非凸张量秩近似在张量补全中的应用
6 浏览量
更新于2024-08-27
收藏 2.22MB PDF 举报
"这篇研究论文探讨了一种非凸张量秩近似方法在张量补全中的应用。文章发表在《应用数学建模》（Applied Mathematical Modelling）期刊上，作者包括Teng-Yu Ji、Ting-Zhu Huang、Xi-Le Zhao、Tian-Hui Ma和Liang-Jian Deng，他们都来自中国电子科技大学数学科学学院/图像与视觉计算研究中心。论文于2016年9月提交，2017年3月修订，同年4月接受，并在线发布于2017年4月。关键词包括张量补全、低秩近似、非凸优化和交替方向乘子法。"
正文：
张量完成是数据恢复领域的一个重要问题，特别是在多维数据处理和分析中。该论文提出了一种新的非凸张量秩近似策略来解决这一问题。传统的张量补全方法通常依赖于低秩假设，即认为原始张量可以被分解成几个低秩因子的乘积，以此减少信息丢失对恢复准确性的影响。
在实际应用中，如图像和视频处理，张量的低秩性来源于其内在的结构和模式。例如，图像的局部相似性和视频的时间连贯性都可以转化为低秩特性。然而，现有的方法，如张量核范数，虽然能够提供一个凸优化问题，从而确保全局最优解，但它对所有奇异值一视同仁，忽略了奇异值可能蕴含的物理意义和不同重要性。
论文的创新之处在于采用非凸优化方法，这允许对张量的奇异值进行更灵活的处理，更好地匹配实际数据的特性。非凸优化虽然可能导致局部最优解，但在某些情况下，它能够捕获数据的更精细结构，从而提高补全的精确度。此外，他们还应用了交替方向乘子法（ADMM），这是一种优化工具，可以有效地处理包含耦合约束的复杂优化问题，将大问题分解为一系列较小的子问题，简化了求解过程。
通过这种方式，作者们旨在找到一个更好的张量秩近似，不仅考虑低秩性，还考虑了奇异值的差异性，从而在保持计算效率的同时，提升张量补全的质量。这种方法对于图像恢复、视频处理、推荐系统和其他依赖于多维数据分析的应用具有潜在的价值，因为它们能够更好地捕捉数据的内在结构并处理缺失数据。
这篇研究论文贡献了一种新的非凸优化框架，以改进张量补全的算法，特别关注于如何利用张量的奇异值信息来提升恢复效果。这一工作对于理解和开发更有效的张量处理技术具有重要的理论和实践意义。
412 T.- Y. Ji et al. / Applied Mathematical Modelling 48 (2017) 410–422
Fig. 1. Mode- n unfoldings of a third-order tensor.
low-rank, if X
( n )
is low-rank for all n . This definition relates to the Tucker decomposition [33] . Please refer to [13] for its
extensive overview.
2.2. Related work
In this section, we consider the problem estimating the missing entries using low-rank prior of the underlying tensor.
Given an observed tensor B whose entries in the index set are known but the remaining are missing, find the tensor X
from the following optimization problem:
min
X
rank (X )
s.t. X
= B
. (1)
Recently, Liu et al. [14] developed the definition for the tensor nuclear norm as a surrogate for the tensor n -rank, and
they solved the following convex problem as a variant of (1) :
min
X
X
∗
:=
N
n =1
α
n
X
(n )
∗
s.t. X
= B
, (2)
where α
n
are constants satisfying α
n
≥ 0 and
N
n =1
α
n
= 1 . One of the algorithms for solving the problem (2) in [14] is
based on ADMM called HaLRTC. Recently, Gandy et al. in [15] studied noisy tensor completion based on nuclear norm and
proposed a Douglas-Rachford splitting method.
3. Proposed model
In this section, we explain the motivation and propose the non-convex but smooth surrogate function for the tensor
n -rank by using the logDet function rather than the nuclear norm. We also establish the completion model based on this
definition.
Our motivation for this model can be divided into three parts. (1) Although the non-convex functions are more
difficult, many works have paid more attention to them because of their effectiveness. The non-convex l
p
-norm (0 <
p < 1) [45,46] performs much better than the l
1
-norm in approximating the l
0
-norm; the low-rank matrix factorization
[16,20,26] and MCP function [18,29,30] also perform better than the convex nuclear norm. Therefore, we conjecture that a
non-convex approximation for the tensor rank could lead to better results.
(2) The n -rank of a tensor denotes the correlation with respect to the corresponding dimension. A simple description for
a third-order tensor unfolding is shown in Fig. 1 . We can see that the rank of the mode- n unfolding matrix X
( n )
denotes the
linear correlation between the mode- n fibers. In [14] , the rank of X
( n )
is approximated by the nuclear norm of X
( n )
. Fortu-
nately, the nuclear norm is the tightest convex approximation, and the nuclear norm minimization problem can be easily
solved by the singular value thresholding (SVT) [21] , which is theoretically sound [47] . The nuclear norm-based methods
treat each singular value equally. However, the larger singular values are generally associated with the major information,
and hence they should better be shrunk less to preserve the major data information. Clearly, tensor nuclear norm-based
completion methods fail to preserve the major data components. (3) To preserve the major data components, the singular
values should be treated differently. One way is to use the weighted nuclear norm, but it has to introduce more parameters
that may lead the method to be less robust. The logarithm operator can make a larger scalar decrease more rapidly than a
smaller one, i.e., the larger singular values are shrunk less using a smaller weight. Given two positive scalars: one is smaller,
denoted as s , and the other is larger, denoted as l , log (q ) = W
q
∗ q, where q ∈ { s, l }. We can see that the weight | W
l
| < | W
s
|,
because of l > s when l, s ∈ (1, e ) where e is the Euler’s number (The interval (1, e ) is narrower than the range of singular
values, thus we can ignore it). Therefore, the larger singular values are shrunk less, and can be treated as more important. In
addition, for scalars, the effective of the logDet can be seen in Fig. 2 , where we can see that the logDet function can better
approximate the rank than the nuclear norm. For vectors, the logDet function leads to the well-known reweighted l
1
-norm.
In [48] it was shown that the reweighted l
1
-norm achieved better performance than the l
1
-norm. For matrices, the logDet
function also performs better than the nuclear norm and often leads to superior image recovery results [41,42,49] .
剩余12页未读，继续阅读
点击了解资源详情
点击了解资源详情
点击了解资源详情
2021-02-22 上传
2021-02-22 上传
2021-05-25 上传
2021-02-21 上传
2021-02-22 上传
2020-01-18 上传
weixin_38679449
- 粉丝: 5
- 资源: 935
我的内容管理
展开
