/156229

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


Louisiana State University

2. Guevara, Alvaro. A regularization technique in dynamic optimization.

Degree: PhD, Applied Mathematics, 2009, Louisiana State University

In this dissertation we discuss certain aspects of a parametric regularization technique which is based on recent work by R. Goebel. For proper, lower semicontinuous, and convex functions, this regularization is self-dual with respect to convex conjugation, and a simple extension of this smoothing exhibits the same feature when applied to proper, closed, and saddle functions. In Chapter 1 we give a introduction to convex and saddle function theory, which includes new results on the convergence of saddle function values that were not previously available in the form presented. In Chapter 2, we define the regularization and extend some of the properties previously shown in the convex case to the saddle one. Furthermore, we investigate the properties of this regularization without convexity assumptions. In particular, we show that for a prox-bounded function the family of infimal values of the regularization converges to the infimal values of the given function, even when the given function might not have a minimizer. Also we show that for a general type of prox-regular functions the regularization is locally convex, even though their Moreau envelope might fail to have this property. Moreover, we apply the regularization technique to Lagrangians of convex optimization problems in two different settings, and describe the convergence of the associated saddle values and the value functions. We also employ the regularization in fully convex problems in calculus of variations, in Chapter 3, in the setting studied by R. Rockafellar and P. Wolenski. In this case, we extend a result by Rockafellar on the Lipschitz continuity of the proximal mapping of the value function jointly in the time and state variables, which in turn implies the same regularity for the gradient of the self-dual regularization. Finally, we attach a software code to use with SCAT (Symbolic Convex Analysis Toolbox) in order to symbolically compute the regularization for functions of one variable.

Subjects/Keywords: saddle functions; epi-convergence; convex functions; convex sets; hypo-epi-convergence; Moreau envelope; prox-regular functions; regularizing approximation; convex optimization; duality theory; saddle value convergence; calculus of variations; value function regularization; SCAT

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

APA (6th Edition):

Guevara, A. (2009). A regularization technique in dynamic optimization. (Doctoral Dissertation). Louisiana State University. Retrieved from etd-07022009-023950 ; https://digitalcommons.lsu.edu/gradschool_dissertations/3915

Chicago Manual of Style (16th Edition):

Guevara, Alvaro. “A regularization technique in dynamic optimization.” 2009. Doctoral Dissertation, Louisiana State University. Accessed July 14, 2020. etd-07022009-023950 ; https://digitalcommons.lsu.edu/gradschool_dissertations/3915.

MLA Handbook (7th Edition):

Guevara, Alvaro. “A regularization technique in dynamic optimization.” 2009. Web. 14 Jul 2020.

Vancouver:

Guevara A. A regularization technique in dynamic optimization. [Internet] [Doctoral dissertation]. Louisiana State University; 2009. [cited 2020 Jul 14]. Available from: etd-07022009-023950 ; https://digitalcommons.lsu.edu/gradschool_dissertations/3915.

Council of Science Editors:

Guevara A. A regularization technique in dynamic optimization. [Doctoral Dissertation]. Louisiana State University; 2009. Available from: etd-07022009-023950 ; https://digitalcommons.lsu.edu/gradschool_dissertations/3915


Université Paris-Sud – Paris XI

3. Bettinelli, Jérémie. Limite d'échelle de cartes aléatoires en genre quelconque : Scaling Limit of Arbitrary Genus Random Maps.

