This paper presents two modifications of the gpICA (geometric post non-linear independent component analysis) algorithm. gpICA algorithm is a novel method to solve the PNL (Post non-linear) scheme. We propose these modifications to improve the mean squared error, the correlation of the recovered signals and algorithm reliability. The first improvement, called compensation, takes advantage of the implicit information given by the point to be linearized. On the other hand, while the original gpICA algorithm uses two sets of two points to make an update, our second modification uses two sets of four points. We present experimental results which validates the effectiveness of each modification. The PNL applications can be seen in sensor array processing, digital satellite, microwave communications, biological systems and nonlinear blind source separation tasks. gpICA can recover the original sources of a nonlinear mixture, unlike some of the other nonlinear BSS algorithms, it does not require any assumption on the distribution of the input signals

Andrés Gaona

Erika Torres

Miguel Torres

Paula Ulloa

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International Journal of Future Generation Communication and Networking 
Vol. 2, No. 3, September, 2009 
 
 
1 
Improvement to Geometric Linearization of gpICA Algorithm by 
Compensation and Multiple Points 
 
 
Erika Torres
1, Paula Ulloa
2, Andrés Gaona
3, and Miguel Torres
4 
 
123Faculty of Engineering, Distrital University FJC, Colombia 
4Faculty of Engineering, Javeriana University, Colombia 
1akane999@gmail.com, 
2paula.ulloa@gmail.com, 
3angaona@gmail.com, 
4metorres@javeriana.edu.coAuthor  
Abstract 
 
This paper presents two modifications of the gpICA (geometric post non-linear 
independent component analysis) algorithm. gpICA algorithm is a novel method to solve the 
PNL (Post non-linear) scheme. We propose these modifications to improve the mean squared 
error, the correlation of the recovered signals and algorithm reliability . The first 
improvement, called compensation, takes advantage of the implicit information given by the 
point to be linearized. On the other hand, while the original gpICA algorithm uses two sets of 
two points to make an update, our second modification uses two sets of four points. We 
present experimental results which validates the effectiveness of each modification. The PNL 
applications can be seen in sensor array processing, digital satellite, microwave 
communications, biological systems and nonlinear blind source separation tasks. gpICA can 
recover the original sources of a nonlinear mixture, unlike some of the other nonlinear BSS 
algorithms, it does not require any assumption on the distribution of the input signals.  
 
Keywords: Blind Source Separation (BSS) , Independent Component Analysis (ICA), Post non-linear mixtures, 
Speech Signal Processing.  
 
1. Introduction 
Blind Source Separation (BSS) is the general problem of determining original sources 
when only their mixtures are available for observation [1]. The BSS problem has been 
resolved from different approaches, one of them is the Independent Component Analysis 
(ICA) [2]. BSS resolved using ICA consist in the recovery of a set of independent signals 
from a set of observations that are unknown linear mixtures of the independent source signals. 
When assuming statistical independence between the sources, ICA has shown to have many 
applications in real world problems, such as in speech recognition systems, 
telecommunications and medical signal processing.  
Many ICA algorithm variations have been proposed to solve the linear case with different 
approaches (JADE [3], Fast ICA [4], Pearson ICA [5], etc.). However, in many situations, the 
basic linear model can not describe the real system adequately. For example all microphones 
can capture only until certain level of power, this phenomenon is called saturation, most 
sensors have this nonlinearity, such disadvantage can be overcome with a suitable nonlinear 
model. Nonlinear BSS is a more realistic model, but is more difficult to solve. 
In orden to solve a more realistic model, Jutten et. al. proposed an important subclass of 
nonlinear BSS models called Post Nonlinear (PNL) scheme [6]. The PNL scheme constrains International Journal of Future Generation Communication and Networking 
Vol. 2, No. 3, September, 2009 
 
 
2 
the transformation to one dimensional invertible nonlinear functions. The main idea behind 
the PNL scheme consists in finding the inverse of such nonlinear functions, in order to reduce 
the problem to a linear case. Hence, this linear case can be solved using any ICA algorithm. 
 
