We consider the unconstrained L2-Lp minimization: find a minimizer of ‖Ax−b‖2+λ‖x‖p for given A ∈ R, b ∈ R and parameters λ > 0, p ∈ [0, 1). This problem has been studied extensively in variable selection and sparse least squares fitting for high dimensional data. Theoretical results show that the minimizers of the L2-Lp problem have various attractive features due to the concavity and non-Lips...

In [3], two interior-point `2-penalty methods with strong global convergence properties were proposed for solving nonlinear programming problems. In this paper we show that under standard assumptions, slight modifications of these methods lead to fast local convergence. Specifically, we show that for each fixed small barrier parameter μ, iterates in a small neighborhood (roughly within o(μ)) of...

Recently, convex solutions to low-rank matrix factorization problems have received increasing attention in machine learning. However, in many applications the data can display other structures beyond simply being low-rank. For example, images and videos present complex spatio-temporal structures, which are largely ignored by current low-rank methods. In this paper we explore a matrix factorizat...

We propose a Bayesian framework for recursively estimating the classifier weights in online learning of a classifier ensemble. In contrast with past methods, such as stochastic gradient descent or online boosting, our framework estimates the weights in terms of evolving posterior distributions. For a specified class of loss functions, we show that it is possible to formulate a suitably defined ...

Given a global 1-homogeneous minimizer U0 to the Alt-Caffarelli energy functional, with sing(F(U0))={0}, we provide foliation of half-space Rn×[0,+∞) dilations graphs minimizers U¯≤U0≤U¯ analytic free boundaries at distance 1 from origin.

A simple inverse problem for the wave equation requires determination of both velocity in a homogenous acoustic material and transient waveform an isotropic point radiator, given time history wavefield at remote space. The duration (support) source source-to-receiver distance are assumed known. least squares formulation this exhibits "cycle-skipping" behaviour observed field scale problems type...

In this paper we obtain secondand first-order optimality conditions of Kuhn-Tucker type and Fritz John one for weak efficiency in the vector problem with inequality constraints. In the necessary conditions we suppose that the objective function and the active constraints are continuously differentiable. We introduce notions of KTSP-invex problem and second-order KTSP-invex one. We obtain that t...

We consider the global optimization problem for d-variate Lipschitz functions which, in a certain sense, do not increase too slowly in a neighborhood of the global minimizer(s). On these functions, we apply optimization algorithms which use only function values. We propose two adaptive deterministic methods. The first one applies in a situation when the Lipschitz constant L is known. The second...

We establish sharp energy estimates for some solutions, such as global minimizers, monotone solutions and saddle-shaped solutions, of the fractional nonlinear equation (−∆)u = f(u) in R. Our energy estimates hold for every nonlinearity f and are sharp since they are optimal for one-dimensional solutions, that is, for solutions depending only on one Euclidian variable. As a consequence, in dimen...