Orbital and strongly orbital spaces

We say that a (countably dimensional) topological vector space X is orbital if there is T∈L(X) and a vector x∈X such that X is the linear span of the orbit {Tnx:n=0,1,…}. We say that X is strongly orbital if, additionally, x can be chosen to be a hypercyclic vector for T. Of course, X can be orbital only if the algebraic dimension of X is finite or infinite countable. We characterize orbital and strongly orbital metrizable locally convex spaces. We also show that every countably dimensional metrizable locally convex space X does not have the invariant subset property. That is, there is T∈L(X) such that every non-zero x∈X is a hypercyclic vector for T. Finally, assuming the Continuum Hypothesis, we construct a complete strongly orbital locally convex space.

As a byproduct of our constructions, we determine the number of isomorphism classes in the set of dense countably dimensional subspaces of any given separable infinite dimensional Fréchet space X. For instance, in X=ℓ2×ω, there are exactly 3 pairwise non-isomorphic (as topological vector spaces) dense countably dimensional subspaces.

Comparing the orthogonal and homotopy functor calculi

Goodwillie’s homotopy functor calculus constructs a Taylor tower of approximations toF , often a functor from spaces to spaces. Weiss’s orthogonal calculus provides a Taylortower for functors from vector spaces to spaces. In particular, there is a Weiss towerassociated to the functor V ÞÑ FpSVq, where SVis the one-point compactification of V .In this paper, we give a comparison of these two towers and show that when F isanalytic the towers agree up to weak equivalence. We include two main applications, oneof which gives as a corollary the convergence of the Weiss Taylor tower of BO. We alsolift the homotopy level tower comparison to a commutative diagram of Quillen functors,relating model categories for Goodwillie calculus and model categories for the orthogonal calculus.

Building Support Vector Machines in the Context of Regularised Least Squares

This paper formulates a linear kernel support vector machine (SVM) as a regularized least-squares (RLS) problem. By defining a set of indicator variables of the errors, the solution to the RLS problem is represented as an equation that relates the error vector to the indicator variables. Through partitioning the training set, the SVM weights and bias are expressed analytically using the support vectors. It is also shown how this approach naturally extends to Sums with nonlinear kernels whilst avoiding the need to make use of Lagrange multipliers and duality theory. A fast iterative solution algorithm based on Cholesky decomposition with permutation of the support vectors is suggested as a solution method. The properties of our SVM formulation are analyzed and compared with standard SVMs using a simple example that can be illustrated graphically. The correctness and behavior of our solution (merely derived in the primal context of RLS) is demonstrated using a set of public benchmarking problems for both linear and nonlinear SVMs.