Copulas are a powerful tool for modeling multivariate distributions as they allow to separately estimate the univariate marginal distributions and the joint dependency structure. However, known parametric copulas offer limited flexibility especially in high dimensions, while commonly used non-parametric methods suffer from the curse of dimensionality. A popular remedy is to construct a tree-based hierarchy of conditional bivariate copulas.In this paper, we propose a flexible, yet conceptually simple alternative based on implicit generative neural networks.The key challenge is to ensure marginal uniformity of the estimated copula distribution.We achieve this by learning a multivariate latent distribution with unspecified marginals but the desired dependency structure.By applying the probability integral transform, we can then obtain samples from the high-dimensional copula distribution without relying on parametric assumptions or the need to find a suitable tree structure.Experiments on synthetic and real data from finance, physics, and image generation demonstrate the performance of this approach.

We propose a novel approach to relational clustering: Given a matrix of pairwise similarity values between objects our algorithm computes a partition of the objects such that similar objects belong to the same cluster and dissimilar objects belong to different clusters. The proposed approach is based on the assumption that the given similarities are products of cluster membership variables. It is based on eigen vector decomposition and minimizes the squared error between the similarities and the products of membership vectors in an efficient, non-iterative way with guaranteed global optimality. In experiments with real world data we show superior performance to conventional iterative clustering approaches.

We study nonparametric regression between Riemannian manifolds based on regularized empirical risk minimization. Regularization functionals for mappings between manifolds should respect the geometry of input and output manifold and be independent of the chosen parametrization of the manifolds. We define and analyze the three most simple regularization functionals with these properties and present a rather general scheme for solving the resulting optimization problem. As application examples we discuss interpolation on the sphere, fingerprint processing, and correspondence computations between three-dimensional surfaces. We conclude with characterizing interesting and sometimes counterintuitive implications and new open problems that are specific to learning between Riemannian manifolds and are not encountered in multivariate regression in Euclidean space.

Multi-modal image registration is a challenging problem in medical imaging. The goal is to align anatomically identical structures; however, their appearance in images acquired with different imaging devices, such as CT or MR, may be very different. Registration algorithms generally deform one image, the floating image, such that it matches with a second, the reference image, by maximizing some similarity score between the deformed and the reference image. Instead of using a universal, but a priori fixed similarity criterion such as mutual information, we propose learning a similarity measure in a discriminative manner such that the reference and correctly deformed floating images receive high similarity scores. To this end, we develop an algorithm derived from max-margin structured output learning, and employ the learned similarity measure within a standard rigid registration algorithm. Compared to other approaches, our method adapts to the specific registration problem at hand and exploits correlations between neighboring pixels in the reference and the floating image. Empirical evaluation on CT-MR/PET-MR rigid registration tasks demonstrates that our approach yields robust performance and outperforms the state of the art methods for multi-modal medical image registration.

Semi-supervised regression based on the graph Laplacian suffers from the fact that the solution is biased towards a constant and the lack of extrapolating power. Outgoing from these observations we propose to use the second-order Hessian energy for semi-supervised regression which overcomes both of these problems, in particular, if the data lies on or close to a low-dimensional submanifold in the feature space, the Hessian energy prefers functions which vary ``linearly with respect to the natural parameters in the data. This property makes it also particularly suited for the task of semi-supervised dimensionality reduction where the goal is to find the natural parameters in the data based on a few labeled points. The experimental result suggest that our method is superior to semi-supervised regression using Laplacian regularization and standard supervised methods and is particularly suited for semi-supervised dimensionality reduction.

This paper discusses non-parametric regression between Riemannian manifolds. This learning problem arises frequently in many application areas ranging from signal processing, computer vision, over robotics to computer graphics. We present a new algorithmic scheme for the solution of this general learning problem based on regularized empirical risk minimization. The regularization functional takes into account the geometry of input and output manifold, and we show that it implements a prior which is particularly natural. Moreover, we demonstrate that our algorithm performs well in a difficult surface registration problem.

Abstract We present a generalization of thin‐plate splines for interpolation and approximation of manifold‐valued data, and demonstrate its usefulness in computer graphics with several applications from different fields. The cornerstone of our theoretical framework is an energy functional for mappings between two Riemannian manifolds which is independent of parametrization and respects the geometry of both manifolds. If the manifolds are Euclidean, the energy functional reduces to the classical thin‐plate spline energy. We show how the resulting optimization problems can be solved efficiently in many cases. Our example applications range from orientation interpolation and motion planning in animation over geometric modelling tasks to color interpolation.

Establishing correspondence between distinct objects is an important and nontrivial task: correctness of the correspondence hinges on properties which are difficult to capture in an a priori criterion. While previous work has used a priori criteria which in some cases led to very good results, the present paper explores whether it is possible to learn a combination of features that, for a given training set of aligned human heads, characterizes the notion of correct correspondence. By optimizing this criterion, we are then able to compute correspondence and morphs for novel heads.

We propose machine learning methods for the estimation of deformation fields that transform two given objects into each other, thereby establishing a dense point to point correspondence. The fields are computed using a modified support vector machine containing a penalty enforcing that points of one object will be mapped to "similar" points on the other one. Our system, which contains little engineering or domain knowledge, delivers state of the art performance. We present application results including close to photorealistic morphs of 3D head models.

We propose statistical learning methods for approximating implicit surfaces and computing dense 3D deformation\nfields. Our approach is based on Support Vector (SV) Machines, which are state of the art in machine learning. It\nis straightforward to implement and computationally competitive; its parameters can be automatically set using\nstandard machine learning methods.\nThe surface approximation is based on a modified Support Vector regression. We present applications to 3D head\nreconstruction, including automatic removal of outliers and hole filling.\nIn a second step, we build on our SV representation to compute dense 3D deformation fields between two objects.\nThe fields are computed using a generalized SVMachine enforcing correspondence between the previously learned\nimplicit SV object representations, as well as correspondences between feature points if such points are available.\nWe apply the method to the morphing of 3D heads and other objects.