20 research outputs found
Homotopy Analysis for Tensor PCA
Developing efficient and guaranteed nonconvex algorithms has been an important challenge in modern machine learning. Algorithms with good empirical performance such as stochastic gradient descent often lack theoretical guarantees. In this paper, we analyze the class of homotopy or continuation methods for global optimization of nonconvex functions. These methods start from an objective function that is efficient to optimize (e.g. convex), and progressively modify it to obtain the required objective, and the solutions are passed along the homotopy path. For the challenging problem of tensor PCA, we prove global convergence of the homotopy method in the “high noise” regime. The signal-to-noise requirement for our algorithm is tight in the sense that it matches the recovery guarantee for the \em best degree-4 sum-of-squares algorithm. In addition, we prove a phase transition along the homotopy path for tensor PCA. This allows us to simplify the homotopy method to a local search algorithm, viz., tensor power iterations, with a specific initialization and a noise injection procedure, while retaining the theoretical guarantees
Spectral Methods from Tensor Networks
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.Comment: 30 pages, 8 figure