Performance Analysis of the nc-FastICA Algorithm

Abstract

The noncircular complex Fast Independent Component Analysis (nc-FastICA) algorithm is one of the most popular algorithms for solving the ICA problems with circular and noncircular complex-valued data. However, the performance of nc-FastICA has not been comprehensively studied in the past. In this paper, this matter is addressed. Based on the global analysis of fixed-point iteration and cost function, we get that there exist undesirable fixed points for nc-FastICA algorithm. To get rid of the undesirable fixed points, a simple check of undesirable points is proposed based on the kurtosis of the estimated sources. Furthermore, theoretical analysis in this paper shows that the nc-FastICA algorithm can work well even when the sources fall into the area of instability. Computer simulations validate the correctness of the theoretical findings in the paper.

Appendix: Analysis of Stationary Points

The detailed analysis of stationary points is presented as follows:

• (1) If $E({{s}^{2}})=0$, then the stationary points are $\large \{ _{\forall \alpha }^{\theta = 0}$, $\large \{ _{\forall \alpha }^{\theta = \pi /2}$, $\large \{ _{\forall \alpha }^{\theta = \pi /4}$;

  • (i) As $T(\theta =0,\forall \alpha )=0$, second derivatives test gives no information. We now make a small perturbation $({{\varepsilon }_{\theta }},{{\varepsilon }_{\alpha }})$ around the point ($\theta =0,\forall \alpha$) to derive

    $$\begin{aligned} J({\varepsilon _\theta },\alpha + {\varepsilon _\alpha }) - J(0,\alpha ) = \frac{1}{4}{\sin ^2}(2{\varepsilon _\theta }) [ - E({\left| s \right| ^4}) + 2{E^2}({\left| s \right| ^2})] \end{aligned}$$

    (35)

    If $-E({{\left| s \right| }^{4}})+2{{E}^{2}}({{\left| s \right| }^{2}})>0$, then $J({{\varepsilon }_{\theta }},\alpha +{{\varepsilon }_{\alpha }})-J(0,\alpha )\ge 0$. In this case, the point ($\theta =0,\forall \alpha$) is a local minimum of cost function. If $-E({{\left| s \right| }^{4}})+2{{E}^{2}}({{\left| s \right| }^{2}})<0$, then $J({{\varepsilon }_{\theta }},\alpha +{{\varepsilon }_{\alpha }})-J(0,\alpha )\le 0$. In this case, the point ($\theta =0,\forall \alpha$) is a local maximum of cost function. If $-E({{\left| s \right| }^{4}})+2{{E}^{2}}({{\left| s \right| }^{2}})=0$, then $J(\theta ,\alpha )$ is a constant function. In this case, the point ($\theta =0,\forall \alpha$) is an ordinary point of cost function.

  • (ii) The detailed analysis of stationary point $(\theta =\pi /2,\forall \alpha )$ is the same as $(\theta =0,\forall \alpha )$ and is omitted here.

  • (iii) Similarly, due to $T(\theta =\pi /4,\forall \alpha )=0$, second derivatives test gives no information. Therefore, we make a small perturbation $({{\varepsilon }_{\theta }},{{\varepsilon }_{\alpha }})$ around the point ($\theta =\pi /4,\alpha$) to derive

    $$\begin{aligned} \begin{array}{l} J\Big (\frac{\pi }{4} + {\varepsilon _\theta },\alpha + {\varepsilon _\alpha }\Big ) - J\Big (\frac{\pi }{4},\alpha \Big )= \frac{1}{4}\Big [ - E\Big ({\left| s \right| ^4}\Big ) + 2{E^2}\Big ({\left| s \right| ^2}\Big )\Big ]\\ \quad \Big [{\sin ^2}\Big (\frac{\pi }{2} + 2{\varepsilon _\theta }\Big ) - 1\Big ] \end{array} \end{aligned}$$

    (36)

    If $- E({\left| s \right| ^4}) + 2{E^2}({\left| s \right| ^2}) > 0$, then $J(\frac{\pi }{4} + {\varepsilon _\theta },\alpha + {\varepsilon _\alpha }) > 0$. In this case, the point ($\theta = \pi /4,\forall \alpha$) is a local minimum of cost function. If $- E({\left| s \right| ^4}) + 2{E^2}({\left| s \right| ^2}) < 0$, then $J(\frac{\pi }{4} + {\varepsilon _\theta },\alpha + {\varepsilon _\alpha }) < 0$. In this case, the point ($\theta = \pi /4,\forall \alpha$) is a local maximum of cost function. If $- E({\left| s \right| ^4}) + 2{E^2}({\left| s \right| ^2}) = 0$, then $J(\theta ,\alpha )$ is a constant function. In this case, the point ($\theta = \pi /4,\forall \alpha$) is an ordinary point of cost function.

