APPROXIMATION OF DISCRETE AND PENALIZED LEAST SQUARES SPLINES OVER SPHERICAL TRIANGULATIONS

Convergence of discrete and penalized least squares spherical splines

We study the convergence of discrete and penalized least squares spherical splines in spaces with stable local bases. We derive a bound for error in the approximation of a sufficiently smooth function by the discrete and penalized least squares splines. The error bound for the discrete least squares splines is explicitly dependent on the mesh size of the underlying triangulation. The error boun...

Penalized Partial Least Squares Based on B-Splines Transformations

On the Approximation Order of Splines on Spherical Triangulations

Bounds are provided on how well functions in Sobolev spaces on the sphere can be approximated by spherical splines, where a spherical spline of degree d is a C r function whose pieces are the restrictions of homogoneous polynomials of degree d to the sphere. The bounds are expressed in terms of appropriate seminorms deened with the help of radial projection, and are obtained using appropriate q...

Penalized Least Squares and Penalized Likelihood

where pλ(·) is the penalty function. Best subset selection corresponds to pλ(t) = (λ/2)I(t 6= 0). If we take pλ(t) = λ|t|, then (1.2) becomes the Lasso problem (1.1). Setting pλ(t) = at + (1 − a)|t| with 0 ≤ a ≤ 1 results in the method of elastic net. With pλ(t) = |t| for some 0 < q ≤ 2, it is called bridge regression, which includes the ridge regression as a special case when q = 2. Some penal...

Uniform approximation by discrete least squares polynomials

We study uniform approximation of differentiable or analytic functions of one or several variables on a compact set K by a sequence of discrete least squares polynomials. In particular, if K satisfies a Markov inequality and we use point evaluations on standard discretization grids with the number of points growing polynomially in the degree, these polynomials provide nearly optimal approximant...