This letter discusses blind separability based on temporal predictability (Stone, 2001 ; Xie, He, & Fu, 2005 ). Our results show that the sources are separable using the temporal predictability method if and only if they have different temporal structures (i.e., autocorrelations). Consequently, the applicability and limitations of the temporal predictability method are clarified. In addition, instead of using generalized eigendecomposition, we suggest using joint approximate diagonalization algorithms to improve the robustness of the method. A new criterion is presented to evaluate the separation results. Numerical simulations are performed to demonstrate the validity of the theoretical results.

Overcomplete representations have greater robustness in noise environment and also have greater flexibility in matching structure in the data. Lewicki and Sejnowski (2000) proposed an efficient extended natural gradient for learning the overcomplete basis and developed an overcomplete representation approach. However, they derived their gradient by many approximations, and their proof is very complicated. To give a stronger theoretical basis, we provide a brief and more rigorous mathematical proof for this gradient in this note. In addition, we propose a more robust constrained Lewicki-Sejnowski gradient.

Stone's method is one of the novel approaches to the blind source separation (BSS) problem and is based on Stone's conjecture. However, this conjecture has not been proved. We present a simple simulation to demonstrate that Stone's conjecture is incorrect. We then modify Stone's conjecture and prove this modified conjecture as a theorem, which can be used a basis for BSS algorithms.