 
 
 
 
 
 
 
 
 
The PNL scheme is not a trivial problem, in order to solve it, Nguyen et. al. proposed the 
gpICA (geometric post nonlinear independent component analysis) algorithm in [7]. The 
gpICA algorithm was inspired from the next simple observation: in a multidimensional space, 
the graph of a nonlinear mixture is a nonlinear surface. On the other hand, when the mixture 
is a linear one, its graph is a plane. Based on this observation, the gpICA algorithm works in 
two stages. The first stage transforms a nonlinear surface (the geometric representation of the 
nonlinear mixture) in to a plane (The geometric representation of a linear mixture). The 
geometric linearization process reduces the nonlinear BSS problem in to a linear one. The 
second stage applies a linear BSS algorithm to extract the original sources. 
The paper is organized as follows: Principles of gpICA algorithm are explained in Section 
2. In Section 3 a scheme in 2D for gpICA Algorithm is presented, with this we developed two 
improvements to the original geometric linearization algorithm (First stage of gpICA): 
Compensation and Linearization with multiple points, their details are shown in Sections 4 
and 5. We provide some computer simulation results in Section 6 in order to validate each 
modification. Finally, we discuss the issues related to the proposed improvements in Section 
7. 
 
2. gpICA: Geometric Post-nonlinear ICA  
The gpICA algorithm was proposed by Nguyen et. al. in [7], and was based on the PNL 
model with a geometric linearization process, followed by an ICA algorithm. The geometric 
approach has some advantages over the other PNL algorithms: (i) the first stage and second 
stage are independent from each other, therefore, any ICA algorithm can be applied.(ii) unlike 
some of the other algorithms, it does not require any assumption on the distribution of the 
input signals [8]. 
In a 3D space, with two sources s1 and s2 where values are in axis x-y, respectively. The 
values of vi (linear mixture) and xi (nonlinear mixture) are in z-axis, forming a plane and a 
nonlinear surface,  respectively. In order to transform a plane from a given nonlinear surface, 
gpICA uses properties of the straight line in 3D space. This is shown in Figure 1. 
Figure 1. Lineal, nonlineal mixture and gpICA algorithm: geometric 
linearization and blind source separation (use ICA). A is a 2x2 Matrix and 
f() is a nonlineal function International Journal of Future Generation Communication and Networking 
Vol. 2, No. 3, September, 2009 
 
 
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The notation " companion points " is given to the points that belong to the 3D plane, which 
have equal position in x and y coordinate axis (unknown values). To recognize such points a 
time index to identify a companion point is used, that is, point p1 = (v1(t1),v2(t1),x1(t1)) has a 
companion point p2 = (v1(t2),v2(t2),x1(t2)). 
The algorithm uses a "fake plane" Sk as reference plane and use it to transform the other 
surfaces Sx. The "fake plane" changes alternatively from one surface to another during the 
transformation process. The transformation scheme for two sources can be explained by the 
following steps: 
1)  Sort the x signals. 
2)  Filter the sorted x signals using equation (1). cont = 0. 
3)  Restore the original order of the smoothed x signals. 
4)  Select fake plane named Sk. Other surface has been called Si . 
5)  Pick up two random pk1=(v1(t1),v2(t1),xk(t1)) and pk2=(v1(t2),v2(t2),xk(t2)) from the 
surface Sk. Locate their respective " companion points " pi1 and pi2 in the surface Si. 
6)  Select an arbitrary point pkc=(v1(tc),v2(tc),xk(tc)) between pk1 and pk2, and its companion 
point pic in Si . Find the value zi(tc) of point qic=(v1(tc),v2(tc), zi(tc)) using (2). 
7)  Compute xnewi(tc) using equation (3). 
8)  if cont ≤ Nk,  then cont=cont +1 and go to step 4. Else: cont=0; go to next step. 
Figure 3. Improvement 1: Linearization with Compensation. Left: Case 1, Right: Case 2. Inside the circle 1 and 2 shown the 
parts where original algorithm fail. 
 
 
 
 
 
 
 
 
 
 
 
Figure 3. Improvement 1: Linearization with Compensation. Left: Case 1, Right: Case 
2. Inside the circle 1 and 2 shown the parts where original algorithm fail 
Figure 3. Improvement 1: Linearization with Compensation. Left: Case 1, Right: Case 2. Inside the circle 1 and 2 shown the 
parts where original algorithm fail. 
Figure 2. Scheme in 2D for gpICA Algorithm. 2D cut of a 3D lineal 
mixture planes. International Journal of Future Generation Communication and Networking 
Vol. 2, No. 3, September, 2009 
 
 
4 
9)  Compute the 
error using 
equation (4). 
10) if e > x go to numeral 2, Else v = x. finish. 
 
