SVM Supervector

An SVM (Support Vector Machine) is a two class classifier. It is constructed by sums of kernel function K(.,.):

$$f(x) = \sum\limits_{i = 1}^L {\alpha_i t_i K(x,x_i ) + d} $$

((1))

t _i are the ideal outputs (−1 for one class and +1 for the other class) and$\sum\limits_{{\rm i = 1}}^{\rm L} {{\alpha}_{\rm i} t_{\rm i} = 0(} {\alpha}_{\rm i} > 0)$ The vectors x _i are the support vectors (belonging to the training vectors) and are obtained by using an optimization algorithm. A class decision is based upon the value of f (x) with respect to a threshold. The kernel function is constrained to verify the Mercer condition: $K(x,y) = b(x)^t b(y),$where b(x) is a mapping from the input space (containing the vectors x) to a possibly infinite-dimensional SVM expansion space.

In the case of speaker verification, given universal background (GMM UBM):

$$g(x) = \sum\limits_{i = 1}^M {\omega_i N(x,\mu _i ,\rm \Sigma_i {} ),} $$

((2))

where, ω _i are the mixture weights, N() is a Gaussian, and\({(\mu...