2008 NOV 17 - (VerticalNews.com) -- "This paper considers the least-square online gradient descent algorithm in a reproducing kernel Hilbert space (RKHS) without an explicit regularization term," investigators in London, the United Kingdom report.

  "We present a novel capacity independent approach to derive error bounds and convergence results for this algorithm. The essential element in our analysis is the interplay between the generalization error and a weighted cumulative error which we define in the paper," wrote Y.M. Ying and colleagues ...read more

  2008 NOV 17 - (VerticalNews.com) -- According to a study from the United States, "Let F-q be the finite field of q elements, where q = p(h). Let f(x) be a polynomial over F-q in n variables with m nonzero terms."

  "Let N(f) denote the number of solutions of f(x) = 0 with coordinates in F-q. In this paper we give a deterministic algorithm which computes the reduction of N(f) modulo p(b) in O(n(8m)((h+b)p)) bit operations," wrote D.Q. Wan and colleagues, University of California ...read more

  2008 NOV 17 - (VerticalNews.com) -- According to recent research from Toulouse, France, "For an ideal I subset of R[x] given by a set of generators, a new semidefinite characterization of its real radical I(V-R(I)) is presented, provided it is zero-dimensional (even if I is not)."

  "Moreover, we propose an algorithm using numerical linear algebra and semidefinite optimization techniques, to compute all (finitely many) points of the real variety V-R(I) as well as a set of generators of the real radical ideal. The latter is obtained in the form of a border or Grobner basis," wrote J.B. Lasserre and colleagues ...read more