Interpretable-by-design models are gaining traction in computer vision because they provide faithful explanations for their predictions. In image classification, these models typically recover human-interpretable concepts from an image and use them for classification. Sparse concept recovery methods leverage the latent space of vision-language models to represent image embeddings as sparse combinations of concept embeddings. However, by ignoring the hierarchical structure of semantic concepts, these methods may produce correct predictions with explanations that are inconsistent with the hierarchy. In this work, we propose Hierarchical Concept Embedding & Pursuit (HCEP), a framework that induces a hierarchy of concept embeddings in the latent space and performs hierarchical sparse coding to recover the concepts present in an image. Given a hierarchy of semantic concepts, we introduce a geometric construction for the corresponding hierarchy of embeddings. Under the assumption that the true concepts form a rooted path in the hierarchy, we derive sufficient conditions for their recovery in the embedding space. We further show that hierarchical sparse coding reliably recovers hierarchical concept embeddings, whereas standard sparse coding fails. Experiments on real-world datasets show that HCEP improves concept precision and recall compared to existing methods while maintaining competitive classification accuracy. Moreover, when the number of samples available for concept estimation and classifier training is limited, HCEP achieves superior classification accuracy and concept recovery. Our results demonstrate that incorporating hierarchical structure into sparse concept recovery leads to more faithful and interpretable image classification models.