A parametric version of herding is formulated. The nonlinear mapping between consecutive time slices is learned by a form of self-supervised training. The resulting dynamical system generates pseudo-samples that resemble the original data. We show how this parametric herding can be successfully used to compress a dataset consisting of binary digits. It is also verified that high compression rat...

Studying a critical value function φ in parametric nonlinear programming, we recall conditions guaranteeing that φ is a C function and derive second order Taylor expansion formulas including second-order terms in the form of certain generalized derivatives of Dφ. Several specializations and applications are discussed. These results are understood as supplements to the well–developed theory of f...

In this paper we address the problem of learning the structure of a Bayesian network in domains with continuous variables. This task requires a procedure for comparing different candidate structures. In the Bayesian framework, this is done by evaluating the marginal likelihood of the data given a candidate structure. This term can be computed in closed-form for standard parametric families (e.g...

The term distribution-free is used quite often in the statistical learning theory community, to refer to an analysis that makes no assumptions on the distribution of training examples (and only assumes these are independent and identically distributed samples from an unknown distribution). For our purposes we will say a model is nonparametric if it is not a parametric model. In the simplest for...

In geometric modelling surfaces can be given essentially in two ways: implicit and parametric form. The automatic transition between the implicit and the parametric representations of surfaces is of fundamental importance. In the literature there are several symbolic/numeric implicitization techniques based on resultants [1], Gröbner–basis [2], moving surfaces [3], linear algebra [4], but the n...

We address the following problem: given a curve in parametric form, compute the implicit representation of another one that approximates the parametric curve on a certain domain of interest. We study this problem from the numerical point of view: what happens with the output curve if the input curve is slightly changed? It is shown that for any approximate parameterization of the given curve, t...

We present here the explicit parametric solutions of second order differential equations invariant under time translation and rescaling and third order differential equations invariant under time translation and the two homogeneity symmetries. The computation of first integrals gives in the most general case, the parametric form of the general solution. For some polynomial functions we obtain a...

In this paper, a particular shape preserving parametric polynomial approximation of conic sections is studied. The approach is based upon a general strategy to the parametric approximation of implicitly defined planar curves. Polynomial approximants derived are given in a closed form and provide the highest possible approximation order. Although they are primarily studied to be of practical use...