SDLS: a Matlab package for solving conic least-squares problems

This document is an introduction to the Matlab package SDLS (Semi-Definite Least-Squares) for solving least-squares problems over convex symmetric cones. The package is shortly presented through the addressed problem, a sketch of the implemented algorithm, the syntax and calling sequences, a simple numerical example and some more advanced features. The implemented method consists in solving the dual problem with a quasi-Newton algorithm. We note that SDLS is not the most competitive implementation of this algorithm: efficient, robust, commercial implementations are available (contact the authors). Our main goal with this Matlab SDLS package is to provide a simple, user-friendly software for solving and experimenting with semidefinite least-squares problems. Up to our knowledge, no such freeware exists at this date

Baby-Boom Aging and Average Living Standards

A calibrated overlapping generations model is used to investigate the effect on living standards of the aging baby boom. The relative scarcity of labor when baby boomers are old raises the wage-rental ratio by an amount that is sufficient to ensure that the post baby-boom generation can enjoy a modest increase in living standards - despite facing higher taxes. Nevertheless, the baby-boom cohort itself suffers a drop in consumption, and when the two generations are considered as a group, overall living standards fall by a modest amount. These results are robust to several changes in specification: the existence of liquidity constraints, alternative assumptions regarding individuals' expectations concerning future interest rates, and different fiscal policies concerning the tax treatment of private saving for retirement. Policy initiatives that bring significant hardship today to avoid a future "crisis" are not supported by the standard overlapping generations model.over-lapping generations model; living standards; baby-boom; aging

Sensitivity Analysis for Mirror-Stratifiable Convex Functions

This paper provides a set of sensitivity analysis and activity identification results for a class of convex functions with a strong geometric structure, that we coined "mirror-stratifiable". These functions are such that there is a bijection between a primal and a dual stratification of the space into partitioning sets, called strata. This pairing is crucial to track the strata that are identifiable by solutions of parametrized optimization problems or by iterates of optimization algorithms. This class of functions encompasses all regularizers routinely used in signal and image processing, machine learning, and statistics. We show that this "mirror-stratifiable" structure enjoys a nice sensitivity theory, allowing us to study stability of solutions of optimization problems to small perturbations, as well as activity identification of first-order proximal splitting-type algorithms. Existing results in the literature typically assume that, under a non-degeneracy condition, the active set associated to a minimizer is stable to small perturbations and is identified in finite time by optimization schemes. In contrast, our results do not require any non-degeneracy assumption: in consequence, the optimal active set is not necessarily stable anymore, but we are able to track precisely the set of identifiable strata.We show that these results have crucial implications when solving challenging ill-posed inverse problems via regularization, a typical scenario where the non-degeneracy condition is not fulfilled. Our theoretical results, illustrated by numerical simulations, allow to characterize the instability behaviour of the regularized solutions, by locating the set of all low-dimensional strata that can be potentially identified by these solutions

Locally symmetric submanifolds lift to spectral manifolds

In this work we prove that every locally symmetric smooth submanifold gives rise to a naturally defined smooth submanifold of the space of symmetric matrices, called spectral manifold, consisting of all matrices whose ordered vector of eigenvalues belongs to the locally symmetric manifold. We also present an explicit formula for the dimension of the spectral manifold in terms of the dimension and the intrinsic properties of the locally symmetric manifold

Model Consistency for Learning with Mirror-Stratifiable Regularizers

Low-complexity non-smooth convex regularizers are routinely used to impose some structure (such as sparsity or low-rank) on the coefficients for linear predictors in supervised learning. Model consistency consists then in selecting the correct structure (for instance support or rank) by regularized empirical risk minimization. It is known that model consistency holds under appropriate non-degeneracy conditions. However such conditions typically fail for highly correlated designs and it is observed that regularization methods tend to select larger models. In this work, we provide the theoretical underpinning of this behavior using the notion of mirror-stratifiable regularizers. This class of regularizers encompasses the most well-known in the literature, including the $\ell_1$ or trace norms. It brings into play a pair of primal-dual models, which in turn allows one to locate the structure of the solution using a specific dual certificate. We also show how this analysis is applicable to optimal solutions of the learning problem, and also to the iterates computed by a certain class of stochastic proximal-gradient algorithms.Comment: 14 pages, 4 figure

Accommodating Empire: Comparing French and American Paths to the Legalization of Gay Marriage [Draft]

Dating back to the revolutionary era, France and the United States have vied, sometimes directly, in a longstanding contest for leadership status in the area of human rights. Where gay marriage is concerned, however, it would be more accurate to describe both nations as followers rather than leaders. In late April 2013, about twelve years after the Netherlands became the world’s first nation to legalize same-sex marriage,1 and on the heels of large and passionate protests by social conservatives, France became the fourteenth such country, eliminating the Civil Code’s gender-specific language barring equal marriage.2 Not to be outdone, United States, acting through judicial rather than legislative channels, followed suit in June 2013 with United States v. Windsor, striking down the Federal Defense of Marriage Act (“DOMA”)