We consider kernel methods on general geodesic metric spaces and provide both negative and positive results. First we show that the common Gaussian kernel can only be generalized to a positive definite kernel on a geodesic metric space if the space i
s flat. As a result, for data on a Riemannian manifold, the geodesic Gaussian kernel is only positive definite if the Riemannian manifold is Euclidean. This implies that any attempt to design geodesic Gaussian kernels on curved Riemannian manifolds is futile. However, we show that for spaces with conditionally negative definite distances the geodesic Laplacian kernel can be generalized while retaining positive definiteness. This implies that geodesic Laplacian kernels can be generalized to some curved spaces, including spheres and hyperbolic spaces. Our theoretical results are verified empirically.
Methodological contributions: This paper introduces a family of kernels for analyzing (anatomical) trees endowed with vector valued measurements made along the tree. While state-of-the-art graph and tree kernels use combinatorial tree/graph structure
with discrete node and edge labels, the kernels presented in this paper can include geometric information such as branch shape, branch radius or other vector valued properties. In addition to being flexible in their ability to model different types of attributes, the presented kernels are computationally efficient and some of them can easily be computed for large datasets (N of the order 10.000) of trees with 30-600 branches. Combining the kernels with standard machine learning tools enables us to analyze the relation between disease and anatomical tree structure and geometry. Experimental results: The kernels are used to compare airway trees segmented from low-dose CT, endowed with branch shape descriptors and airway wall area percentage measurements made along the tree. Using kernelized hypothesis testing we show that the geometric airway trees are significantly differently distributed in patients with Chronic Obstructive Pulmonary Disease (COPD) than in healthy individuals. The geometric tree kernels also give a significant increase in the classification accuracy of COPD from geometric tree structure endowed with airway wall thickness measurements in comparison with state-of-the-art methods, giving further insight into the relationship between airway wall thickness and COPD. Software: Software for computing kernels and statistical tests is available at http://image.diku.dk/aasa/software.php.
In order to develop statistical methods for shapes with a tree-structure, we construct a shape space framework for tree-like shapes and study metrics on the shape space. This shape space has singularities, corresponding to topological transitions in
the represented trees. We study two closely related metrics on the shape space, TED and QED. QED is a quotient Euclidean distance arising naturally from the shape space formulation, while TED is the classical tree edit distance. Using Gromovs metric geometry we gain new insight into the geometries defined by TED and QED. We show that the new metric QED has nice geometric properties which facilitate statistical analysis, such as existence and local uniqueness of geodesics and averages. TED, on the other hand, does not share the geometric advantages of QED, but has nice algorithmic properties. We provide a theoretical framework and experimental results on synthetic data trees as well as airway trees from pulmonary CT scans. This way, we effectively illustrate that our framework has both the theoretical and qualitative properties necessary to build a theory of statistical tree-shape analysis.
We study the structure of classical groups of equivalences for smooth multigerms $f colon (N,S) to (P,y)$, and extend several known results for monogerm equivalences to the case of mulitgerms. In particular, we study the group $A$ of source- and targ
et diffeomorphism germs, and its stabilizer $A_f$. For monogerms $f$ it is well-known that if $f$ is finitely $A$-determined, then $A_f$ has a maximal compact subgroup $MC(A_f)$, unique up to conjugacy, and $A_f/MC(A_f)$ is contractible. We prove the same result for finitely $A$-determined multigerms $f$. Moreover, we show that for a ministable multigerm $f$, the maximal compact subgroup $MC(A_f)$ decomposes as a product of maximal compact subgroups $MC(A_{g_i})$ for suitable representatives $g_i$ of the monogerm components of $f$. We study a product decomposition of $MC(A_f)$ in terms of $MC(mathscr{R}_f)$ and a group of target diffeomorphisms, and conjecture a decomposition theorem. Finally, we show that for a large class of maps, maximal compact subgroups are small and easy to compute.
Hanners theorem is a classical theorem in the theory of retracts and extensors in topological spaces, which states that a local ANE is an ANE. While Hanners original proof of the theorem is quite simple for separable spaces, it is rather involved for
the general case. We provide a proof which is not only short, but also elementary, relying only on well-known classical point-set topology.
