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On the performance of tensor methods for solving ill-conditioned problems.
This paper investigates the performance of tensor methods for solving small- and large-scale systems of nonlinear equations where the Jacobian matrix at the root is ill-conditioned or singular. This condition occurs on many classes of problems, such as identifying or approaching turning points in path following problems. The singular case has been studied more than the highly ill-conditioned case, for both Newton and tensor methods. It is known that Newton-based methods do not work well with singular problems because they converge linearly to the solution and, in some cases, with poor accuracy. On the other hand, direct tensor methods have performed well on singular problems and have superlinear convergence on such problems under certain conditions. This behavior originates from the use of a special, restricted form of the second-order term included in the local tensor model that provides information lacking in a (nearly) singular Jacobian. With several implementations available for large-scale problems, tensor methods now are capable of solving larger problems. We compare the performance of tensor methods and Newton-based methods for both small- and large-scale problems over a range of conditionings, from well-conditioned to ill-conditioned to singular. Previous studies with tensor methods only concerned the ends of this spectrum. Our results show that tensor methods are increasingly superior to Newton-based methods as the problem grows more ill-conditioned
A Riemannian Trust Region Method for the Canonical Tensor Rank Approximation Problem
The canonical tensor rank approximation problem (TAP) consists of
approximating a real-valued tensor by one of low canonical rank, which is a
challenging non-linear, non-convex, constrained optimization problem, where the
constraint set forms a non-smooth semi-algebraic set. We introduce a Riemannian
Gauss-Newton method with trust region for solving small-scale, dense TAPs. The
novelty of our approach is threefold. First, we parametrize the constraint set
as the Cartesian product of Segre manifolds, hereby formulating the TAP as a
Riemannian optimization problem, and we argue why this parametrization is among
the theoretically best possible. Second, an original ST-HOSVD-based retraction
operator is proposed. Third, we introduce a hot restart mechanism that
efficiently detects when the optimization process is tending to an
ill-conditioned tensor rank decomposition and which often yields a quick escape
path from such spurious decompositions. Numerical experiments show improvements
of up to three orders of magnitude in terms of the expected time to compute a
successful solution over existing state-of-the-art methods
Preconditioned low-rank Riemannian optimization for linear systems with tensor product structure
The numerical solution of partial differential equations on high-dimensional
domains gives rise to computationally challenging linear systems. When using
standard discretization techniques, the size of the linear system grows
exponentially with the number of dimensions, making the use of classic
iterative solvers infeasible. During the last few years, low-rank tensor
approaches have been developed that allow to mitigate this curse of
dimensionality by exploiting the underlying structure of the linear operator.
In this work, we focus on tensors represented in the Tucker and tensor train
formats. We propose two preconditioned gradient methods on the corresponding
low-rank tensor manifolds: A Riemannian version of the preconditioned
Richardson method as well as an approximate Newton scheme based on the
Riemannian Hessian. For the latter, considerable attention is given to the
efficient solution of the resulting Newton equation. In numerical experiments,
we compare the efficiency of our Riemannian algorithms with other established
tensor-based approaches such as a truncated preconditioned Richardson method
and the alternating linear scheme. The results show that our approximate
Riemannian Newton scheme is significantly faster in cases when the application
of the linear operator is expensive.Comment: 24 pages, 8 figure
Objective acceleration for unconstrained optimization
Acceleration schemes can dramatically improve existing optimization
procedures. In most of the work on these schemes, such as nonlinear Generalized
Minimal Residual (N-GMRES), acceleration is based on minimizing the
norm of some target on subspaces of . There are many numerical
examples that show how accelerating general purpose and domain-specific
optimizers with N-GMRES results in large improvements. We propose a natural
modification to N-GMRES, which significantly improves the performance in a
testing environment originally used to advocate N-GMRES. Our proposed approach,
which we refer to as O-ACCEL (Objective Acceleration), is novel in that it
minimizes an approximation to the \emph{objective function} on subspaces of
. We prove that O-ACCEL reduces to the Full Orthogonalization
Method for linear systems when the objective is quadratic, which differentiates
our proposed approach from existing acceleration methods. Comparisons with
L-BFGS and N-CG indicate the competitiveness of O-ACCEL. As it can be combined
with domain-specific optimizers, it may also be beneficial in areas where
L-BFGS or N-CG are not suitable.Comment: 18 pages, 6 figures, 5 table
Dictionary-based Tensor Canonical Polyadic Decomposition
To ensure interpretability of extracted sources in tensor decomposition, we
introduce in this paper a dictionary-based tensor canonical polyadic
decomposition which enforces one factor to belong exactly to a known
dictionary. A new formulation of sparse coding is proposed which enables high
dimensional tensors dictionary-based canonical polyadic decomposition. The
benefits of using a dictionary in tensor decomposition models are explored both
in terms of parameter identifiability and estimation accuracy. Performances of
the proposed algorithms are evaluated on the decomposition of simulated data
and the unmixing of hyperspectral images
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