With the fast development of science and information technology, data analysis
and processing have become ubiquitous in modern life and society. Learning theory
as a brand new research ¯eld develops rapidly bene¯tting from the wide applications
of computers for data analysis in scienti¯c and practical problems. In this talk, we
will talk about some algorithms in learning theory for the purpose of regression and
manifold learning. Their approximation and sparsity behaviors will be discussed in
detail.
We will ¯rst talk about the `1-regularized algorithm for regression in a data
dependent hypothesis space. Sparsity of the algorithm is studied based on our error
analysis. Second, we will present an Hermite learning algorithm for data involving
both function values and gradients. Normal estimation is an important topic in
computer graphics. At the end of this talk, we will discuss estimating normals for
a (unknown) submanifold of a Euclidean space from random points, which can be
used for processing point cloud data and implicit surface reconstruction.