7 research outputs found
Non-Gaussian Component Analysis using Entropy Methods
Non-Gaussian component analysis (NGCA) is a problem in multidimensional data
analysis which, since its formulation in 2006, has attracted considerable
attention in statistics and machine learning. In this problem, we have a random
variable in -dimensional Euclidean space. There is an unknown subspace
of the -dimensional Euclidean space such that the orthogonal
projection of onto is standard multidimensional Gaussian and the
orthogonal projection of onto , the orthogonal complement
of , is non-Gaussian, in the sense that all its one-dimensional
marginals are different from the Gaussian in a certain metric defined in terms
of moments. The NGCA problem is to approximate the non-Gaussian subspace
given samples of .
Vectors in correspond to `interesting' directions, whereas
vectors in correspond to the directions where data is very noisy. The
most interesting applications of the NGCA model is for the case when the
magnitude of the noise is comparable to that of the true signal, a setting in
which traditional noise reduction techniques such as PCA don't apply directly.
NGCA is also related to dimension reduction and to other data analysis problems
such as ICA. NGCA-like problems have been studied in statistics for a long time
using techniques such as projection pursuit.
We give an algorithm that takes polynomial time in the dimension and has
an inverse polynomial dependence on the error parameter measuring the angle
distance between the non-Gaussian subspace and the subspace output by the
algorithm. Our algorithm is based on relative entropy as the contrast function
and fits under the projection pursuit framework. The techniques we develop for
analyzing our algorithm maybe of use for other related problems
ICA based on a Smooth Estimation of the Differential Entropy
In this paper we introduce the MeanNN approach for estimation of main information theoretic measures such as differential entropy, mutual information and divergence. As opposed to other nonparametric approaches the MeanNN results in smooth differentiable functions of the data samples with clear geometrical interpretation. Then we apply the proposed estimators to the ICA problem and obtain a smooth expression for the mutual information that can be analytically optimized by gradient descent methods. The improved performance of the proposed ICA algorithm is demonstrated on several test examples in comparison with state-ofthe-art techniques.