An Overview of the Asymptotic Performance of the Family of the FastICA Algorithms
Tianwen~Wei
This contribution summarizes the results on the asymptotic performance of several variants of the FastICA algorithm. A number of new closed-form expressions %on the asymptotic covariance matrix of the FastICA estimator are presented. %By comparing the asymptotic error of the FastICA under different data preprocessing scenarios, the impact of the latter procedure on the asymptotic performance of the algorithms is revealed.
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\title{An Overview of the Asymptotic Performance of the Family of the FastICA Algorithms}
\author{Tianwen~Wei}
\institute{Laboratoire de Math\'ematiques de Besancon \\ Universit\'e de Franche-Comt\'e \\
 16 Route de Gray \\
 25000 Besancon, France}

%\thanks{
%T. Wei is with
%Laboratoire de Math\'ematiques de Besan\c{c}on, University of Franche-Comt\'e, Besan\c{c}%on, France. He is also with the Department of Statistics and Mathematics, Zhongnan %University of Economics and Law, Wuhan, China.
%Corresponding Author: tianwen.wei.2014@ieee.org.
%Part of this work was presented \cite{WEISSP2} at IEEE workshop on Statistical Signal %Processing 2014, Gold Coast, Australia.}
%}
\begin{document}
\maketitle

\begin{abstract}
This contribution summarizes the results on the asymptotic performance of several variants of the FastICA algorithm. A number of new closed-form expressions 
%on the asymptotic covariance matrix of the FastICA estimator
are presented. 
%By comparing the asymptotic error of the FastICA under different data preprocessing scenarios, the impact of the latter procedure on the asymptotic performance of the algorithms is revealed.
\end{abstract}

%\begin{IEEEkeywords}
\begin{keywords}
Independent component analysis, symmetric FastICA, deflationary FastICA, data whitening, data centering, asymptotic performance.
\end{keywords}
%\end{IEEEkeywords}

\section{Introduction}

In what follows, we denote scalars by lowercase letters $(a,b,c,\ldots)$, vectors by
boldface lowercase letters $(\a,\b,\cc, \ldots)$ and matrices by
 boldface uppercase letters $(\A,\B,\C,\ldots)$. Greek letters $(\alpha,\beta,\gamma,\ldots)$ are reserved for particular scalar quantities. We denote by $\A^\TR$ the matrix transpose of $\A$ and by $\|\cdot\|$ the Euclidean norm.

\subsection{ICA Data Model\label{sectionIIA}}
We consider the following noiseless linear ICA model:
\be
\y(t)=\H\s(t),\quad t=1,\ldots,N,\label{ICAmodel1}
\ee
where
\begin{enumerate}
\item $\s(t)\defeq(s_1(t),\ldots, s_d(t))^\TR$ denotes the $t$th realization of the unknown \emph{source signal}. The components $s_1(t),\ldots, s_d(t)$ are mutually statistically independent, have unit variance and at most one of them is Gaussian. 
Furthermore, $\s(1),\ldots, \s(N)$ denote $N$ independent realizations of $\s$.
\item $\y(t)\defeq(y_1(t),\ldots, y_d(t))^\TR$ denotes the $t$th realization of the \emph{observed signal}.
\item $\H\in\Rb^{d\times d}$ is a full rank square matrix, called the mixing matrix.
%\item  
\end{enumerate}
%Note that we do not assume $\s(t)$ has zero mean.

\subsection{Data Preprocessing}
Most ICA methods require the observed signal $\{\y(t)\}$ to be standardized 
\cite{COMO94,CARD1993,HYVA99}. The standardization of $\{\y(t)\}$ consists of the data centering and data whitening, which involve the estimation of $\Eb[\y]$ and $\cov(\y)$. 
%Depending on the hypothesis on the length of the sample size, the estimation of these %statistics to be exact or not. If the sample size is assumed infinite, then we can infer %the exact value of both $\Eb[\y]$ and $\cov(\y)$.
In practice, $\Eb[\y]$ and $\cov(\y)$ are usually estimated by the sample mean and sample variance:
\be
\bar{\y}\defeq \sum_{t=1}^N \frac{1}{N}\y(t), \quad
\widehat{\C} \defeq \frac{1}{N}\sum_{t=1}^N (\y(t) - \bar{\y})(\y(t) - \bar{\y})^\TR.
\ee  
%In general, $\bar{\y}\neq \Eb[\y]$ and $\widehat{\C}\neq \cov(\y)$. 
%It has been noticed \cite{CARD94} that the empirical data whitening procedure imposes a bound on the asymptotic performance of the algorithm; whereas the impact of empirical data centering was rarely discussed.
 In this work, we shall consider several different data preprocessing scenarios. 
 %we intend to reveal explicitly the influence of the data standardization procedure on the asymptotic performance of the algorithm. 
 Denote
 \be
 \widetilde{\C}= \frac{1}{N}\sum_{t=1}^N (\y(t) - \Eb[\y])(\y(t) - \Eb[\y])^\TR.
 \ee
The following data preprocessing scenarios will be studied: 
\begin{enumerate}
\item[1).] Theoretical whitening and theoretical centering.
 \ben\label{Preprocess1}
 \x(t)  \defeq \cov(\y)^{-\frac{1}{2}}(\y(t)-\Eb[\y]).
 \een
\item[2).] Theoretical whitening and empirical centering.
 \ben\label{Preprocess2}
 \x(t)  \defeq \cov(\y)^{-\frac{1}{2}}(\y(t)-\bar{\y}).
 \een
\item[3).] Empirical whitening and theoretical centering.
 \ben\label{Preprocess3}
 \x(t)  \defeq \widetilde{\C}^{-\frac{1}{2}}(\y(t)-\Eb[\y]).
 \een
\item[4).] Empirical whitening and empirical centering.
 \ben\label{Preprocess4}
 \x(t)  \defeq \widehat{\C}^{-\frac{1}{2}}(\y(t)-\bar{\y}).
 \een
\end{enumerate}
In the sequel, $\x(t)$ will always stand for the standardized signal under one of the  scenarios defined above. The specific data preprocessing scenario will be stated explicitly when necessary. 

