Department of Mathematics and Statistics
                          The University at Albany
                                      
                                 COLLOQUIUM
                                      
                               Vitaly Maiorov
                                      
                              Technion, Israel
                                      
           Geometrical properties of the ridge functions manifold
                                      
                         Friday, September 28, 2007
                                      
                             4:15 p.m., ES 143
                                      
                          Tea at 3:45 p.m., ES 152
                                      
                                  ABSTRACT
                                      
   We consider geometrical properties of the manifold G_{n} formed by
   possible linear combinations of ridge functions of the form g((a,x)),
   where (a,x) is the inner product in Euclidean space. We estimate the
   entropy of the manifold G_{n} in terms of smaller covering number of
   the intersection of G_{n} with the unit ball in the polynomial space.
   Also we estimate the pseudo-dimension of the manifold G_{n}. Analogous
   properties we study for the manifolds formed by linear combinations of
   the functions of the form g(P(a,x)), where P is a fixed polynomial
   (for example radial function g(|x-a|^{2})).