Can back-projection fully resolve polarity indeterminacy of independent component analysis in study of event-related potential?

In the study of event-related potentials (ERPs) using independent component analysis (ICA), it is a traditional way to project the extracted ERP component back to electrodes for correcting its scaling (magnitude and polarity) indeterminacy. However, ICA tends to be locally optimized in practice, and then, the back-projection of a component estimated by the ICA can possibly not fully correct its polarity at every electrode. We demonstrate this phenomenon from the view of the theoretical analysis and numerical simulations and suggest checking and modifying the abnormal polarity of the projected component in the electrode field before further analysis. Moreover, when several components are to be projected, instead of the parallel projection of those components simultaneously, the sequential projection of component by component permits the correction of the abnormal polarity of a certain projected component at a certain electrode, which can improve the accuracy of the back-projection. Furthermore, after one extracted component by the ICA is projected back to electrodes under the global optimization, we cannot achieve the real source yet, but the determined scaled source, i.e., the multiplication between the real source and the mapping coefficient from the source to the point at the scalp.