• (2) if $E({{s}^{2}})\ne 0$, then the stationary points are $\large \{ _{\forall \alpha }^{\theta = 0}$, $\large \{ _{\forall \alpha }^{\theta = \pi /2}$, $\large \{ _{\alpha = 0}^{\theta = \pi /4}$, $\large \{ _{\alpha = \pi /2}^{\theta = \pi /4}$, $\large \{ _{\alpha = \pi }^{\theta = \pi /4}$.

  • (i) As $T(\theta = 0,\forall \alpha ) = 0$, second derivatives test gives no information. We now make a small perturbation $({\varepsilon _\theta },{\varepsilon _\alpha })$ around the point ($\theta = 0,\forall \alpha$) to derive

    $$\begin{aligned} \begin{array}{l} J({\varepsilon _\theta },\alpha + {\varepsilon _\alpha }) - J(0,\alpha ) = \frac{1}{4}{\sin ^2}(2{\varepsilon _\theta }) \left[ {\cos (2\alpha + 2{\varepsilon _\alpha })E({s^2})E({s^{*2}})} \right. \\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \left. { - E({{\left| s \right| }^4}) + 2{E^2}({{\left| s \right| }^2})} \right] \end{array} \end{aligned}$$

    (37)

    If $-E({{\left| s \right| }^{4}})+2{{E}^{2}}({{\left| s \right| }^{2}})\pm E({{s}^{2}})E({{s}^{*2}})>0$, then $J({{\varepsilon }_{\theta }},\alpha +{{\varepsilon }_{\alpha }})-J(0,\alpha )\ge 0$. In this case, the point ($\theta =0,\forall \alpha$) is a local minimum of cost function. If $-E({{\left| s \right| }^{4}})+2{{E}^{2}}({{\left| s \right| }^{2}})\pm E({{s}^{2}})E({{s}^{*2}})<0$, then $J({{\varepsilon }_{\theta }},\alpha +{{\varepsilon }_{\alpha }})-J(0,\alpha )\le 0$. In this case, the point ($\theta =0,\forall \alpha$) is a local maximum of cost function. If $\left| 2{{E}^{2}}({{\left| s \right| }^{2}})-E({{\left| s \right| }^{4}}) \right| \le \left| E({{s}^{2}})E({{s}^{*2}}) \right|$, then the sources fall into the area of instability defined in [9]. In this case, whether the point ($\theta =0,\forall \alpha$) is an extreme of cost function or not depends on $\alpha$. If $\cos (2{{\alpha }_{0}})>\frac{2{{E}^{2}}({{\left| s \right| }^{2}})-E({{\left| s \right| }^{4}})}{E({{s}^{2}})E({{s}^{*2}})}$, then there exists a deleted disk neighborhood area ${{D}_{0}}$ with center ($\theta =0,\alpha ={{\alpha }_{0}}$) such that $J({{\varepsilon }_{\theta }},\alpha +{{\varepsilon }_{\alpha }})-J(0,{{\alpha }_{0}})>0(\forall {{\varepsilon }_{\theta }},{{\varepsilon }_{\alpha }}\in {{D}_{0}})$. Thus, the point ($\theta =0,\alpha ={{\alpha }_{0}}$) is a local minimum of cost function. If $\cos (2{{\alpha }_{0}})<\frac{2{{E}^{2}}({{\left| s \right| }^{2}})-E({{\left| s \right| }^{4}})}{E({{s}^{2}})E({{s}^{*2}})}$, then there exists a deleted disk neighborhood area ${{D}_{0}}$ with center ($\theta =0,\alpha ={{\alpha }_{0}}$) such that $J({{\varepsilon }_{\theta }},\alpha +{{\varepsilon }_{\alpha }})-J(0,{{\alpha }_{0}})<0(\forall {{\varepsilon }_{\theta }},{{\varepsilon }_{\alpha }}\in {{D}_{0}})$. Thus, the point ($\theta =0,\alpha ={{\alpha }_{0}}$) is a local maximum of cost function. If $\cos (2{{\alpha }_{0}})=\frac{2{{E}^{2}}({{\left| s \right| }^{2}})-E({{\left| s \right| }^{4}})}{E({{s}^{2}})E({{s}^{*2}})}$, then every deleted disk neighborhood area with center ($\theta =0,\alpha ={{\alpha }_{0}}$) contains points where $J({{\varepsilon }_{\theta }},{{\alpha }_{0}}+{{\varepsilon }_{\alpha }})-J(0,{{\alpha }_{0}})$ take positive values as well as points where $J({{\varepsilon }_{\theta }},{{\alpha }_{0}}+{{\varepsilon }_{\alpha }})-J(0,{{\alpha }_{0}})$ take negative values. Thus, the point ($\theta =0,{{\alpha }_{0}}$) is a saddle point of cost function. Based on the above analysis, it can be easily obtained that the point ($\theta =0,{{\alpha }_{0}}\ne \frac{1}{2}\arccos \left( \frac{2{{E}^{2}}({{\left| s \right| }^{2}})-E({{\left| s \right| }^{4}})}{E({{s}^{2}})E({{s}^{*2}})} \right)$) is a local extreme of cost function even when the sources fall into the area of instability.