\subsection{Variants of the FastICA Algorithm}
Before proceeding further, we need to introduce some notations first. We denote by $\Sc$ the unit sphere in $\Rb^d$. We denote by $g(\cdot): \Rb\to\Rb$ the nonlinearity function, and by
$G(\cdot)$ its primitive. The nonlinearity function $g$ is usually supposed to be
 non-linear, non-quadratic and smooth.
For any function $f:\Rb^d\to\Rb^m$, we write
$\EE_{\x}[f(\x)]\defeq \frac{1}{N} \sum_{t=1}^N f(\x(t))$ for conciseness.

\subsubsection{The Deflationary FastICA Algorithm}
This version of the FastICA algorithm extracts the sources sequentially. It consists of the following steps \cite{HYVA99}:
\begin{enumerate}
\item[-] {\bf Input:} $\x(1),\ldots,\x(N)$.
\item[1).] Set $p=1$.
\item[2).] Choose  an arbitrary  initial iterate $\w\in\Sc$;
\item[3).] Run iteration
\ben
\w & \leftarrow & \EE_{\x}[g'(\w^{\TR}\x)\w - g(\w^{\TR}\x)\x] \label{52} \\
\w &\leftarrow &\w-\sum_{i=1}^{p-1}(\w_i^{DFL})^\TR\w \label{52c} \\
\w & \leftarrow & \frac{\w}{\|\w\|} \label{52a}
\een
 until convergence\footnote{We impose the number of iterations to be even, so that the well known sign-flipping phenomenon disappears.  }. The limit is stored as $\w^{DFL}_p$.
 \item[4)] Break if $p=d$. Otherwise $p \leftarrow p+1$
  then go to step 2).
 \item[-] {\bf Output:} $\W^{DFL}=(\w_1^{DFL},\ldots,\w_d^{DFL})$.
\end{enumerate}
%\begin{rema}
%This is the asymptotic error of the ``one-unit'' FastICA, which is equivalent to the %deflationary version of FastICA subject to the extraction of only one source.
%\end{rema}

\subsubsection{The Symmetric FastICA Algorithm}
The symmetric version of FastICA extracts all the sources simultaneously. 
%It consists of parallel implementations of (\ref{52}) followed by a matrix orthogonalization step.
It can be described as follows:
\begin{enumerate}
\item[-] {\bf Input:} $\x(1),\ldots,\x(N)$.
\item[1).] Choose an arbitrary orthonormal matrix $\W=(\w_1,\ldots,\w_d)\in\Rb^{d\times d}$.
\item[2).] Run
\ben
\w_1 & \leftarrow & \EE_\x[g'(\w_1^{\TR}\x)\w_1 - g(\w_1^{\TR}\x)\x]  \label{127c1} \\
 & \vdots &  \nonumber \\
\w_d & \leftarrow & \EE_\x[g'(\w_d^{\TR}\x)\w_1 - g(\w_1^{\TR}\x)\x]  \label{127c2} \\
\W   & \leftarrow  &    \Big(\W\W^{\TR}  \Big)^{-1/2}\W\label{127b}
\een
 until convergence. The limit is denoted by $\W^{SYM}$.
\item[-] {\bf Output:} $\W^{SYM}=(\w_1^{SYM},\ldots,\w_d^{SYM})$.
\end{enumerate}


\section{Asymptotic Performance}
%\subsection{Objective}
Let us introduce the notion of gain matrix:
\be
\G^{DFL}\defeq (\W^{DFL})^\TR\C^{-1/2}\H, \quad \G^{SYM}\defeq (\W^{SYM})^\TR\C^{-1/2}\H,
\ee
where $\C^{-1/2}$ stands for the sphering matrix used in the data preprocessing stage,
i.e. $\C=\cov(\y)$ in scenarios (\ref{Preprocess1}) and (\ref{Preprocess2}),
$\C=\widetilde{\C}$ 
in scenario (\ref{Preprocess3})
and  $\C=\widehat{\C}$ in scenario (\ref{Preprocess4}).
Without loss of generality, we shall omit the permutation and sign ambiguities of ICA.
 Then,
% 
%We have
%$\G^{DFL} \approx \P\Lbd$ and
%$\G^{SYM} \approx \P\Lbd$, where $\P$ is a permutation matrix and $\Lbd$ is a diagonal %matrix whose diagonal entries are either $1$ or $-1$. 
%Matrices $\P$ and $\Lbd$ are due to the inherent permutation and sign ambiguities of the %ICA.
%Without any loss of generality, we will omit these ambiguities by assuming
%$\P=\Lbd=\I$
%in the subsequent analysis, where $\I$ denotes the identity matrix.
% Now that 
% 
 $\G^{DFL} \approx \I$ and $\G^{SYM} \approx \I$, hence $\C^{-1/2}\W^{DFL}$
 and $\C^{-1/2}\W^{SYM}$ can be considered as estimators of $\B\defeq(\H^{-1})^\TR$.
In the sequel, we will study the asymptotic errors of
$N^{1/2}(\C^{-1/2}\W^{DFL} - \B)$ and $N^{1/2}(\C^{-1/2}\W^{SYM} - \B)$ under proposed data preprocessing scenarios.

