999 research outputs found
Characterizing Distances of Networks on the Tensor Manifold
At the core of understanding dynamical systems is the ability to maintain and
control the systems behavior that includes notions of robustness,
heterogeneity, or regime-shift detection. Recently, to explore such functional
properties, a convenient representation has been to model such dynamical
systems as a weighted graph consisting of a finite, but very large number of
interacting agents. This said, there exists very limited relevant statistical
theory that is able cope with real-life data, i.e., how does perform analysis
and/or statistics over a family of networks as opposed to a specific network or
network-to-network variation. Here, we are interested in the analysis of
network families whereby each network represents a point on an underlying
statistical manifold. To do so, we explore the Riemannian structure of the
tensor manifold developed by Pennec previously applied to Diffusion Tensor
Imaging (DTI) towards the problem of network analysis. In particular, while
this note focuses on Pennec definition of geodesics amongst a family of
networks, we show how it lays the foundation for future work for developing
measures of network robustness for regime-shift detection. We conclude with
experiments highlighting the proposed distance on synthetic networks and an
application towards biological (stem-cell) systems.Comment: This paper is accepted at 8th International Conference on Complex
Networks 201
Algorithms to automatically quantify the geometric similarity of anatomical surfaces
We describe new approaches for distances between pairs of 2-dimensional
surfaces (embedded in 3-dimensional space) that use local structures and global
information contained in inter-structure geometric relationships. We present
algorithms to automatically determine these distances as well as geometric
correspondences. This is motivated by the aspiration of students of natural
science to understand the continuity of form that unites the diversity of life.
At present, scientists using physical traits to study evolutionary
relationships among living and extinct animals analyze data extracted from
carefully defined anatomical correspondence points (landmarks). Identifying and
recording these landmarks is time consuming and can be done accurately only by
trained morphologists. This renders these studies inaccessible to
non-morphologists, and causes phenomics to lag behind genomics in elucidating
evolutionary patterns. Unlike other algorithms presented for morphological
correspondences our approach does not require any preliminary marking of
special features or landmarks by the user. It also differs from other seminal
work in computational geometry in that our algorithms are polynomial in nature
and thus faster, making pairwise comparisons feasible for significantly larger
numbers of digitized surfaces. We illustrate our approach using three datasets
representing teeth and different bones of primates and humans, and show that it
leads to highly accurate results.Comment: Changes with respect to v1, v2: an Erratum was added, correcting the
references for one of the three datasets. Note that the datasets and code for
this paper can be obtained from the Data Conservancy (see Download column on
v1, v2
A Sparse Multi-Scale Algorithm for Dense Optimal Transport
Discrete optimal transport solvers do not scale well on dense large problems
since they do not explicitly exploit the geometric structure of the cost
function. In analogy to continuous optimal transport we provide a framework to
verify global optimality of a discrete transport plan locally. This allows
construction of an algorithm to solve large dense problems by considering a
sequence of sparse problems instead. The algorithm lends itself to being
combined with a hierarchical multi-scale scheme. Any existing discrete solver
can be used as internal black-box.Several cost functions, including the noisy
squared Euclidean distance, are explicitly detailed. We observe a significant
reduction of run-time and memory requirements.Comment: Published "online first" in Journal of Mathematical Imaging and
Vision, see DO
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