In this paper, a compressive sensing (CS) perspective to exemplar-based speech processing is proposed. Relying on an analytical relationship between CS formulation and statistical speech recognition (Hidden Markov Models HMM), the automatic speech recognition (ASR) problem is cast as recovery of high-dimensional sparse word representation from the observed low-dimensional acoustic features. The acoustic features are exemplars obtained from (deep) neural network sub-word conditional posterior probabilities. Low-dimensional word manifolds are learned using these sub-word posterior exemplars and exploited to construct a linguistic dictionary for sparse representation of word posteriors. Dictionary learning has been found to be a principled way to alleviate the need of having huge collection of exemplars as required in conventional exemplar-based approaches, while still improving the performance. Context appending and collaborative hierarchical sparsity are used to exploit the sequential and group structure underlying word sparse representation. This formulation leads to a posterior-based sparse modeling approach to speech recognition. The potential of the proposed approach is demonstrated on isolated word (Phonebook corpus) and continuous speech (Numbers corpus) recognition tasks.