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The generalized linear discriminant sequence (GLDS) kernel has been shown to provide very good performance and efficiency at the NIST Speaker Recognition Evaluations (SRE) in the last few years. This kernel is based on an explicit map of polynomial expansions of input frames which, because of practical limitations, have to be of a degree less or equal to three. In this paper, we consider an extension of the GLDS kernel to allow not only any polynomial degree but also any embedding, including infinite-dimensional ones associated with Mercer kernels (such as Gaussian kernels). It turns out that the resulting kernels belong to the family of posterior covariance kernels. However, their exact ldquokernelizedrdquo form involves the computation of the Gram matrix on background data, and may be intractable when the background corpus is very large (which is the case in speaker verification). To overcome this problem, we use a low-rank approximation of the Gram matrix to provide an approximate but tractable form of these kernels. We then present comparative experiments on NIST SRE 2005. The results show that our sequence kernel outperforms the GLDS one, and gives similar (individual) performances to the traditional universal background model-Gaussiam mixture model (UBM-GMM) system. As expected, the fusion of both improves the scores.