# Why does the training time of SVMs dramatically decrease after applying dimensionality reduction to the features?

Training an SVM with an RBF kernel model with c = 5.5 and gamma = 1.06, for a 5-class classification problem on the NSL-KDD train data-set with 122 features using one vs rest strategy takes $$2162$$ seconds. Also, considering binary classification (c = 10, gamma = 4), it takes $$520.56$$ seconds.

After dimensionality reduction, from 122 to 30, using a sparse auto-encoder, the training time falls dramatically, from $$2162$$ to $$240$$ and $$520$$ to $$170$$, while using the same hyperparameters for the RBF-kernel.

What is the reason for that? Is it not true that using kernel neutralized the effect of high dimensions?

SVM complexity is $$O(\max(n,d)\min(n,d)^2)$$ according to Chapelle, Olivier. "Training a support vector machine in the primal." Neural Computation 19.5 (2007): 1155-1178.
$$n$$ is the number of instances and $$d$$ is the number of dimensions. I'm assuming that you have more instances than dimensions giving a complexity of $$O(nd^2)$$. Hopefully this explains fully why reducing the number of dimensions will reduce the training time.