Figure 1.
The excess kurtosis of a source as a function of the relative size of the active region.
A Gaussian has zero excess kurtosis. Here as in Example 2 of the original paper [8]. The four vertical lines at correspond to the relative sizes of the small box, the medium box, the large box, and a very large box corresponding to the maximal kurtosis case. Note that the medium and large box experiments have near zero excess kurtosis, i.e., kurtosis value matching that of a Gaussian. In addition, the pdfs of these sources are bimodal (see inset figures), ensuring that ICA algorithms designed for unimodal super-Gaussian distributions such as Infomax and FastICA with standard parameter settings, will likely fail. At the bottom of the figure are the ISI values (see Equation (2)) for the various algorithms at those four points (see Table 1 for full list). Also note the best separation performance of Infomax and FastICA for the maximum kurtosis case, which corresponds to almost the lowest level of sparsity.
Table 1.
Source estimates for the four cases indicated in Figure 1.
Figure 2.
The distribution of sources and mixtures for (
).
We plot (A–C) the distribution of sources, and (D) the contour plot of mixtures for the case of (
). Contrary to the claim made in Daubechies et al., the sources have in fact very peaky and heavy-tailed distributions and are not at all close to a Gaussian distribution. For comparison purposes we also present Gaussian distribution curves (blue, A–B).
Table 2.
Tabulated results for the so-called [8] ICA “promotional material.”
Figure 3.
Sparsity measures for three different coordinate system origins ().
Sparsity as measured with respect to different coordinate system origins (), as a function of the relative size of the active region. Remark that for a relative size of zero, only background samples are present and, thus, the mean of the mixture model coincides with the mean of the background (and the two sparsity measures correspond at this point). An analogous observation can be made for a relative size of one, now with respect to the activity (signal samples).