Degree: Docteur es, Mathématiques, 2011, Université Paris-Sud – Paris XI

Au cours de ce travail, nous nous intéressons aux limites d'échelle de deux classes de cartes. Dans un premier temps, nous regardons les quadrangulations biparties de genre strictement positif g fixé et, dans un second temps, les quadrangulations planaires à bord dont la longueur du bord est de l'ordre de la racine carrée du nombre de faces. Nous voyons ces objets comme des espaces métriques, en munissant leurs ensembles de sommets de la distance de graphe, convenablement renormalisée. Nous montrons qu'une carte prise uniformément parmi les cartes ayant n faces dans l'une de ces deux classes tend en loi, au moins à extraction près, vers un espace métrique limite aléatoire lorsque n tend vers l'infini. Cette convergence s'entend au sens de la topologie de Gromov – Hausdorff. On dispose de plus des informations suivantes sur l'espace limite que l'on obtient. Dans le premier cas, c'est presque sûrement un espace de dimension de Hausdorff 4 homéomorphe à la surface de genre g. Dans le second cas, c'est presque sûrement un espace de dimension 4 avec une frontière de dimension 2, homéomorphe au disque unité de R2. Nous montrons en outre que, dans le second cas, si la longueur du bord est un petit~o de la racine carrée du nombre de faces, on obtient la même limite que pour les quadrangulations sans bord, c'est-à-dire la carte brownienne, et l'extraction n'est plus requise.

In this work, we discuss the scaling limits of two particular classes of maps. In a first time, we address bipartite quadrangulations of fixed positive genus g and, in a second time, planar quadrangulations with a boundary whose length is of order the square root of the number of faces. We view these objects as metric spaces by endowing their sets of vertices with the graph metric, suitably rescaled.We show that a map uniformly chosen among the maps having n faces in one of these two classes converges in distribution, at least along some subsequence, toward a limiting random metric space as n tends to infinity. This convergence holds in the sense of the Gromov – Hausdorff topology on compact metric spaces. We moreover have the following information on the limiting space. In the first case, it is almost surely a space of Hausdorff dimension 4 that is homeomorphic to the genus g surface. In the second case, it is almost surely a space of Hausdorff dimension 4 with a boundary of Hausdorff dimension 2 that is homeomorphic to the unit disc of R2. We also show that in the second case, if the length of the boundary is little-o of the square root of the number of faces, the same convergence holds without extraction and the limit is the same as for quadrangulations without boundary, that is the Brownian map.

Advisors/Committee Members: Miermont, Grégory (thesis director).

Subjects/Keywords: Cartes aléatoires; Arbres aléatoires; Limite d'échelle; Processus conditionnés; Convergence régulière; Topologie de Gromov; Dimension de Hausdorff; Arbre continu brownien; Espaces métriques aléatoires; Random maps; Random trees; Scaling limits; Conditioned processes; Regular convergence; Gromov topology; Hausdorff dimension; Brownian CRT; Random metric spaces

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

APA (6th Edition):

Bettinelli, J. (2011). Limite d'échelle de cartes aléatoires en genre quelconque : Scaling Limit of Arbitrary Genus Random Maps. (Doctoral Dissertation). Université Paris-Sud – Paris XI. Retrieved from http://www.theses.fr/2011PA112213

Chicago Manual of Style (16th Edition):

Bettinelli, Jérémie. “Limite d'échelle de cartes aléatoires en genre quelconque : Scaling Limit of Arbitrary Genus Random Maps.” 2011. Doctoral Dissertation, Université Paris-Sud – Paris XI. Accessed July 14, 2020. http://www.theses.fr/2011PA112213.

MLA Handbook (7th Edition):

Bettinelli, Jérémie. “Limite d'échelle de cartes aléatoires en genre quelconque : Scaling Limit of Arbitrary Genus Random Maps.” 2011. Web. 14 Jul 2020.

Vancouver:

Bettinelli J. Limite d'échelle de cartes aléatoires en genre quelconque : Scaling Limit of Arbitrary Genus Random Maps. [Internet] [Doctoral dissertation]. Université Paris-Sud – Paris XI; 2011. [cited 2020 Jul 14]. Available from: http://www.theses.fr/2011PA112213.

Council of Science Editors:

Bettinelli J. Limite d'échelle de cartes aléatoires en genre quelconque : Scaling Limit of Arbitrary Genus Random Maps. [Doctoral Dissertation]. Université Paris-Sud – Paris XI; 2011. Available from: http://www.theses.fr/2011PA112213

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