Global optimization for low-dimensional switching linear regression and bounded-error estimation

The paper provides global optimization algorithms for two particularly difficult nonconvex problems raised by hybrid system identification: switching linear regression and bounded-error estimation. While most works focus on local optimization heuristics without global optimality guarantees or with guarantees valid only under restrictive conditions, the proposed approach always yields a solution with a certificate of global optimality. This approach relies on a branch-and-bound strategy for which we devise lower bounds that can be efficiently computed. In order to obtain scalable algorithms with respect to the number of data, we directly optimize the model parameters in a continuous optimization setting without involving integer variables. Numerical experiments show that the proposed algorithms offer a higher accuracy than convex relaxations with a reasonable computational burden for hybrid system identification. In addition, we discuss how bounded-error estimation is related to robust estimation in the presence of outliers and exact recovery under sparse noise, for which we also obtain promising numerical results

On the complexity of switching linear regression

This technical note extends recent results on the computational complexity of globally minimizing the error of piecewise-affine models to the related problem of minimizing the error of switching linear regression models. In particular, we show that, on the one hand the problem is NP-hard, but on the other hand, it admits a polynomial-time algorithm with respect to the number of data points for any fixed data dimension and number of modes.Comment: Automatica, Elsevier, 201

Error Bounds for Piecewise Smooth and Switching Regression

The paper deals with regression problems, in which the nonsmooth target is assumed to switch between different operating modes. Specifically, piecewise smooth (PWS) regression considers target functions switching deterministically via a partition of the input space, while switching regression considers arbitrary switching laws. The paper derives generalization error bounds in these two settings by following the approach based on Rademacher complexities. For PWS regression, our derivation involves a chaining argument and a decomposition of the covering numbers of PWS classes in terms of the ones of their component functions and the capacity of the classifier partitioning the input space. This yields error bounds with a radical dependency on the number of modes. For switching regression, the decomposition can be performed directly at the level of the Rademacher complexities, which yields bounds with a linear dependency on the number of modes. By using once more chaining and a decomposition at the level of covering numbers, we show how to recover a radical dependency. Examples of applications are given in particular for PWS and swichting regression with linear and kernel-based component functions.Comment: This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice,after which this version may no longer be accessibl

Education, Gender and Earnings in France and Germany: Level and Dispersion Effects

This paper analyses the relationship between education, gender and earnings in France and Germany. The model chosen here enables to estimate the impact of education not only on the expected earnings level but also on their dispersion, taking gender-specific sample selectivity into account. The results indicate that the completion of a minimum level of general instruction yields an earnings premium that cannot compensated by a vocational degree. Moreover, education affects the uncertainty of earnings. General qualifications are found to increase the earnings risk, vocational one to reduce it. More education, especially tertiary education, yields a high earnings premium but is associated with the highest earnings uncertainty. Women enjoy a higher earnings premium for education than men and though they face overall a higher earnings uncertainty, they can - more than men - reduce this risk by investing in their education. --education,earnings,heteroscedasticity

Education and Unemployment: A French-German Comparison

This paper analyses the link between educational attainment and unemployment risk in a French-German comparison, based on a discrete time competing risks hazard rate model applied to comparable microdata sets. The unemployment risk is broken down into the risk of entering unemployment and the risk, once unemployed, of not getting reemployed. The paper examines the impact of education on both risk components. France faces a higher unemployment rate than West-Germany, due to a higher risk of entering unemployment whereas the risk, when unemployed, of not getting reemployed is lower than in Germany. The risk of entering unemployment is particularly high for French employees with poor education, but higher education graduates face a higher risk of getting unemployed in Germany than in France. Concerning the reemployment prospects of the unemployed, they are better in France than in West-Germany at all education levels, but particularly for the unemployed with a low education level. The effect of education on both risk components does not differ significantly across genders, all else equal. --Education,unemployment,hazard rate model

Family background, cohort and education: A French-German comparison

This paper analyses the impact of family background, gender and cohort on educational attainment in France and Germany, relying on a theoretical model imbedded in the human capital theory. In a second step, the educational process is decomposed into school and post-school achievement. The same conceptual framework applies at both stages, but a correlation is permitted between them. Empirically, this boils down to estimating a multivariate ordered probit model. The results show that in spite of huge differences in the distribution of education in France and Germany, these countries prove surprisingly similar with respect to the impact of family background and cohorts. However, there are significant dissimilarities depending on the stage observed in the educational career, in particular with respect to gender differences. --Educational attainment,Multivariate ordered probit

Finding sparse solutions of systems of polynomial equations via group-sparsity optimization

The paper deals with the problem of finding sparse solutions to systems of polynomial equations possibly perturbed by noise. In particular, we show how these solutions can be recovered from group-sparse solutions of a derived system of linear equations. Then, two approaches are considered to find these group-sparse solutions. The first one is based on a convex relaxation resulting in a second-order cone programming formulation which can benefit from efficient reweighting techniques for sparsity enhancement. For this approach, sufficient conditions for the exact recovery of the sparsest solution to the polynomial system are derived in the noiseless setting, while stable recovery results are obtained for the noisy case. Though lacking a similar analysis, the second approach provides a more computationally efficient algorithm based on a greedy strategy adding the groups one-by-one. With respect to previous work, the proposed methods recover the sparsest solution in a very short computing time while remaining at least as accurate in terms of the probability of success. This probability is empirically analyzed to emphasize the relationship between the ability of the methods to solve the polynomial system and the sparsity of the solution.Comment: Journal of Global Optimization (2014) to appea