The proofs of the results presented below are based on the method of M-estimators.
However, all proofs will be omitted due to the lack of space.  
A complete version of this work can be provided upon request.
 The readers are also referred to \cite{WEI4} for a more detailed account of this subject. 
\subsection{The Asymptotic Error of Deflationary FastICA}
Assume that the  following mathematical expectations exist for $i=1,\ldots,d$:
\ben
\alpha_{i}&\defeq& \Eb[g'(z_{i}) - g(z_{i})z_{i}] \nonumber \\
%\alpha_i & \defeq & \Eb[g_i'(s_{\sigma(i)}) - g_i(s_{\sigma(i)})s_{\sigma(i)}]\\
\beta_{i} & \defeq & \Eb[g(z_{i})^2]  \nonumber  \\
\gamma_{i} & \defeq & \Eb[g(z_{i})z_{i}]  \nonumber \\
\eta_{i} & \defeq & \Eb[g(z_{i})] \nonumber \\
\tau_{i} & \defeq & (\Eb[z_{i}^4]-1)/4, \nonumber
\een
where $z_i=s_i-\Eb[s_i]$ for $i=1,\ldots,d$.
\begin{theo}\label{main1}
Let $\b_i$ denote the $i$th column of $\B$. Under some mild regularity conditions, we have
\be
N^{1/2}(\C^{-1/2}\w_i^{DFL} - \b_i)\CD \Nc(0, \R_{(k)}^{DFL}),
\ee
where $k\in\{1,2,3,4\}$ is the label of the underlying data preprocessing scenario (see (\ref{Preprocess1})-(\ref{Preprocess4})) and $\R_{(k)}^{DFL}$ is given as follows:
\ben
\R_{(1)}^{DFL}& =& \sum_{j=1}^{i-1} \frac{\beta_j^2}{\alpha_j^2}\b_j\b_j^\TR
+ \sum_{ p,q=1\atop p\neq q}^{i-1}\frac{\eta_p\eta_q}{\alpha_p\alpha_{q}}\b_p\b_q^\TR
+   \frac{\beta_i^2}{\alpha_i^2}\sum_{j=i+1}^{d} \b_j\b_j^\TR, \label{DFL1} \\
%&&+\frac{\beta_i-\gamma_i^2-\eta_i^2}{\alpha_i^2 } \sum_{j=i+1}^d\b_j\b_j^\TR.\\
\R_{(2)}^{DFL}& =&  \sum_{j=1}^{i-1}\frac{\beta_j-\eta_j^2}{\alpha_j^2 }\b_j\b_j^\TR +\frac{\beta_i-\eta_i^2}{\alpha_i^2 } \sum_{j=i+1}^d\b_j\b_j^\TR, \label{DFL2}\\
\R_{(3)}^{DFL}& =&  \sum_{j=1}^{i-1}\frac{\beta_j - \gamma_j^2  + \alpha_j^2}{\alpha_j^{2}} \b_j\b_j^\TR
 +  \sum_{p,q=1\atop p\neq q}^{i-1} \frac{\eta_p\eta_q}{ \alpha_p\alpha_q}\b_p\b_q^\TR  
 \nonumber   + \tau_i  \b_i\b_i^\TR  \\
 && +\frac{\beta_i - \gamma_i^2}{\alpha^2_{i}}\sum_{j=i+1}^{d} \b_j\b_j^\TR   -  \sum_{j=1}^{i-1}\frac{\Eb[s_i^3]\eta_j}{\alpha_j}(\b_j{\b_i^\TR} + \b_i{\b_j^\TR} ) \label{DFL3}, \\
\R_{(4)}^{DFL}&=& \sum_{j=1}^{i-1}\frac{\beta_j - \gamma_j^2  + \alpha_j^2 - \eta_j^2}{\alpha_j^{2}} \b_j\b_j^\TR   + \tau_i \b_i\b_i^\TR 
  +\frac{\beta_i - \gamma_i^2 -\eta_i^2}{\alpha^2_{i}}\sum_{j=i+1}^{d} \b_j\b_j^\TR 
  \nonumber \\
&& -  \sum_{j=1}^{i-1}\frac{\Eb[s_i^3]\eta_j}{\alpha_j}(\b_j{\b_i^\TR} + \b_i{\b_j^\TR} ).\label{DFL4}
\een
\end{theo}
%{\bf Proof.} See Appendix for a sketch of the proof. \hfill$\Box$

\begin{comment}
\begin{coro}
Taking $i=1$ in (\ref{DFL1})-(\ref{DFL4}) gives
\ben
\R^{1U}_{(1)}& =&   \frac{\beta_1}{\alpha_1^2 } \sum_{j=2}^d\b_j\b_j^\TR,  \label{U1}\\
\R^{1U}_{(2)}& =&   \frac{\beta_1-\eta_1^2}{\alpha_1^2 } \sum_{j=2}^d\b_j\b_j^\TR,\label{U2}\\
\R^{1U}_{(3)}& =&   \frac{\beta_1 - \gamma_1^2}{\alpha_1^2 } \sum_{j=2}^d\b_j\b_j^\TR + \tau_1\b_1\b_1^\TR, \label{U3}\\
\R^{1U}_{(4)}& =&   \frac{\beta_1-\gamma_1^2-\eta_1^2}{\alpha_1^2 } \sum_{j=2}^d\b_j\b_j^\TR + \tau_1\b_1\b_1^\TR. \label{U4}
\een
%This is the asymptotic error of the deflationary FastICA when one intends to extract only one source.
\end{coro}
\end{comment}