 
 
 
 
Figure 4. Improvement 2: Geometric linearization with Multiple Points. PNL 
mixture vs. sources 
 
 
Where μ Є [0,1], ξ is a threshold to stop the linearization and Nk is the number of points to be 
updated in each iteration. For more sources the same transformation scheme is applied, but 
executes the actualization (equation (2) and (3)) in every plane different to the fake plane. 
 
3. Scheme in 2D for gpICA Algorithm  
The linearization algorithm is easier to understand in a transversal cut over line   of 
the 3D plane, obtaining a 2D plane shown in Figure 2, (with the values of xi in Y axis, (being 
the only known values). Points pk1, pk2, . . ., are transformed in a 2D plane. 
Table 1. Improvement 1: Geometric linearization with 
Compensation cases International Journal of Future Generation Communication and Networking 
Vol. 2, No. 3, September, 2009 
 
 
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We have λ2=kδ2, where we can compute k=λ2/δ2. Then we can obtain λ1=kδ1, and 
zc=xi(t1)−  λ1, this is the same equation (2), with δ1=xk(tc)−xk(t1),  δ2=xk(t2)−xk(t1), 
λ1=xi(t1)−xi(tc) and λ2= xi(t1)−xi(t2).  
 
4. Improvement 1: Linearization with compensation 
The geometric linearization algorithm presents problems when qic moves away from line 
, an explanation of this is shown in Figure 3 and Table 1. 
The variable k2 is k2 = λ1/δ1 (see Figure 3), this is the same relationship between k and 
points pk2−pi2, but in points pkc− pic. There are three zones in line  : i) k2<k, ii) k2>k, 
and iii) k2=k when we have a line (perfect linearization); this information will be used to 
compensate the problem with the wrong position for qic, show in circles in Figure 3. 
Figure 3 shows two cases for the point pkc, in the first case (pkc under  ). we have that 
if the original point pic belongs to the zone k2<k, those cases are shown in detail in the Table 
1. Wrong zones are depicted in the Figure 3. A Wrong position occurs when gpICA does not 
correct the position, on the contrary, damaging information. 
The wrong position for qic is compensated by adding or subtracting the difference 
(zi(tc)−xi(tc)). The compensation in a zone around pic is greater than the compensation applied 
to a point outside this zone, then we multiply the difference by k2/k. The new zi(tc) 
compensated zi(tc) named zi(tc)
cp shown in equation (5) is obtained of the zi(tc) calculated in 
(2). 
 
 
 
 
5. Improvement 2: Linearization with Multiple Points 
The usual geometric linearization [7] procedure uses two sets of points (pk1, pi1); (pk2, pi2), 
then the algorithm updates one point pic, in this case we use four sets of points (pk1, pi1); (pk2, 
pi2); (pk4, pi4); (pk5, pi5) in order to update one point pi3, this leads to more precision, reliability 
and to reduce ambiguity. 
Figure 4 shows the same scheme of Figure 2, but with five points in every line, they form 
triangles where it is possible to find a relationship between the cathetuses, to find the location 
of the unknown point pi3. To relate the points, we use the triangles formed by (δ1,c1), (δ2,c1 
+c2), (δ3,c1+c2+c3), (d4,c1+c2+c3+c4) and triangles at the bigger triangle top (a−b− δ1,c1), (a−c− 
δ2,c1+c2), (a−d−d3,c1+c2+c3), (a−e−d4,c1+c2+c3+c4), we do not have any information about (c1, 
. . . ,c4); because we only have PNL mixtures, we made enough equations in order to 
eliminate the need to use such variables. The resulting equations are shown below. 
 
 
 
 
 
 
 International Journal of Future Generation Communication and Networking 
Vol. 2, No. 3, September, 2009 
 
 
6 
by elimination of (6) and (7) 
 
 
 
We average the results of equation (8) to calculate c, in order to obtain a new zi with multiple 
points, called zi
mp to be replaced in equation (2). 
 
