Most tensor problems are NP-hard

We show that multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NP-hard. Our list here includes: determining the feasibility of a system of bilinear equations, deciding whether a tensor possesses a given eigenvalue, singular value, or spectral norm; approximating an eigenvalue, eigenvector, singular vector, or spectral norm; determining a best … Read more

PARNES: A rapidly convergent algorithm for accurate recovery of sparse and approximately sparse signals

In this article we propose an algorithm, NESTA-LASSO, for the LASSO problem (i.e., an underdetermined linear least-squares problem with a one-norm constraint on the solution) that exhibits linear convergence under the restricted isometry property (RIP) and some other reasonable assumptions. Inspired by the state-of-the-art sparse recovery method, NESTA, we rely on an accelerated proximal gradient … Read more

Quasi-Newton methods on Grassmannians and multilinear approximations of tensors

In this paper we proposed quasi-Newton and limited memory quasi-Newton methods for objective functions defined on Grassmannians or a product of Grassmannians. Specifically we defined BFGS and L-BFGS updates in local and global coordinates on Grassmannians or a product of these. We proved that, when local coordinates are used, our BFGS updates on Grassmannians share … Read more