In this talk, I will first present a signal representation framework called the Sparse Manifold Transform that combines key ideas from sparse coding, manifold learning, and slow feature analysis. It turns non-linear transformations in the primary sensory signal space into linear interpolations in a representational embedding space while maintaining approximate invertibility. The sparse manifold transform is an unsupervised and generative framework that explicitly and simultaneously models the sparse discreteness and low-dimensional manifold structure found in natural scenes. When stacked, it also models hierarchical composition. The proposed framework also provides new geometric insights in understanding the simple cells, complex cells and beyond.

Signal representation is a broad and comprehensive topic. Besides this major quest to build a principled representation for natural signals, I will also present a series of exciting projects I have worked on along the way to understand the representation of different signal modalities from different angles. This involves explainable word embeddings visualization, energy-based generative models, model superpositions, etc.