Choosing a deep neural network architecture is a fundamental problem in applications that require balancing performance and parameter efficiency. Standard approaches rely on ad-hoc engineering or computationally expensive validation on a specific dataset. We instead attempt to quantify networks by their intrinsic capacity for unique and robust representations, enabling efficient architecture comparisons without requiring any data. Building upon theoretical connections between deep learning and sparse approximation, we propose the deep frame potential: a measure of coherence that is approximately related to representation stability but has minimizers that depend only on network structure. This provides a framework for jointly quantifying the contributions of architectural hyper-parameters such as depth, width, and skip connections. We validate its use as a criterion for model selection and demonstrate correlation with generalization error on a variety of common residual and densely connected network architectures.