Ernesto De Vito

Are loss functions all the same

In this letter, we investigate the impact of choosing different loss functions from the viewpoint of statistical learning theory. We introduce a convexity assumption, which is met by all loss functions commonly used in the literature, and study how the bound on the estimation error changes with the loss. We also derive a general result on the minimizer of the expected risk for a convex loss function in the case of classification. The main outcome of our analysis is that for classification, the hinge loss appears to be the loss of choice. Other things being equal, the hinge loss leads to a convergence rate practically indistinguishable from the logistic loss rate and much better than the square loss rate. Furthermore, if the hypothesis space is sufficiently rich, the bounds obtained for the hinge loss are not loosened by the thresholding stage.

Edges and Corners With Shearlets

Shearlets are a relatively new and very effective multi-scale framework for signal analysis. Contrary to the traditional wavelets, shearlets are capable to efficiently capture the anisotropic information in multivariate problem classes. Therefore, shearlets can be seen as the valid choice for multi-scale analysis and detection of directional sensitive visual features like edges and corners. In this paper, we start by reviewing the main properties of shearlets that are important for edge and corner detection. Then, we study algorithms for multi-scale edge and corner detection based on the shearlet representation. We provide an extensive experimental assessment on benchmark data sets which empirically confirms the potential of shearlets feature detection.

Wavelet transforms and discrete frames associated to semidirect products

We consider a semidirect product G=Rn×′H and its unitary representations U of the form IndG0G(p0m) where Ind is the unitary induction, p0 is in the dual group of Rn, G0 is the stability group of p0, and m is a unitary representation of G0∩H. We give sufficient conditions such that U defines a wavelet transform and a discrete frame.

DISCRETIZATION ERROR ANALYSIS FOR TIKHONOV REGULARIZATION

We study the discretization of inverse problems defined by a Carleman operator. In particular, we develop a discretization strategy for this class of inverse problems and we give a convergence analysis. Learning from examples, as well as the discretization of integral equations, can be analyzed in our setting.

Quantum homodyne tomography as an informationally complete positive-operator-valued measure

We define a positive-operator-valued measure E on describing the measurement of randomly sampled quadratures in quantum homodyne tomography, and we study its probabilistic properties. Moreover, we give a mathematical analysis of the relation between the description of a state in terms of E and the description provided by its Wigner transform.

A consistent algorithm to solve Lasso, elastic-net and Tikhonov regularization

In the framework of supervised learning, we prove that the iterative algorithm introduced in Umanita and Villa (2010) [22] allows us to estimate in a consistent way the relevant features of the regression function under the a priori assumption that it admits a sparse representation on a fixed dictionary.

On discrete frames associated with semidirect products

For groups which are the semidirect product of some vector group with a unimodular group we prove that the existence of a discrete frame obtained from an at-most countable set of vectors through the action of a given unitary representation implies that the representation in use has to be square-integrable.

Phase space observables and isotypic spaces

We give necessary and sufficient conditions for the set of Neumark projections of a countable set of phase space observables to constitute a resolution of the identity, and we give a criteria for a phase space observable to be informationally complete. The results will be applied to the phase space observables arising from an irreducible representation of the Heisenberg group.

Symmetries of the Quantum State Space and Group Representations

The homomorphisms of a connected Lie group G into the symmetry group of a quantum system are classified in terms of unitary representations of a simply connected Lie group associated with G. Moreover, an explicit description of the T-multipliers of G is obtained in terms of the ℝ-multipliers of the universal covering G* of G and the characters of G*. As an application, the Poincare group and the Galilei group, both in 3+1 and 2+1 dimensions, are considered.

Different Faces of the Shearlet Group

Recently, shearlet groups have received much attention in connection with shearlet transforms applied for orientation sensitive image analysis and restoration. The square integrable representations of the shearlet groups provide not only the basis for the shearlet transforms but also for a very natural definition of scales of smoothness spaces, called shearlet coorbit spaces. The aim of this paper is twofold: first we discover isomorphisms between shearlet groups and other well-known groups, namely extended Heisenberg groups and subgroups of the symplectic group. Interestingly, the connected shearlet group with positive dilations has an isomorphic copy in the symplectic group, while this is not true for the full shearlet group with all nonzero dilations. Indeed we prove the general result that there exist, up to adjoint action of the symplectic group, only one embedding of the extended Heisenberg algebra into the Lie algebra of the symplectic group. Having understood the various group isomorphisms it is natural to ask for the relations between coorbit spaces of isomorphic groups with equivalent representations. These connections are examined in the second part of the paper. We describe how isomorphic groups with equivalent representations lead to isomorphic coorbit spaces. In particular we apply this result to square integrable representations of the connected shearlet groups and metaplectic representations of subgroups of the symplectic group. This implies the definition of metaplectic coorbit spaces. Besides the usual full and connected shearlet groups we also deal with Toeplitz shearlet groups.