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Publications

Published in refereed international journals

• Flores, Salvador.
  On the efficient computation of robust regression estimators. (preprint here)
  
  Computational Statistics & Data Analysis, 54 (12) (2010),pp. 3044--3056. [+bibtex]

  @Article{ Flores2010, author = "Salvador Flores",
  title = "On the efficient computation of robust regression estimators",
  journal = "Computational Statistics \& Data Analysis",
  volume = "54",
  number = "12",
  pages = "3044--3056",
  year = "2010",
  doi = "DOI: 10.1016/j.csda.2010.03.020", }

  show abstract [+]

  The problem of providing efficient and reliable robust regression algorithms is considered. The impact of global optimization methods, such as stopping conditions and clustering techniques, in the calculation of robust regression estimators is investigated. The use of stopping conditions permits us to devise new algorithms that perform as well as the existing algorithms in less time and with adaptive algorithm parameters. Clustering global optimization is shown to be a general framework encompassing many of the existing algorithms. [-]
• Alvarez, Felipe; Flores, Salvador.
  Remarks on Lipschitz solutions to measurable differential inclusions and an existence result for some nonconvex variational problems. (preprint here)
  
  J. Convex Anal. 15 (2008), no. 2, 349-363. [+bibtex]

  @article {Alvarez-Flores2008,
  AUTHOR = {Alvarez, Felipe and Flores, Salvador},
  TITLE = {Remarks on {L}ipschitz solutions to measurable differential inclusions and an
  existence result for some nonconvex variational problems},
  JOURNAL = {J. Convex Anal.},
  VOLUME = {15},
  YEAR = {2008},
  NUMBER = {2},
  PAGES = {349--363}
  }

  show abstract [+]

  In this paper we examine the problem of finding a Lipschitz function on an open domain with prescribed boundary values and whose gradient is required to satisfy some nonhomogeneous pointwise constraints a.e. in the domain, $$\nabla u(x) \in C(x)\ \text{ a.e. in }\Omega,\quad u|_{\partial \Omega}=g \in {\rm Lip}(\partial \Omega).$$ These constraints are supposed to be given by a measurable set-valued mapping $C:\mathbb{R}^n\rightrightarrows \mathbb{R}^n$   with convex, uniformly compact and nonempty-interior values. We discuss existence and metric properties of maximal solutions of such a problem. We exploit some connections with weak solutions to discontinuous Hamilton-Jacobi equations, and we provide a variational principle that characterizes maximal solutions. We investigate the case where the original problem is supplemented with bilateral obstacle constraints on the function values. Finally, as an application of these results, we prove existence for a specific class of nonconvex problems from the calculus of variations, with and without obstacle constraints, under mild regularity hypotheses on the data. [-]

Submitted for Publication

• Flores, Salvador.
  An improved Branch and Bound algorithm for an optimal feature subset selection problem from robust statistics.
  

  show abstract [+]

  We present a Branch and Bound algorithm for computing the Least Trimmed Squares estimator for performing strongly robust linear regression. The main novelty of our approach is the use of lower bounds obtained from a second order cone programming relaxation of the mixed-integer nonlinear programming problem that defines the estimator, which allows pruning at top levels of the enumeration tree. Numerical results show the gains in computing time, notably when the number of explicative variables is close to the number of cases. [-]
• Flores, Salvador.
  Sharp non-asymptotic performance bounds for $\ell_1$ and Huber robust regression estimators
  

  show abstract [+]

  A quantitative study of the robustness properties of the $\ell_1$ and the Huber M-estimator on finite samples is presented. The focus is on the linear model involving a fixed design matrix and additive errors restricted to the dependent variables consisting of noise and sparse outliers. We derive sharp error bounds for the $\ell_1$ estimator in terms of the leverage constants of a design matrix introduced here. A similar analysis is performed for Huber's estimator usign an equivalent problem formulation of independent interest. Our analysis considers outliers of arbitrary magnitude, and we recover breakdown point results as particular cases when outliers diverge. A Montecarlo simulation illustrates the ideas previously developed. [-]
• Alvarez, Felipe; Flores, Salvador.
  Existence And Approximation Results For Variational Problems Under Uniform Constraints On The Gradient By Power Penalty.
  

  show abstract [+]

  Variational problems under uniform quasiconvex constraints on the gradient are studied. Our technique consists in approximating the original problem by a one\textminus parameter family of smooth unconstrained optimization problems. Existence of solutions to such problems is proved as well as existence of lagrange multipliers associated to the uniform constraint; no constraint qualification condition is required. The solution-multiplier pairs are shown to satisfy an Euler-Lagrange equation and a complementarity property. Numerical experiments confirm the ability of our method to accurately compute solutions and Lagrange multipliers. [-]

Thesis

Subject : "Global optimization problems in robust statistics "
Advisors : Anne Ruiz-Gazen, Université Toulouse 1 Capitole
     Marcel Mongeau, Université Paul Sabatier
Defended : february 25, 2011
Committee : Jean-Baptiste Hiriart-Urruty, Université Paul Sabatier
       Marcel Mongeau, Université Paul Sabatier
       Fabio Schoen, Università di Firenze, Italy
       Michael Schyns, Université de Liège, Belgium
       Roy Welsch, Massachusetts Institute of Technology, USA





show abstract [+]

Robust statistics is a branch of statistics dealing with the analysis of data containing contaminated observations. The robustness of an estimator is measured notably by means of the breakdown point. High-breakdown point estimators are usually defined as global minima of a non-convex scale of the errors, hence their computation is a challenging global optimization problem. The objective of this dissertation is to investigate the potential contributions of modern global optimization methods to this class of problems.
The first part of this thesis is devoted to the $\tau$-estimator for linear regression, which is defined as a global minimum of a nonconvex differentiable function. We investigate the impact of incorporating clustering techniques and stopping conditions in existing stochastic algorithms. The consequences of some phenomena involving the nearest neighbor in high dimension on clustering global optimization algorithms is thoroughly discussed as well.
The second part is devoted to deterministic algorithms for computing the least trimmed squares regression estimator, which is defined through a nonlinear mixed-integer program. Due to the combinatorial nature of this problem, we concentrated on obtaining lower bounds to be used in a branch-and-bound algorithm. In particular, we propose a second-order cone relaxation that can be complemented with concavity cuts that we obtain explicitly. Global optimality conditions are also provided. [-]

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© 2011 Salvador Flores

Last update: 19/08/2011