This paper consider the penalized least squares estimators with convex penalties or regularization norms.We provide sparsity oracle inequalities for the prediction error for a general convex penalty and for the partic...This paper consider the penalized least squares estimators with convex penalties or regularization norms.We provide sparsity oracle inequalities for the prediction error for a general convex penalty and for the particular cases of Lasso and Group Lasso estimators in a regression setting.The main contribution is that our oracle inequalities are established for the more general case where the observations noise is issued from probability measures that satisfy a weak spectral gap(or Poincaré)inequality instead of Gaussian distributions.We illustrate our results on a heavy tailed example and a sub Gaussian one;we especially give the explicit bounds of the oracle inequalities for these two special examples.展开更多
Poincaréinequality has been studied by Bobkov for radial measures,but few are known about the logarithmic Sobolev inequality in the radial case.We try to fill this gap here using different methods:Bobkov's ar...Poincaréinequality has been studied by Bobkov for radial measures,but few are known about the logarithmic Sobolev inequality in the radial case.We try to fill this gap here using different methods:Bobkov's argument and super-Poincaréinequalities,direct approach via L_(1)-logarithmic Sobolev inequalities.We also give various examples where the obtained bounds are quite sharp.Recent bounds obtained by Lee–Vempala in the log-concave bounded case are refined for radial measures.展开更多
Moran or Wright–Fisher processes are probably the most well known models to study the evolution of a population under environmental various effects.Our object of study will be the Simpson index which measures the lev...Moran or Wright–Fisher processes are probably the most well known models to study the evolution of a population under environmental various effects.Our object of study will be the Simpson index which measures the level of diversity of the population,one of the key parameters for ecologists who study for example,forest dynamics.Following ecological motivations,we will consider,here,the case,where there are various species with fitness and immigration parameters being random processes(and thus time evolving).The Simpson index is difficult to evaluate when the population is large,except in the neutral(no selection)case,because it has no closed formula.Our approach relies on the large population limit in the“weak”selection case,and thus to give a procedure which enables us to approximate,with controlled rate,the expectation of the Simpson index at fixed time.We will also study the long time behavior(invariant measure and convergence speed towards equilibrium)of the Wright–Fisher process in a simplified setting,allowing us to get a full picture for the approximation of the expectation of the Simpson index.展开更多
基金This work has been(partially)supported by the Project EFI ANR-17-CE40-0030 of the French National Research Agency.
文摘This paper consider the penalized least squares estimators with convex penalties or regularization norms.We provide sparsity oracle inequalities for the prediction error for a general convex penalty and for the particular cases of Lasso and Group Lasso estimators in a regression setting.The main contribution is that our oracle inequalities are established for the more general case where the observations noise is issued from probability measures that satisfy a weak spectral gap(or Poincaré)inequality instead of Gaussian distributions.We illustrate our results on a heavy tailed example and a sub Gaussian one;we especially give the explicit bounds of the oracle inequalities for these two special examples.
基金Supported by ANR(Grant No.EFI ANR-17-CE40-0030)。
文摘Poincaréinequality has been studied by Bobkov for radial measures,but few are known about the logarithmic Sobolev inequality in the radial case.We try to fill this gap here using different methods:Bobkov's argument and super-Poincaréinequalities,direct approach via L_(1)-logarithmic Sobolev inequalities.We also give various examples where the obtained bounds are quite sharp.Recent bounds obtained by Lee–Vempala in the log-concave bounded case are refined for radial measures.
文摘Moran or Wright–Fisher processes are probably the most well known models to study the evolution of a population under environmental various effects.Our object of study will be the Simpson index which measures the level of diversity of the population,one of the key parameters for ecologists who study for example,forest dynamics.Following ecological motivations,we will consider,here,the case,where there are various species with fitness and immigration parameters being random processes(and thus time evolving).The Simpson index is difficult to evaluate when the population is large,except in the neutral(no selection)case,because it has no closed formula.Our approach relies on the large population limit in the“weak”selection case,and thus to give a procedure which enables us to approximate,with controlled rate,the expectation of the Simpson index at fixed time.We will also study the long time behavior(invariant measure and convergence speed towards equilibrium)of the Wright–Fisher process in a simplified setting,allowing us to get a full picture for the approximation of the expectation of the Simpson index.