\begin{coro}\label{coroDFL2}There holds
$
N^{1/2}({\G}_{ij}^{DFL} - \delta_{ij})\CD \Nc(0, V_{(k)}^{DFL}),
$
where ${\G}_{ij}^{DFL}$ denotes the $(i,j)$th entry of $\G^{DFL}$ and $V_{(k)}^{DFL}$ is given as follows:
\begin{enumerate}
\item Case $j< i$:
\be
V^{DFL}_{(1)} &=& \frac{\beta_j^2}{\alpha_j^2}   \\
V^{DFL}_{(2)} &=& \frac{\beta_j - \eta_j^2}{\alpha_j^2} \\
V^{DFL}_{(3)} &=& \frac{\beta_j - \gamma_j^2 + \alpha_j^2}{\alpha_j^2} \\
V^{DFL}_{(4)} &=& \frac{\beta_j - \gamma_j^2 + \alpha_j^2 - \eta_j^2}{\alpha_j^2}.
\ee
\item Case $j= i$:
\be
V^{DFL}_{(1)} =V^{DFL}_{(2)} =0 ,\quad\quad V^{DFL}_{(3)} =V^{DFL}_{(4)} = \tau_i .
\ee
\item  Case $j> i$:
\ben
V^{DFL}_{(1)} &=& \frac{\beta_i}{\alpha_i^2}  \label{dflgain1} \\
V^{DFL}_{(2)} &=& \frac{\beta_i - \eta_i^2}{\alpha_i^2}  \label{dflgain2} \\
V^{DFL}_{(3)} &=& \frac{\beta_i - \gamma_i^2}{\alpha_i^2}  \label{dflgain3} \\
V^{DFL}_{(4)} &=& \frac{\beta_i - \gamma_i^2 - \eta_i^2}{\alpha_i^2}. \label{dflgain4}
\een
\end{enumerate}
\end{coro}



\subsection{The Asymptotic Error of Symmetric FastICA}
\begin{theo}\label{main2}
Under some mild regularity conditions, we have
$
N^{1/2}(\C^{-1/2}\w_i^{SYM} - \b_i)\CD \Nc(0, \R_{(k)}^{SYM}),
$
where
\ben
\R^{SYM}_{(1)}&=&\sum_{j\neq i}^d \frac{\beta_i + \beta_j -2\gamma_i\gamma_j -2 \eta_j^2}{(|	\alpha_i|+|\alpha_j|)^2}\b_j\b_j^\TR  + 2\sum_{j\neq i}^d \frac{\eta_j\b_j}{|\alpha_i| + |\alpha_j|} \sum_{j\neq i}^d 	
	\frac{\eta_j\b_j^\TR}{|\alpha_i| + |\alpha_j|}, \label{SYM1}\\
\R^{SYM}_{(2)}&=& \sum_{j\neq i}^d \frac{\beta_i  + \beta_j - 2\gamma_i\gamma_j-2\eta_i^2 }
	{(|\alpha_i| 		+	|\alpha_j|)^2} \b_j\b_j^\TR , \label{SYM2} \\
\R^{SYM}_{(3)} &=&\sum_{j\neq i}^d \frac{\beta_i -\gamma_i^2 + \beta_j -\gamma_j^2 + 	
	\alpha_j^2 - \eta_j^2}{(|\alpha_i| 
 	+ |\alpha_j|)^2} \b_j\b_j^\TR  + \sum_{j\neq i}^d \frac{\eta_j \b_j }{(|\alpha_{i}|+|\alpha_{j})|} \sum_{j\neq i}^d 	\frac{ \eta_j\b_j^\TR }{(|\alpha_{i}|+|\alpha_{j}|)}
	\nonumber \\
&&+ \tau_i\b_i\b_i^\TR 	 -\sum_{j\neq i}^d\frac{ \Eb[s_i^3]\eta_j }{2(|\alpha_{i}|+|\alpha_{j}|)}  (\b_j
	\b_i^\TR + \b_i\b_j^\TR),  \label{SYM3}\\
\R^{SYM}_{(4)}&=&\sum_{j\neq i}^d \frac{\beta_i - \gamma_i^2 + \beta_j - \gamma_j^2 + 
	\alpha_j^2 -\eta_i^2 - \eta_j^2
 	}{(|\alpha_i| + |\alpha_j|)^2}\b_j\b_j^\TR + \tau_i\b_i\b_i^\TR.  \label{SYM4}
\een
\end{theo}
%{\bf Proof.} See Appendix for a sketch of the proof. \hfill$\Box$
\begin{coro}\label{coroSYM}For $i,j=1,\ldots,d$, there holds
$
N^{1/2}({\G}_{ij}^{SYM} - \delta_{ij})\CD \Nc(0, V_{(k)}^{SYM}),
$
 where 
\begin{enumerate}
\item Case $j=i$:
 \be
V^{SYM}_{(1)}=V^{SYM}_{(2)}=0 ,\quad\quad V^{SYM}_{(3)}=V^{SYM}_{(4)}=\tau_i. 
\ee
\item Case $j\neq i$:
\ben
V_{(1)}^{SYM}&=&\frac{\beta_i  + \beta_j - 2\gamma_i\gamma_j  }{(|\alpha_i| + |\alpha_j|)^2}, \quad\label{gain1} \\
V_{(2)}^{SYM}&=&\frac{\beta_i  + \beta_j - 2\gamma_i\gamma_j -2\eta_i^2  }{(|\alpha_i| + |\alpha_j|)^2}, \quad\label{gain2} \\
V_{(3)}^{SYM}&=&\frac{\beta_i - \gamma_i^2 + \beta_j - \gamma_j^2 + \alpha_j^2  }{(|\alpha_i| + |\alpha_j|)^2}, \quad\label{gain3} \\
V_{(4)}^{SYM}&=&\frac{\beta_i - \gamma_i^2 + \beta_j - \gamma_j^2 + \alpha_j^2 -\eta_i^2 - \eta_j^2
 }{(|\alpha_i| + |\alpha_j|)^2}. \quad\label{gain4}
\een
\end{enumerate}
\end{coro}
\begin{rema}
Although the asymptotic error of the FastICA algorithm has already been studied by quite a few researchers \cite{HYVA97,HYVA2006,OLLI2011,TICHOJA},
many of the results presented in this contribution, notably expressions (\ref{DFL1})-(\ref{DFL3}) established in Theorem \ref{main1} and (\ref{SYM1})-(\ref{SYM4}) 
in Theorem \ref{main2}, are new.
\end{rema}

