Sparling: End-to-End Spatial Concept Learning via Extremely Sparse Activations

Real-world processes often contain intermediate state that can be modeled as an extremely sparse activation tensor. In this work, we analyze the identifiability of such sparse and local latent intermediate variables, which we call motifs. We prove our Motif Identifiability Theorem, stating that under certain assumptions it is possible to precisely identify these motifs exclusively by reducing end-to-end error. Additionally, we provide the Sparling algorithm, which uses a new kind of informational bottleneck that enforces levels of activation sparsity unachievable using other techniques. We find that extreme sparsity is necessary to achieve good intermediate state modeling empirically. On our synthetic DigitCircle domain as well as the LaTeX-OCR and Audio-MNIST-Sequence domains, we are able to precisely localize the intermediate states up to feature permutation with $>90\%$ accuracy, even though we only train end-to-end.