Learning Sets with Separating Kernels

We consider the problem of learning a set from random samples. We show how relevant geometric and topological properties of a set can be studied analytically using concepts from the theory of reproducing kernel Hilbert spaces. A new kind of reproducing kernel, that we call separating kernel, plays a crucial role in our study and is analyzed in detail. We prove a new analytic characterization of the support of a distribution, that naturally leads to a family of provably consistent regularized learning algorithms and we discuss the stability of these methods with respect to random sampling. Numerical experiments show that the approach is competitive, and often better, than other state of the art techniques.Comment: final versio

The Use of Representations in Applied Harmonic Analysis

The role of unitary group representations in applied mathematics is manifold and has been frequently pointed out and exploited. In this chapter, we first review the basic notions and constructs of Lie theory and then present the main features of some of the most useful unitary representations, such as the wavelet representation of the affine group, the Schr\uf6dinger representation of the Heisenberg group, and the metaplectic representation. The emphasis is on reproducing formulae. In the last section we discuss a promising class of unitary representations arising by restricting the metaplectic representation to triangular subgroups of the symplectic group. This class includes many known important examples, like the shearlet representation, and others that have not been looked at from the point of view of possible applications, like the so-called Schr\uf6dingerlets

Scale Invariant Interest Points with Shearlets

Shearlets are a relatively new directional multi-scale framework for signal analysis, which have been shown effective to enhance signal discontinuities such as edges and corners at multiple scales. In this work we address the problem of detecting and describing blob-like features in the shearlets framework. We derive a measure which is very effective for blob detection and closely related to the Laplacian of Gaussian. We demonstrate the measure satisfies the perfect scale invariance property in the continuous case. In the discrete setting, we derive algorithms for blob detection and keypoint description. Finally, we provide qualitative justifications of our findings as well as a quantitative evaluation on benchmark data. We also report an experimental evidence that our method is very suitable to deal with compressed and noisy images, thanks to the sparsity property of shearlets