\begin{exam}
The validity of formulas (\ref{dflgain1})-(\ref{dflgain4}) and (\ref{gain1})-(\ref{gain4}) is verified in computer simulations, see Fig. \ref{DFLplot} and Fig. \ref{SYMplot}. The simulations are configured as follows: $d=3$, $N=5000$, all three sources have identical bimodal Gaussian distribution with asymmetrical density. Both deflationary FastICA and symmetric FastICA have been tested with different data preprocessing (\ref{Preprocess1})-(\ref{Preprocess4}) in 5000 independent trials.
\end{exam}

\begin{comment}
The asymptotic error of the deflationary FastICA was first tackled by Hyv\"{a}rinen \cite{HYVA97} and later by Tichavsky et al. \cite{TICHOJA}.
Assuming the data preprocessing scenario (\ref{Preprocess3}),
Hyv\"{a}rinen and Tichavsky et al. studied the special case where the source of interest is extracted during the first run
 and obtained expression (\ref{dflgain3}).
  %i.e. the deflationary step  is not involved.
%, under the data preprocessing scenario of exact centering and empirical whitening. 
%Some expressions equivalent to (\ref{dflgain3}) were derived therein. 
The general case of deflationary FastICA under the data preprocessing scenario (\ref{Preprocess4})
was addressed by Ollila et al. \cite{OLLI, OLLI2011}, 
in which 
%the deflationary step (\ref{52c}) was taken into account and
 expression (\ref{DFL4}) was derived. 
 
 The asymptotic error of the symmetric FastICA was first tackled by Shimizu \cite{HYVA2006} and Tichavsky et al. \cite{TICHOJA}. In \cite{HYVA2006} no closed-form expression was given while in
 \cite{TICHOJA} the authors obtained (\ref{gain3}).
 Recently, in a separate paper \cite{WEI4}, the author of the current work studied the symmetric FastICA  under the data preprocessing scenario (\ref{Preprocess4}) and derived expression (\ref{SYM4}). This contribution can be considered as a following-up of \cite{WEI4}.
\end{comment}

\subsection{Discussion}
First, comparing the expressions
in Corollary \ref{coroDFL2} and Corollary \ref{coroSYM},
 we find that for the $(i,j)$th entry of the gain matrix,
\be
V^{DFL}_{(1)} - V^{DFL}_{(2)}&=& V^{DFL}_{(3)} - V^{DFL}_{(4)}
=\frac{\eta_j^2}{\alpha_j^2}, \quad j<i,\\
V^{DFL}_{(1)} - V^{DFL}_{(2)}&=& V^{DFL}_{(3)} - V^{DFL}_{(4)}
=\frac{\eta_i^2}{\alpha_i^2}, \quad j>i,\\
V^{SYM}_{(1)} - V^{SYM}_{(2)}&=& 
\frac{2\eta_i^2}{(|\alpha_i| + |\alpha_j|)^2},\quad i\neq j,\\
V^{SYM}_{(3)} - V^{SYM}_{(4)}&=& 
\frac{\eta_i^2 + \eta_j^2}{(|\alpha_i| + |\alpha_j|)^2},\quad i\neq j.
\ee
Since all the differences above are non-negative\footnote{They become zero if 
$\eta_i$ and/or $\eta_j$ vanish. This is the case
 if, e.g. $g$ is pair and the involved sources have symmetric distributions. 
% We also point out that $\alpha_j$ cannot be zero, otherwise FastICA will not work \cite{HYVA99}.
 }, we assert that the empirical data centering generally leads to a better asymptotic performance.

\begin{comment} 
Second, notice that for the $(i,i)$th entry of the gain matrix we have
\be
G^{DFL}_{(1)} &=& G^{DFL}_{(2)}=0, \\
G^{DFL}_{(3)} &=& G^{DFL}_{(4)}=\tau_i, \\
G^{SYM}_{(1)} &=& G^{SYM}_{(2)}=0, \\
G^{SYM}_{(3)} &=& G^{SYM}_{(4)}=\tau_i,
\ee
where $\tau_i>0$. This means that
for diagonal entries of the gain matrices,
their asymptotic error with empirical whitening is always larger than that with theoretical whitening. 
As for the off-diagonal entries of the gain matrices, we see from Fig. \ref{DFLplot} and Fig. \ref{SYMplot} that, the corresponding asymptotic error under empirical data whitening can be either lower or larger than that under exact data whitening.
\end{comment}









