Sparse representation has been studied within the field of signal processing as a means of providing a compact form of signal representation. This paper introduces a sparse representation based framework named Sparse Probabilistic Linear Discriminant Analysis in speaker recognition. In this latent variable model, probabilistic linear discriminant analysis is modified to obtain an algorithm for learning overcomplete sparse representations by replacing the Gaussian prior on the factors with Laplace prior that encourages sparseness. For a given speaker signal, the dictionary obtained from this model has good representational power while supporting optimal discrimination of the classes. An expectation-maximization algorithm is derived to train the model with a variational approximation to a range of heavy-tailed distributions whose limit is the Laplace. The variational approximation is also used to compute the likelihood ratio score of all trials of speakers. This approach performed well on the core-extended conditions of the NIST 2010 Speaker Recognition Evaluation, and is competitive compared to the Gaussian Probabilistic Linear Discriminant Analysis, in terms of normalized Decision Cost Function and Equal Error Rate.