Learning Asymmetric Laplacian Representations in Hilbert Space

Laplacian representation learning is a highly promising generally and in reinforcement learning, where an important challenge is to retain the asymmetries of directed and stochastic dynamics in the environment. In the self-adjoint setting, prior methods use ordered orthogonality and a smoothness objective to learn a set of spectral representations corresponding to lowest-magnitude eigenvalues. Such ordering dynamics does not extend to the asymmetric Laplacian. Unlike a self-adjoint Laplacian, an asymmetric Laplacian can have complex eigenvalues and non-orthogonal complex eigenfunctions; that is, its right and adjoint-left eigenfunctions can differ and form bi-orthogonality. These properties can complicate approaches that attempt to learn an ordered eigenbasis. One might therefore expect an asymmetric solution to require a substantially more complicated approach. We instead keep the representation learning problem to a similar complexity of prior symmetric approaches and find that, by targeting an unordered subspace, we can learn vectors that approximate mixtures of the asymmetric Laplacian eigenvectors we seek. Careful unmixing then yields good approximations of low magnitude asymmetric eigenfunctions. We compare the obtained asymmetric eigenfunctions with oracle solutions using cosine similarity across multiple domains. Additionally, we show that the learnt eigenfunctions can be used in domains to obtain hitting time estimates and compare them to the prior symmetric Laplacian based commute-time distance, showing improved distance learning in directed and stochastic settings.