9,300 research outputs found
Simultaneous Tensor Completion and Denoising by Noise Inequality Constrained Convex Optimization
Tensor completion is a technique of filling missing elements of the
incomplete data tensors. It being actively studied based on the convex
optimization scheme such as nuclear-norm minimization. When given data tensors
include some noises, the nuclear-norm minimization problem is usually converted
to the nuclear-norm `regularization' problem which simultaneously minimize
penalty and error terms with some trade-off parameter. However, the good value
of trade-off is not easily determined because of the difference of two units
and the data dependence. In the sense of trade-off tuning, the noisy tensor
completion problem with the `noise inequality constraint' is better choice than
the `regularization' because the good noise threshold can be easily bounded
with noise standard deviation. In this study, we tackle to solve the convex
tensor completion problems with two types of noise inequality constraints:
Gaussian and Laplace distributions. The contributions of this study are
follows: (1) New tensor completion and denoising models using tensor total
variation and nuclear-norm are proposed which can be characterized as a
generalization/extension of many past matrix and tensor completion models, (2)
proximal mappings for noise inequalities are derived which are analytically
computable with low computational complexity, (3) convex optimization algorithm
is proposed based on primal-dual splitting framework, (4) new step-size
adaptation method is proposed to accelerate the optimization, and (5) extensive
experiments demonstrated the advantages of the proposed method for visual data
retrieval such as for color images, movies, and 3D-volumetric data
Robust Low-Rank Tensor Ring Completion
Low-rank tensor completion recovers missing entries based on different tensor
decompositions. Due to its outstanding performance in exploiting some
higher-order data structure, low rank tensor ring has been applied in tensor
completion. To further deal with its sensitivity to sparse component as it does
in tensor principle component analysis, we propose robust tensor ring
completion (RTRC), which separates latent low-rank tensor component from sparse
component with limited number of measurements. The low rank tensor component is
constrained by the weighted sum of nuclear norms of its balanced unfoldings,
while the sparse component is regularized by its l1 norm. We analyze the RTRC
model and gives the exact recovery guarantee. The alternating direction method
of multipliers is used to divide the problem into several sub-problems with
fast solutions. In numerical experiments, we verify the recovery condition of
the proposed method on synthetic data, and show the proposed method outperforms
the state-of-the-art ones in terms of both accuracy and computational
complexity in a number of real-world data based tasks, i.e., light-field image
recovery, shadow removal in face images, and background extraction in color
video
Noisy Tensor Completion via the Sum-of-Squares Hierarchy
In the noisy tensor completion problem we observe entries (whose location
is chosen uniformly at random) from an unknown
tensor . We assume that is entry-wise close to being rank . Our goal
is to fill in its missing entries using as few observations as possible. Let . We show that if then there is a
polynomial time algorithm based on the sixth level of the sum-of-squares
hierarchy for completing it. Our estimate agrees with almost all of 's
entries almost exactly and works even when our observations are corrupted by
noise. This is also the first algorithm for tensor completion that works in the
overcomplete case when , and in fact it works all the way up to .
Our proofs are short and simple and are based on establishing a new
connection between noisy tensor completion (through the language of Rademacher
complexity) and the task of refuting random constant satisfaction problems.
This connection seems to have gone unnoticed even in the context of matrix
completion. Furthermore, we use this connection to show matching lower bounds.
Our main technical result is in characterizing the Rademacher complexity of the
sequence of norms that arise in the sum-of-squares relaxations to the tensor
nuclear norm. These results point to an interesting new direction: Can we
explore computational vs. sample complexity tradeoffs through the
sum-of-squares hierarchy?Comment: 24 page
Frequency-Weighted Robust Tensor Principal Component Analysis
Robust tensor principal component analysis (RTPCA) can separate the low-rank
component and sparse component from multidimensional data, which has been used
successfully in several image applications. Its performance varies with
different kinds of tensor decompositions, and the tensor singular value
decomposition (t-SVD) is a popularly selected one. The standard t-SVD takes the
discrete Fourier transform to exploit the residual in the 3rd mode in the
decomposition. When minimizing the tensor nuclear norm related to t-SVD, all
the frontal slices in frequency domain are optimized equally. In this paper, we
incorporate frequency component analysis into t-SVD to enhance the RTPCA
performance. Specially, different frequency bands are unequally weighted with
respect to the corresponding physical meanings, and the frequency-weighted
tensor nuclear norm can be obtained. Accordingly we rigorously deduce the
frequency-weighted tensor singular value threshold operator, and apply it for
low rank approximation subproblem in RTPCA. The newly obtained
frequency-weighted RTPCA can be solved by alternating direction method of
multipliers, and it is the first time that frequency analysis is taken in
tensor principal component analysis. Numerical experiments on synthetic 3D
data, color image denoising and background modeling verify that the proposed
method outperforms the state-of-the-art algorithms both in accuracy and
computational complexity
Tensor-Ring Nuclear Norm Minimization and Application for Visual Data Completion
Tensor ring (TR) decomposition has been successfully used to obtain the
state-of-the-art performance in the visual data completion problem. However,
the existing TR-based completion methods are severely non-convex and
computationally demanding. In addition, the determination of the optimal TR
rank is a tough work in practice. To overcome these drawbacks, we first
introduce a class of new tensor nuclear norms by using tensor circular
unfolding. Then we theoretically establish connection between the rank of the
circularly-unfolded matrices and the TR ranks. We also develop an efficient
tensor completion algorithm by minimizing the proposed tensor nuclear norm.
