300 CHAPTER 8 Inner Products for Representation and Learning appropriate value of g max is chosen, as shown in Figure 8.7 and noticed in Figure 8.8(c). Finally, using the inner product in Eq. (8.20) yields the best results. As with the PCA these results are considerably better than the previ- ous ones because the latter inner product nonlinearly measures the similarity between functional representations. This avoids the ambiguity in the inner product in equation (8.18), 8 adding to the expressiveness of the inner product (Figure 8.8(b)). Also, in the latter inner product we have to choose the band- width parameter of the nonlinear kernel K . However, unlike the former, the sensitivity on this parameter is very low. We tried values between 0.1 and 10, but the results changed only ±0.1%. 8.5 DISCUSSION A general framework to develop spike train machine learning methods was presented. The most important contribution of this work is in defining spike train inner products for building the framework. Unlike the top-down approach