\begin{figure}[t]
\centerline{
\includegraphics[width=1.3\textwidth]{deflation12_N5k_T5k.eps}}
\caption{Asymptotic error of the deflationary FastICA in each preprocessing scenario.
We plotted the histograms of an (upper) off-diagonal entry of $N^{1/2}{\G}^{DFL}$
in 5000 independent trials versus
the theoretical curves of the Gaussian PDFs with variances given by
 (\ref{dflgain1})-(\ref{dflgain4}). \label{DFLplot}}
\centerline{
\includegraphics[width=1.3\textwidth]{symmetric12_N5k_T5k.eps}}
\caption{Asymptotic error of the symmetric FastICA in each preprocessing scenario.
We plotted the histograms of an off-diagonal entry of $N^{1/2}{\G}^{SYM}$
in 5000 independent trials versus
the theoretical curves of the Gaussian PDFs with variances given by
 (\ref{gain1})-(\ref{gain4}). \label{SYMplot}}
\end{figure}



\section{Conclusion}
The contribution of this work is twofold. First, we derived explicit formulas for the asymptotic error of the two most important variants of the FastICA algorithm, the deflationary FastICA and the symmetric FastICA, under four different data preprocessing scenarios. Many of the presented formulas are novel. 
Second, we assessed the impact of empirical data preprocessing procedure on the
asymptotic performance of the algorithms.
We showed that, compared to the theoretical data centering, the empirical data centering generally leads to a better separation performance.%, while the influence of the empirical data whitening is quite subtle. 
%For diagonal entries of the gain matrix, the asymptotic error of both versions of FastICA %under empirical data whitening is always larger than that under theoretical data %whitening. 
%However, for off-diagonal entries, the asymptotic error under empirical data whitening %can be either lower or larger than that under exact data whitening.

\begin{comment}
\section{Appendices}
Our approach is based on the method of M-estimators \cite{VAAR}.
% is a powerful tool to solve problems of this kind, see\cite{HYVA2006, OLLI2011, %REYH2012, WEI2} for some earlier attempts.
Suppose that the unknown distribution of random vector $\y$ depends on some parameter $\teta$ of interest. Suppose also that the true parameter $\teta^*$ satisfies equation $\Eb[\PSI(\teta^*,\y)]=0$, where $\PSI$ is some vector valued function. Let
$\y(1),\ldots,\y(N)$ be an i.i.d. sample of $\y$. The estimator $\hat{\teta}$ obtained by solving the following equation
\ben\label{219a}
\frac{1}{N}\sum_{t=1}^N\PSI\big({\teta}, \y(t)\big)=\EE_{\y}[\PSI(\teta,\y)]=0.
\een
 is called an M-estimator, and (\ref{219a}) is called the estimating equation.
Under some mild regularity conditions \cite{VAAR},  there holds
\be
N^{1/2}(\hat{\teta}-\teta^*)\CD \Nc\Big(0,   \Q^{-1}\Eb[\PSI(\teta^*,\y) \PSI(\teta^*,\y)^\TR]\Q^{-\TR}\Big),
\ee
where $\CD$ denotes the convergence in distribution and
\be
\Q\defeq \Eb\Big[\frac{\partial}{\partial{\teta}}\PSI(\teta,\y)\Big|_{\teta=\teta^*}\Big].
\ee

To derive the estimating equation, we need to characterize the limit of the underlying algorithm. For the deflationary FastICA, it is easily seen \cite{OLLI2011} that
for $i=1,\ldots,d$
\be
\w^{DFL}_{i}= \argmin{\|\w\|=1; 
\w^\TR\w^{DFL}_k=0, \atop k=1,\ldots, i-1}
 \EE_{\x}[G(\w^\TR\x)].
\ee
Then it follows from the first order condition that $\w^{DFL}_i$ must satisfies
\be
(\w^{DFL}_i)^\TR\w^{DFL}_i&=&1; \\
(\w^{DFL}_i)^\TR\w^{DFL}_j&=&0, \quad j=1,\ldots, i-1
\ee
and
\be
\EE_{\x}[g((\w_i^{DFL})^\TR\x)\x] = \sum_{j=1}^i \lambda_{ij}\w_{j}^{DFL},
\ee
for some $(\lambda_{i1},\ldots,\lambda_{ii})$. Substituting $\x$ by one of (\ref{Preprocess1})-(\ref{Preprocess4}) and using the change of variable
$\w_i=\C^{-1/2}\w^{DFL}_i$, where $\C$ equals $\cov(\y)$, $\widetilde{\C}$ or $\widehat{\C}$ depending on the preprocessing scenario, 
 we get (\ref{DFLPHI1})-(\ref{DFLPHI4}). Then the estimating equation for the
 deflationary FastICA follows.
 
 
As for the symmetric FastICA, we have \cite{WEI4}
\be
(\w_1^{SYM},\ldots,\w_d^{SYM})= \argmin{\w_i^\TR\w_j=\delta_{ij},
\atop  i,j=1,\ldots,d}
  \sum_{i=1}^d \EE_{\x}[\sign(\alpha_i)G(\w_i^\TR\x)].
\ee 
Thus, by the first order condition, $(\w^{SYM}_1,\ldots,\w^{SYM}_d)$ must satisfy
\be
(\w_i^{SYM})^\TR\w_j^{SYM}=\delta_{ij}
\ee
and
\be
\EE_{\x}[\sign(\alpha_i)g(\w_i^\TR\x)\w_j^\TR\x]
=\EE_{\x}[\sign(\alpha_j)g(\w_j^\TR\x)\w_i^\TR\x]
\ee
for all $i,j=1,\ldots,d$.
Writing $\tilde{g}_i=\sign(\alpha_i)g$ and applying the same change of variable as in the case of deflationary FastICA, we obtain (\ref{SYMPHI1})-(\ref{SYMPHI4}).

