Changes in bulk transcriptional profiles of heterogeneous samples often reflect changes in proportions of individual cell types. Several robust techniques have been developed to dissect the composition of such mixed samples given transcriptional signatures of the pure components or their proportions. These approaches are insufficient, however, in situations when no information about individual mixture components is available. This problem is known as the complete deconvolution problem, where the composition is revealed without any a priori knowledge about cell types and their proportions. Here, we identify a previously unrecognized property of tissue-specific genes - their mutual linearity - and use it to reveal the structure of the topological space of mixed transcriptional profiles and provide a noise-robust approach to the complete deconvolution problem. Furthermore, our analysis reveals systematic bias of all deconvolution techniques due to differences in cell size or RNA-content, and we demonstrate how to address this bias at the experimental design level.