Statistical speech recognition has been cast as a natural realization of the compressive sensing problem in this work. The compressed acoustic observations are sub-word posterior probabilities obtained from a deep neural network. Dictionary learning and sparse recovery are exploited for inference of the high-dimensional sparse word posterior probabilities. This formulation amounts to realization of a \textit{sparse} hidden Markov model where each state is characterized by a dictionary learned from training exemplars and the emission probabilities are obtained from sparse representations of test exemplars. This new dictionary-based speech processing paradigm alleviates the need for a huge collection of exemplars as required in the conventional exemplar-based methods. We study the performance of the proposed approach for continuous speech recognition using Phonebook and Numbers'95 database.