Is it possible for a trained support vector machine (SVM) to take an input of a different length (say during the testing phase) than the length used when it was trained?
e.g. training data input: vector $\bf x_r \in R^n$ ; test data input: vector $\bf x_t \in R^m$ where $ \bf m \ne\bf n$
This would depend upon the kernel function. In Euclidean space (i.e. a linear SVM), this does not work because the kernel is defined as an inner product between the example and the training examples.
If you generalize this to a non-linear SVM, the kernel is defined as an inner product in some higher dimensional Hilbert space. Strictly speaking, the mapping into this higher dimensional space can be any non-linearity you like and hence could potentially accommodate examples of differing dimensionality. However, in order for this to work, the non-linearity needs to make sense for the type of data you're using. Just because you can define some space where there exists an inner product between two data points of different dimensions, doesn't mean you should.
One such non-linearity that comes to mind, is the Pseudo Dynamic Time Warping and Global Alignment Kernels of Cuturi. However, these are specialized kernels and would not be a sensible choice for most problems.
