9,431 research outputs found
Using state space differential geometry for nonlinear blind source separation
Given a time series of multicomponent measurements of an evolving stimulus,
nonlinear blind source separation (BSS) seeks to find a "source" time series,
comprised of statistically independent combinations of the measured components.
In this paper, we seek a source time series with local velocity cross
correlations that vanish everywhere in stimulus state space. However, in an
earlier paper the local velocity correlation matrix was shown to constitute a
metric on state space. Therefore, nonlinear BSS maps onto a problem of
differential geometry: given the metric observed in the measurement coordinate
system, find another coordinate system in which the metric is diagonal
everywhere. We show how to determine if the observed data are separable in this
way, and, if they are, we show how to construct the required transformation to
the source coordinate system, which is essentially unique except for an unknown
rotation that can be found by applying the methods of linear BSS. Thus, the
proposed technique solves nonlinear BSS in many situations or, at least,
reduces it to linear BSS, without the use of probabilistic, parametric, or
iterative procedures. This paper also describes a generalization of this
methodology that performs nonlinear independent subspace separation. In every
case, the resulting decomposition of the observed data is an intrinsic property
of the stimulus' evolution in the sense that it does not depend on the way the
observer chooses to view it (e.g., the choice of the observing machine's
sensors). In other words, the decomposition is a property of the evolution of
the "real" stimulus that is "out there" broadcasting energy to the observer.
The technique is illustrated with analytic and numerical examples.Comment: Contains 14 pages and 3 figures. For related papers, see
http://www.geocities.com/dlevin2001/ . New version is identical to original
version except for URL in the bylin
Performing Nonlinear Blind Source Separation with Signal Invariants
Given a time series of multicomponent measurements x(t), the usual objective
of nonlinear blind source separation (BSS) is to find a "source" time series
s(t), comprised of statistically independent combinations of the measured
components. In this paper, the source time series is required to have a density
function in (s,ds/dt)-space that is equal to the product of density functions
of individual components. This formulation of the BSS problem has a solution
that is unique, up to permutations and component-wise transformations.
Separability is shown to impose constraints on certain locally invariant
(scalar) functions of x, which are derived from local higher-order correlations
of the data's velocity dx/dt. The data are separable if and only if they
satisfy these constraints, and, if the constraints are satisfied, the sources
can be explicitly constructed from the data. The method is illustrated by using
it to separate two speech-like sounds recorded with a single microphone.Comment: 8 pages, 3 figure
Diagonality Measures of Hermitian Positive-Definite Matrices with Application to the Approximate Joint Diagonalization Problem
In this paper, we introduce properly-invariant diagonality measures of
Hermitian positive-definite matrices. These diagonality measures are defined as
distances or divergences between a given positive-definite matrix and its
diagonal part. We then give closed-form expressions of these diagonality
measures and discuss their invariance properties. The diagonality measure based
on the log-determinant -divergence is general enough as it includes a
diagonality criterion used by the signal processing community as a special
case. These diagonality measures are then used to formulate minimization
problems for finding the approximate joint diagonalizer of a given set of
Hermitian positive-definite matrices. Numerical computations based on a
modified Newton method are presented and commented
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