LOCA: LOcal Conformal Autoencoder for standardized data coordinates

We propose a deep-learning based method for obtaining standardized data coordinates from scientific measurements.Data observations are modeled as samples from an unknown, non-linear deformation of an underlying Riemannian manifold, which is parametrized by a few normalized latent variables. By leveraging a repeated measurement sampling strategy, we present a method for learning an embedding in $\mathbb{R}^d$ that is isometric to the latent variables of the manifold. These data coordinates, being invariant under smooth changes of variables, enable matching between different instrumental observations of the same phenomenon. Our embedding is obtained using a LOcal Conformal Autoencoder (LOCA), an algorithm that constructs an embedding to rectify deformations by using a local z-scoring procedure while preserving relevant geometric information. We demonstrate the isometric embedding properties of LOCA on various model settings and observe that it exhibits promising interpolation and extrapolation capabilities. Finally, we apply LOCA to single-site Wi-Fi localization data, and to $3$-dimensional curved surface estimation based on a $2$-dimensional projection

Hyperbolic Interaction Model For Hierarchical Multi-Label Classification

Different from the traditional classification tasks which assume mutual exclusion of labels, hierarchical multi-label classification (HMLC) aims to assign multiple labels to every instance with the labels organized under hierarchical relations. Besides the labels, since linguistic ontologies are intrinsic hierarchies, the conceptual relations between words can also form hierarchical structures. Thus it can be a challenge to learn mappings from word hierarchies to label hierarchies. We propose to model the word and label hierarchies by embedding them jointly in the hyperbolic space. The main reason is that the tree-likeness of the hyperbolic space matches the complexity of symbolic data with hierarchical structures. A new Hyperbolic Interaction Model (HyperIM) is designed to learn the label-aware document representations and make predictions for HMLC. Extensive experiments are conducted on three benchmark datasets. The results have demonstrated that the new model can realistically capture the complex data structures and further improve the performance for HMLC comparing with the state-of-the-art methods. To facilitate future research, our code is publicly available

Calculating Sparse and Dense Correspondences for Near-Isometric Shapes

Comparing and analysing digital models are basic techniques of geometric shape processing. These techniques have a variety of applications, such as extracting the domain knowledge contained in the growing number of digital models to simplify shape modelling. Another example application is the analysis of real-world objects, which itself has a variety of applications, such as medical examinations, medical and agricultural research, and infrastructure maintenance. As methods to digitalize physical objects mature, any advances in the analysis of digital shapes lead to progress in the analysis of real-world objects. Global shape properties, like volume and surface area, are simple to compare but contain only very limited information. Much more information is contained in local shape differences, such as where and how a plant grew. Sadly the computation of local shape differences is hard as it requires knowledge of corresponding point pairs, i.e. points on both shapes that correspond to each other. The following article thesis (cumulative dissertation) discusses several recent publications for the computation of corresponding points: - Geodesic distances between points, i.e. distances along the surface, are fundamental for several shape processing tasks as well as several shape matching techniques. Chapter 3 introduces and analyses fast and accurate bounds on geodesic distances. - When building a shape space on a set of shapes, misaligned correspondences lead to points moving along the surfaces and finally to a larger shape space. Chapter 4 shows that this also works the other way around, that is good correspondences are obtain by optimizing them to generate a compact shape space. - Representing correspondences with a “functional map” has a variety of advantages. Chapter 5 shows that representing the correspondence map as an alignment of Green’s functions of the Laplace operator has similar advantages, but is much less dependent on the number of eigenvectors used for the computations. - Quadratic assignment problems were recently shown to reliably yield sparse correspondences. Chapter 6 compares state-of-the-art convex relaxations of graphics and vision with methods from discrete optimization on typical quadratic assignment problems emerging in shape matching

Learning shape correspondence with anisotropic convolutional neural networks

Establishing correspondence between shapes is a fundamental problem in geometry processing, arising in a wide variety of applications. The problem is especially difficult in the setting of non-isometric deformations, as well as in the presence of topological noise and missing parts, mainly due to the limited capability to model such deformations axiomatically. Several recent works showed that invariance to complex shape transformations can be learned from examples. In this paper, we introduce an intrinsic convolutional neural network architecture based on anisotropic diffusion kernels, which we term Anisotropic Convolutional Neural Network (ACNN). In our construction, we generalize convolutions to non-Euclidean domains by constructing a set of oriented anisotropic diffusion kernels, creating in this way a local intrinsic polar representation of the data (`patch'), which is then correlated with a filter. Several cascades of such filters, linear, and non-linear operators are stacked to form a deep neural network whose parameters are learned by minimizing a task-specific cost. We use ACNNs to effectively learn intrinsic dense correspondences between deformable shapes in very challenging settings, achieving state-of-the-art results on some of the most difficult recent correspondence benchmarks

Stable Teichmueller quasigeodesics and ending laminations

We characterize which cobounded quasigeodesics in the Teichmueller space T of a closed surface are at bounded distance from a geodesic. More generally, given a cobounded lipschitz path gamma in T, we show that gamma is a quasigeodesic with finite Hausdorff distance from some geodesic if and only if the canonical hyperbolic plane bundle over gamma is a hyperbolic metric space. As an application, for complete hyperbolic 3-manifolds N with finitely generated, freely indecomposable fundamental group and with bounded geometry, we give a new construction of model geometries for the geometrically infinite ends of N, a key step in Minsky's proof of Thurston's ending lamination conjecture for such manifolds.Comment: Published in Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol7/paper2.abs.htm

Kernel learning over the manifold of symmetric positive definite matrices for dimensionality reduction in a BCI application

In this paper, we propose a kernel for nonlinear dimensionality reduction over the manifold of Symmetric Positive Definite (SPD) matrices in a Motor Imagery (MI)-based Brain Computer Interface (BCI) application. The proposed kernel, which is based on Riemannian geometry, tries to preserve the topology of data points in the feature space. Topology preservation is the main challenge in nonlinear dimensionality reduction (NLDR). Our main idea is to decrease the non-Euclidean characteristics of the manifold by modifying the volume elements. We apply a conformal transform over data-dependent isometric mapping to reduce the negative eigen fraction to learn a data dependent kernel over the Riemannian manifolds. Multiple experiments were carried out using the proposed kernel for a dimensionality reduction of SPD matrices that describe the EEG signals of dataset IIa from BCI competition IV. The experiments show that this kernel adapts to the input data and leads to promising results in comparison with the most popular manifold learning methods and the Common Spatial Pattern (CSP) technique as a reference algorithm in BCI competitions. The proposed kernel is strong, particularly in the cases where data points have a complex and nonlinear separable distribution