Spectral Methods from Tensor Networks
This addresses a challenging inference problem in applications such as cryo-electron microscopy, providing a novel algorithmic approach for infinite groups where no general methods existed, though it is incremental in leveraging existing tensor network abstractions.
The paper tackles continuous tensor decomposition problems, specifically orbit recovery over infinite groups like SO(2), by developing a new spectral algorithm based on tensor networks, which extends to heterogeneous cases.
A tensor network is a diagram that specifies a way to "multiply" a collection of tensors together to produce another tensor (or matrix). Many existing algorithms for tensor problems (such as tensor decomposition and tensor PCA), although they are not presented this way, can be viewed as spectral methods on matrices built from simple tensor networks. In this work we leverage the full power of this abstraction to design new algorithms for certain continuous tensor decomposition problems. An important and challenging family of tensor problems comes from orbit recovery, a class of inference problems involving group actions (inspired by applications such as cryo-electron microscopy). Orbit recovery problems over finite groups can often be solved via standard tensor methods. However, for infinite groups, no general algorithms are known. We give a new spectral algorithm based on tensor networks for one such problem: continuous multi-reference alignment over the infinite group SO(2). Our algorithm extends to the more general heterogeneous case.