Bilinear approximation of a matrix is a powerful paradigm of unsupervised learning. In some applications, however, there is a natural hierarchy of concepts that ought to be reflected in the unsupervised analysis. For example, in the neurosciences image sequence considered here, there are the semantic concepts of pixel $\rightarrow$ neuron $\rightarrow$ assembly that should find their counterpart in the unsupervised analysis. Driven by this concrete problem, we propose a decomposition of the matrix of observations into a product of more than two sparse matrices, with the rank decreasing from lower to higher levels. In contrast to prior work, we allow for both hierarchical and heterarchical relations of lower-level to higher-level concepts. In addition, we learn the nature of these relations rather than imposing them. Finally, we describe an optimization scheme that allows to optimize the decomposition over all levels jointly, rather than in a greedy level-by-level fashion. The proposed bilevel SHMF (sparse heterarchical matrix factorization) is the first formalism that allows to simultaneously interpret a calcium imaging sequence in terms of the constituent neurons, their membership in assemblies, and the time courses of both neurons and assemblies. Experiments show that the proposed model fully recovers the structure from difficult synthetic data designed to imitate the experimental data. More importantly, bilevel SHMF yields plausible interpretations of real-world Calcium imaging data.
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