17 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
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
Multivariate Spectral Estimation based on the concept of Optimal Prediction
In this technical note, we deal with a spectrum approximation problem arising
in THREE-like multivariate spectral estimation approaches. The solution to the
problem minimizes a suitable divergence index with respect to an a priori
spectral density. We derive a new divergence family between multivariate
spectral densities which takes root in the prediction theory. Under mild
assumptions on the a priori spectral density, the approximation problem, based
on this new divergence family, admits a family of solutions. Moreover, an upper
bound on the complexity degree of these solutions is provided
A new family of high-resolution multivariate spectral estimators
In this paper, we extend the Beta divergence family to multivariate power
spectral densities. Similarly to the scalar case, we show that it smoothly
connects the multivariate Kullback-Leibler divergence with the multivariate
Itakura-Saito distance. We successively study a spectrum approximation problem,
based on the Beta divergence family, which is related to a multivariate
extension of the THREE spectral estimation technique. It is then possible to
characterize a family of solutions to the problem. An upper bound on the
complexity of these solutions will also be provided. Simulations suggest that
the most suitable solution of this family depends on the specific features
required from the estimation problem
Representation and Characterization of Non-Stationary Processes by Dilation Operators and Induced Shape Space Manifolds
We have introduce a new vision of stochastic processes through the geometry
induced by the dilation. The dilation matrices of a given processes are
obtained by a composition of rotations matrices, contain the measure
information in a condensed way. Particularly interesting is the fact that the
obtention of dilation matrices is regardless of the stationarity of the
underlying process. When the process is stationary, it coincides with the
Naimark Dilation and only one rotation matrix is computed, when the process is
non-stationary, a set of rotation matrices are computed. In particular, the
periodicity of the correlation function that may appear in some classes of
signal is transmitted to the set of dilation matrices. These rotation matrices,
which can be arbitrarily close to each other depending on the sampling or the
rescaling of the signal are seen as a distinctive feature of the signal. In
order to study this sequence of matrices, and guided by the possibility to
rescale the signal, the correct geometrical framework to use with the
dilation's theoretic results is the space of curves on manifolds, that is the
set of all curve that lies on a base manifold. To give a complete sight about
the space of curve, a metric and the derived geodesic equation are provided.
The general results are adapted to the more specific case where the base
manifold is the Lie group of rotation matrices. The notion of the shape of a
curve can be formalized as the set of equivalence classes of curves given by
the quotient space of the space of curves and the increasing diffeomorphisms.
The metric in the space of curve naturally extent to the space of shapes and
enable comparison between shapes.Comment: 19 pages, draft pape