Now that we have the estimating equations, it remains to calculate the asymptotic covariance matrices. 
Due to the lack of space,  this part has to be omitted and
 the readers are referred to \cite{WEI4} for a more detailed account. A complete proof can also be provided upon request.


\begin{figure*}
\ben
\PSI_{(1)}^{DFL}(\teta,\y)
&=&\begin{pmatrix}
\w_1^\TR \H\H^\TR\w_1 - \delta_{11} \\
\w_2^\TR\H\H^\TR\w_1 - \delta_{12} \\
\vdots \\
\w_d^\TR\H\H^\TR\w_d - \delta_{dd} \\
g(\w_1^\TR\y)\y - \lambda_{11}\H\H^\TR\w_1 \\
\vdots \\
g(\w_d^\TR\y)\y - \sum_{j=1}^d\lambda_{dj} \H\H^\TR\w_j 
\end{pmatrix}, 
\quad \teta=(\w_1,\ldots,\w_d, \lambda_{11},\ldots,\lambda_{dd}); \nonumber \\
&&  \label{DFLPHI1} \\
\PSI_{(2)}^{DFL}(\teta,\y)
&=&\begin{pmatrix}
\y-\muu \\
\w_1^\TR \H\H^\TR\w_1 - \delta_{11} \\
\w_2^\TR\H\H^\TR\w_1 - \delta_{12} \\
\vdots \\
\w_d^\TR\H\H^\TR\w_d - \delta_{dd} \\
g\Big(\w_1^\TR(\y-\muu)\Big)(\y-\muu) - \lambda_{11}\H\H^\TR\w_1 \\
\vdots \\
g\Big(\w_d^\TR(\y-\muu)\Big)(\y-\muu) - \sum_{j=1}^d\lambda_{dj}\H\H^\TR\w_j
\end{pmatrix}, 
\quad \teta=(\w_1,\ldots,\w_d, \lambda_{11},\ldots,\lambda_{dd},\muu); \nonumber \\
&&  \label{DFLPHI2} \\
\PSI_{(3)}^{DFL}(\teta,\y)
&=&\begin{pmatrix}
\w_1^\TR\y\y^\TR\w_1 - \delta_{11} \\
\w_2^\TR\y\y^\TR\w_1 - \delta_{12} \\
\vdots \\
\w_d^\TR\y\y^\TR\w_d - \delta_{dd} \\
g(\w_1^\TR\y)\y - \lambda_{11}\y\y^\TR\w_1 \\
\vdots \\
g(\w_d^\TR\y)\y - \sum_{j=1}^d \lambda_{dj}\y\y^\TR\w_j
\end{pmatrix}, 
\quad \teta=(\w_1,\ldots,\w_d, \lambda_{11},\ldots,\lambda_{dd}); \nonumber \\
&& \label{DFLPHI3}\\
\PSI_{(4)}^{DFL}(\teta,\y)
&=&\begin{pmatrix}
\y-\muu \\
\w_1^\TR(\y-\muu)(\y-\muu)^\TR\w_1 - \delta_{11} \\
\w_2^\TR(\y-\muu)(\y-\muu)^\TR\w_1 - \delta_{12} \\
\vdots \\
\w_d^\TR(\y-\muu)(\y-\muu)^\TR\w_d - \delta_{dd} \\
g\Big(\w_1^\TR(\y-\muu)\Big)(\y-\muu) - \lambda_{11}(\y-\muu)(\y-\muu)^\TR\w_1 \\
\vdots \\
g\Big(\w_d^\TR(\y-\muu)\Big)(\y-\muu) - \sum_{j=1}^d \lambda_{dj}(\y-\muu)(\y-\muu)^\TR\w_j
\end{pmatrix}, 
\quad \teta=(\w_1,\ldots,\w_d, \lambda_{11},\ldots,\lambda_{dd},\muu); \nonumber \\
&&  \label{DFLPHI4}
\een
\end{figure*}





