Learning sparse feature representations is a useful instrument for solving an unsupervised learning problem. In this paper, we present three labeled handwritten digit datasets, collectively called n-MNIST by adding noise to the MNIST dataset, and three labeled datasets formed by adding noise to the offline Bangla numeral database. Then we propose a novel framework for the classification of handwritten digits that learns sparse representations using probabilistic quadtrees and Deep Belief Nets. On the MNIST, n-MNIST and noisy Bangla datasets, our framework shows promising results and outperforms traditional Deep Belief Networks.