21,649 research outputs found
Convergence of Alternating Least Squares Optimisation for Rank-One Approximation to High Order Tensors
The approximation of tensors has important applications in various
disciplines, but it remains an extremely challenging task. It is well known
that tensors of higher order can fail to have best low-rank approximations, but
with an important exception that best rank-one approximations always exists.
The most popular approach to low-rank approximation is the alternating least
squares (ALS) method. The convergence of the alternating least squares
algorithm for the rank-one approximation problem is analysed in this paper. In
our analysis we are focusing on the global convergence and the rate of
convergence of the ALS algorithm. It is shown that the ALS method can converge
sublinearly, Q-linearly, and even Q-superlinearly. Our theoretical results are
illustrated on explicit examples.Comment: tensor format, tensor representation, alternating least squares
optimisation, orthogonal projection metho
A literature survey of low-rank tensor approximation techniques
During the last years, low-rank tensor approximation has been established as
a new tool in scientific computing to address large-scale linear and
multilinear algebra problems, which would be intractable by classical
techniques. This survey attempts to give a literature overview of current
developments in this area, with an emphasis on function-related tensors
Rank-1 Tensor Approximation Methods and Application to Deflation
Because of the attractiveness of the canonical polyadic (CP) tensor
decomposition in various applications, several algorithms have been designed to
compute it, but efficient ones are still lacking. Iterative deflation
algorithms based on successive rank-1 approximations can be used to perform
this task, since the latter are rather easy to compute. We first present an
algebraic rank-1 approximation method that performs better than the standard
higher-order singular value decomposition (HOSVD) for three-way tensors.
Second, we propose a new iterative rank-1 approximation algorithm that improves
any other rank-1 approximation method. Third, we describe a probabilistic
framework allowing to study the convergence of deflation CP decomposition
(DCPD) algorithms based on successive rank-1 approximations. A set of computer
experiments then validates theoretical results and demonstrates the efficiency
of DCPD algorithms compared to other ones
On convergence of the maximum block improvement method
Abstract. The MBI (maximum block improvement) method is a greedy approach to solving optimization problems where the decision variables can be grouped into a finite number of blocks. Assuming that optimizing over one block of variables while fixing all others is relatively easy, the MBI method updates the block of variables corresponding to the maximally improving block at each iteration, which is arguably a most natural and simple process to tackle block-structured problems with great potentials for engineering applications. In this paper we establish global and local linear convergence results for this method. The global convergence is established under the Lojasiewicz inequality assumption, while the local analysis invokes second-order assumptions. We study in particular the tensor optimization model with spherical constraints. Conditions for linear convergence of the famous power method for computing the maximum eigenvalue of a matrix follow in this framework as a special case. The condition is interpreted in various other forms for the rank-one tensor optimization model under spherical constraints. Numerical experiments are shown to support the convergence property of the MBI method
- …