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Identifying functionally important cell states and structure within heterogeneous tumors remains a significant biological and computational challenge. Current clustering or trajectory-based models are ill-equipped to address the notion that cancer cells reside along a phenotypic continuum. We present Archetypal Analysis network (AAnet), a neural network that learns archetypal states within a phenotypic continuum in single-cell data. Unlike traditional archetypal analysis, AAnet learns archetypes in simplex-shaped neural network latent space. Using pre-clinical models and clinical breast cancers, AAnet resolves distinct cell states and processes, including cell proliferation, hypoxia, metabolism and immune interactions. Primary tumor archetypes are recapitulated in matched liver, lung and lymph node metastases. Spatial transcriptomics reveal archetypal organization within the tumor, and, intra-archetypal mirroring between cancer and adjacent stromal cells. AAnet identifies GLUT3 within the hypoxic archetype that proves critical for tumor growth and metastasis. AAnet is a powerful tool, capturing complex, functional cell states from multimodal data. 

Directed graphs are a natural model for many phenomena, in particular scientific knowledge graphs such as molecular interaction or chemical reaction networks that define cellular signaling relationships. In these situations, source nodes typically have distinct biophysical properties from sinks. Due to their ordered and unidirectional relationships, many such networks also have hierarchical and multiscale structure. However, the majority of methods performing node- and edge-level tasks in machine learning do not take these properties into account, and thus have not been leveraged effectively for scientific tasks such as cellular signaling network inference. We propose a new framework called Directed Scattering Autoencoder (DSAE) which uses a directed version of a geometric scattering transform, combined with the non-linear dimensionality reduction properties of an autoencoder and the geometric properties of the hyperbolic space to learn latent hierarchies. We show this method outperforms numerous others on tasks such as embedding directed graphs and learning cellular signaling networks.