A unified framework for solving a general class of conditional and robust set-membership estimation problems

In this paper we present a unified framework for solving a general class of problems arising in the context of set-membership estimation/identification theory. More precisely, the paper aims at providing an original approach for the computation of optimal conditional and robust projection estimates in a nonlinear estimation setting where the operator relating the data and the parameter to be estimated is assumed to be a generic multivariate polynomial function and the uncertainties affecting the data are assumed to belong to semialgebraic sets. By noticing that the computation of both the conditional and the robust projection optimal estimators requires the solution to min-max optimization problems that share the same structure, we propose a unified two-stage approach based on semidefinite-relaxation techniques for solving such estimation problems. The key idea of the proposed procedure is to recognize that the optimal functional of the inner optimization problems can be approximated to any desired precision by a multivariate polynomial function by suitably exploiting recently proposed results in the field of parametric optimization. Two simulation examples are reported to show the effectiveness of the proposed approach.Comment: Accpeted for publication in the IEEE Transactions on Automatic Control (2014

A Geometric View on Constrained M-Estimators

We study the estimation error of constrained M-estimators, and derive explicit upper bounds on the expected estimation error determined by the Gaussian width of the constraint set. Both of the cases where the true parameter is on the boundary of the constraint set (matched constraint), and where the true parameter is strictly in the constraint set (mismatched constraint) are considered. For both cases, we derive novel universal estimation error bounds for regression in a generalized linear model with the canonical link function. Our error bound for the mismatched constraint case is minimax optimal in terms of its dependence on the sample size, for Gaussian linear regression by the Lasso

A Family of Subgradient-Based Methods for Convex Optimization Problems in a Unifying Framework

We propose a new family of subgradient- and gradient-based methods which converges with optimal complexity for convex optimization problems whose feasible region is simple enough. This includes cases where the objective function is non-smooth, smooth, have composite/saddle structure, or are given by an inexact oracle model. We unified the way of constructing the subproblems which are necessary to be solved at each iteration of these methods. This permitted us to analyze the convergence of these methods in a unified way compared to previous results which required different approaches for each method/algorithm. Our contribution rely on two well-known methods in non-smooth convex optimization: the mirror-descent method by Nemirovski-Yudin and the dual-averaging method by Nesterov. Therefore, our family of methods includes them and many other methods as particular cases. For instance, the proposed family of classical gradient methods and its accelerations generalize Devolder et al.'s, Nesterov's primal/dual gradient methods, and Tseng's accelerated proximal gradient methods. Also our family of methods can partially become special cases of other universal methods, too. As an additional contribution, the novel extended mirror-descent method removes the compactness assumption of the feasible region and the fixation of the total number of iterations which is required by the original mirror-descent method in order to attain the optimal complexity.Comment: 31 pages. v3: Major revision. Research Report B-477, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, February 201

Low Complexity Regularization of Linear Inverse Problems

Inverse problems and regularization theory is a central theme in contemporary signal processing, where the goal is to reconstruct an unknown signal from partial indirect, and possibly noisy, measurements of it. A now standard method for recovering the unknown signal is to solve a convex optimization problem that enforces some prior knowledge about its structure. This has proved efficient in many problems routinely encountered in imaging sciences, statistics and machine learning. This chapter delivers a review of recent advances in the field where the regularization prior promotes solutions conforming to some notion of simplicity/low-complexity. These priors encompass as popular examples sparsity and group sparsity (to capture the compressibility of natural signals and images), total variation and analysis sparsity (to promote piecewise regularity), and low-rank (as natural extension of sparsity to matrix-valued data). Our aim is to provide a unified treatment of all these regularizations under a single umbrella, namely the theory of partial smoothness. This framework is very general and accommodates all low-complexity regularizers just mentioned, as well as many others. Partial smoothness turns out to be the canonical way to encode low-dimensional models that can be linear spaces or more general smooth manifolds. This review is intended to serve as a one stop shop toward the understanding of the theoretical properties of the so-regularized solutions. It covers a large spectrum including: (i) recovery guarantees and stability to noise, both in terms of $\ell^2$-stability and model (manifold) identification; (ii) sensitivity analysis to perturbations of the parameters involved (in particular the observations), with applications to unbiased risk estimation ; (iii) convergence properties of the forward-backward proximal splitting scheme, that is particularly well suited to solve the corresponding large-scale regularized optimization problem