Inspired by the use of random projections in biological sensing systems, we present a new algorithm for processing data in classification problems. This is based on observations of the human brain and the fruit fly's olfactory system and involves randomly projecting data into a space of greatly increased dimension before applying a cap operation to truncate the smaller entries. This leads to a simple algorithm that is very computationally efficient and can be used to either give a sparse representation with minimal loss in classification accuracy or give improved robustness, in the sense that classification accuracy is improved when noise is added to the data. This is demonstrated with numerical experiments, which supplement theoretical results demonstrating that the resulting signal transform is continuous and invertible, in an appropriate sense.