3 research outputs found
Tensor N-tubal rank and its convex relaxation for low-rank tensor recovery
As low-rank modeling has achieved great success in tensor recovery, many
research efforts devote to defining the tensor rank. Among them, the recent
popular tensor tubal rank, defined based on the tensor singular value
decomposition (t-SVD), obtains promising results. However, the framework of the
t-SVD and the tensor tubal rank are applicable only to three-way tensors and
lack of flexibility to handle different correlations along different modes. To
tackle these two issues, we define a new tensor unfolding operator, named
mode- tensor unfolding, as the process of lexicographically stacking
the mode- slices of an -way tensor into a three-way tensor, which is
a three-way extension of the well-known mode- tensor matricization. Based on
it, we define a novel tensor rank, the tensor -tubal rank, as a vector whose
elements contain the tubal rank of all mode- unfolding tensors, to
depict the correlations along different modes. To efficiently minimize the
proposed -tubal rank, we establish its convex relaxation: the weighted sum
of tensor nuclear norm (WSTNN). Then, we apply WSTNN to low-rank tensor
completion (LRTC) and tensor robust principal component analysis (TRPCA). The
corresponding WSTNN-based LRTC and TRPCA models are proposed, and two efficient
alternating direction method of multipliers (ADMM)-based algorithms are
developed to solve the proposed models. Numerical experiments demonstrate that
the proposed models significantly outperform the compared ones
Hierarchical Tensor Ring Completion
Tensor completion can estimate missing values of a high-order data from its
partially observed entries. Recent works show that low rank tensor ring
approximation is one of the most powerful tools to solve tensor completion
problem. However, existing algorithms need predefined tensor ring rank which
may be hard to determine in practice. To address the issue, we propose a
hierarchical tensor ring decomposition for more compact representation. We use
the standard tensor ring to decompose a tensor into several 3-order sub-tensors
in the first layer, and each sub-tensor is further factorized by tensor
singular value decomposition (t-SVD) in the second layer. In the low rank
tensor completion based on the proposed decomposition, the zero elements in the
3-order core tensor are pruned in the second layer, which helps to
automatically determinate the tensor ring rank. To further enhance the recovery
performance, we use total variation to exploit the locally piece-wise
smoothness data structure. The alternating direction method of multiplier can
divide the optimization model into several subproblems, and each one can be
solved efficiently. Numerical experiments on color images and hyperspectral
images demonstrate that the proposed algorithm outperforms state-of-the-arts
ones in terms of recovery accuracy
Tensor completion via nonconvex tensor ring rank minimization with guaranteed convergence
In recent studies, the tensor ring (TR) rank has shown high effectiveness in
tensor completion due to its ability of capturing the intrinsic structure
within high-order tensors. A recently proposed TR rank minimization method is
based on the convex relaxation by penalizing the weighted sum of nuclear norm
of TR unfolding matrices. However, this method treats each singular value
equally and neglects their physical meanings, which usually leads to suboptimal
solutions in practice. In this paper, we propose to use the logdet-based
function as a nonconvex smooth relaxation of the TR rank for tensor completion,
which can more accurately approximate the TR rank and better promote the
low-rankness of the solution. To solve the proposed nonconvex model
efficiently, we develop an alternating direction method of multipliers
algorithm and theoretically prove that, under some mild assumptions, our
algorithm converges to a stationary point. Extensive experiments on color
images, multispectral images, and color videos demonstrate that the proposed
method outperforms several state-of-the-art competitors in both visual and
quantitative comparison. Key words: nonconvex optimization, tensor ring rank,
logdet function, tensor completion, alternating direction method of
multipliers