Smoothness | EPFL Graph Search

In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called differentiability class. At the very minimum, a function could be considered smooth if it is differentiable everywhere (hence continuous). At the other end, it might also possess derivatives of all orders in its domain, in which case it is said to be infinitely differentiable and referred to as a C-infinity function (or C^{\infty} function). Differentiability classes Differentiability class is a classification of functions according to the properties of their derivatives. It is a measure of the highest order of derivative that exists and is continuous for a function. Consider an open set U on the real line and a function f defined on U with real values. Let k be a non-negative integer. The function f is said to be of differentiability class C^k if t

Hypomonotonicity of the Normal Cone and Proximal Smoothness

Grigory Ivanov

We study the properties of the normal cone to a proximally smooth set. We give a complete characterization of a proximally smooth set through the monotonicity properties of its normal cone in an arbitrary uniformly convex and uniformly smooth Banach space. We also give the exact bounds for the right-hand side in the monotonicity inequality for the normal cone in terms of the moduli of smoothness and convexity of a Banach space.

Heldermann Verlag2017

New Moduli For Banach Spaces

Grigory Ivanov

Modifying the moduli of supporting convexity and supporting smoothness, we introduce new moduli for Banach spaces which occur, for example, as lengths of catheti of right-angled triangles (defined via so-called quasiorthogonality). These triangles have two boundary points of the unit ball of a Banach space as endpoints of their hypotenuse, and their third vertex lies in a supporting hyperplane of one of the two other vertices. Among other things, it is our goal to quantify via such triangles the local deviation of the unit sphere from its supporting hyperplanes. We prove respective Day-Nordlander-type results involving generalizations of the modulus of convexity and the modulus of Banas.

Tusi Mathematical Research Group2017

Automatic L2 Regularization for Multiple Generalized Additive Models

Yousra El Bachir

Multiple generalized additive models are a class of statistical regression models wherein parameters of probability distributions incorporate information through additive smooth functions of predictors. The functions are represented by basis function expansions, whose coefficients are the regression parameters. The smoothness is induced by a quadratic roughness penalty on the functionsâ curvature, which is equivalent to a weighted

L_2

regularization controlled by smoothing parameters. Regression fitting relies on maximum penalized likelihood estimation for the regression coefficients, and smoothness selection relies on maximum marginal likelihood estimation for the smoothing parameters. Owing to their nonlinearity, flexibility and interpretability, generalized additive models are widely used in statistical modeling, but despite recent advances, reliable and fast methods for automatic smoothing in massive datasets are unavailable. Existing approaches are either reliable, complex and slow, or unreliable, simpler and fast, so a compromise must be made. A bridge between these categories is needed to extend use of multiple generalized additive models to settings beyond those possible in existing software. This thesis is one step in this direction. We adopt the marginal likelihood approach to develop approximate expectation-maximization methods for automatic smoothing, which avoid evaluation of expensive and unstable terms. This results in simpler algorithms that do not sacrifice reliability and achieve state-of-the-art accuracy and computational efficiency. We extend the proposed approach to big-data settings and produce the first reliable, high-performance and distributed-memory algorithm for fitting massive multiple generalized additive models. Furthermore, we develop the underlying generic software libraries and make them accessible to the open-source community.

EPFL2019

Automatic L2 Regularization for Multiple Generalized Additive Models

Yousra El Bachir

L_2

EPFL2019