Blending Independent Components and Principal Components Analysis

2.8 Gradient ascent

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2.8          Gradient ascent

All of the above approaches require us to maximise (or minimise) some function (the kurtosis, the log likelihood, the Kolmogorov complexity etc.) with respect to different unmixing vectors (or unmixing matrices, i.e. simultaneously for several unmixing vectors all at once). Whilst brute force could be applied for simple problems this rapidly becomes impractical as the number of signals increases. Instead, we typically use gradient ascent, in which we head up the (possibly hyper-dimensional) surface formed by plotting the value of the function for different unmixing vectors in the direction of steepest ascent. The direction of steepest ascent can be found from the first partial derivative of the function with respect to the different components of the unmixing vector/matrix. Second order methods can be used to estimate how far to go along that gradient before next evaluating the function and its derivatives, see e.g. Press et al. (2007).

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