On non-asymptotic bounds for estimation in generalized linear models with highly correlated design

We study a high-dimensional generalized linear model and penalized empirical risk minimization with $\ell_1$ penalty. Our aim is to provide a non-trivial illustration that non-asymptotic bounds for the estimator can be obtained without relying on the chaining technique and/or the peeling device.Comment: Published at http://dx.doi.org/10.1214/074921707000000319 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

The Smooth-Lasso and other ℓ1+ℓ2\ell_1+\ell_2-penalized methods

We consider a linear regression problem in a high dimensional setting where the number of covariates $p$ can be much larger than the sample size $n$. In such a situation, one often assumes sparsity of the regression vector, \textit i.e., the regression vector contains many zero components. We propose a Lasso-type estimator $\hat{\beta}^{Quad}$ (where '$Quad$' stands for quadratic) which is based on two penalty terms. The first one is the $\ell_1$ norm of the regression coefficients used to exploit the sparsity of the regression as done by the Lasso estimator, whereas the second is a quadratic penalty term introduced to capture some additional information on the setting of the problem. We detail two special cases: the Elastic-Net $\hat{\beta}^{EN}$, which deals with sparse problems where correlations between variables may exist; and the Smooth-Lasso $\hat{\beta}^{SL}$, which responds to sparse problems where successive regression coefficients are known to vary slowly (in some situations, this can also be interpreted in terms of correlations between successive variables). From a theoretical point of view, we establish variable selection consistency results and show that $\hat{\beta}^{Quad}$ achieves a Sparsity Inequality, \textit i.e., a bound in terms of the number of non-zero components of the 'true' regression vector. These results are provided under a weaker assumption on the Gram matrix than the one used by the Lasso. In some situations this guarantees a significant improvement over the Lasso. Furthermore, a simulation study is conducted and shows that the S-Lasso $\hat{\beta}^{SL}$ performs better than known methods as the Lasso, the Elastic-Net $\hat{\beta}^{EN}$, and the Fused-Lasso with respect to the estimation accuracy. This is especially the case when the regression vector is 'smooth', \textit i.e., when the variations between successive coefficients of the unknown parameter of the regression are small. The study also reveals that the theoretical calibration of the tuning parameters and the one based on 10 fold cross validation imply two S-Lasso solutions with close performance

On the conditions used to prove oracle results for the Lasso

Oracle inequalities and variable selection properties for the Lasso in linear models have been established under a variety of different assumptions on the design matrix. We show in this paper how the different conditions and concepts relate to each other. The restricted eigenvalue condition (Bickel et al., 2009) or the slightly weaker compatibility condition (van de Geer, 2007) are sufficient for oracle results. We argue that both these conditions allow for a fairly general class of design matrices. Hence, optimality of the Lasso for prediction and estimation holds for more general situations than what it appears from coherence (Bunea et al, 2007b,c) or restricted isometry (Candes and Tao, 2005) assumptions.Comment: 33 pages, 1 figur

Including Limited Partners in the Diversity Jurisdiction Analysis

This paper presents the results of the Dynamic Pricing Challenge, held on the occasion of the 17th INFORMS Revenue Management and Pricing Section Conference on June 29–30, 2017 in Amsterdam, The Netherlands. For this challenge, participants submitted algorithms for pricing and demand learning of which the numerical performance was analyzed in simulated market environments. This allows consideration of market dynamics that are not analytically tractable or can not be empirically analyzed due to practical complications. Our findings implicate that the relative performance of algorithms varies substantially across different market dynamics, which confirms the intrinsic complexity of pricing and learning in the presence of competition

A dwarf elephant and a rock mouse on Naxos (Cyclades, Greece) with a revision of the palaeozoogeography of the Cycladic Islands (Greece)during the Pleistocene

During the Late Pleistocene, Naxos and adjacent areas, including Delos and Paros, constituted a mega-island, here referred to as palaeo-Cyclades. The extensive low-lying plainswith lakes and rivers provided a suitable habitat for elephants. Due to long-term isolation from the mainland and mainland populations, these elephants evolved miniature size. The species found on Naxos had a body size of about ten percent of that of the mainland ancestor, Palaeoloxodon antiquus. During the glacial periods of the Late Pleistocene, P. antiquus may have migrated eastwards and southwards in search of better conditions and reached the islands. The dwarf species of the various Southern Aegean islands (e.g. Crete, Tilos, Rhodos, palaeo-Cyclades) are each the result of independent colonisation events. The very small size of the Naxos species respective to the dwarf elephants from Crete is explained as due to the lack of competitors. The only other elements of the contemporaneous fauna were a rock mouse (Apodemus cf. mystacinus) and a shrew (Crocidura sp.). Submergence of the area, climate change, volcanism, hunting by humans or a combination of these factors during the terminal Pleistocene may have caused the extinction of this endemic fauna

Influence of Deep Margin Elevation and preparation design on the fracture strength of indirectly restored molars

The objectives of this in-vitro study were to investigate the influence of Deep Margin Elevation (DME) and the preparation design (cusp coverage) on the fracture strength and repairability of CAD/CAM manufactured lithium disilicate (LS2) restorations on molars. Sound extracted human molars (n = 60) were randomly divided into 4 groups (n = 15) (inlay without DME (InoD); inlay with DME (IWD); onlay without DME (OnoD); onlay with DME (OnWD)). All samples were aged (1.2 × 106 cycles of 50N, 8000 cycles of 5–55 °C) followed by oblique static loading until fracture. Fracture strength was measured in Newton and the fracture analysis was performed using a (scanning electron) microscope. Data was statistically analyzed using two-way ANOVA and contingency tables. DME did not affect the fracture strength of LS2 restorations to a statistically significant level (p =.15). Onlays were stronger compared to inlays (p =.00). DME and preparation design did not interact (p =.97). However, onlays with DME were significantly stronger than inlays without DME (p =.00). More repairable fractures were observed among inlays (p =.00). Catastrophic, crown-root fractures were more prevalent in onlays (p =.00). DME did not influence repairability of fractures or fracture types to a statistically significant level (p &gt;.05). Within the limitations of this in-vitro study, DME did not statistical significantly affect the fracture strength, nor the fracture type or repairability of LS2 restorations in molars. Cusp coverage did increase the fracture strength. However, oblique forces necessary to fracture both inlays and onlays, either with or without DME, by far exceeded the bite forces that can be expected under physiological clinical conditions. Hence, both inlays and onlays are likely to be fracture resistant during clinical service.</p