 
 
 
6. Experiment Results: Mixture of Four Speech Signals 
 
 
 
 
 
 
 
 
 
 
We executed the same experiments proposed in [7] in order to evaluate the performance of 
our improvements to gpICA. The experiment used four speech signal with 5000 samples each 
(taken from [9]). The linear and non linear mixtures are given by equations (10) and (11). We 
use  μ=0.2 in three cases. L=150 in original linearization and improvement 1. L=50 in 
improvement 2. We use Fast-ICA Algorithm for the blind source separation presented by 
Hyvarinen and Oja[4]. 
We performed 100 executions, the correlation coefficient, quadratic error and variation 
coefficient are shown in Table 2.The performance indicators were measured for the following 
cases (see Figure 1): 
Case 1: between the linearized signal and the lineally mix sources. Case 2: between 
original sources and estimated signals. The variation coefficient, correlation and quadratic 
error were better for Improvement 2 in both cases. 
The correlation was computed with (12). The variation coefficient (Standard deviation 
divide by mean) was used to measure data dispersion. 
 
 
Table 2. Table of results, correlation between linear mixtures and linearization output, correlation 
between sources and their estimated, quadratic error, and improvement. International Journal of Future Generation Communication and Networking 
Vol. 2, No. 3, September, 2009 
 
 
7 
The correlation mean for case 2, was greater than 0.85 for all algorithms, while for case 1 
was greater than 0.93, the variation also got worse, it was less than 4%, with this information 
we can say that the separation process scattered the algorithms results, and declines the 
performance after the linearization process. Hence, the algorithms (gpICA, Improvement 1 
and 2) performance does not depends on the linearization process, it also depends on the 
selection of a suitable linear BSS algorithm. Improvement 2 has 41% less error than original 
geometric linearization algorithm. 
 
7. Conclusions 
The geometric linearization algorithm with Compensation (Improvement 1), shows a new 
approach to find the correct position of qic (calculated point), the geometric linearization 
performance was increased only by adding some compensation to the point which was 
calculated with the usual gpICA equations, using information of the point to linearize that 
was not used before by the original algorithm. With the linearization with Multiple Points, 
performance was improved, because a larger set of points (a set of four points and a set of 
five points) to update every surface point position were used. This improvement is the more 
reliable than usual gpICA as shown by the results of experiments. 
 
References 
 
[1] Jean-Francois Cardoso. Blind signal separation: statistical principles. Proceedings of the IEEE, OCT. 1998. 
[2] Aapo Hyvärinen and Erkki Oja. Independent component analysis: Algorithms and applications. 2000. 
[3] J.F. Cardoso and Souloumiac. Blind beamforming for non-gaussian signals. 1993. 
[4] A. Hyvarinen and E. Oja. A fast fixed-point algorithm for independent component analysis. Journal In Neural 
Computation, pages 1483,1492. 
[5] Juha Karvanen, Jan Eriksson, and Visa Koivunen. Maximum likelihood estimation of ica model for wide class 
of source distributions. Signal Processing Laboratory. 
[6] A. Taleb and C. Jutten. Source separation in post-nonlinear mixtures. IEEE Transactions on Signal Processing, 
47, 1999. 
[7] Thang Viet Nguyen, Jagdish Chandra Patra, and Sabu Emmanuel. gpica: A novel nonlinear ica algorithm using 
geometric linearization. EURASIP Journal on Advances in Signal Processing, 2007. 
[8] Thang Viet Nguyen, Jagdish Chandra Patra, Amitabha Das, and Geok See Ng. Post nonlinear blind source 
separation by geometric linearization. Proceeding of international Conference on Neural Neural Networks, I, 
2005. 
[9] A Cichocky, S. Amari, K.Siwek, and all. Icalab toolbox. http://www.bsp.brain.riken.jp/ICALAB, 2003. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 International Journal of Future Generation Communication and Networking 
Vol. 2, No. 3, September, 2009 
 
 
8 
 
Authors 
 
 
 
Erika Torres, Electronic Engineer from Distrital University of Colombia (2008). Master of 
control in course of University of Science and Technology of Beijing. 
 
Paula Ulloa, Electronic Engineer from Distrital University of Colombia (2008). 
 
Andres Gaona, Electronic Engineer from Distrital University of Colombia (2003), Master of 
Electronic Engineer in from Andes University of Colombia (2006), and Assistant Professor at 
the Faculty of Engineering at Distrital University. 
 
Miguel Torres, Systems Engineer from National University of Colombia (1999), Master of 
Science in Computer Science from Mississippi State University (2003), and Assistant 
Professor at the Faculty of Engineering at Javeriana University. 
 

Improvement to Geometric Linearization of gpICA Algorithm by Compensation and Multiple Points

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Improvement to Geometric Linearization of gpICA Algorithm by Compensation and Multiple Points

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