67 research outputs found
Metrics for matrix-valued measures via test functions
It is perhaps not widely recognized that certain common notions of distance
between probability measures have an alternative dual interpretation which
compares corresponding functionals against suitable families of test functions.
This dual viewpoint extends in a straightforward manner to suggest metrics
between matrix-valued measures. Our main interest has been in developing
weakly-continuous metrics that are suitable for comparing matrix-valued power
spectral density functions. To this end, and following the suggested recipe of
utilizing suitable families of test functions, we develop a weakly-continuous
metric that is analogous to the Wasserstein metric and applies to matrix-valued
densities. We use a numerical example to compare this metric to certain
standard alternatives including a different version of a matricial Wasserstein
metric developed earlier
Minimum-entropy causal inference and its application in brain network analysis
Identification of the causal relationship between multivariate time series is
a ubiquitous problem in data science. Granger causality measure (GCM) and
conditional Granger causality measure (cGCM) are widely used statistical
methods for causal inference and effective connectivity analysis in
neuroimaging research. Both GCM and cGCM have frequency-domain formulations
that are developed based on a heuristic algorithm for matrix decompositions.
The goal of this work is to generalize GCM and cGCM measures and their
frequency-domain formulations by using a theoretic framework for minimum
entropy (ME) estimation. The proposed ME-estimation method extends the
classical theory of minimum mean squared error (MMSE) estimation for stochastic
processes. It provides three formulations of cGCM that include Geweke's
original time-domain cGCM as a special case. But all three frequency-domain
formulations of cGCM are different from previous methods. Experimental results
based on simulations have shown that one of the proposed frequency-domain cGCM
has enhanced sensitivity and specificity in detecting network connections
compared to other methods. In an example based on in vivo functional magnetic
resonance imaging, the proposed frequency-domain measure cGCM can significantly
enhance the consistency between the structural and effective connectivity of
human brain networks
Geometric methods for estimation of structured covariances
We consider problems of estimation of structured covariance matrices, and in
particular of matrices with a Toeplitz structure. We follow a geometric
viewpoint that is based on some suitable notion of distance. To this end, we
overview and compare several alternatives metrics and divergence measures. We
advocate a specific one which represents the Wasserstein distance between the
corresponding Gaussians distributions and show that it coincides with the
so-called Bures/Hellinger distance between covariance matrices as well. Most
importantly, besides the physically appealing interpretation, computation of
the metric requires solving a linear matrix inequality (LMI). As a consequence,
computations scale nicely for problems involving large covariance matrices, and
linear prior constraints on the covariance structure are easy to handle. We
compare this transportation/Bures/Hellinger metric with the maximum likelihood
and the Burg methods as to their performance with regard to estimation of power
spectra with spectral lines on a representative case study from the literature.Comment: 12 pages, 3 figure
Matrix-valued Monge-Kantorovich Optimal Mass Transport
We formulate an optimal transport problem for matrix-valued density
functions. This is pertinent in the spectral analysis of multivariable
time-series. The "mass" represents energy at various frequencies whereas, in
addition to a usual transportation cost across frequencies, a cost of rotation
is also taken into account. We show that it is natural to seek the
transportation plan in the tensor product of the spaces for the two
matrix-valued marginals. In contrast to the classical Monge-Kantorovich
setting, the transportation plan is no longer supported on a thin zero-measure
set.Comment: 11 page
Convex Clustering via Optimal Mass Transport
We consider approximating distributions within the framework of optimal mass
transport and specialize to the problem of clustering data sets. Distances
between distributions are measured in the Wasserstein metric. The main problem
we consider is that of approximating sample distributions by ones with sparse
support. This provides a new viewpoint to clustering. We propose different
relaxations of a cardinality function which penalizes the size of the support
set. We establish that a certain relaxation provides the tightest convex lower
approximation to the cardinality penalty. We compare the performance of
alternative relaxations on a numerical study on clustering.Comment: 12 pages, 12 figure
Matricial Wasserstein-1 Distance
In this note, we propose an extension of the Wasserstein 1-metric () for
matrix probability densities, matrix-valued density measures, and an unbalanced
interpretation of mass transport. The key is using duality theory, in
particular, a "dual of the dual" formulation of . This matrix analogue of
the Earth Mover's Distance has several attractive features including ease of
computation.Comment: 8 page
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