11 research outputs found
Algorithms for distance problems in planar complexes of global nonpositive curvature
CAT(0) metric spaces and hyperbolic spaces play an important role in
combinatorial and geometric group theory. In this paper, we present efficient
algorithms for distance problems in CAT(0) planar complexes. First of all, we
present an algorithm for answering single-point distance queries in a CAT(0)
planar complex. Namely, we show that for a CAT(0) planar complex K with n
vertices, one can construct in O(n^2 log n) time a data structure D of size
O(n^2) so that, given a point x in K, the shortest path gamma(x,y) between x
and the query point y can be computed in linear time. Our second algorithm
computes the convex hull of a finite set of points in a CAT(0) planar complex.
This algorithm is based on Toussaint's algorithm for computing the convex hull
of a finite set of points in a simple polygon and it constructs the convex hull
of a set of k points in O(n^2 log n + nk log k) time, using a data structure of
size O(n^2 + k)
Sliced-Wasserstein on Symmetric Positive Definite Matrices for M/EEG Signals
When dealing with electro or magnetoencephalography records, many supervised
prediction tasks are solved by working with covariance matrices to summarize
the signals. Learning with these matrices requires using Riemanian geometry to
account for their structure. In this paper, we propose a new method to deal
with distributions of covariance matrices and demonstrate its computational
efficiency on M/EEG multivariate time series. More specifically, we define a
Sliced-Wasserstein distance between measures of symmetric positive definite
matrices that comes with strong theoretical guarantees. Then, we take advantage
of its properties and kernel methods to apply this distance to brain-age
prediction from MEG data and compare it to state-of-the-art algorithms based on
Riemannian geometry. Finally, we show that it is an efficient surrogate to the
Wasserstein distance in domain adaptation for Brain Computer Interface
applications
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