19,155 research outputs found
Independent components in spectroscopic analysis of complex mixtures
We applied two methods of "blind" spectral decomposition (MILCA and SNICA) to
quantitative and qualitative analysis of UV absorption spectra of several
non-trivial mixture types. Both methods use the concept of statistical
independence and aim at the reconstruction of minimally dependent components
from a linear mixture. We examined mixtures of major ecotoxicants (aromatic and
polyaromatic hydrocarbons), amino acids and complex mixtures of vitamins in a
veterinary drug. Both MICLA and SNICA were able to recover concentrations and
individual spectra with minimal errors comparable with instrumental noise. In
most cases their performance was similar to or better than that of other
chemometric methods such as MCR-ALS, SIMPLISMA, RADICAL, JADE and FastICA.
These results suggest that the ICA methods used in this study are suitable for
real life applications. Data used in this paper along with simple matlab codes
to reproduce paper figures can be found at
http://www.klab.caltech.edu/~kraskov/MILCA/spectraComment: 22 pages, 4 tables, 6 figure
Understanding Slow Feature Analysis: A Mathematical Framework
Slow feature analysis is an algorithm for unsupervised learning of invariant representations from data with temporal correlations. Here, we present a mathematical analysis of slow feature analysis for the case where the input-output functions are not restricted in complexity. We show that the optimal functions obey a partial differential eigenvalue problem of a type that is common in theoretical physics. This analogy allows the transfer of mathematical techniques and intuitions from physics to concrete applications of slow feature analysis, thereby providing the means for analytical predictions and a better understanding of simulation results. We put particular emphasis on the situation where the input data are generated from a set of statistically independent sources.\ud
The dependence of the optimal functions on the sources is calculated analytically for the cases where the sources have Gaussian or uniform distribution
An Extension of Slow Feature Analysis for Nonlinear Blind Source Separation
We present and test an extension of slow feature analysis as a novel approach to nonlinear blind source separation. The algorithm relies on temporal correlations and iteratively reconstructs a set of statistically independent sources from arbitrary nonlinear instantaneous mixtures. Simulations show that it is able to invert a complicated nonlinear mixture of two audio signals with a reliability of more than \%. The algorithm is based on a mathematical analysis of slow feature analysis for the case of input data that are generated from statistically independent sources
Symmetric tensor decomposition
We present an algorithm for decomposing a symmetric tensor, of dimension n
and order d as a sum of rank-1 symmetric tensors, extending the algorithm of
Sylvester devised in 1886 for binary forms. We recall the correspondence
between the decomposition of a homogeneous polynomial in n variables of total
degree d as a sum of powers of linear forms (Waring's problem), incidence
properties on secant varieties of the Veronese Variety and the representation
of linear forms as a linear combination of evaluations at distinct points. Then
we reformulate Sylvester's approach from the dual point of view. Exploiting
this duality, we propose necessary and sufficient conditions for the existence
of such a decomposition of a given rank, using the properties of Hankel (and
quasi-Hankel) matrices, derived from multivariate polynomials and normal form
computations. This leads to the resolution of polynomial equations of small
degree in non-generic cases. We propose a new algorithm for symmetric tensor
decomposition, based on this characterization and on linear algebra
computations with these Hankel matrices. The impact of this contribution is
two-fold. First it permits an efficient computation of the decomposition of any
tensor of sub-generic rank, as opposed to widely used iterative algorithms with
unproved global convergence (e.g. Alternate Least Squares or gradient
descents). Second, it gives tools for understanding uniqueness conditions, and
for detecting the rank
Information Theoretical Estimators Toolbox
We present ITE (information theoretical estimators) a free and open source,
multi-platform, Matlab/Octave toolbox that is capable of estimating many
different variants of entropy, mutual information, divergence, association
measures, cross quantities, and kernels on distributions. Thanks to its highly
modular design, ITE supports additionally (i) the combinations of the
estimation techniques, (ii) the easy construction and embedding of novel
information theoretical estimators, and (iii) their immediate application in
information theoretical optimization problems. ITE also includes a prototype
application in a central problem class of signal processing, independent
subspace analysis and its extensions.Comment: 5 pages; ITE toolbox: https://bitbucket.org/szzoli/ite
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