Convergence analysis of a Lasserre hierarchy of upper bounds for polynomial minimization on the sphere

We study the convergence rate of a hierarchy of upper bounds for polynomial minimization problems, proposed by Lasserre [SIAM J. Optim. 21(3) (2011), pp. 864-885], for the special case when the feasible set is the unit (hyper)sphere. The upper bound at level r of the hierarchy is defined as the minimal expected value of the polynomial over all probability distributions on the sphere, when the probability density function is a sum-of-squares polynomial of degree at most 2r with respect to the surface measure. We show that the exact rate of convergence is Theta(1/r^2), and explore the implications for the related rate of convergence for the generalized problem of moments on the sphere.Comment: 14 pages, 2 figure

Does generalization performance of lql^q regularization learning depend on qq? A negative example

$l^q$-regularization has been demonstrated to be an attractive technique in machine learning and statistical modeling. It attempts to improve the generalization (prediction) capability of a machine (model) through appropriately shrinking its coefficients. The shape of a $l^q$ estimator differs in varying choices of the regularization order $q$. In particular, $l^1$ leads to the LASSO estimate, while $l^{2}$ corresponds to the smooth ridge regression. This makes the order $q$ a potential tuning parameter in applications. To facilitate the use of $l^{q}$-regularization, we intend to seek for a modeling strategy where an elaborative selection on $q$ is avoidable. In this spirit, we place our investigation within a general framework of $l^{q}$-regularized kernel learning under a sample dependent hypothesis space (SDHS). For a designated class of kernel functions, we show that all $l^{q}$ estimators for $0< q < \infty$ attain similar generalization error bounds. These estimated bounds are almost optimal in the sense that up to a logarithmic factor, the upper and lower bounds are asymptotically identical. This finding tentatively reveals that, in some modeling contexts, the choice of $q$ might not have a strong impact in terms of the generalization capability. From this perspective, $q$ can be arbitrarily specified, or specified merely by other no generalization criteria like smoothness, computational complexity, sparsity, etc..Comment: 35 pages, 3 figure

Optimal polynomial meshes and Caratheodory-Tchakaloff submeshes on the sphere

Using the notion of Dubiner distance, we give an elementary proof of the fact that good covering point configurations on the 2-sphere are optimal polynomial meshes. From these we extract Caratheodory-Tchakaloff (CATCH) submeshes for compressed Least Squares fitting

Inversion of noisy Radon transform by SVD based needlet

A linear method for inverting noisy observations of the Radon transform is developed based on decomposition systems (needlets) with rapidly decaying elements induced by the Radon transform SVD basis. Upper bounds of the risk of the estimator are established in $L^p$ ($1\le p\le \infty$) norms for functions with Besov space smoothness. A practical implementation of the method is given and several examples are discussed