Understanding Implicit Regularization in Over-Parameterized Single Index Model.

In this paper, we leverage over-parameterization to design regularization-free algorithms for the high-dimensional single index model and provide theoretical guarantees for the induced implicit regularization phenomenon. Specifically, we study both vector and matrix single index models where the link function is nonlinear and unknown, the signal parameter is either a sparse vector or a low-rank symmetric matrix, and the response variable can be heavy-tailed. To gain a better understanding of the role played by implicit regularization without excess technicality, we assume that the distribution of the covariates is known a priori. For both the vector and matrix settings, we construct an over-parameterized least-squares loss function by employing the score function transform and a robust truncation step designed specifically for heavy-tailed data. We propose to estimate the true parameter by applying regularization-free gradient descent to the loss function. When the initialization is close to the origin and the stepsize is sufficiently small, we prove that the obtained solution achieves minimax optimal statistical rates of convergence in both the vector and matrix cases. In addition, our experimental results support our theoretical findings and also demonstrate that our methods empirically outperform classical methods with explicit regularization in terms of both ℓ2-statistical rate and variable selection consistency.

Reducing conservatism in robust stability analysis of fractional-order-polytopic systems.

This paper studies the robust stability of the fractional-order (FO) LTI systems with polytopic uncertainty. Generally, the characteristic polynomial of the system dynamic matrix is not an affine function of the uncertain parameters. Consequently, the robust stability of the uncertain system cannot be evaluated by well-known approaches including LMIs or exposed edges theorem. Here, an over-parameterization technique is developed to convert the main characteristic polynomial into a set of local over-parameterized characteristic polynomials (LOPCPs). It is proved that the robust stability of LOPCPs implies the robust stability of the uncertain system. Then, an algorithm is proposed to explore system's robust stability through investigating the robust stability of these LOPCPs based on the exposed edges idea. For the sake of feasibility comparison, extensive examples are elaborated that reveal the superiority of the proposed algorithm.

Too many covariates and too few cases? - a comparative study.

Prior research indicates that 10-15 cases or controls, whichever fewer, are required per parameter to reliably estimate regression coefficients in multivariable logistic regression models. This condition may be difficult to meet even in a well-designed study when the number of potential confounders is large, the outcome is rare, and/or interactions are of interest. Various propensity score approaches have been implemented when the exposure is binary. Recent work on shrinkage approaches like lasso were motivated by the critical need to develop methods for the p >> n situation, where p is the number of parameters and n is the sample size. Those methods, however, have been less frequently used when p≈n, and in this situation, there is no guidance on choosing among regular logistic regression models, propensity score methods, and shrinkage approaches. To fill this gap, we conducted extensive simulations mimicking our motivating clinical data, estimating vaccine effectiveness for preventing influenza hospitalizations in the 2011-2012 influenza season. Ridge regression and penalized logistic regression models that penalize all but the coefficient of the exposure may be considered in these types of studies. Copyright © 2016 John Wiley & Sons, Ltd.

Layer- and region-specific material characterization of ascending thoracic aortic aneurysms by microstructure-based models.

Material characterization of ascending thoracic aortic aneurysms is indispensable for the determination of stress distributions across wall thickness and the different aneurysm regions that may be responsible for their catastrophic rupture or dissection, but only few studies have addressed this issue hitherto. In this article, we are presenting our findings of implementing microstructure-based formulations for characterizing layer- and region-specific variations in wall properties, which is a reasonable consensus today. Together, we performed image-based analysis to derive collagen-fiber orientation angles that may serve as validation of the preferred candidate for a fiber-reinforced constitutive descriptor. We considered a four-fiber model with dispersions of fiber angles about the main directions, based on our histological observations, demonstrating a wide distribution of fiber orientations spanning circumferential to longitudinal directions, and its successful implementation to our biomechanical data from tensile testing. However, an in-depth parametric analysis showed that a condensed model without longitudinal-fiber family described the data just as well and did not omit essential histological organization of collagen fibers, while reserving a smaller number of parameters, which makes it advantageous for computational applications. A major aberration from almost all existing models in the literature is the hypothesis made that fibers can support compressive stresses. Such a hypothesis needs further examination but it has the benefits of allowing improved fits to the vanishing transverse stresses under uniaxial test conditions and of properly reflecting the exponential nature of the compressive stress-strain response of aortic tissue, being consistent with observations of collagen being under compression in the unloaded wall.