  • (ii) The detailed analysis of stationary point $(\theta =\pi /2,\forall \alpha )$ is the same as $(\theta =0,\forall \alpha )$ and is omitted here.

  • (iii) $T(\theta =\pi /4,\alpha =0)=2E({{s}^{2}})E({{s}^{*2}})[2{{E}^{2}}({{\left| s \right| }^{2}})-E({{\left| s \right| }^{4}})+E({{s}^{2}})E({{s}^{*2}})]$ If $2{{E}^{2}}({{\left| s \right| }^{2}})-E({{\left| s \right| }^{4}})+E({{s}^{2}})E({{s}^{*2}})>0$, then $\frac{{{\partial }^{2}}J(\theta ,\alpha )}{\partial {{\theta }^{2}}}{{|}_{\theta =\pi /4,\alpha =0}}<0~and~T(\theta =\pi /4,\alpha =0)>0$. In this case, the point $(\theta =\pi /4,\alpha =0)$ is a local maximum of cost function. If $2{{E}^{2}}({{\left| s \right| }^{2}})-E({{\left| s \right| }^{4}})+E({{s}^{2}})E({{s}^{*2}})<0$, then $T(\theta =\pi /4,\alpha =0)<0$. In this case, the point $(\theta =\pi /4,\alpha =0)$ is a saddle point of cost function. If $2{{E}^{2}}({{\left| s \right| }^{2}})-E({{\left| s \right| }^{4}})+E({{s}^{2}})E({{s}^{*2}})=0$, then

    $$\begin{aligned} \begin{array}{l} J(\frac{\pi }{4} + {\varepsilon _\theta },{\varepsilon _\alpha }) - J(\frac{\pi }{4},0) = \frac{1}{4} [{\sin ^2}(\frac{\pi }{2} + 2{\varepsilon _\theta }) ] [\cos (2{\varepsilon _\alpha }) - 1]E({s^2})E({s^{*2}})] \end{array}\nonumber \\ \end{aligned}$$

    (38)

    In this case, the point $(\theta =\pi /4,\alpha =0)$ is a local maximum of cost function.

  • (iv) $T(\theta =\pi /4,\alpha =\pi /2)=-2E({{s}^{2}})E({{s}^{*2}})[2{{E}^{2}}({{\left| s \right| }^{2}})-E({{\left| s \right| }^{4}})-E({{s}^{2}})E({{s}^{*2}})]$ Similarly, it can be easily obtained that, if $2{{E}^{2}}({{\left| s \right| }^{2}})-E({{\left| s \right| }^{4}})-E({{s}^{2}})E({{s}^{*2}})>0$, the point $(\theta =\pi /4,\alpha =\pi /2)$ is a saddle point of cost function; if $2{{E}^{2}}({{\left| s \right| }^{2}})-E({{\left| s \right| }^{4}})-E({{s}^{2}})E({{s}^{*2}})<0$, the point ($\theta =\pi /4,\alpha =\pi /2$) is a local minimum of cost function; if $2{{E}^{2}}({{\left| s \right| }^{2}})-E({{\left| s \right| }^{4}})-E({{s}^{2}})E({{s}^{*2}})=0$, the point ($\theta =\pi /4,\alpha =\pi /2$) is a local minimum of cost function.

  • (v) The detailed analysis of stationary point $(\theta =\pi /4,\alpha =0)$ is the same as the point $(\theta =\pi /4,\alpha =\pi )$ and is omitted here.