The Invariant Refinement Method

Formalization of Invariant Patterns for the Invariant Refinement Method

This report provides formalization of the Invariant Refinement Method. In particular, it formally describes the individual invariant patterns w.r.t. (soft) real-time constraints and provides formulation, as well as proofs of theorems related to refinement among these patterns. This work was partially supported by the Grant Agency of the Czech Republic project P202/11/0312. This work was also pa...

Refinement of Operator-valued Reproducing Kernels

This paper studies the construction of a refinement kernel for a given operator-valued reproducing kernel such that the vector-valued reproducing kernel Hilbert space of the refinement kernel contains that of the given kernel as a subspace. The study is motivated from the need of updating the current operator-valued reproducing kernel in multi-task learning when underfitting or overfitting occu...

Empirical validation of the Invariant Refinement Method – a controlled experiment

Refinement of Reproducing Kernels

We continue our recent study on constructing a refinement kernel for a given kernel so that the reproducing kernel Hilbert space associated with the refinement kernel contains that with the original kernel as a subspace. To motivate this study, we first develop a refinement kernel method for learning, which gives an efficient algorithm for updating a learning predictor. Several characterization...

A Note on the Knot Floer Homology of Fibered Knots

We prove that the knot Floer homology of a fibered knot is nontrivial in its nextto-top Alexander grading. Immediate applications include new proofs of Krcatovich’s result that knots with L-space surgeries are prime and Hedden and Watson’s result that the rank of knot Floer homology detects the trefoil among knots in the 3-sphere. We also generalize the latter result, proving a similar theorem ...

Efficient manifold tracing for planar maps

Invariant manifolds of unstable periodic orbits organize the dynamics of chaotic orbits in phase space. They provide insight into the mechanisms of transport and chaotic advection and have important applications in physical situations involving three-dimensional flows. The numerical determination of invariant manifolds for planar maps is a problem on its own. Efficient and practical techniques ...