BSS error can be characterized by the source and input dimensionalities. In all panels, $s$ is generated by an identical uniform distribution; $Nf=Nx$ is fixed; and $A,B,a$ are sampled from gaussian distributions as in Figure 2. Encoders $u˜=(u˜1,…,u˜Ns)T$ are obtained by applying the PCA–ICA cascade to sensory inputs. For visualization purpose, elements of $u˜$ are permuted and sign-flipped to ensure that $u˜i$ encodes $si$⁠. (A) Each element of $u˜$ encodes a hidden source, wherein an error in representing a source monotonically decreases when $Nx$ increases. The generative process is characterized with $Ns=100$ and $f(•)=sign(•)$⁠. Shaded areas represent theoretical values of the standard deviation computed using equation 2.13. As elements of $B$ are gaussian distributed, $Δ=I$ holds. Inset panels depict the absolute value of covariance matrix $|Cov[u˜,s]|$ with gray-scale values ranging from 0 (white) to 1 (black). A diagonal covariance matrix indicates the successful identification of all the true hidden sources. (B, C) Nonlinear BSS when the generative process is characterized by $f(•)=(•)3$ or $f(•)=ReLU(•)$⁠. $ReLU(•)$ outputs $•$ for $•>0$ or 0 otherwise. In panel C, the shaded area is computed using equation 2.12. (D) The quantitative relationship between the source and input dimensionalities and element-wise BSS error is scored by the mean squared error $E[|s-u˜|2]/Ns$⁠. Here, $f(•)=sign(•)$ is supposed. Circles and error bars indicate the means and areas between maximum and minimum values obtained with 20 different realizations of $s,A,B,a$⁠, where some error bars are hidden by the circles. Solid and dashed lines represent theoretical values computed using equations 2.12 and 2.13, respectively. These lines fit the actual BSS errors when $Nx≫Ns≫1$⁠, although some deviations occur when $Ns$ or $Nx$ is small. Blue cross marks indicate actual BSS errors for $Ns=10$ when hidden sources are sampled from a symmetric truncated normal distribution.