\begin{figure*}[!t]
\ben\label{PSI}
\PSI^{\small{SYM}}_{(1)}(\teta,\y)&\defeq&
\begin{pmatrix}
\w_1^\TR\H\H^\TR\w_1 - \delta_{11} \\
\w_1^\TR \H\H^\TR\w_2 - \delta_{12} \\
\vdots \\
\w_d^\TR\H\H^\TR\w_d - \delta_{dd} \\
\tilde{g}_1\big(\w_1^\TR\y\big)\w_2^\TR\y - \tilde{g}_2\big(\w_2^\TR\y \big)\w_1^\TR\y \\
\vdots\\
\tilde{g}_{d-1}\big(\w_{d-1}^\TR\y\big)\w_{d}^\TR\y- \tilde{g}_{d}\big(\w_{d}^\TR\y\big)\w_{d-1}^\TR\y
\end{pmatrix}, \quad \teta=(\w_1,\ldots,\w_d);   \label{SYMPHI1} \\
\PSI^{SYM}_{(2)}(\teta, \y) & \defeq &
\begin{pmatrix}
\y-\muu\\
\w_1^\TR \H\H^\TR\w_1 - \delta_{11} \\
\w_1^\TR \H\H^\TR\w_2 - \delta_{12} \\
\vdots \\
\w_d^\TR \H\H^\TR\w_d - \delta_{dd} \\
\tilde{g}_1\big(\w_1^\TR(\y-\muu)\big)\w_2^\TR(\y-\muu) - \tilde{g}_2\big(\w_2^\TR(\y-\muu)\big)\w_1^\TR(\y-\muu) \\
\tilde{g}_1\big(\w_1^\TR(\y-\muu)\big)\w_3^\TR(\y-\muu) - \tilde{g}_3\big(\w_3^\TR(\y-\muu)\big)\w_1^\TR(\y-\muu) \\
\vdots\\
\tilde{g}_{d-1}\big(\w_{d-1}^\TR(\y-\muu)\big)\w_{d}^\TR(\y-\muu) - \tilde{g}_{d}\big(\w_{d}^\TR(\y-\muu)\big)\w_{d-1}^\TR(\y-\muu)
\end{pmatrix} , \quad \teta=(\w_1,\ldots,\w_d, \muu);  \label{SYMPHI2} \\
\PSI^{\small{SYM}}_{(3)}(\teta,\y)&\defeq&
\begin{pmatrix}
\w_1^\TR\y\y^\TR\w_1 - \delta_{11} \\
\w_1^\TR\y\y^\TR\w_2 - \delta_{12} \\
\vdots \\
\w_d^\TR\y\y^\TR\w_d - \delta_{dd} \\
\tilde{g}_1\big(\w_1^\TR\y\big)\w_2^\TR\y - \tilde{g}_2\big(\w_2^\TR\y \big)\w_1^\TR\y \\
\vdots\\
\tilde{g}_{d-1}\big(\w_{d-1}^\TR\y\big)\w_{d}^\TR\y- \tilde{g}_{d}\big(\w_{d}^\TR\y\big)\w_{d-1}^\TR\y
\end{pmatrix}, \quad \teta=(\w_1,\ldots,\w_d);  \label{SYMPHI3} \\
\PSI^{SYM}_{(4)}(\teta \y) & \defeq &
\begin{pmatrix}
\y-\muu\\
\w_1^\TR(\y-\muu)(\y-\muu)^\TR\w_1 - \delta_{11} \\
\w_1^\TR(\y-\muu)(\y-\muu)^\TR\w_2 - \delta_{12} \\
\vdots \\
\w_d^\TR(\y-\muu)(\y-\muu)^\TR\w_d - \delta_{dd} \\
\tilde{g}_1\big(\w_1^\TR(\y-\muu)\big)\w_2^\TR(\y-\muu) - \tilde{g}_2\big(\w_2^\TR(\y-\muu)\big)\w_1^\TR(\y-\muu) \\
\tilde{g}_1\big(\w_1^\TR(\y-\muu)\big)\w_3^\TR(\y-\muu) - \tilde{g}_3\big(\w_3^\TR(\y-\muu)\big)\w_1^\TR(\y-\muu) \\
\vdots\\
\tilde{g}_{d-1}\big(\w_{d-1}^\TR(\y-\muu)\big)\w_{d}^\TR(\y-\muu) - \tilde{g}_{d}\big(\w_{d}^\TR(\y-\muu)\big)\w_{d-1}^\TR(\y-\muu)
\end{pmatrix}, \quad \teta=(\w_1,\ldots,\w_d,\muu);  \label{SYMPHI4}
\een
%\setcounter{equation}{\value{mytempeqncnt}}
\hrulefill
\vspace*{4pt}
\end{figure*}
\end{comment}



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\bibitem{COMO94}
P.~Comon, ``Independent component analysis: a new concept?'' \emph{Signal
  Processing}, vol.~36, no.~3, pp. 287--314, Apr. 1994.

\bibitem{CARD1993}
J.~F. Cardoso and A.~Souloumiac, ``Blind beamforming for non-gaussian
  signals,'' \emph{IEEE Proceedings-F}, vol. 140, no.~6, pp. 362--370, Dec.
  1993.

\bibitem{HYVA99}
A.~Hyv{\"a}rinen, ``Fast and robust fixed-point algorithms for independent
  component analysis,'' \emph{IEEE Transactions on Neural Networks}, vol.~10,
  no.~3, pp. 626--634, 1999.

\bibitem{WEI4}
T.~Wei, ``A convergence and asymptotic analysis of the generalized symmetric
  fastica algorithm (submitted),'' \emph{ArXiv}, 2015.

\bibitem{HYVA97}
A.~Hyv{\"a}rinen, ``One-unit contrast functions for independent component
  analysis: A statistical analysis,'' in \emph{Proc. IEEE NNSP Workshop
  '97}.\hskip 1em plus 0.5em minus 0.4em\relax Neural Networks for Signal
  Processing VII, 1997.

\bibitem{HYVA2006}
A.~Shimizu, A.~Hyv{\"{a}}rinen, K.~Yutaka, P.~Hoyer, and A.~J. Kerminen,
  ``Testing signifcance of mixing and demixing coefficients in {ICA},'' in
  \emph{Int. Conf. Independent Component Analysis ({ICA} 2006)}, 2006.

\bibitem{OLLI2011}
K.~Nordhausen, P.~Ilmonen, A.~Mandal, H.~Oja, and E.~Ollila, ``Deflation-based
  {FastICA} reloaded,'' in \emph{19th European Signal Processing Conference
  (EUSIPCO 2011)}, Barcelona, Spain, Sep. 2011.

\bibitem{TICHOJA}
P.~Tichavsky, Z.~Koldovsky, and E.~Oja, ``Performance analysis of the {FastICA}
  algorithm and cramer-rao bounds for linear independent component analysis,''
  \emph{IEEE transactions on Signal Processing}, vol.~54, no.~4, pp.
  1189--1203, Apr. 2006.

\end{thebibliography}