Extensive experimental results demonstrate that our proposed tensor completion
method outperforms the conventional tensor completion methods in the
image/video in-painting problem with striped missing values.Comment: This paper has been accepted by ICASSP 201
An Iterative Reweighted Method for Tucker Decomposition of Incomplete Multiway Tensors
We consider the problem of low-rank decomposition of incomplete multiway
tensors. Since many real-world data lie on an intrinsically low dimensional
subspace, tensor low-rank decomposition with missing entries has applications
in many data analysis problems such as recommender systems and image
inpainting. In this paper, we focus on Tucker decomposition which represents an
Nth-order tensor in terms of N factor matrices and a core tensor via
multilinear operations. To exploit the underlying multilinear low-rank
structure in high-dimensional datasets, we propose a group-based log-sum
penalty functional to place structural sparsity over the core tensor, which
leads to a compact representation with smallest core tensor. The method for
Tucker decomposition is developed by iteratively minimizing a surrogate
function that majorizes the original objective function, which results in an
iterative reweighted process. In addition, to reduce the computational
complexity, an over-relaxed monotone fast iterative shrinkage-thresholding
technique is adapted and embedded in the iterative reweighted process. The
proposed method is able to determine the model complexity (i.e. multilinear
rank) in an automatic way. Simulation results show that the proposed algorithm
offers competitive performance compared with other existing algorithms
Tensor Ring Decomposition with Rank Minimization on Latent Space: An Efficient Approach for Tensor Completion
In tensor completion tasks, the traditional low-rank tensor decomposition
models suffer from the laborious model selection problem due to their high
model sensitivity. In particular, for tensor ring (TR) decomposition, the
number of model possibilities grows exponentially with the tensor order, which
makes it rather challenging to find the optimal TR decomposition. In this
paper, by exploiting the low-rank structure of the TR latent space, we propose
a novel tensor completion method which is robust to model selection. In
contrast to imposing the low-rank constraint on the data space, we introduce
nuclear norm regularization on the latent TR factors, resulting in the
optimization step using singular value decomposition (SVD) being performed at a
much smaller scale. By leveraging the alternating direction method of
multipliers (ADMM) scheme, the latent TR factors with optimal rank and the
recovered tensor can be obtained simultaneously. Our proposed algorithm is
shown to effectively alleviate the burden of TR-rank selection, thereby greatly
reducing the computational cost. The extensive experimental results on both
synthetic and real-world data demonstrate the superior performance and
efficiency of the proposed approach against the state-of-the-art algorithms
Near-optimal sample complexity for convex tensor completion
We analyze low rank tensor completion (TC) using noisy measurements of a
subset of the tensor. Assuming a rank-, order-, tensor where , the best sampling complexity that was achieved
is , which is obtained by solving a tensor nuclear-norm
minimization problem. However, this bound is significantly larger than the
number of free variables in a low rank tensor which is . In this paper,
we show that by using an atomic-norm whose atoms are rank- sign tensors, one
can obtain a sample complexity of . Moreover, we generalize the matrix
max-norm definition to tensors, which results in a max-quasi-norm (max-qnorm)
whose unit ball has small Rademacher complexity. We prove that solving a
constrained least squares estimation using either the convex atomic-norm or the
nonconvex max-qnorm results in optimal sample complexity for the problem of
low-rank tensor completion. Furthermore, we show that these bounds are nearly
minimax rate-optimal. We also provide promising numerical results for max-qnorm
constrained tensor completion, showing improved recovery results compared to
matricization and alternating least squares
Optimal Low-Rank Tensor Recovery from Separable Measurements: Four Contractions Suffice
Tensors play a central role in many modern machine learning and signal
processing applications. In such applications, the target tensor is usually of
low rank, i.e., can be expressed as a sum of a small number of rank one
tensors. This motivates us to consider the problem of low rank tensor recovery
from a class of linear measurements called separable measurements. As specific
examples, we focus on two distinct types of separable measurement mechanisms
(a) Random projections, where each measurement corresponds to an inner product
of the tensor with a suitable random tensor, and (b) the completion problem
where measurements constitute revelation of a random set of entries. We present
a computationally efficient algorithm, with rigorous and order-optimal sample
complexity results (upto logarithmic factors) for tensor recovery. Our method
is based on reduction to matrix completion sub-problems and adaptation of
Leurgans' method for tensor decomposition. We extend the methodology and sample
complexity results to higher order tensors, and experimentally validate our
theoretical results
Enhanced nonconvex low-rank approximation of tensor multi-modes for tensor completion
Higher-order low-rank tensor arises in many data processing applications and
has attracted great interests. Inspired by low-rank approximation theory,
researchers have proposed a series of effective tensor completion methods.
However, most of these methods directly consider the global low-rankness of
underlying tensors, which is not sufficient for a low sampling rate; in
addition, the single nuclear norm or its relaxation is usually adopted to
approximate the rank function, which would lead to suboptimal solution deviated
from the original one. To alleviate the above problems, in this paper, we
propose a novel low-rank approximation of tensor multi-modes (LRATM), in which
a double nonconvex norm is designed to represent the underlying
joint-manifold drawn from the modal factorization factors of the underlying
tensor. A block successive upper-bound minimization method-based algorithm is
designed to efficiently solve the proposed model, and it can be demonstrated
that our numerical scheme converges to the coordinatewise minimizers. Numerical
results on three types of public multi-dimensional datasets have tested and
shown that our algorithm can recover a variety of low-rank tensors with
significantly fewer samples than the compared methods.Comment: arXiv admin note: substantial text overlap with arXiv:2004.0874
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