NONLINEAR GEOMETRIC ICA Fabian J. Theis 1 , Carlos G. Puntonet 2 , Elmar W. Lang 1 1 Institute of Biophysics, University of Regensburg, D-93040 Regensburg, Germany 2 Dept. Arquitectura y Tecnologia de Computadores, Universidad de Granada, E-18071 Granada, Spain ABSTRACT We present a new algorithm for nonlinear blind source sepa- ration, which is based on the geometry of the mixture space. This space is decomposed in a set of concentric rings, in which we perform ordinary linear ICA after central trans- formation; we show that this transformation can be left out if we use linear geometric ICA. In any case, we get a set of images of ring points under the original mixing mapping. Putting those together we can reconstruct the mixing map- ping. Indeed, this approach contains linear ICA and post- nonlinear ICA after whitening. The paper finishes with var- ious examples on toy and speech data. 1. INTRODUCTION Independent component analysis (ICA) tries to transform a random vector X such that f (X) is as independent as pos- sible. This is applied to the blind source separation (BSS) case, where X is already the mixture of some underlying independent sources S. Classically, linear BSS, where X = AS, has been treated most thoroughly. Several linear blind separation algorithms have been developed starting with maximum mutual infor- mation algorithms by Linsker [1] and Jutten’s Hebbian learn- ing algorithm [2]. With the growing popularity of ICA, more and more nonlinear algorithms have been proposed, like for example algorithms for postnonlinear ICA [3] [4], ideas based on various clustering algorithms to approximate the mixing or unmixing models [5] [6] or Almeida’s pattern repulsion using density uniformization [7]. We present a generalization of geometric ICA to special nonlinear models. This nonlinear geometric algorithm is based on the fact that when performing linear ICA, already any restriction of the mixture random vector contains all the information about the mixture matrix, lemma 2.3. Here, we will subdivide the mixture space and perform linear approx- imations in each subdivision. The first ideas of this kind of algorithm have been pre- sented by Puntonet et al. [8] and improved using genetic algorithms and simulated annealing [9]. 2. ALGORITHM We want to introduce a special kind of nonlinear symmet- ric ICA called nonlinearGeo that encompasses linear and postnonlinear ICA. The mixing mapping can hereby be a mapping f with |f (x)| = |x| for all x ∈ R n . Indeed, we could also allow any increasing function in each coordinate, but for simplicity we will restrict ourselves to this case for now. Then f (0) = 0, and f maps centered random vec- tors to centered ones. So in spherical coordinates (r, Φ) := (r, ϕ, ϑ 1 ,...,ϑ n-2 ), we have f (r, Φ) = (r, g(r, Φ)) for some possibly nonlinear function g : R + 0 ×[0, 2π)×[0,π) n-2 → [0, 2π) × [0,π) n-2 . Given a independent random variable S :Ω → R n , set as usual X = f ◦ S. The basic idea of the presented al- gorithm then is to divide the mixture space R n into a set of concentric rings with growing radii r 1 ,...,r s . In each of the rings so defined perform a linear ICA after central trans- formation in order to get f (r i e j ). These images of the basis vectors are then used to reconstruct the original signals. This indeed encompasses linear ICA, because if we are in the standard linear model all restrictions have the same BSSs as the whole model: 2.1. Restriction of a random variable We want to construct a new random vector out of a given one by restricting the random vector in the sense that only samples from a given region are taken into account. This notion is formalized in the next lemma: Lemma 2.1. Let X :Ω → R n be a random vector on the probability space (Ω, A), and let U ⊂ R n be measurable with P X (U )= P (X -1 (U )) > 0. Then X|U : X -1 (U ) -→ R n ω -→ X(ω) defines a new random vector on ( X -1 (U ), A ′ ) with σ-algebra A ′ := {A ∈ A| A ⊂ X -1 (U )} and probability measure P ′ (A)= P (A) PX(U) for A ∈A ′ . It is called the restriction of X to U . 275 4th International Symposium on Independent Component Analysis and Blind Signal Separation (ICA2003), April 2003, Nara, Japan