Principal Quadratic Functionals

Principal Quadratic Functionals and Self-Adjoint Second-Order Differential Equations.

Quantization Functionals and Regularized Principal Manifolds

Many settings of unsupervised learning can be viewed as quantization problems, namely of minimizing the expected quantization error subject to some restrictions. This has the advantage that tools known from the theory of (supervised) risk minimization like regularization can be readily applied to unsupervised settings. Moreover, one may show that this setting is very closely related to both, pr...

Minimax Estimation of Quadratic Fourier Functionals

We study estimation of (semi-)inner products between two nonparametric probability distributions, given IID samples from each distribution. These products include relatively well-studied classical L and Sobolev inner products, as well as those induced by translation-invariant reproducing kernels, for which we believe our results are the first. We first propose estimators for these quantities, a...

Fair Surface Reconstruction Using Quadratic Functionals

An algorithm for surface reconstruction from a polyhedron with arbitrary topology consisting of triangular faces is presented. The rst variant of the algorithm constructs a curve network consisting of cubic B ezier curves meeting with tangent plane continuity at the vertices. This curve network is extended to a smooth surface by replacing each of the networks facets with a split patch consistin...

Singular quadratic functionals of one dependent variable

Singular quadratic functionals of one dependent variable with nonseparated boundary conditions are investigated. Necessary and sufficient conditions for nonnegativity of these functionals are derived using the concept of coupled point and singularity condition. The paper also includes two comparison theorems for coupled points with respect to the various boundary conditions.