Differences in SVM performance

Question

Why would a polynomial SVM have better performance than a linear SVM but the same performance as a radial SVM?

Answer 1

If the performance with a linear kernel is bad, it means the data is not separable in input space (assuming a classication context). The feature spaces induced by more complex kernels (such as polynomial or RBF) can lead to linear separability.

The feature space of an RBF kernel is more complex than both linear and polynomial (in fact, both linear and degree-2 polynomial are degenerate versions of RBF). Hence, if a polynomial kernel can make the data separable, an appropriate RBF kernel can too.