It is still challenging to cluster multi-view data since existing methods can
only assign an object to a specific (singleton) cluster when combining
different view information. As a result, it fails to characterize imprecision
of objects in overlapping regions of different clusters, thus leading to a high
risk of errors. In this paper, we thereby want to answer the question: how to
characterize imprecision in multi-view clustering? Correspondingly, we propose
a multi-view low-rank evidential c-means based on entropy constraint (MvLRECM).
The proposed MvLRECM can be considered as a multi-view version of evidential
c-means based on the theory of belief functions. In MvLRECM, each object is
allowed to belong to different clusters with various degrees of support (masses
of belief) to characterize uncertainty when decision-making. Moreover, if an
object is in the overlapping region of several singleton clusters, it can be
assigned to a meta-cluster, defined as the union of these singleton clusters,
to characterize the local imprecision in the result. In addition,
entropy-weighting and low-rank constraints are employed to reduce imprecision
and improve accuracy. Compared to state-of-the-art methods, the effectiveness
of MvLRECM is demonstrated based on several toy and UCI real datasets.

提出了一种基于熵约束的多视图低秩证据 c 均值（MvLRECM）方法，用于多视图数据聚类中不确定性和不精确性的表征，并通过熵加权和低秩约束来提高准确性。模型应用于多个数据集，展示了其在实验中的有效性。

如何对多视角聚类中的不精确性进行表征？

How to characterize imprecision in multi-view clustering?

We introduce a novel optimization algorithm for image recovery under learned
sparse and low-rank constraints, which we parameterize as weighted extensions
of the $\ell_p^p$-vector and $\mathcal S_p^p$ Schatten-matrix quasi-norms for
$0\!<p\!\le1$, respectively. Our proposed algorithm generalizes the Iteratively
Reweighted Least Squares (IRLS) method, used for signal recovery under $\ell_1$
and nuclear-norm constrained minimization. Further, we interpret our overall
minimization approach as a recurrent network that we then employ to deal with
inverse low-level computer vision problems. Thanks to the convergence
guarantees that our IRLS strategy offers, we are able to train the derived
reconstruction networks using a memory-efficient implicit back-propagation
scheme, which does not pose any restrictions on their effective depth. To
assess our networks' performance, we compare them against other existing
reconstruction methods on several inverse problems, namely image deblurring,
super-resolution, demosaicking and sparse recovery. Our reconstruction results
are shown to be very competitive and in many cases outperform those of existing
unrolled networks, whose number of parameters is orders of magnitude higher
than that of our learned models.

提出了一种基于学习的稀疏和低秩约束的图像恢复新优化算法，并将其和现有算法进行了比较和评估。

通过迭代加权最小二乘法实现图像恢复的稀疏和低秩先验学习

Learning Sparse and Low-Rank Priors for Image Recovery via Iterative  Reweighted Least Squares Minimization

Several recent applications of optimal transport (OT) theory to machine
learning have relied on regularization, notably entropy and the Sinkhorn
algorithm. Because matrix-vector products are pervasive in the Sinkhorn
algorithm, several works have proposed to \textit{approximate} kernel matrices
appearing in its iterations using low-rank factors. Another route lies instead
in imposing low-rank constraints on the feasible set of couplings considered in
OT problems, with no approximations on cost nor kernel matrices. This route was
first explored by Forrow et al., 2018, who proposed an algorithm tailored for
the squared Euclidean ground cost, using a proxy objective that can be solved
through the machinery of regularized 2-Wasserstein barycenters. Building on
this, we introduce in this work a generic approach that aims at solving, in
full generality, the OT problem under low-rank constraints with arbitrary
costs. Our algorithm relies on an explicit factorization of low rank couplings
as a product of \textit{sub-coupling} factors linked by a common marginal;
similar to an NMF approach, we alternatively updates these factors. We prove
the non-asymptotic stationary convergence of this algorithm and illustrate its
efficiency on benchmark experiments.

本研究基于低秩约束探索了一种求解任意代价的最优输运问题的泛用性算法，并通过基准实验验证其效率。

低秩 Sinkhorn 分解

Low-Rank Sinkhorn Factorization

Convolutional neural networks typically consist of many convolutional layers
followed by one or more fully connected layers. While convolutional layers map
between high-order activation tensors, the fully connected layers operate on
flattened activation vectors. Despite empirical success, this approach has
notable drawbacks. Flattening followed by fully connected layers discards
multilinear structure in the activations and requires many parameters. We
address these problems by incorporating tensor algebraic operations that
preserve multilinear structure at every layer. First, we introduce Tensor
Contraction Layers (TCLs) that reduce the dimensionality of their input while
preserving their multilinear structure using tensor contraction. Next, we
introduce Tensor Regression Layers (TRLs), which express outputs through a
low-rank multilinear mapping from a high-order activation tensor to an output
tensor of arbitrary order. We learn the contraction and regression factors
end-to-end, and produce accurate nets with fewer parameters. Additionally, our
layers regularize networks by imposing low-rank constraints on the activations
(TCL) and regression weights (TRL). Experiments on ImageNet show that, applied
to VGG and ResNet architectures, TCLs and TRLs reduce the number of parameters
compared to fully connected layers by more than 65% while maintaining or
increasing accuracy. In addition to the space savings, our approach's ability
to leverage topological structure can be crucial for structured data such as
MRI. In particular, we demonstrate significant performance improvements over
comparable architectures on three tasks associated with the UK Biobank dataset.

本研究通过引入张量代数操作来替代全连接层，提出了张量收缩层和张量回归层，以减少参数数量并保留激活的多线性结构，同时通过在张量收缩层和张量回归层中加入低秩约束，改进了神经网络的性能，并在 MRI 图像分析任务上取得了显著的性能提升。