SVM and Neural Network

There are two parts to this question. The first part is "what is the form of function learned by these methods?" For NN and SVM this is typically the same. For example, a single hidden layer neural network uses exactly the same form of model as an SVM. That is:

Given an input vector x, the output is: output(x) = sum_over_all_i weight_i * nonlinear_function_i(x)

Generally the nonlinear functions will also have some parameters. So these methods need to learn how many nonlinear functions should be used, what their parameters are, and what the value of all the weight_i weights should be.

Therefore, the difference between a SVM and a NN is in how they decide what these parameters should be set to. Usually when someone says they are using a neural network they mean they are trying to find the parameters which minimize the mean squared prediction error with respect to a set of training examples. They will also almost always be using the stochastic gradient descent optimization algorithm to do this. SVM's on the other hand try to minimize both training error and some measure of "hypothesis complexity". So they will find a set of parameters that fits the data but also is "simple" in some sense. You can think of it like Occam's razor for machine learning. The most common optimization algorithm used with SVMs is sequential minimal optimization.

Another big difference between the two methods is that stochastic gradient descent isn't guaranteed to find the optimal set of parameters when used the way NN implementations employ it. However, any decent SVM implementation is going to find the optimal set of parameters. People like to say that neural networks get stuck in a local minima while SVMs don't.

NNs are heuristic, while SVMs are theoretically founded. A SVM is guaranteed to converge towards the best solution in the PAC (probably approximately correct) sense. For example, for two linearly separable classes SVM will draw the separating hyperplane directly halfway between the nearest points of the two classes (these become support vectors). A neural network would draw any line which separates the samples, which is correct for the training set, but might not have the best generalization properties.

So no, even for linearly separable problems NNs and SVMs are not same.

In case of linearly non-separable classes, both SVMs and NNs apply non-linear projection into higher-dimensional space. In the case of NNs this is achieved by introducing additional neurons in the hidden layer(s). For SVMs, a kernel function is used to the same effect. A neat property of the kernel function is that the computational complexity doesn't rise with the number of dimensions, while for NNs it obviously rises with the number of neurons.

Running a simple out-of-the-box comparison between support vector machines and neural networks (WITHOUT any parameter-selection) on several popular regression and classification datasets demonstrates the practical differences: an SVM becomes a very slow predictor if many support vectors are being created while a neural network's prediction speed is much higher and model-size much smaller. On the other hand, the training time is much shorter for SVMs. Concerning the accuracy/loss - despite the aforementioned theoretical drawbacks of neural networks - both methods are on par - especially for regression problems, neural networks often outperform support vector machines. Depending on your specific problem, this might help to choose the right model.